Quasiparticles and vortices in the high temperature cuprate ...

QUASIPARTICLES AND VORTICES IN THE HIGHTEMPERATURE CUPRATE SUPERCONDUCTORSbyOskar VafekA dissertation submitted to The Johns Hopkins University in conformity with therequirements for the degree of Doctor of Philosophy.Baltimore, MarylandAugust, 2003c○ Oskar Vafek 2003All rights reserved

AbstractIn this thesis I present a study of the interplay of vortices and the fermionic quasiparticlestates in a quasi 2-dimensional d-wave superconductors, such as high temperaturecuprate superconductors. In the first part, I start by analyzing the quasiparticlestates in the presence of the magnetic field induced vortex lattice. Unlike in the s-wave superconductor, there are no vortex bound states, rather all the quasiparticlestates are extended. By using general arguments based on symmetry principles andby direct (numerical) computation, I show that these extended quasiparticle statesare gapped. In addition, they are characterized by a topological invariant with regardto their spin Hall conduction. Finally, I relate the spin Hall conductivity tensor σ spinxyto the thermal Hall conductivity κ xy by deriving the appropriate “Wiedemann-Franz”law. In the second part, I analyze the fluctuations around an ordered d-wave superconductor(in the absence of the external magnetic field), focusing on the interactionbetween the low energy nodal fermions and the vortex-antivortex phase excitations.It is shown that the corresponding low energy effective theory for the nodal fermionsin the normal (non-superconducting) state is QED 3 or quantum electrodynamics in2+1 space time dimensions. The massless U(1) gauge field encodes the topologicalinteraction between the quasiparticles and the hc/2e vortices at long wavelengths.I analyze the symmetries, correlations and stability of this state.In particular, Istudy the role of Dirac cone anisotropy and residual interactions in QED 3 as well asthe physical meaning of the chiral symmetry breaking. Remarkably, the spin densitywave corresponds to an instability of a phase fluctuating d-wave superconductor.Advisor: Zlatko B. Tesanovic.ii

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AcknowledgementsI wish to express my deep gratitude to Prof. Zlatko Tešanovic for sharing hisinsights and library of knowledge with me, and for constant and uncompromisingquest for the profound that he inspires.I also wish to thank Ashot Melikyan for many invaluable discussions and debates,and for his exceptional selflessness in helping me and others.iv

ContentsAbstractAcknowledgementsList of Figuresiiivvii1 Introduction 11.1 Phenomenology of High Temperature Cuprate Superconductors . . . 32 Quasiparticles in the mixed state of HTS 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Quasiparticle excitation spectrum of a d-wave superconductor in themixed state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Particle-Hole Symmetry . . . . . . . . . . . . . . . . . . . . . 192.2.2 Franz-Tešanović Transformation and Translation Symmetry . 192.2.3 Vortex Lattice Inversion Symmetry and Level Crossing . . . . 222.3 Topological quantization of spin and thermal Hall conductivities . . . 252.3.1 Spin Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . 312.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 333 QED 3 theory of the phase disordered d-wave superconductors 363.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Vortex quasiparticle interaction . . . . . . . . . . . . . . . . . . . . . 383.2.1 Protectorate of the pairing pseudogap . . . . . . . . . . . . . . 383.2.2 Topological frustration . . . . . . . . . . . . . . . . . . . . . . 393.2.3 “Coarse-grained” Doppler and Berry U(1) gauge fields . . . . 433.2.4 Further remarks on FT gauge . . . . . . . . . . . . . . . . . . 463.2.5 Maxwellian form of the Jacobian L 0 [v µ , a µ ] . . . . . . . . . . . 483.3 Low energy effective theory . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 Effective Lagrangian for the pseudogap state . . . . . . . . . . 583.3.2 Berryon propagator . . . . . . . . . . . . . . . . . . . . . . . . 60v

3.3.3 TF self energy and propagator . . . . . . . . . . . . . . . . . . 623.4 Effects of Dirac anisotropy in symmetric QED 3 . . . . . . . . . . . . 643.4.1 Anisotropic QED 3 . . . . . . . . . . . . . . . . . . . . . . . . 653.4.2 Gauge field propagator . . . . . . . . . . . . . . . . . . . . . . 663.4.3 TF self energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4.4 Dirac anisotropy and its β function . . . . . . . . . . . . . . . 703.5 Finite temperature extensions of QED 3 . . . . . . . . . . . . . . . . . 723.5.1 Specific heat and scaling of thermodynamics . . . . . . . . . . 753.5.2 Uniform spin susceptibility . . . . . . . . . . . . . . . . . . . . 763.6 Chiral symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 783.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 85A Quasiparticle correlation functions in the mixed state 87A.1 Green’s functions: definitions and identities . . . . . . . . . . . . . . 87A.2 Spin current and spin conductivity tensor . . . . . . . . . . . . . . . . 89A.3 Thermal current and thermal conductivity tensor . . . . . . . . . . . 91B Field theory of quasiparticles and vortices 98B.1 Jacobian L 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.1.1 2D thermal vortex-antivortex fluctuations . . . . . . . . . . . 98B.1.2 Quantum fluctuations of (2+1)D vortex loops . . . . . . . . . 101B.2 Feynman integrals in QED 3 . . . . . . . . . . . . . . . . . . . . . . . 106B.2.1 Vacuum polarization bubble . . . . . . . . . . . . . . . . . . . 106B.2.2 TF self energy: Lorentz gauge . . . . . . . . . . . . . . . . . . 107B.3 Feynman integrals for anisotropic case . . . . . . . . . . . . . . . . . 108B.3.1 Self energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108B.4 Physical modes of a finite T QED 3 . . . . . . . . . . . . . . . . . . . 110B.5 Finite Temperature Vacuum Polarization . . . . . . . . . . . . . . . . 113B.6 Free energy scaling of QED 3 . . . . . . . . . . . . . . . . . . . . . . . 118Bibliography 120Vita 127vi

List of Figures1.1 (A) The unit cell of La 2−x Sr x CuO 4 family of high temperature superconductors.It is believed that most of the interesting physics happensin the Cu-O 2 plane, which extends in the a − b directions. The c−axiselectronic coupling is very small. In this family of materials, doping isachieved by replacing some of the La atoms for Sr, or by adding interstitialoxygen atoms. This crystalline structure is slightly modifiedin different high-T c cuprates, but all share the weakly coupled Cu-O 2planes. (B) Arrows indicate spin alignment in the antiferromagenticstate, the parent state of HTS. (Taken from [3]) . . . . . . . . . . . . 41.2 Phase diagram of electron and hole doped High Temperature superconductorsshowing the superconducting (SC), antiferromagnetic (AF),pseudogap and metallic phases. (Taken from [4]). . . . . . . . . . . . 51.3 The magnetic flux threading through a polycrystalline YBCO ring,monitored with a SQUID magnetometer as a function of time. The fluxjumps occur in integral multiples of the superconducting flux quantumΦ 0 = hc/2e. This experiment demonstrates that the superconductivityin high temperature cuprates is due to Cooper pairing. (taken from [5]) 61.4 The pairing gap of HTS has d x 2 −y2-wave symmetry (left). The pointson the Fermi surface at which the gap disappears (the nodal points)can be identified in the angle resolved photoemission spectra (right) [4]. 71.5 Contours of the constant Nernst coefficient in the Temperature vs. dopingphase diagram. The units of the Nernst coefficient are nanoVolt perKelvin per Tesla. At low temperature and in the strongly underdopedregion the Nernst signal is almost three orders of magnitude greaterthan it would be in a normal metal. Such a large signal is typicallyobserved in the vortex liquid state, where it is associated with Josephsonphase-slips produced across the sample due to thermally diffusingvortices. In this experiment, however, this signal is seen several tensof Kelvins above the superconducting transition temperature T c ! [8] 8vii

Chapter 1IntroductionThe properties of matter at low temperature amplify nature’s most fascinatingand least comprehensible laws: the laws of quantum mechanics. This amplificationprocess occurs via a principle of symmetry breaking: when the available phase spaceis sufficiently restricted, interacting systems with a macroscopically large number ofdegrees of freedom tend to seek a highly organized state. In other words, the lowenergy state of a many particle system possesses less symmetry than the laws whichare obeyed by its constituents. A commonplace example are solids. Despite thefact that the laws of quantum mechanics are invariant under spatial translation, theatoms in a solid occupy a regular array which is only disrupted by small amplitudevibrations and occasional dislocation defects. It is the translational symmetry thatis broken by the solid state of matter. This symmetry breaking has an importantramification: the emergence of Goldstone modes, or long lived degrees of freedom,which tend to restore the symmetry. In the case of a solid, this is embodied by theemergence of transverse elastic waves. In addition, symmetry breaking generally leadsto stable topological defects; in solids these are lattice dislocations and disclinations.While such topological defects are quite rare deep inside an ordered state, they canbe crucial in destroying the order near a phase transition.A less commonplace example of symmetry breaking is a superconductor. Whilethe laws of quantum mechanics are invariant under a local U(1) transformation, i.e.multiplication of the wavefunction by a space dependent single valued phase factor,1

the superconducting state is not. In a superconductor the many body wavefunctionacquires rigidity under a phase twist. This leads to most of the fascinating propertiesof superconductors such as the passage of current without any voltage drop, theMeissner effect, or even the magnetic levitation. The density of low energy fermionicexcitations is generically appreciably reduced compared to the non-superconductingstate. The Goldstone mode that accompanies the symmetry breaking corresponds tosmooth (longitudinal) space-time variations of the phase of the many body wavefunction.On the other hand, 2π phase twists of the wavefunction, or vortices, are theanalog of the dislocations in the solid. They form the topological defects in the type-IIsuperconductors. At a phase transition, temperature T = T c , such a superconductorlooses its phase rigidity due to the free motion of vortex defects. The temperatureinterval around T c in which this physics applies is typically barely detectable in theconventional low T c superconductors, but there is growing experimental evidencethat in the high T c cuprate superconductors (HTS) vortex unbinding is the dominantmechanism of the loss of long range order. In particular, the underdoped materialsappear to exhibit many characteristics associated with vortex physics, in some casesall the way down to zero temperature! Such observations suggest that the quantumanalog of vortex unbinding occurs at the superconductor-insulator quantum phasetransition.The underlying theme in this work is the interplay of vortices and the fermionicquasiparticles, the single particle excitations of the superconductor. I start by brieflyreviewing the phenomenology of HTS. In Chapter 2, I analyze the quasiparticle statesin the presence of the magnetic field induced vortex lattice in a planar d-wave superconductor.Several interesting and non-trivial properties of such a state are derived.In Chapter 3, I then go on to analyze the nature of the fermionic excitations in aquantum phase disordered superconductor. It is shown that quantum unbinding ofhc/2e vortex defects produces gauge interactions between quasiparticles. The lowenergy effective field theory for such a state is argued to be QED 3 (2+1 dimensionalelectrodynamics with massless fermions). This theory has a very rich structure: ithas a phase characterized by very strong interaction between fermions and the gaugefield, but in which no symmetries are broken. In addition it also has a broken symme-2

try phase in which the fermions gain mass. Remarkably, this state can be associatedwith a spin density wave, a state adiabatically connected to an antiferromagnet, theparent state of all cuprate superconductors.1.1 Phenomenology of High Temperature CuprateSuperconductorsHigh temperature cuprate superconductors were discovered by Bednorz and Müllerin 1986 [1]. Soon thereafter, Anderson pointed out three essential features of the newmaterials [2]. First, they are quasi 2-dimensional with weakly coupled CuO 2 planes(Fig. 1.1). Second, the superconductor is created by doping a Mott insulator, andthird, Anderson proposed that the combination of proximity to a Mott insulator andlow dimensionality would cause the doped material to exhibit fundamentally newbehavior, impossible to explain by conventional paradigms of metal physics [2, 3].In a Mott insulator, charge transport is prohibited not merely by the Pauli exclusionprinciple (as in a band insulator), but also by strong electron-electron repulsionthat pins the electrons to the lattice sites. This insulating state is stable when thereis exactly one electron per each site in the valence band, since motion of electrons requiresenergetically expensive double occupancy. Virtual charge excitations generatea ”super-exchange” interaction that favors antiferromagnetic alignment of spins (Fig.1.1).Upon doping, the system becomes a weak conductor and eventually a superconductor,Fig. (1.2). However, the cuprates are the only Mott insulators whichbecome superconducting when doped [3]. It was established soon after the discoveryof HTS that the superconductivity is due to the electrons forming Cooper pairs (seeFig. 1.3). However, unlike in conventional s-wave superconductors, the gap functionchanges sign upon 90 degree rotation i.e. it has d x 2 −y2 symmetry. As such, it vanishesat four points on the Fermi surface, leading to gapless Fermi points. These pointswere identified by angle–resolved photoemission spectroscopy (ARPES) (Fig. 1.4). Inaddition, several ingenious phase sensitive tests, based on Josephson tunneling, were3

Figure 1.1: (A) The unit cell of La 2−x Sr x CuO 4 family of high temperature superconductors.It is believed that most of the interesting physics happens in the Cu-O 2plane, which extends in the a − b directions. The c−axis electronic coupling is verysmall. In this family of materials, doping is achieved by replacing some of the Laatoms for Sr, or by adding interstitial oxygen atoms. This crystalline structure isslightly modified in different high-T c cuprates, but all share the weakly coupled Cu-O 2 planes. (B) Arrows indicate spin alignment in the antiferromagentic state, theparent state of HTS. (Taken from [3])developed to demonstrate the change of sign of the pairing amplitude (for a reviewsee [5]). The low energy properties of such d-wave superconductors are governed bythe excitations around the four nodal points. These correspond to nodal BCS quasiparticleswhose existence was demonstrated by several techniques, but most directlyby ARPES. These low energy BCS quasiparticles are responsible, for example, for thelinear in temperature depletion of superfluid density [6] or ”universal” longitudinalthermal conductivity, which was demonstrated experimentally [7] to depend only onthe ratio of the quasiparticle velocity perpendicular and parallel to the Fermi surfaceat the Fermi points.The boundary in the phase diagram between the antiferromagnet and the superconductorcorresponds to a fascinating phase: the pseudogap state. This state is nota superconductor, but spectroscopically it is nearly indistinguishable from the superconductoras there is a suppression of single particle states at the Fermi surface aswell as the nodal points. While the spin fluctuations are gapped, the in-plane charge4

Figure 1.2: Phase diagram of electron and hole doped High Temperature superconductorsshowing the superconducting (SC), antiferromagnetic (AF), pseudogap andmetallic phases. (Taken from [4]).transport seems unaffected, whereas the c-axis transport is suppressed. Moreover,strong superconducting fluctuations seem to be prominent in this state. Especiallystriking are the recent observations of anomalously large Nernst signal [8] above T c insingle crystal underdoped cuprates (see Fig. 1.5). In this measurement, a magneticfield is applied perpendicular to the CuO 2 planes, along with a weak thermal gradientswithin the plane. One then measures the electrical voltage drop perpendicular tothe thermal current flow. The signal seen is nearly three orders of magnitude largerthan in conventional metals, not unlike in vortex liquid state. However, it is observedseveral tens on Kelvin above T c ! This suggests that the physics of pseudogap isdominated by strong pairing fluctuations.In what follows, I shall first study the BCS quasiparticles of a d-wave superconductorwith magnetic field induced array of Abrikosov vortices. I shall assume that weare at T = 0 far away from any phase transition so that fluctuations can be neglected(e.g. the region between optimally doped and slightly overdoped in the phase diagram(1.2)). Later, I shall concentrate on the pseudogap region, and assume that it isdominated by phase fluctuations, in particular that the pseudogap region is nothing5

Figure 1.3: The magnetic flux threading through a polycrystalline YBCO ring, monitoredwith a SQUID magnetometer as a function of time. The flux jumps occur inintegral multiples of the superconducting flux quantum Φ 0 = hc/2e. This experimentdemonstrates that the superconductivity in high temperature cuprates is dueto Cooper pairing. (taken from [5])but a phase disordered d-wave superconductor.6

−k F+ +−Figure 1.4: The pairing gap of HTS has d x 2 −y2-wave symmetry (left). The points onthe Fermi surface at which the gap disappears (the nodal points) can be identified inthe angle resolved photoemission spectra (right) [4].7

120Bi 2Sr 2-yLa yCuO 61008010nV/KTT (K)6040202040100200T onsetT c400100001.0 0.8 0.6 0.4 0.2 0.0La content yFigure 1.5: Contours of the constant Nernst coefficient in the Temperature vs. dopingphase diagram. The units of the Nernst coefficient are nanoVolt per Kelvin per Tesla.At low temperature and in the strongly underdoped region the Nernst signal is almostthree orders of magnitude greater than it would be in a normal metal. Such a largesignal is typically observed in the vortex liquid state, where it is associated withJosephson phase-slips produced across the sample due to thermally diffusing vortices.In this experiment, however, this signal is seen several tens of Kelvins above thesuperconducting transition temperature T c ! [8]8

Chapter 2Quasiparticles in the mixed stateof HTS2.1 IntroductionIn this section we give a brief review of the fermionic quasiparticle excitations inthe superconductors, both conventional and unconventional.In conventional (s-wave) superconductors the effective attraction between electronsis mainly due to exchange of phonons [9]. The evidence of the phonon mediatedinteraction comes from the isotope effect i.e. the shift of the transition temperatureupon the replacement of the crystal ions with their isotopes. The electrons pair inthe lowest angular momentum channel, l = 0 or s-wave, and the pairing amplitudedoes not vary appreciably around the Fermi surface. As a result, the single particlefermionic excitations (quasiparticles) are fully gapped everywhere on the Fermisurface and the quasiparticle density of states vanishes below a specific energy. Thishas profound consequences for the traditional phenomenology of superconductors.The gap in the fermionic spectrum leads to the well known activated form of thequasiparticle contribution to various thermodynamic and transport properties, andcan be directly observed in tunneling spectroscopy (see e.g. [5]). Furthermore, evenbeyond the mean-field theory, the presence of the pairing gap in the superconductingstate cuts off the infra-red singularities which allows perturbative treatment of various9

types of quasiparticle interactions.High temperature cuprate superconductors (HTS), however, are different. Whilethe origin of pairing remains an unsolved problem, the cuprates appear to be accuratelydescribed by the d x 2 −y2-wave order parameter [10], i.e. the pairing happensin l = 2 channel whose degeneracy is further split by the crystal field in favor ofd x 2 −y2. As a result, quasiparticle excitations occur at arbitrarily low energy near thenodal points. These low energy fermionic excitations appear to govern much of thethermodynamics and transport in the HTS materials. This represents a new intellectualchallenge [11]: one must devise methods that can incorporate the low energyfermionic excitations into the phenomenology of superconductors, both within themean-field BCS-like theory and beyond.This challenge is not trivial and has many diverse components. Low energy quasiparticlesare scattered by impurities in novel and unusual ways, depending on thelow energy density of states [12]. They interact with external perturbations in waysnot encountered in conventional superconductors, and these interactions give rise tonew phenomena [13, 14]. The low energy quasiparticles are thus expected to qualitativelyaffect the quantum critical behavior of HTS (see Chapter 3). Among themany aspects of this new quasiparticle phenomenology a particularly prominent roleis played by the low lying quasiparticle excitations in the mixed (or vortex) state [15].All HTS are extreme type-II systems and have a huge mixed phase extending fromthe lower critical field H c1 which is in the range of 10-100 Gauss to the upper criticalfield H c2 which can be as large as 100-200 T. In this large region the interactionsbetween quasiparticles and vortices play the essential role in defining the nature ofthermodynamic and transport properties.Thermodynamic and transport properties are expected to be rather different fordistinct classes of unconventional superconductors. The difference stems from a complexmotion of the quasiparticles under the combined effects of both the magneticfield B and the local drift produced by chiral supercurrents of the vortex state. Forexample, in HTS the d x 2 −y2-wave nature of the gap function results in its vanishingalong nodal directions. Along these nodal directions the pair-breaking induced bysupercurrents has a particularly strong effect. On the other hand, in unconventional10

superconductors with the p x±iy pairing, Sr 2 RuO 4 being a possible candidate [16], thespectrum is fully gapped but the order parameter is chiral even in the absence ofexternal magnetic field. This leads to two different types of vortices for two differentfield orientations [17, 18].Still, in all these different situations, the quantum dynamics of quasiparticlesin the vortex state contains two essential common ingredients. First, there is thepurely classical effect of the Doppler shift [13, 14]: quasiparticles’ energy is shiftedby a locally drifting superfluid, E(k) → E(k) − v s (r) · k, where v s (r) is the localsuperfluid velocity. v s (r) contains information about vortex configurations, allowingus to connect quasiparticle spectral properties to various cooperative phenomena inthe system of vortices [19, 20, 21]. The Doppler shift effect is not peculiar to the vortexstate. It also occurs in the Meissner phase [14] and is generally present whenever aquasiparticle experiences a locally uniform drift in the superfluid velocity. Second,there is also a purely quantum effect which is intimately tied to the vortex state:as a quasiparticle circles around a vortex while maintaining its quantum coherence,the accumulated phase through a Doppler shift is ±π. This implies that there mustbe an additional compensating ±π contribution to the phase on top of the one dueto the Doppler shift in order for the wavefunction to remain single-valued [22]. Therequired ±π contribution is supplied by a “Berry phase” effect and can be built in atthe Hamiltonian level as a half-flux Aharonov-Bohm scattering of quasiparticles byvortices [22]. This interplay between the classical (Doppler shift) and purely quantumeffect (“Berry phase”) is what makes the problem of quasiparticle-vortex interactionparticularly fascinating.Let us briefly review what is already known about the quasiparticles in the vortexstate. The initial theoretical investigations of gapped and gapless superconductorsin the vortex state were directed along rather separate lines. The low energy quasiparticlespectrum of an s-wave superconductor in the mixed state was originallystudied by Caroli, de Gennes and Matricon (CdGM) [23] within the framework of theBogoliubov-de Gennes equations [24]. Their solution yields well known bound statesin the vortex cores. These states are localized in the core and have an exponentialenvelope, the scale of which is set by the BCS coherence length. The low energy11

end of the spectrum is given by ɛ µ ∼ µ(∆ 2 0/E F ), where µ = 1/2, 3/2, . . . , ∆ 0 is theoverall BCS gap, and E F is the Fermi energy. This solution can be relatively straightforwardlygeneralized to a fully gapped, chiral p-wave superconductor. In this casethe low energy quasiparticle spectrum also displays bound vortex core states, whoseenergy quantization is, however, modified relative to its s-wave counterpart, preciselybecause of the chiral character of a p x±iy -wave superconductor and the ensuing shiftin the angular momentum. For example, the low energy spectrum of quasiparticlesin the singly quantized vortex of the p x±iy -wave superconductor, possesses a state atexactly zero energy [17, 18].By comparison, the spectrum of a gapless d-wave superconductor in the mixedphase has become the subject of an active debate only relatively recently, fueled bythe interest in HTS. Naturally, the first question that arises is what is the analogof the CdGM solution for a single vortex. It is important to realize here that thesituation in a d x 2 −y 2superconductor is qualitatively different from the classic s-wavecase [25]: when the pairing state has a finite angular momentum and is not a globaleigenstate of the angular momentum L z (a d x 2 −y2 superconductor is an equal admixtureof L z = ±2 states), the problem of fermionic excitations in the core cannotbe reduced to a collection of decoupled 1D dimensional eigenvalue equations for eachangular momentum channel, the key feature of the CdGM solution. Instead, all channelsremain coupled and one must solve a full 2D problem. The fully self-consistentnumerical solution of the BdG equations [25, 26] reveals the most important physicalconsequence of this qualitatively new situation: the vortex core quasiparticle states ina pure d x 2 −y 2superconductor are delocalized with wave functions extended along thenodal directions. The low lying states have a continuous spectrum and, in a broadrange of parameters, do not seem to exhibit strong resonant behavior.This is insharp contrast with a discrete spectrum and true bound quasiparticle states of theCdGM s-wave solution.A particularly important issue in this context is the nature of the quasiparticleexcitations at very low fields, in the presence of a vortex lattice.This is a novelchallenge since the spectrum starts as gapless at zero field and at issue is the interactionof these low lying quasiparticles with the vortex lattice.12This problem has

een addressed via numerical solution of the tight binding model [27], a numericaldiagonalization of the continuum model [28] and a semiclassical analysis [13]. Gorkov,Schrieffer [29] and, in a somewhat different context, Anderson [61], predicted that thequasiparticle spectrum is described by a Dirac-like Landau quantization of energylevelsE n = ±ω H√ n, n = 0, 1, ... (2.1)where ω H = √ 2ω c ∆ 0 /, ω c = eB/mc is the cyclotron frequency and ∆ 0 is themaximum superconducting gap. The Dirac-like spectrum of Landau levels arises fromthe linear dispersion of nodal quasiparticles at zero field. This argument neglects theeffect of spatially varying supercurrents in the vortex array which were shown tostrongly mix individual Landau levels [31].Recently, Franz and Tešanović (FT) [22] pointed out that the low energy quasiparticlestates of a d x 2 −y2-wave superconductor in the vortex state are most naturallydescribed by strongly dispersive Bloch waves. This conclusion was based on the particularchoice of a singular gauge transformation, which allows for the treatment ofthe uniform external magnetic field and the effects produced by chiral supercurrentson equal footing. The starting point was the Bogoliubov-de Gennes (BdG) equationlinearized around a Dirac node. By employing the singular gauge transformation FTmapped the original problem onto that of a Dirac Hamiltonian in periodic vectorand scalar potentials, comprised of an array of an effective magnetic Aharonov-Bohmhalf-fluxes, and with a vanishing overall magnetic flux per unit cell. The FT gaugetransformation allows use of standard band structure and other zero-field techniquesto study the quasiparticle dynamics in the presence of vortex arrays, ordered or disordered.Its utility was illustrated in Ref. [22] through computation of the quasiparticlespectra of a square vortex lattice. A remarkable feature of these spectra is the persistenceof the massless Dirac node at finite fields and the appearance of the “linesof nodes” in the gap at large values of the anisotropy ratio α D = v F /v ∆ , starting atα D ≃ 15. Furthermore, the FT transformation directly reveals that a quasiparticlemoving coherently through a vortex array experiences not only a Doppler shift causedby circulating supercurrents but also an additional, “Berry phase” effect: the latter13

is a purely quantum mechanical phenomenon and is absent from a typical semiclassicalapproach. Interestingly, the cyclotron motion in Dirac cones is entirely causedby such “Berry phase” effect, which takes the form of a half-flux Aharonov-Bohmscattering of quasiparticles by vortices, and does not explicitly involve the externalmagnetic field. It is for this reason that the Dirac-like Landau level quantization isabsent from the exact quasiparticle spectrum.Further progress was achieved by Marinelli, Halperin and Simon [32] who presenteda detailed perturbative analysis of the linearized Hamiltonian of Ref. [22].They showed that the presence of the particle-hole symmetry is of key importance indetermining the nature of the spectrum of low energy excitations. If the vortices arearranged in a Bravais lattice, they showed that, to all orders in perturbation theory,the Dirac node is preserved at finite fields, i.e the quasiparticle spectrum remainsgapless at the Γ point. This result masks intense mixing of individual basis vectors(in the case of Ref. [32] these are Dirac plane waves), including strong mixing ofstates far removed in energy. The continuing presence of the massless Dirac node atthe Γ point after the application of the external field is thus not due to the lack ofscattering which is actually remarkably strong. Rather, it is dictated by symmetry:Marinelli et al. demonstrated that the crucial agent responsible for the presence ofthe Dirac node is the particle-hole symmetry, present at every point in the Brillouinzone. The fact that it is the particle-hole symmetry rather than the lack of scatteringthat protects the Dirac node is clearly revealed in the related problem of aSchrödinger electron in the presence of a single Aharonov-Bohm half-flux, where thedensity of states acquires a δ function depletion at k = 0 [33], thus shifting part ofthe spectral weight to infinity due to remarkably strong scattering. The authors ofRef. [32] also corrected Ref. [22] by showing that the “lines of nodes” must actuallybe the “lines of near nodes” since true zeroes of the energy away from Dirac nodeare prohibited on symmetry grounds. Still, these “lines” will act as true nodes in allrealistic circumstances, due to extraordinarily small excitation energies.In this work I extend the original analysis which was based solely on the continuumdescription by introducing a tight binding “regularization” of the full lattice BdGHamiltonian, to which we then apply the FT gauge transformation. The lattice for-14

mulation allows us to study what, if any, role is played by internodal scattering whichis simply not a part of the linearized description. This is important and necessarysince the straightforward linearization of BdG equations drops curvature terms andresults in the thermal Hall conductivity: κ xy = 0[15, 34]. We employ the Franz andTešanović (FT) transformation so that we can use the familiar Bloch representationof the translation group in which the overall chirality of the problem vanishes. Thisshould be contrasted with the original problem where the overall chirality is finiteand the magnetic translation group states must be used instead. Naively, it mightappear that after an FT singular gauge transformation the effects of the magneticfield have somehow been transformed away since the new problem is found to havezero average effective magnetic field. Of course, this is not true. The presence ofmagnetic field in the original problem reveals itself fully in the FT transformed quasiparticlewavefunctions. Alternatively, there is an “intrinsic” chirality imposed on thesystem which cannot be transformed away by a choice of the basis. One manifestationof this chirality is the Hall effect. The utility of the singular gauge transformationin the calculation of the electrical Hall conductivity in the normal 2D electron gasin a (non-uniform) magnetic field was realized by Nielsen and Hedegård [35]. Theydemonstrated that using singularly gauge transformed wavefunctions one still obtainsthe correct result, giving the electrical Hall conductance quantized in units of e 2 /h ifthe chemical potential lies in the energy gap. In a superconductor, the question ofHall response becomes rather interesting as there is a strong mixing between particlesand holes. Evidently, the electrical Hall response is very different from the normalstate, since charge is not conserved in the state with broken U(1) symmetry. Therefore,as pointed out in Ref. [36], charge cannot be transported by diffusion. On theother hand, the spin is still a good quantum number[36] and it is natural to ask whatis the spin Hall conductivity in the vortex state of an extreme type-II superconductor[37]. Moreover, every channel of spin conduction simultaneously transports entropy[36, 38, 39] and we would expect some variation of the Wiedemann-Franz law to holdbetween spin and thermal conductivity.As one of our main results, we derive the Wiedemann-Franz law connecting thespin and thermal Hall transport in the vortex state of a d-wave superconductor. In15

the process, we show that the spin Hall conductivity, σxy, s just like the electricalHall conductivity of a normal state in a magnetic field, is topological in nature andcan be explicitly evaluated as a first Chern number characterizing the eigenstatesof our singularly gauge transformed problem [37, 40, 41]. Consequently, as T →0, the spin Hall conductivity is quantized in the units of /8π when the energyspectrum is gapped, which, combined with the Wiedemann-Franz law, implies thequantization of κ xy /T . We then explicitly compute the quantized values of σxy s fora sequence of gapped states using our lattice d-wave superconductor model in thecase of an inversion-symmetric vortex lattice. Within this model one is naturallyled to consider the BCS-Hofstadter problem: the BCS pairing problem defined on auniformly frustrated tight-binding lattice. We find a sequence of plateau transitions,separating gapped states characterized by different quantized values of σxy s . At aplateau transition, level crossings take place and σxy s changes by an even integer [42].Both the origin of the gaps in quasiparticle spectra and the sequence of values forσxy s are rather different than in the normal state, i.e. in the standard Hofstadterproblem [43]. In a superconductor, the gaps are strongly affected by the pairing andthe interactions of quasiparticles with a vortex array. The sequence of σxy s changesas a function of the pairing strength (and therefore interactions), measured by themaximum value of the gap function ∆ [44].2.2 Quasiparticle excitation spectrum of a d-wavesuperconductor in the mixed stateThe experimental evidence points towards well defined d-wave quasiparticles incuprate superconductors in the absence of the external magnetic field. This suggeststhat to zeroth order fluctuations can be ignored and that one can think in terms of aneffective BCS Hamiltonian, the simplest of which is written on the 2-D tight-bindinglattice with the nearest neighbor interaction thus naturally implementing d x 2 −y 2 pairing.In question is then the response of such a superconductor to an externally appliedmagnetic field B. All high temperature superconductors are extreme type-II form-16

ing a vortex state in a wide range of magnetic fields. This immediately sets up thecontrast between B = 0 and B ≠ 0 situations: first, the problem is not spatiallyuniform and therefore momentum is not a good quantum number and second, thearray of hc vortex fluxes poses topological constraint on the quasi-particles encircling2ethe vortices. Therefore, despite ignoring any fluctuations, the problem is far fromtrivial and demands careful examination.The natural starting point is therefore the mean-field BCS Hamiltonian writtenin second quantized form [45]:∫ ( 1H = dx ψ α † (x) 2m (p − e )∗ c A)2 − µ ψ α (x)+∫ ∫dx dy[∆(x, y)ψ † ↑ (x)ψ† ↓ (y) + ∆∗ (x, y)ψ ↓ (y)ψ ↑ (x)] (2.2)where A(x) is the vector potential associated with the uniform external magneticfield B, single electron energy is measured relative to the chemical potential µ, ψ α (x)is the fermion field operator with spin index α, and ∆(x, y) is the pairing field. Forconvenience we will define an integral operator ˆ∆ such that:∫ˆ∆ψ(x) = dy∆(x, y)ψ(y). (2.3)In the strictest sense, on the mean field level this problem must be solved selfconsistentlywhich renders any analytical solution virtually intractable. On the otherhand, in the case at hand the vortex lattice is dilute for a wide range of magneticfields, and by the very nature of cuprate superconductors having short coherencelength, the size of the vortex core can be ignored relative to the distance betweenthe vortices. Thus, to the first approximation, all essential physics is captured byfixing the amplitude of the order parameter ∆ while allowing vortex defects in itsphase. Moreover, on a tight-binding lattice the vortex flux is concentrated inside theplaquette and thus the length-scale associated with the core is implicitly the latticespacing δ of the underlying tight-binding lattice. As shown in Ref.[45], under theseapproximations the d-wave pairing operator in the vortex state can be written as adifferential operator:∑ˆ∆ = ∆ 0 η δ e iφ(x)/2 e iδ·p e iφ(x)/2 . (2.4)δ17

The sums are over nearest neighbors and on the square tight-binding lattice δ =±ˆx, ±ŷ; the vortex phase fields satisfy ∇×∇φ(x) = 2πẑ ∑ i δ(x−x i) with x i denotingthe vortex positions and δ(x − x i ) being a 2D Dirac delta function; p is a momentumoperator, andη δ ={1 if δ = ±ˆx−1if δ = ±ŷ.(2.5)The operator η δ follows from the d-wave pairing: ∆ = 2∆ 0 [cos(k x δ x ) − cos(k y δ y )]. Fornotational convenience we will use units where = 1 and return to the conventionalunits when necessary.It is straightforward to derive the continuum version of the tight binding latticeoperator ˆ∆ (see Ref.[45]):ˆ∆ = 1 {∂2p 2 x , {∂ x , ∆(x)}} − 1 {∂F2p 2 y , {∂ y , ∆(x)}} +F+ i ∆(x) [ (∂8pxφ) 2 − (∂yφ) ] 2 , (2.6)2Fbut for convenience we will keep the lattice definition (2.4) throughout.One canalways define continuum as an appropriate limit of the tight-binding lattice theory.With the above definitions, the Hamiltonian (2.2) can now be written in the Nambuformalism as∫H =dx Ψ † (x) Ĥ0 Ψ(x) (2.7)where the Nambu spinor Ψ † = (ψ † ↑ , ψ ↓) and the matrix differential operatorĤ 0 =( )ĥ ˆ∆. (2.8)ˆ∆ ∗ −ĥ∗In the continuum formulation ĥ = 12m ∗ (p − e c A)2 − µ, while on the tight-bindinglattice:ĥ = −t ∑ e i x+δx (p− e c A)·dl − µ. (2.9)δt is the hopping constant and µ is the chemical potential. The equations of motionof the Nambu fields Ψ are then:i ˙Ψ = [Ψ, H] = Ĥ0Ψ. (2.10)18

2.2.1 Particle-Hole SymmetryThe equations of motion (2.10) for stationary states lead to Bogoliubov-de Gennesequations [45]Ĥ 0 Φ n = ɛ n Φ n . (2.11)The solution of these coupled differential equations are quasi-particle wavefunctionsthat are rank two spinors in the Nambu space, Φ T (r) = (u(r), v(r)).The singleparticle excitations of the system are completely specified once the quasi-particlewavefunctions are given, and as discussed later, transverse transport coefficients canbe computed solely on the basis of Φ’s. It is a general symmetry of the BdG equationsthat if (u n (r), v n (r)) is a solution with energy ɛ n , then there is always another solution(−v ∗ n (r), u∗ n (r)) with energy −ɛ n (see for example Ref. [24]).In addition, on the tight-binding lattice, if the chemical potential µ = 0 in theabove BdG Hamiltonian (2.8), then there is a particle-hole symmetry in the followingsense: if (u n (r), v n (r)) is a solution with energy ɛ n , then there is always anothersolution e iπ(rx+ry) (u n (r), v n (r)) with energy −ɛ n . Thus we can choose:((−) u n (r)v n(−) (r))((+) u= e iπ(rx+ry) n (r)v n(+) (r)), (2.12)where + (−) corresponds to a solution with positive (negative) energy eigenvalue.We will refer to this as particle hole transformation ˆP H .2.2.2 Franz-Tešanović Transformation and Translation SymmetryIn order to elucidate another important symmetry of the Hamiltonian (2.8), wefollow FT [22, 45] and perform a “bipartite” singular gauge transformation on theBogoliubov-de Gennes Hamiltonian (2.11),( )eiφ e(r)Ĥ 0 → U −1 0Ĥ 0 U, U =, (2.13)0 e −iφ h(r)19

magnetic unit cellABlδFigure 2.1: Example of A and B sublattices for the square vortex arrangement. Theunderlying tight-binding lattice, on which the electrons and holes are allowed to move,is also indicated.where φ e (r) and φ h (r) are two auxiliary vortex phase functions satisfyingφ e (r) + φ h (r) = φ(r). (2.14)This transformation eliminates the phase of the order parameter from the pairingterm of the Hamiltonian. The phase fields φ e (r) and φ h (r) can be chosen in a waythat avoids multiple valuedness of the wavefunctions. The way to accomplish this isto assign the singular part of the phase field generated by any given vortex to eitherφ e (r) or φ h (r), but not both. Physically, a vortex assigned to φ e (r) will be seen byelectrons and be invisible to holes, while vortex assigned to φ h (r) will be seen by holesand be invisible to electrons. For periodic Abrikosov vortex array, we implement theabove transformation by dividing vortices into two groups A and B, positioned at{r A i } and {rB i } respectively (see Fig. 2.1). We then define two phase fields φA (r) andφ B (r) such that∇ × ∇φ α (r) = 2πẑ ∑ iδ(r − r α i ), α = A, B, (2.15)20

and identify φ e = φ A and φ h = φ B . On the tight-binding lattice the transformedHamiltonian becomesĤ N = ∑ δ{σ 3(−teir+δ(a−σ r 3 v)·dl e iδ·p − µ ) }+ σ 1 ∆ 0 η δ e i r+δr a·dl e iδ·p(2.16)wherev = 1 2 ∇φ − e c A; a = 1 2 (∇φA − ∇φ B ), (2.17)σ 1 and σ 3 are Pauli matrices operating in Nambu space, and the sum is again overthe nearest neighbors. Note that the integrand of Eq. (2.16) is proportional to thesuperfluid velocitiesvs α = 1 m ∗ (∇φα − e A), α = A, B. (2.18)cand is therefore explicitly gauge invariant as are the off-diagonal pairing terms.From the perspective of quasiparticles v A s and vB s can be thought of as effectivevector potentials acting on electrons and holes respectively. Corresponding effectivemagnetic field seen by the quasiparticles is B α eff = − m∗ ce(∇×vα s ), and can be expressedusing Eqs. (2.15) and (2.16) asB α eff = B − φ 0 ẑ ∑ iδ(r − r α i ), α = A, B, (2.19)where B = ∇ × A is the physical magnetic field and φ 0 = hc/e is the flux quantum.We observe that quasi-electrons and quasi-holes propagate in the effective field whichconsists of (almost) uniform physical magnetic field B and an array of opposing deltafunction “spikes” of unit fluxes associated with vortex singularities. The latter aredifferent for electrons and holes. As discussed in [22, 45] this choice guarantees thatthe effective magnetic field vanishes on average, i.e. 〈B α eff 〉 = 0 since we have preciselyone flux spike (of A and B type) per flux quantum of the physical magnetic field.Flux quantization guarantees that the right hand side of Eq. (2.19) vanishes whenaveraged over a vortex lattice unit cell containing two physical vortices. It also impliesthat there must be equal numbers of A and B vortices in the system.The essential advantage of the choice with vanishing 〈B α eff 〉 is that vA s and v B s canbe chosen periodic in space with periodicity of the magnetic unit cell containing an21

integer number of electronic flux quanta hc/e. Notice that vector potential of a fieldthat does not vanish on average can never be periodic in space. Condition 〈B α eff 〉 = 0 istherefore crucial in this respect. The singular gauge transformation (2.13) thus mapsthe original Hamiltonian of fermionic quasiparticles in finite magnetic field onto anew Hamiltonian which is formally in zero average field and has only ”neutralized”singular phase windings in the off-diagonal components.The resulting new Hamiltonian now commutes with translations spanned by themagnetic unit cell i.e.[ ˆT R , ĤN] = 0, (2.20)where the translation operator ˆT R = exp(iR · p). We can therefore label eigenstateswith a “vortex” crystal momentum quantum number k and use the familiar Blochstates as the natural basis for the eigen-problem. In particular we seek the eigensolutionof the BdG equation ĤNψ = ɛψ in the Bloch form( )ψ nk (r) = e ik·r Φ nk (r) = e ik·r Unk (r), (2.21)V nk (r)where (U nk , V nk ) are periodic on the corresponding unit cell, n is a band index and kis a crystal wave vector. Bloch wavefunction Φ nk (r) satisfies the “off-diagonal” Blochequation Ĥ(k)Φ nk = ɛ nk Φ nk with the Hamiltonian of the formĤ(k) = e −ik·r Ĥ N e ik·r . (2.22)Note, that the dependence on k, which is bounded to lie in the first Brillouin zone, iscontinuous. This will become important when topological properties of spin transportare discussed.2.2.3 Vortex Lattice Inversion Symmetry and Level CrossingGeneral features of the quasi-particle spectrum can be understood on the basisof symmetry alone. Since the time-reversal symmetry is broken, the BogoliubovdeGennes Hamiltonian H 0 (2.11) must be, in general, complex. According to the“non-crossing” theorem of von Neumann and Wigner [46], a complex Hamiltonian can22

have degenerate eigenvalues unrelated to symmetry only if there are at least threeparameters which can be varied simultaneously.Since the system is two dimensional, with the vortices arranged on the lattice,there are two parameters in the HamiltonianĤ(k) (2.22): vortex crystal momentak x and k y which vary in the first Brillouin zone. Therefore, we should not expectany degeneracy to occur, in general, unless there is some symmetry which protectsit. Away from half-filling (µ ≠ 0) and with unspecified arrangement of vortices inthe magnetic unit cell there is not enough symmetry to cause degeneracy. There isonly global Bogoliubov-de Gennes symmetry relating quasi-particle energy ɛ k at somepoint k in the first Brillouin zone to −ɛ −k .In order for every quasiparticle band to be either completely below or completelyabove the Fermi energy, it is sufficient for the vortex lattice to have inversion symmetry.This can be readily seen by the following argument: Consider a vortex lattice withinversion symmetry. Then, by the very nature of the superconducting vortex carryinghc2eflux, there must be even number of vortices per magnetic unit cell and we are thenfree to choose Franz-Tesanovic labels A and B in such a way that v A (−r) = −v B (r).To see this note that the explicit form of the superfluid velocities can be written as[45]:where α = A or B and r α i∫vs α 2π(r) =m ∗d 2 k ik × ẑ(2π) 2 k 2vortex lattice has inversion symmetry then for every r A isuch that r A i∑e ik·(r−rα i ) , (2.23)idenotes the position of the vortex with label α. If the= −r B i . Therefore, under space inversion Ithere is a corresponding −r B iIv A (r) = v A (−r) = −v B (r). (2.24)Recall that the tight-binding lattice Bogoliubov-de Gennes Hamiltonian written inthe Bloch basis (2.22) reads:Ĥ(k) = ∑ { ( r+δσ 3 −tei r (a−σ 3 v)·dl e iδ·(k+p) − µ ) }+ σ 1 ∆ 0 η δ e i r+δr a·dl e iδ·(k+p)δ(2.25)wherev(r) ≡ 1 2(v A (r) + v B (r) ) ; a(r) ≡ 1 2(v A (r) − v B (r) ) . (2.26)23

As before σ 1 and σ 3 are Pauli matrices operating in Nambu space and the sum isagain over the nearest neighbors. It can be easily seen that upon applying the spaceinversion I to Ĥ(k) followed by complex conjugation C and iσ 2 we have a symmetrythat for every ɛ k there is −ɛ k , that is:−iσ 2 CI Ĥ(k) ICiσ 2 = −Ĥ(k) (2.27)which holds for every point in the Brillouin zone. Therefore, in order for the spectrumnot to be gapped, we would need band crossing at the Fermi level. But by the noncrossingtheorem this cannot happen in general. Thus, the quasi-particle spectrum ofan inversion symmetric vortex lattice is gapped, unless an external parameter otherthan k x and k y is fine-tuned.As will be established in the next section, gapped quasi-particle spectrum impliesquantization of the transverse spin conductivity σxy s as well as κ xy/T for T sufficientlylow. Precisely at half-filling (µ = 0) σxy s must vanish on the basis of particle-holesymmetry (see the next section). We can then vary the chemical potential so thatµ ≠ 0 and break particle-hole symmetry. Hence, the chemical potential µ can serveas the third parameter necessary for creating the accidental degeneracy, i.e. at somespecial values of µ ∗ the gap at the Fermi level will close (see Fig. 2.2). This resultsin a possibility of changing the quantized value of σxy s by an integer in units of /8π(Fig. 2.2) [42]. By the very nature of the superconducting state, we achieve plateaudependence on the chemical potential. This is to be contrasted to the plateaus in the“ordinary” integer quantum Hall effect which are due to the presence of disorder. Inour case, the system is clean and the plateaus are due to the magnetic field inducedgap and superconducting pairing.Similarly, we can change the strength of the electron-electron attraction, whichis proportional to the maximum value of the superconducting order parameter ∆ 0while keeping the chemical potential µ fixed. Again, as can be seen in Fig. 2.3, atsome special values ∆ ∗ 0undergoes a transition.the spectrum is gapless and the quantized Hall conductance24

σ (s)xy [h - /8π]121086420-2-4∆ m[10 -2 t]864200.0 0.2 0.4 0.6 0.8 1.0chemical potential µ[t]Figure 2.2: The mechanism for changing the quantized spin Hall conductivity isthrough exchanging the topological quanta via (“accidental”) gap closing. The upperpanel displays spin Hall conductivity σxy s as a function of the chemical potential µ. Thelower panel shows the magnetic field induced gap ∆ m in the quasiparticle spectrum.Note that the change in the spin Hall conductivity occurs precisely at those values ofchemical potential at which the gap closes. Hence the mechanism behind the changesof σxy s is the exchange of the topological quanta at the band crossings. The parametersfor the above calculation were: square vortex lattice, magnetic length l = 4δ, ∆ = 0.1tor equivalently the Dirac anisotropy α D = 10.2.3 Topological quantization of spin and thermalHall conductivitiesNote, that the Hamiltonian in Eq. (2.2) is our starting unperturbed Hamiltonian.In order to compute the linear response to externally applied perturbations we willhave to add terms to Eq. (2.2). In particular, we will consider two types of perturbationsin the later sections: First, partly for theoretical convenience, we will considera weak gradient of magnetic field (∇B) on top of the uniform B already taken intoaccount fully by Eq. (2.2). The ∇B term induces spin current in the superconductor[47]. The response is then characterized by spin conductivity tensor σ s which25

σ (s)xy [h - /8π]86420-2-4∆ m[10 -2 t]64200.0 0.1 0.2 0.3 0.4 0.5∆ 0[t]Figure 2.3: The upper panel displays spin Hall conductivity σxy s as a function of themaximum superconducting order parameter ∆ 0 . The lower panel shows the magneticfield induced gap in the quasiparticle spectrum. The change in the spin Hall conductivityoccurs at those values of ∆ 0 at which the gap closes. The parameters for theabove calculation were: square vortex lattice, magnetic length l = 4δ, µ = 2.2t.in general has non-zero off-diagonal components. Second, we consider perturbingthe system by pseudo-gravitational field, which formally induces flow of energy (see[48, 49, 50, 51]) and allows us to compute thermal conductivity κ xy via linear response.2.3.1 Spin ConductivityWithin the framework of linear-response theory [52], spin dc conductivity can berelated to the spin current-current retarded correlation function D R µν through :σµν s = lim lim − 1 ()Dµν R Ω→0 q 1 ,q 2 →0 iΩ(q 1, q 2 , Ω) − Dµν R (q 1, q 2 , 0) . (2.28)The retarded correlation function Dµν R (Ω) can in turn be related to the Matsubarafinite temperature correlation function∫ βD µν (iΩ) = − e iτΩ 〈T τ jµ(τ)j s ν(0)〉dτ s (2.29)026

aslimq 1 ,q 2 →0 DR µν(q 1 , q 2 , Ω) = D µν (iΩ → Ω + i0). (2.30)In the Eq. (2.29) the spatial average of the spin current j s (τ) is implicit, since we arelooking for dc response of spatially inhomogeneous system. In the next section wederive the spin current and evaluate the above formulae.Spin CurrentIn order to find the dc spin conductivity, we must first find the spin current. Moreprecisely, since we are looking only for the spatial average of the spin current j s (τ)we just need its k → 0 component. In direct analogy with the B = 0 situation [38],we can define the spin current by the continuity equation:˙ ρ s + ∇ · j s = 0 (2.31)where ρ s = 2 (ψ† ↑x ψ ↑x − ψ † ↓x ψ ↓x) is the spin density projected onto z-axis. We can thenuse equations of motion for the ψ fields (2.10) and compute the current density j sfrom (2.31).In the limit of q → 0 the spin current can be written as (see A.2)j s µ = 2 Ψ† V µ Ψ, (2.32)where the Nambu field Ψ † = (ψ † ↑ , ψ ↓) and the generalized velocity matrix operator V µsatisfies the following commutator identityV µ = 1i [x µ, Ĥ0]. (2.33)The equation (2.33) is a direct restatement of the fact that spin can be transported bydiffusion, i.e. it is a good quantum number in a superconductor, and that the averagevelocity of its propagation is just the group velocity of the quantum mechanical wave.In the clean limit, the transverse spin conductivity σ s xy defined in Eq. (2.28) isσxy s 2(T ) =4i∑Vymn Vxnm(f n − f m )(ɛ n − ɛ m + i0) , (2.34)2m,n27

where f m = ( 1 + exp(βɛ n ) ) −1 is the Fermi-Dirac distribution function evaluated atenergy ɛ m . For details of the derivation see A.2. The indices m and n label quantumnumbers of particular states. The matrix elements V mnµ areV mnµ = 〈 m ∣ ∣ Vµ∣ ∣n〉=∫dx (u ∗ m, v ∗ m)V µ(unv n)(2.35)where the particle-hole wavefunctions u m , v n satisfy the Bogoliubov-deGennes equation(2.11). Note that unlike the longitudinal dc conductivity, transverse conductivityis well defined even in the absence of impurity scattering. This demonstrates the factthat the transverse conductivity is not dissipative in origin.topological.In the limit of T → 0 the expression (2.34) for σ s xy becomesσ s xy = 24i∑Vxmnɛ m

not straightforward, since the quasiparticle motion is irreducibly quantum mechanical.Here we present an argument for the full quantum mechanical problem, withoutrelying on the semiclassical analysis. We show that spin conductivity tensor (2.34)vanishes at µ = 0 due to particle hole symmetry (2.12). First note that the Fermi-Dirac distribution function satisfies f(ɛ) = 1 − f(−ɛ). Therefore, the factor f m − f nchanges sign under the particle-hole transformation ˆP H (2.12) while the denominator(ɛ m − ɛ n ) 2 clearly remains unchanged. In addition, each of the matrix elements V mnµchanges sign under ˆP H . Thus the double summation over all states in Eq. (2.34)yields zero.Consequently the spin transport vanishes for a clean strongly type-II BCS d-wavesuperconductor on a tight binding lattice at half filling. Due to Wiedemann-Franzlaw, which we derive in the next section, thermal Hall conductivity also vanishes athalf filling at sufficiently low temperatures. Note that this result is independent ofthe vortex arrangement i.e. it holds even for disordered vortex array and does notrely on any approximation regarding inter- or intra- nodal scattering.Topological Nature of Spin Hall Conductivity at T=0In order to elucidate the topological nature of σxy s , we make use of the translationalsymmetry discussed in Section 2.2.2 and formally assume that the vortex arrangementis periodic. However, the detailed nature of the vortex lattice will not be specified andthus any vortex arrangement is allowed within the magnetic unit cell. The conclusionswe reach are therefore quite general.We will first rewrite the velocity matrix elements V mnµ using the singularly gaugetransformed basis as discussed in Section 2.2.2. Inserting unity in the form of the FTgauge transformation (2.13)V mnµ = 〈 m ∣ ∣ Vµ∣ ∣n〉=〈m∣ ∣U U −1 V µ U U −1∣ ∣ n〉. (2.37)The transformed basis states U −1∣ ∣ n〉can now be written in the Bloch form as eik·r ∣ ∣ nk〉and therefore the matrix element becomesV mnµ = 〈 m k∣ ∣e −ik·r U −1 V µ Ue ik·r∣ ∣ nk〉=〈mk∣ ∣Vµ (k) ∣ ∣ nk〉. (2.38)29

We used the same symbol k for both bra and ket because the crystal momentumin the first Brillouin zone is conserved. The resulting velocity operator can now besimply expressed asV µ (k) = 1 ∂Ĥ(k) , (2.39) ∂k µwhere Ĥ0(k) was defined in (2.22). Furthermore the matrix elements of the partialderivatives of Ĥ(k) can be simplified according to〈 ∣mk ∂Ĥ(k) ∣ 〉 ∣nk = (ɛn∂k k − ɛ m k )〈 ∣m k ∂n k〉= −(ɛnµ∂k k − ɛ m k )〈 ∂m k∣ 〉 ∣nk , (2.40)µ ∂k µfor m ≠ n. Utilizing Eqs. (2.39) and (2.40), Eq. (2.36) for σxy s can now be written asσxy s = ∫dk ∑ 〈∂m k∣ 〉〈 ∣ ∣nk nk ∂m k〉 〈∂m k∣ 〉〈 ∣− ∣nk nk ∂m k〉. (2.41)4i (2π) 2 ∂kɛ m

where C 1 is a first Chern number that is an integer. Therefore, a contribution of eachfilled band to σ s xy isσ s,mxy= 8π N (2.46)where N is an integer. The assumption that the band must be separated from the restof the spectrum can be relaxed. If two or more fully filled bands cross each other thesum total of their contributions to spin Hall conductance is quantized even thoughnothing guarantees the quantization of the individual contributions. The quantizationof the total spin Hall conductance requires a gap in the single particle spectrum at theFermi energy. As discussed in the Section 2.2.3, the general single particle spectrumof the d-wave superconductor in the vortex state with inversion-symmetric vortexlattice is gapped and therefore the quantization of σ s xyis guaranteed.2.3.2 Thermal ConductivityBefore discussing the nature of the quasi-particle spectrum, we will establish aWiedemann-Franz law between spin conductivity and thermal conductivity for a d-wave superconductor. This relation is naturally expected to hold for a very generalsystem in which the quasi-particles form a degenerate assembly i.e. it holds even inthe presence of elastically scattering impurities.Following Luttinger [48], and Smrčka and Středa [50] we introduce a pseudogravitationalpotential χ = x · g/c 2 into the Hamiltonian (2.7) where g is a constantvector. The purpose is to include a coupling to the energy density on the Hamiltonianlevel. This formal trick allows us to equate statistical (T ∇(1/T )) and mechanical (g)forces so that the thermal current j Q , in the long wavelength limit given byj Q = L Q (T )(T ∇ 1 )T − ∇χ , (2.47)will vanish in equilibrium. Therefore it is enough to consider only the dynamical forceg to calculate the phenomenological coefficient L Q µν. Note that thermal conductivityκ xy isκ µν (T ) = 1 T LQ µν(T ). (2.48)31

When the BCS Hamiltonian H introduced in Eq.(2.2) becomes perturbed by thepseudo-gravitational field, the resulting Hamiltonian H T has the formH T = H + F (2.49)where F incorporates the interaction with the perturbing field:F = 1 ∫dx Ψ † (x) (Ĥ0χ + χĤ0) Ψ(x). (2.50)2Since χ is a small perturbation, to the first order in χ the Hamiltonian H Twritten as∫H T =can bedx (1 + χ 2 )Ψ† (x) Ĥ0 (1 + χ )Ψ(x) (2.51)2i.e. the application of the pseudo-gravitational field results in rescaling of the fermionoperators:Ψ → ˜Ψ = (1 + χ )Ψ. (2.52)2If we measure the energy relative to the Fermi level, the transport of heat isequivalent to the transport of energy. In analogy with the Section 2.3.1, we definethe heat current j Q through diffusion of the energy-density h T . From conservation ofthe energy-density the continuity equation followsIn the limit of q → 0 the thermal current isḣ T + ∇ · j Q = 0. (2.53)jµ Q = i ( )˜Ψ† V µ˙˜Ψ − ˙˜Ψ† V µ ˜Ψ . (2.54)2For details see A.3. Note that the quantum statistical average of the current has twocontributions, both linear in χ,〈j Q µ 〉 = 〈j Q 0µ〉 + 〈j Q 1µ〉≡ − (K Q µν + M Q µν)∂ ν χ. (2.55)The first term is the usual Kubo contribution to L Q µνwhile the second term is relatedto magnetization of the sample [54] for transverse components of κ µν and vanishesfor the longitudinal components. In A.3 we show that at T = 0 the term related32

to magnetization cancels the Kubo term and therefore the transverse component ofκ µν is zero at T = 0. To obtain finite temperature response, we perform Sommerfeldexpansion and derive Wiedemann-Franz law for spin and thermal Hall conductivity.whereAs shown in the A.3L Q µν (T ) = − ( 2˜σ s xy(ξ) = 24i∑) 2 ∫Vxmnɛ m

conductivity σ s xy in units of /8π [42]. The topological nature of this phenomenon isdiscussed. The size of the magnetic field induced gap ∆ m in unconventional d-wavesuperconductors is not universal and in principle can be as large as several percentof the maximum superconducting gap ∆ 0 . By virtue of the Wiedemann-Franz law,which we derive for the d-wave Bogoliubov-de( )Gennes equation in the vortex state,2k BT σxy s as T → 0. Thus at T ≪ ∆ m thethe thermal conductivity κ xy= 4π23quantization of κ xy /T will be observable in clean samples with negligible Lande g factorand with well ordered Abrikosov vortex lattice. In conventional superconductors,the size of ∆ m is given by Matricon-Caroli-deGennes vortex bound states ∼ ∆ 2 /ɛ F .In real experimental situations, Lande g factor is not necessarily small. In fact itis close to 2 in cuprates [55]. The Zeeman effect must therefore be included in theanalysis. Furthermore, the Zeeman splitting, wherein the magnetic field acts as achemical potential for the (spinful) quasiparticles, shifts the levels that are populatedbelow the gap by ±µ spin B = ±g e4mc B or in the tight binding units by ±gπt ( a 0l) 2.In the absence of any Zeeman effect the spectrum is gapped. Since this (mini)gap∆ mis associated with the curvature terms in the continuum BdG equation, it isexpected to scale as B.Since the Zeeman effect also scales as B the topologicalquantization studied depends on the magnitude of the coefficients. While this coefficientis known for the Zeeman effect (g ≈ 2), the exact form of ∆ m is not known.However, quite generally, we expect that ∆ m increases with increasing the maximumzero field gap ∆ 0 . Therefore we could (at least theoretically) always arrange for thequantization to persist despite the competition from the Zeeman splitting.Detailed numerical examination of the quasiparticle spectra reveals that the spectrumremains gapped for some parameters even if g = 2. Thus, although the Zeemansplitting is a competing effect, in some situations it is not enough to prevent thequantization of σ s xy and consequently of κ xy/T .We have explicitly evaluated the quantized values of σ s xyon the tight-bindinglattice model of d x 2 −y2-wave BCS superconductor in the vortex state and showed thatin principle a wide range of integer values can be obtained. This should be contrastedwith the notion that the effect of a magnetic field on a d-wave superconductor is34

solely to generate a d + id state for the order parameter, as in that case σxy s = ±2 inunits of /8π. In the presence of a vortex lattice, the situation appears to be morecomplex.By varying some external parameter, for instance the strength of the electronelectronattraction which is proportional to ∆ 0 or the chemical potential µ, the gapclosing is achieved i.e. ∆ m = 0. The transition between two different values of σxysoccurs precisely when the gap closes and topological quanta are exchanged. Theremarkable new feature is the plateau dependence of σxy s on the ∆ 0 or µ. It is qualitativelydifferent from the plateaus in the ordinary integer quantum Hall effect whichare essentially due to disorder. In superconductors, the plateaus happen in a cleansystem because the gap in the quasiparticle spectrum is generated by the superconductingpairing interactions.35

Chapter 3QED 3 theory of the phasedisordered d-wave superconductors3.1 IntroductionWe will now turn our attention to the fluctuations around the broken symmetrystate. As is well known, the fluctuations of the superconducting order parameter,a U(1) field, are crucial in describing the critical behavior below the upper criticaldimension d u = 4, and destroy the long range order altogether at and below thelower critical dimension d l = 2. This can be seen by assuming the ordered stateand computing the correlations between two fields separated by large distance. In2 dimensions the correlations vanish algebraically at low temperatures due to thepresence of a soft phase mode, thereby destroying the (true) long range order. In 3dimensions the long range order persist for a finite range of temperatures.Generically, we would expect that when the measured superconducting coherencelength is long, then the mean-field description should be accurate. However, whenthe coherence length is short, we expect the fluctuations to play a significant role. Incuprate superconductors, the coherence length is very short ≈ 10Å, and moreover,they are highly anisotropic layered materials, which suggests that there are strongfluctuations of the order parameter.The question we shall try to address in this chapter is what is the nature of the36

fermionic excitations in a fluctuating d-wave superconductor. Inspired by experimentson the underdoped cuprates [8, 57], we shall assume that the long range order isdestroyed by phase fluctuations while the amplitude of the order parameter remainsfinite. Such a state, without being a superconductor, would appear to have a d-wavelikegap: a pseudogap. We shall not try to give a very detailed account of the origin ofthe phase fluctuations, rather we shall simply assume, phenomenologically, presenceor absence of the long range order. The d-wave superconductor, with well definedBCS quasiparticles, then serves as a point of departure.We thus assume that the most important effect of strong correlations at the basicmicroscopic level is to form a large pseudogap of d-wave symmetry, which is predominantlypairing in origin, i.e. arises form the particle-particle (p-p) channel. We canthen study the renormalization group fate of various residual interactions among theBCS quasiparticles, only to discover that short range interactions are irrelevant bypower counting and thus do not affect the d-wave pseudogap fixed point. Thus, if onlyshort range interactions are taken into account, the low energy BCS quasiparticlesremain well defined.However, there are important additional interactions: BCS quasiparticles coupleto the collective modes of the pairing pseudogap, in particular they couple to the(soft) phase mode. We analyze this coupling to show that while the fluctuations in thelongitudinal mode at T=0 are not sufficient to destroy the long lived low energy BCSquasiparticles, the transverse fluctuations (vortices) do introduce a novel topologicalfrustration to quasiparticle propagation. We shall argue that this frustration can beencoded by a low energy effective field theory which takes the form of (an anisotropic)QED 3 or quantum electrodynamics in 2+1 dimensions [58]. In its symmetric phase,QED 3 is governed by the interacting critical fixed point, with quasiparticles whoselifetime decays algebraically with energy, leading to a non-Fermi liquid behavior for itsfermionic excitations. This “algebraic” Fermi liquid (AFL) [58] displaces conventionalFermi liquid as the underlying theory of the pseudogap state.The AFL (symmetric QED 3 ) suffers an intrinsic instability when vortex-antivortexfluctuations and residual interactions become too strong. The topological frustrationis relieved by the spontaneous generation of mass for fermions, while the Berry37

gauge field remains massless. In the field theory literature on QED 3 this instabilityis known as the dynamical chiral symmetry breaking (CSB) and is a well-studiedand established phenomenon [59], although some uncertainty remains about its morequantitative aspects [60]. In cuprates, the region of such strong vortex fluctuationscorresponds to heavily underdoped samples. When reinterpreted in electron coordinates,the CSB leads to spontaneous creation of a whole plethora of nearly degenerateordered and gapped states from within the AFL. We shall discuss this in Section 3.6.Notably, the manifold of CSB states contains an incommensurate antiferromagneticinsulator (spin-density wave (SDW)). It is a manifestation of a remarkable richnessof the QED 3 theory that both the “algebraic” Fermi liquid and the SDW and otherCSB insulating states arise from the one and the same starting point.3.2 Vortex quasiparticle interaction3.2.1 Protectorate of the pairing pseudogapThe starting point is the assumption that the pseudogap is predominantly particleparticleor pairing (p-p) in origin and that it has a d x 2 −y2 symmetry. This assumptionis given mathematical expression in the partition function:∫ ∫ ∫Z = DΨ † (r, τ) DΨ(r, τ) Dϕ(r, τ) exp [−S],∫ ∫S = dτ d 2 r{Ψ † ∂ τ Ψ + Ψ † HΨ + (1/g)∆ ∗ ∆}, (3.1)where τ is the imaginary time, r = (x, y), g is an effective coupling constant in thed x 2 −y 2 channel, and Ψ† = ( ¯ψ ↑ , ψ ↓ ) are the standard Grassmann variables representingcoherent states of the Bogoliubov-de Gennes (BdG) fermions. The effective HamiltonianH is given by:H =(Ĥe ˆ∆ˆ∆ ∗−Ĥ∗ e)+ H res (3.2)38

with Ĥe = 12m (ˆp − e c A)2 − ɛ F , ˆp = −i∇ (we take = 1), and ˆ∆ the d-wave pairingoperator (see Eq. 2.6),ˆ∆ = 1 {ˆpkF2 x , {ˆp y , ∆}} −i4k 2 F∆ ( ∂ x ∂ y ϕ ) , (3.3)where ∆(r, τ) = |∆| exp(iϕ(r, τ)) is the center-of-mass gap function and {a, b} ≡(ab + ba)/2. As discussed in Chapter 2, the second term in Eq. (3.3) is necessaryto preserve the overall gauge invariance. Notice that we have rotated ˆ∆ from d x 2 −y 2to d xy to simplify the continuum limit. ∫ Dϕ(r, τ) denotes the integral over smooth(“spin wave”) and singular (vortex) phase fluctuations. Amplitude fluctuations of ˆ∆are suppressed at or just below T ∗ and the amplitude itself is frozen at 2∆ ∼ 3.56T ∗for T ≪ T ∗ .The fermion fields ψ ↑ and ψ ↓ appearing in Eqs. (3.1,3.2) do not necessarily referto the bare electrons.Rather, they represent some effective low-energy fermionsof the theory, already fully renormalized by high-energy interactions, expected tobe strong in cuprates due to Mott-Hubbard correlations near half-filling [61]. Theprecise structure of such fermionic effective fields follows from a more microscopictheory and is not of our immediate concern here – we are only relying on the absenceof true spin-charge separation which allows us to write the effective pairing term (3.2)in the BCS-like form. The experimental evidence that supports this reasoning, at leastwithin the superconducting state and its immediate vicinity, is rather overwhelming[8, 57, 62, 63]. Furthermore, H res represents the “residual” interactions, i.e. the partdominated by the effective interactions in the p-h channel. Our main assumption isequivalent to stating that such interactions are in a certain sense “weak” and lessimportant part of the effective Hamiltonian H than the large pairing interactionsalready incorporated through ˆ∆. This notion of “weakness” of H res will be definedmore precisely and with it the region of validity of our theory.3.2.2 Topological frustrationThe global U(1) gauge invariance mandates that the partition function (3.1) mustbe independent of the overall choice of phase for ˆ∆. We should therefore aim to elimi-39

nate the phase ϕ(r, τ) from the pairing term (3.2) in favor of ∂ µ ϕ terms [µ = (x, y, τ)]in the fermionic action. For the regular (“spin-wave”) piece of ϕ this is easily accomplishedby absorbing a phase factor exp(i 1 ϕ) into both spin-up and spin-down2fermionic fields. This amounts to “screening” the phase of ∆(r, τ) (“XY phase”) by a“half-phase” field (“half-XY phase”) attached to ψ ↑ and ψ ↓ . However, as discussed inthe Chapter 2, when dealing with the singular part of ϕ, such transformation “screens”physical singly quantized hc/2e superconducting vortices with “half-vortices” in thefermionic fields. Consequently, this “half-angle” gauge transformation must be accompaniedby branch cuts in the fermionic fields which originate and terminate atvortex positions and across which the quasiparticle wavefunction must switch its sign.These branch cuts are mathematical manifestation of a fundamental physical effect:in presence of hc/2e vortices the motion of quasiparticles is topologically frustratedsince the natural elementary flux associated with the quasiparticles is hc/e i.e. twiceas large . The physics of this topological frustration is at the origin of all non-trivialdynamics discussed in this work.Dealing with branch cuts in a fluctuating vortex problem is very difficult due totheir non-local character and defeats the original purpose of reducing the problemto that of fermions interacting with local fluctuating superflow field, i.e. with ∂ µ ϕ.Instead, in order to avoid the branch cuts, non-locality and non-single valued wavefunctions,we employ the singular gauge transformation introduced in the Chapter 2,hereafter referred to as ‘FT’ transformation:¯ψ ↑ → exp (−iϕ A ) ¯ψ ↑ , ¯ψ↓ → exp (−iϕ B ) ¯ψ ↓ , (3.4)where ϕ A + ϕ B = ϕ. Here ϕ A(B) is the singular part of the phase due to A(B) vortexdefects:∇ × ∇ϕ A(B) = 2πẑ ∑ i(q i δr − r A(B)i)(τ) , (3.5)with q i = ±1 denoting the topological charge of i-th vortex and r A(B)i (τ) its position.The labels A and B represent some convenient but otherwise arbitrary division of vortexdefects (loops or lines in ϕ(r, τ)) into two sets. The transformation (3.4) “screens”the original superconducting phase ϕ (or “XY phase”) with two ordinary “XY phases”40

ϕ A and ϕ B attached to fermions. Both ϕ A and ϕ B are simply phase configurations of∆(r, τ) but with fewer vortex defects. The key feature of the transformation (3.4) isthat it accomplishes “screening” of the physical hc/2e vortices without introducingany branch cuts into the wavefunctions. The topological frustration still remains, butnow it is expressed through local fields. The resulting fermionic part of the action isthen:L ′ = ¯ψ ↑ [∂ τ + i(∂ τ ϕ A )]ψ ↑ + ¯ψ ↓ [∂ τ + i(∂ τ ϕ B )]ψ ↓ + Ψ † H ′ Ψ, (3.6)where the transformed Hamiltonian H ′ is:(1(ˆπ + )2m v)2 − ɛ FˆDˆD − 1 (ˆπ − 2m v)2 + ɛ Fwith ˆD = (∆ 0 /2k 2 F )(ˆπ xˆπ y + ˆπ yˆπ x ) and ˆπ = ˆp + a.The singular gauge transformation (3.4) generates a “Berry” gauge potentiala µ = 1 2 (∂ µϕ A − ∂ µ ϕ B ) , (3.7)which describes half-flux Aharonov-Bohm scattering of quasiparticles on vortices.a µcouples to BdG fermions minimally and mimics the effect of branch cuts inquasiparticle-vortex dynamics. Closed fermion loops in the Feynman path-integralrepresentation of (3.1) acquire the (−1) phase factors due to this half-flux Aharonov-Bohm effect just as they would from a branch cut attached to a vortex defect.The above (±1) prefactors of the BdG fermion loops come on top of general andever-present U(1) phase factors exp(iδ), where the phase δ depends on spacetimeconfiguration of vortices. These U(1) phase factors are supplied by the “Doppler”gauge fieldv µ = 1 2 (∂ µϕ A + ∂ µ ϕ B ) ≡ 1 2 ∂ µϕ , (3.8)which denotes the classical part of the quasiparticle-vortex interaction. The couplingof v µ to fermions is the same as that of the usual electromagnetic gauge field A µand is therefore non-minimal, due to the pairing term in the original Hamiltonian H(3.2). It is this non-minimal interaction with v µ , which we call the Meissner coupling,that is responsible for the U(1) phase factors exp(iδ). These U(1) phase factors are,41

“random”, in the sense that they are not topological in nature – their values dependon detailed distribution of superfluid fields of all vortices and “spin-waves” as well ason the internal structure of BdG fermion loops, i.e. what is the sequence of spin-upand spin-down portions along such loops. In this respect, while its minimal couplingto BdG fermions means that, within the lattice d-wave superconductor model, oneis naturally tempted to represent the Berry gauge field a µ as a compact U(1) gaugefield, the Doppler gauge field v µ is decidedly non-compact, lattice or no lattice [64]. a µand v µ as defined by Eqs. (3.7,3.8) are not independent since the discrete spacetimeconfigurations of vortex defects {r i (τ)} serve as sources for both.All choices of the sets A and B in transformation (3.4) are completely equivalent– different choices represent different singular gauges and v µ , and therefore exp(iδ),are invariant under such transformations. a µ , on the other hand, changes but onlythrough the introduction of (±) unit Aharonov-Bohm fluxes at locations of thosevortex defects involved in the transformation. Consequently, the Z 2 style (±1) phasefactors associated with a µ that multiply the fermion loops remain unchanged. In orderto symmetrize the partition function with respect to this singular gauge, we define ageneralized transformation (3.4) as the sum over all possible choices of A and B, i.e.,over the entire family of singular gauge transformations. This is an Ising sum with 2 N lmembers, where N l is the total number of vortex defects in ϕ(r, τ), and is itself yetanother choice of the singular gauge. The many benefits of such symmetrized gaugewill be discussed shortly but we stress here that its ultimate function is calculationalconvenience. What actually matters for the physics is that the original ϕ be splitinto two XY phases so that the vortex defects of every distinct topological class areapportioned equally between ϕ A and ϕ B (3.4) [58].The above symmetrization leads to the new partition function: Z → ˜Z:∫˜Z =D ˜Ψ † ∫∫D ˜ΨDv µ∫∫ β ∫Da µ exp [− dτ0d 2 r ˜L] , (3.9)in which the half-flux-to-minus-half-flux (Z 2 ) symmetry of the singular gauge transformation(3.4) is now manifest:˜L = ˜Ψ † [(∂ τ + ia τ )σ 0 + iv τ σ 3 ] ˜Ψ + ˜Ψ † H ′ ˜Ψ + L0 [v µ , a µ ], (3.10)42

where L 0 is the “Jacobian” of the transformation given by∑∫e − β 0 dτ d2 rL 0= 2 −N lDϕ(r, τ) (3.11)×δ[v µ − 1(∂ 2 µϕ A + ∂ µ ϕ B )]δ[a µ − 1(∂ 2 µϕ A − ∂ µ ϕ B )].Here σ µ are the Pauli matrices, β = 1/T , H ′ is given in Eq. (3.6) and, for later convenience,L 0 will also include the energetics of vortex core overlap driven by amplitudefluctuations and thus independent of long range superflow (and of A). We call thetransformed quasiparticles ˜Ψ † = ( ¯˜ψ↑ , ˜ψ ↓ ) appearing in (3.10) “topological fermions”(TF). TF are the natural fermionic excitations of the pseudogapped normal state.They are electrically neutral and are related to the original quasiparticles by theinversion of transformation (3.4).By recasting the original problem in terms of topological fermions we have accomplishedour original goal: the interactions between quasiparticles and vortices arenow described solely in terms of two local superflow fields [58]:v Aµ = ∂ µ ϕ A ; v Bµ = ∂ µ ϕ B , (3.12)which we can think of as superfluid velocities associated with the phase configurationsϕ A(B) (r, τ) of a (2+1)-dimensional XY model with periodic boundary conditions alongthe τ-axis. Our Doppler and Berry gauge fields v µ and a µ are linear combinationsof v Aµ and v Bµ . Note that a µ is produced exclusively by vortex defects since the“spin-wave” configurations of ϕ can be fully absorbed into v µ . All that remains is toperform the sum in (3.9) over all the “spin-wave” fluctuations and all the spacetimeconfigurations of vortex defects {r i (τ)} of this (2+1)-dimensional XY model.A,B3.2.3 “Coarse-grained” Doppler and Berry U(1) gauge fieldsThe exact integration over the phase ϕ(r, τ) is prohibitively difficult. To proceedby analytic means we devise an approximate procedure to integrate over vortexantivortexpositions {r i (τ)} in (3.9) which will capture qualitative features of thelong-distance, low-energy physics of the original problem. A hint of how to devisesuch an approximation comes from examining the role of the Doppler gauge field v µ43

in the physics of this problem. For simplicity, we consider the finite T case wherewe can ignore the τ-dependence of ϕ(r, τ). The results are easily generalized, withappropriate modifications, to include quantum fluctuations.We start by noting that V s = 2v − (2e/c)A is just the physical superfluid velocity[65], invariant under both A ↔ B singular gauge transformations (3.4) andordinary electromagnetic U(1) gauge symmetry. The superfluid velocity (Doppler)field, swirling around each (anti)vortex defect, is responsible for the vast majority ofphenomena that are associated with vortices: long range interactions between vortexdefects, coupling to external magnetic field and the Abrikosov lattice of the mixedstate, Kosterlitz-Thouless transition, etc. Remarkably, its role is essential even forthe physics discussed here, although it now occupies a supporting role relative tothe Berry gauge field a µ .Note that if a (a µ ) were absent – then, upon integrationover topological fermions in (3.10), we obtain the following term in the effectiveLagrangian:M 2 (∇ϕ − 2ec A ) 2+ (· · · ) , (3.13)where (· · · ) denotes higher order powers and derivatives of 2∇ϕ − (2e/c)A. In theabove we have replaced v → (1/2)∇ϕ to emphasize that the leading term, withthe coefficient M 2 proportional to the bare superfluid density, is just the standardsuperfluid-velocity-squared term of the continuum XY model – the notation M 2 forthe coefficient will become clear in a moment. We can now write ∇ϕ = ∇ϕ vortex +∇ϕ spin−wave and re-express the transverse portion of (3.13) in terms of (anti)vortexpositions {r i } to obtain a familiar form:→ M 2 ∑ln |r i − r j | (3.14)2π〈i,j〉or, by using ∇ · ∇ϕ vortex = 0 and ∇ × ∇ϕ vortex = 2πρ(r), equivalently as:→ M 2 ∫ ∫d 2 r d 2 r ′ ρ(r)ρ(r ′ ) ln |r − r ′ | , (3.15)2πwhere ρ(r) = 2π ∑ i q iδ(r − r i ) is the vortex density.The Meissner coupling of v to fermions is very strong – it leads to familiar longrange interactions between vortices which constrain vortex fluctuations to a remarkabledegree. To make this statement mathematically explicit we introduce the Fourier44

transform of the vortex density ρ(q) = ∑ i exp(iq · r i) and observe that its variancesatisfies:〈ρ(q)ρ(−q)〉 ∝ q 2 /M 2 . (3.16)Vortex defects form an incompressible liquid – the long distance vorticity fluctuationsare strongly suppressed. This “incompressibility constraint” is naturally enforced in(3.9) by replacing the integral over discrete vortex positions {r i } with the integralover continuously distributed field ¯ρ(r) with 〈¯ρ(r)〉 = 0.The Kosterlitz-Thoulesstransition and other vortex phenomenology are still maintained in the non-trivialstructure of L 0 [¯ρ(r)]. But the long wavelength form of (3.13) now reads:M 2 (2¯v − 2ec A ) 2+ (· · · ) , (3.17)and can be interpreted as a massive action for a U(1) gauge field ¯v. The latter is thecoarse grained Doppler gauge field defined by∇ × ¯v(r) = π¯ρ(r) . (3.18)The coarse-graining procedure has made v into a massive U(1) gauge field ¯v whoseinfluence on TF disappears in the long wavelength limit. We can therefore drop itfrom low energy fermiology.The full problem also contains the Berry gauge field a (a µ ) which now must berestored. However, having replaced the Doppler field v (defined by discrete vortexsources) with the distributed quantity (3.18), we cannot simply continue to keep a (a µ )specified by half-flux Dirac (Aharonov-Bohm) strings located at discrete vortex positions{r i }. Instead, a (a µ ) must be replaced by a new gauge field which reflects the“coarse-graining” that has been applied to v – simply put, the Z 2 -valued Berry gaugefield (3.7) of the original problem (3.10) must be “dressed” in such a way as to bestcompensate for the error introduced by “coarse-graining” v.Note that if we insist on replacing v by its “coarse-grained” form (3.18), the onlyway to achieve this is to “coarse-grain” both v A and v B in the same manner:∇ × v A → 2πρ A ; ∇ × v B → 2πρ B , (3.19)45

where ρ A,B (r) are now continuously distributed densities of A(B) vortex defects (thereader should contrast this with (3.7,3.8)). This is because the elementary vortexvariables of our problem are not the sources of v and a; rather, they are the sourcesof v A and v B . We cannot separately fluctuate or “coarse-grain” the Doppler andthe Berry “halves” of a given vortex defect – they are permanently confined into aphysical (hc/2e) vortex. On the other hand, we can independently fluctuate A and Bvortices – this is why it was important to use the singular gauge (3.4) to rewrite theoriginal problem solely in terms of A and B vortices and associated superflow fields(3.12).We can now reassemble the coarse-grained Doppler and Berry gauge fields as:v = 1 2 (v A + v B ) ; a = 1 2 (v A − v B ) , (3.20)with the straightforward generalization to (2+1)D:v = 1 2 (v A + v B ) ; a = 1 2 (v A − v B ) . (3.21)The coupling of v A (v A ) and v B (v B ) to fermions is a hybrid of Meissner and minimalcoupling [64]. They contribute a product of U(1) phase factors exp(iδ A ) × exp(iδ B )to the BdG fermion loops with both exp(iδ A ) and exp(iδ B ) being “random” in thesense of previous subsection. Upon coarse-graining v A (v A ) and v B (v B ) turn intonon-compact U(1) gauge fields and therefore v(v) and a(a) must as well.3.2.4 Further remarks on FT gaugeThe above “coarse-grained” theory must have the following symmetry: it has tobe invariant under the exchange of spin-up and spin-down labels, ψ ↑ ↔ ψ ↓ , withoutany changes in v (v). This symmetry insures that (an arbitrarily preselected) S zcomponent of the spin, which is the same for TF and real electrons, decouples fromthe physical superfluid velocity which naturally must couple only to charge. Whendealing with discrete vortex defects this symmetry is guaranteed by the singular gaugesymmetry defined by the family of transformations (3.4). However, if we replace v Aand v B by their distributed “coarse-grained” versions, the said symmetry is preserved46

only in the FT gauge. This is seen by considering the effective Lagrangian expressedin terms of coarse-grained quantities:L → ˜Ψ † ∂ τ ˜Ψ + ˜Ψ† H ′ ˜Ψ + L0 [ρ A , ρ B ] , (3.22)where H ′ is given by (3.6), v A , v B are connected to ρ A , ρ B via (3.19), and L 0 isindependent of τ.The problem lurks in L 0 [ρ A , ρ B ] – this is just the entropy functional of fluctuatingfree A(B) vortex-antivortex defects and has the following symmetry: ρ A(B) → −ρ A(B)with ρ B(A) kept unchanged. This symmetry reflects the fact that the entropic “interactions”do not depend on vorticity. Above the Kosterlitz-Thouless transition we canexpand:L 0 → K A4 (∇ × v A) 2 + K B4 (∇ × v B) 2 + (· · · ), (3.23)where the ellipsis denote higher order terms and the coefficients K −1A(B) → nA(B) lAppendix B.1 for details).The above discussed symmetry of Hamiltonian H ′ (3.6) demands that v and a(3.20,3.21) be the natural choice for independent distributed vortex fluctuation gaugefields which should appear in our ultimate effective theory. L 0 , however, collides withthis symmetry of H ′ – if we replace v A(B) → v ± a in (3.23), we realize that v and aare coupled through L 0 in the general case K A ≠ K B :(seeL 0 → K A + K B4((∇ × v) 2 + (∇ × a) 2)+ K A − K B(∇ × a) · (∇ × v) . (3.24)2Therefore, via its coupling to a, the superfluid velocity v couples to the “spin” oftopological fermions and ultimately to the true spin of the real electrons.This isan unacceptable feature for the effective theory and seriously handicaps the general“A − B” gauge, in which the original phase is split into ϕ → ϕ A + ϕ B , with ϕ A(B)each containing some arbitrary fraction of the original vortex defects. In contrast, thesymmetrized transformation (3.4) which apportions vortex defects equally between ϕ Aand ϕ B [58] leads to K A = K B and to decoupling of v and a at quadratic order, thuseliminating the problem at its root. Furthermore, even if we start with the general47

“A − B” gauge, the renormalization of L 0 arising from integration over fermionswill ultimately drive K A − K B → 0 and make the coupling of v and a irrelevant inthe RG sense [67]. This argument is actually quite rigorous in the case of quantumfluctuations where the symmetrized gauge (3.4) represents a fixed point in the RGanalysis [58]. Consequently, it appears that the symmetrized singular transformation(3.4) employed in (3.10) is the preferred gauge for the construction of the effectivelow-energy theory. In this respect, while all the singular A − B gauges are createdequal some are ultimately more equal than others.The above discussion provides the rationale behind using the FT transformation inour quest for the effective theory. At low energies, the interactions between quasiparticlesand vortices are represented by two U(1) gauge fields v and a (3.20,3.21). Theconversion of a from a Z 2 -valued to a non-compact U(1) field with Maxwellian actionis effected by the confinement of the Doppler to the Berry half of a singly quantizedvortex – in the coarse-graining process the phase factors exp(iδ) of the non-compactDoppler part “contaminate” the original (±1) factors supplied by a (3.7). This contaminationdiminishes as doping x → 0 since then v F /v ∆ → 0 and “vortices” areeffectively liberated of their Doppler content. In this limit, the pure Z 2 nature of ais recovered and one enters the realm of the Z 2 gauge theory of Senthil and Fisher[68]. In contrast, in the pairing pseudogap regime of this paper where v F /v ∆ > 1and singly quantized (hc/2e) vortices appear to be the relevant excitations [69], weexpect the effective theory to take the U(1) form described by v and a (3.20,3.21).3.2.5 Maxwellian form of the Jacobian L 0 [v µ , a µ ]We shall now derive an expression for the long distance, low energy form of the“Jacobian” L 0 [v µ , a µ ] (see Eqn. 3.11) which serves as the “bare action” for the gaugefields v µ and a µ of our effective theory. As shown below, this form is a non-compactMaxwellian whose stiffness K (or inverse “charge” 1/e 2 = K) stands in intimaterelation to the helicity modulus tensor of a dSC and, in the pseudogap regime ofstrong superconducting fluctuations, can be expressed in terms of a finite physicalsuperconducting correlation length ξ sc , K ∝ ξsc 2 (2D); K ∝ ξ sc ((2+1)D). Upon48

entering the superconducting phase and ξ sc → ∞, K → ∞ as well (or e 2 → 0),implying that v µ and a µ have become massive. The straightforward relationshipbetween the massless (or massive) character of L 0 [v µ , a µ ] and the superconductingphase disorder (or order) is a consequence of rather general physical and symmetryprinciples.For the sake of simplicity, but without loss of generality, we consider first anexample of an s-wave superconductor with a large gap ∆ extending over all of theFermi surface. The action takes the form similar to (3.10):˜L = ˜Ψ † [(∂ τ + ia τ )σ 0 + iv τ σ 3 ] ˜Ψ + ˜Ψ † H s ′ ˜Ψ + L 0 [v µ , a µ ], (3.25)but with H ′ sdefined as:(1(ˆπ + )2m v)2 − ɛ F ∆∆ − 1 (ˆπ − 2m v)2 + ɛ F, (3.26)with ˆπ = ˆp + a. When the the electromagnetic field is included, v µ in the fermionicpart action is substituted by v µ − (e/c)A µ , while there is no substitution in L 0 .Thermal vortex-antivortex fluctuationsIn the language of BdG fermions the system (3.25) is a large gap “semiconductor”and the Berry gauge field couples to it minimally through “BdG” vector and scalarpotentials a and a τ . Such BdG semiconductor is a poor dielectric diamagnet withrespect to a µ . We proceed to ignore its “diamagnetic susceptibility” and also set a τ =0 to concentrate on thermal fluctuations. The Berry gauge field part of the couplingbetween quasiparticles and vortices in a large gap s-wave superconductor influencesthe vortices only through weak short-range interactions which are unimportant inthe region of strong vortex fluctuations near Kosterlitz-Thouless transition. We cantherefore drop a from the fermionic part of the action and integrate over it to obtainL 0 in terms of physical vorticity ρ(r) = (∇ × v)/π. Additional integration over thefermions produces the effective free energy functional for vortices:F[ρ] = M 2 (2v − 2ec Aext ) 2 + (· · · ) + L 0 [ρ] , (3.27)49

where we have used our earlier notation and have introduced a small external transversevector potential A ext . The ellipsis denotes higher order contributions to vortexinteractions. As discussed earlier in this Section, the familiar long range interactionsbetween vortices lead directly to the standard Coulomb gas representation of thevortex-antivortex fluctuation problem and Kosterlitz-Thouless transition.The presence of these long range interactions implies that the vortex system isincompressible (3.16) and long distance vortex density fluctuations are suppressed.When studying the coupling of BdG quasiparticles to these fluctuations it thereforesuffices to expand the “entropic” part:L 0 [ρ] ∼ = 1 2 π2 Kδρ 2 + · · · = 1 2 K(∇ × v)2 + . . . . (3.28)The above expansion is justified above T c since we know that at T ≫ T c we mustmatch the purely entropic form of a non-interacting particle system, L 0 ∝ δρ 2 (seeAppendix B.1).To uncover the physical meaning of the coefficient K we expand the free energyF of the vortex system to second order in A ext . In the pseudogap state the gaugeinvariance demands that F depends only on ∇ × A ext :∫F [∇ × A ext ] = F [0] + (2e)22c χ d 2 r(∇ × A ext ) 2 + . . . (3.29)2Note that ((2e) 2 /c 2 )χ is just the diamagnetic susceptibility in the pseudogap state.χ determines the long wavelength form of the helicity modulus tensor Υ µν (q) definedas:δ 2 F ∣Υ µν (q) = Ω∣AδA ext µ (q)δAext ν (−q) ext. (3.30)µ →0The above is the more general form of Υ µν (q) applicable to uniaxially symmetric 3Dand (2+1)D XY or Ginzburg-Landau models; in 2D only Υ xy (q) appears. In the longwavelength limit Υ µν (q) vanishes as q 2 χ:c 2(2e) 2 Υ µν(q) = χɛ ραµ ɛ ρβν q α q β + . . . , (3.31)for the isotropic case, while for the anisotropic situation χ ⊥ ≠ χ ‖ :c 2(2e) 2 Υ µν(q) = (χ ‖ − χ ⊥ )ɛ zαµ ɛ zβν q α q β + χ ⊥ ɛ ραµ ɛ ρβν q α q β . (3.32)50

ɛ αβγ is the Levi-Civita symbol, summation over repeated indices is understood andindex z of the anisotropic 3D GL or XY model is replaced by τ for the (2+1)D case.(2e) 2 /c 2 is factored out for later convenience.Let us compute the helicity modulus of the problem explicitly. This is done byabsorbing the small transverse vector potential A ext into v in (3.25,3.6) and integratingover the new variable v − (e/c)A ext . The hecility modulus tensor measuresthe screening properties of the vortex system. In a superconductor, with topologicaldefects bound in vortex-antivortex dipoles, there is no screening at long distances.This translates into Meissner effect for A ext . When the dipoles unbind and some freevortex-antivortex excitations appear the screening is now possible over all lengthscalesand there is no Meissner effect for A ext . The information on presence or absence ofsuch screening is actually stored entirely in L 0 , where A ext re-emerges after the abovechange of variables. We finally obtain:4χ = K − K2 π 2T∫lim|q|→0 +d 2 re iq·r 〈δρ(r)δρ(0)〉 , (3.33)where the thermal average 〈· · · 〉 is over the free energy (3.27) with A ext= 0 or,equivalently, (3.25). In the normal phase only the first term contributes in the longwavelength limit, the second being down by an extra power of q 2 courtesy of longrange vortex interactions. Consequently, K = 4χ (the factor of 4 is due to the factthat the true superfluid velocity is 2v [65]).We see that in the pseudogap phase, with free vortex defects available to screen,L 0 [v] takes on a massless Maxwellian form, the stiffness of which is given by thediamagnetic susceptibility of a strongly fluctuating superconductor. In the fluctuationregion χ is given by the superconducting correlation length ξ sc [70]:K = 4χ = 4C 2 T ξ 2 sc , (3.34)where C 2 is a numerical constant, intrinsic to a 2D GL, XY or some other model ofsuperconducting fluctuations.As we approach T c , ξ sc → ∞ and the stiffness of Maxwellian term (3.28) diverges.This can be interpreted as the Doppler gauge field becoming massive. Indeed, immediatelybellow the Kosterlitz-Thouless transition at T c , L 0∼ = m2v v 2 + . . . , where51

m v ≪ M and m v → 0 as T → T − c . This is just a reflection of the helicity modulustensor now becoming finite in the long wavelength limit,Υ xy → 4e2 m 2 v M 2.c 2 4M 2 + m 2 vTopological defects are now bound in vortex-antivortex pairs and cannot screen resultingin the Meissner effect for A ext . The system is a superconductor and v hadbecome massive.Returning to a d-wave superconductor, we can retrace the steps in the aboveanalysis but we must replace H ′ s in (3.25) with H′ (3.6). Now, instead of a large gapBdG “semiconductor”, we are dealing with a narrow gap “semiconductor” or BdG“semimetal” because of the low energy nodal quasiparticles.This means that wemust restore the Berry gauge field a to the fermionic action since the contributionfrom nodal quasiparticles makes its BdG “diamagnetic susceptibility” very large,χ BdG ∼ 1/T ≫ 1/T ∗ (see the next Section). The long distance fluctuations of vand a are now both strongly suppressed, the former through incompressibility of thevortex system and the latter through χ BdG . This allows us to expand (3.23):L 0∼ =K A4 (∇ × v A) 2 + K B4 (∇ × v B) 2 + (· · · ) , (3.35)where K A = K B = K is mandated by the FT singular gauge. Since in our gauge thefermion spin and charge channels decouple A ext still couples only to v and the abovearguments connecting K to the helicity modulus and diamagnetic susceptibility χfollow through. This finally gives the Maxwellian form of Ref. [58]:L 0 → K 2 (∇ × v)2 + K 2 (∇ × a)2 , (3.36)where K is still given by Eq. (3.34). Note however that ξ sc of a d-wave and of ans-wave superconductor are two rather different functions of T , x and other parametersof the problem, due to strong Berry gauge field renormalizations of vortex interactionsin the d-wave case. Nonetheless, as T → T c , the Kosterlitz-Thouless critical behaviorremains unaffected since χ BdG , while large, is still finite at all finite T .To summarize, we have shown that L 0 [ρ A , ρ B ] = L 0 [v, a] in the pseudogap statetakes on the massless Maxwellian form (3.36), with the stiffness K set by the true52

superconducting correlation length ξ sc (3.34). This result holds as a general feature ofour theory irrespective of whether one employs a Ginzburg-Landau theory, XY model,vortex-antivortex Coulomb plasma, or any other description of strongly fluctuatingdSC, as long as such description properly takes into account vortex-antivortex fluctuationsand reproduces Kosterlitz-Thouless phenomenology. In the Appendix B.1 weshow that within the continuum vortex-antivortex Coulomb plasma model:L 0 →T2π 2 n l[(∇ × v) 2 + (∇ × a) 2] , (3.37)where n l is the average density of free vortex and antivortex defects. Comparisonwith (3.36) allows us to identify ξ −2sc ↔ 4π 2 C 2 n l [58].Quantum fluctuations of (2+1)D vortex loopsThe above results can be generalized to quantum fluctuations of spacetime vortexloops. The superflow fields v A(B)µ satisfy (∂ ×v A(B) ) µ = 2πj A(B)µ , where j A(B)µ (x) arethe coarse-grained vorticities associated with A(B) vortex defects and 〈j A(B)µ 〉 = 0.The topology of vortex loops dictates that j A(B)µ (x) be a purely transverse field, i.e.∂ · j A(B) = 0, reflecting the fact that loops have no starting nor ending point. Again,we begin with a large gap s-wave superconductor and use its poor BdG diamagnetic/dielectricnature to justify dropping the Berry gauge field a µ from the fermionicpart of (3.25) and integrating over a µ in the “entropic” part containing L 0 [v µ , a µ ].The integration over the fermions contains an important novelty specific to the(2+1)D case: the appearance of the Berry phase terms for quantum vortices as theywind around fermions. Such Berry phase is the consequence of the first order timederivative in the original fermionic action (3.1). If we think of spacetime vortex loopsas worldlines of some relativistic quantum bosons dual to the Cooper pair field ∆(r, τ),as we do in the Appendix B.1, then these bosons see Cooper pairs and quasiparticlesas sources of “magnetic” flux [71]. At the mean field level, this translates into a dual“Abrikosov lattice” or a Wigner crystal of holes in a dual superfluid. Accordingly,the non-superconducting ground state in the pseudogap regime will likely contain aweak charge modulation. We will ignore it for the rest of this paper. This is justifiedby the fact that a µ does not couple to charge directly.53

Hereafter, we assume that the transition from a dSC into a pseudogap phaseproceeds via unbinding of vortex loops of a (2+1)D XY model or its GL counterpartor, equivalently, an anisotropic 3D XY or GL model where the role of imaginary timeis taken on by a third spatial axis z [72]. We can now integrate out the fermions toobtain the effective Lagrangian for coarse-grained spacetime loops (πj µ = (∂ × v) µ ):L[j µ ] = M 2 µ (2v µ − 2ec Aext µ )2 + (· · · ) + L 0 [j µ ] , (3.38)where M x = M y = M and M τ = M/c s , with c s ∼ v Fbeing the effective “speed oflight” in the vortex loop spacetime. The incompressibility condition reads 〈j µ (q)j ν (−q)〉 ∼q 2 [δ µν − (q µ q ν /q 2 )] and in the pseudogap state permits the expansion:L 0 [j µ ] ∼ = 1 2 π2 K τ jτ 2 + 1 ∑2 π2 K i ji 2 , (3.39)where K x = K y = K ≠ K τ . Using the analogy with the uniaxially symmetricanisotropic 3D XY (or GL) model we can expand the ground state energy in themanner of (3.29):E[∂ × A ext ] = E[0] + (2e)22c 2with χ ⊥ = χ x = χ y = χ and χ ‖ = χ τi=x,y∑∫χ ⊥,‖⊥,‖d 3 x(∂ × A ext ) 2 ⊥,‖ , (3.40)≠ χ. The form of L 0 (3.39) follows directlyfrom the requirement that there are infinitely large vortex loops, resulting in vorticityfluctuations over all distances. χ and χ τsusceptibilities of this insulating pseudogap state.determine the diamagnetic and dielectricThe explicit computation of Υ µν (q) (3.30,3.32) leads to:∫4χ i,τ = K i,τ − Ki,τ 2 π2 lim d 3 xe iq·x 〈j(x) i,τ j(0) i,τ 〉 , (3.41)q→0 +where the second term is again eliminated by the incompressibility of the vortexsystem. This results in:K = 4χ = 4C 3 ξ τ ; K τ = 4χ τ = 4C 3ξ 2 scξ τ, (3.42)where we used the result for the anisotropic 3D XY or GL models: χ ⊥ = C 3 T ξ ‖ ,χ ‖ = C 3 T ξ 2 ⊥ /ξ ‖, C 3 being the intrinsic numerical constant for those models (C 3 ≠ C 2 ).54

In the case of (2+1)D vortex loops ξ τ ∝ ξ sc since our adopted model has the dynamicalcritical exponent z = 1.The application to a d-wave superconductor is straightforward: the nodal structureof BdG quasiparticles in dSC helps along by the way of providing the anomalousstiffness for the Berry gauge field a µ – upon integration over the nodal fermions thefollowing term emerges in the effective action:12 χ BdG(q)(q 2 δ µν − q µ q ν )a µ (q)a ν (−q) ∼ |q|[δ µν − (q µ q ν /q 2 )]a µ (q)a ν (−q). (3.43)In the terminology of the BdG “semimetal”, the “susceptibility” χ BdG is not merelyvery large, it diverges: χ BdG ∼ 1/q, as q → 0. This allows us to expand L 0 [j Aµ , j Bµ ]and retain only quadratic terms:L 0∼ K Aµ=4 (∂ × v A) 2 µ + K Bµ4 (∂ × v B) 2 µ + (· · · ) , (3.44)where K Aµ = K Bµ = K µ is again assured by our choice of the FT singular gauge(3.4,3.11).Finally, we rewrite (3.44) in terms of v µ , a µ = (1/2)(v Aµ ± v Bµ ):L 0 → K µ2 (∂ × v)2 µ + K µ2 (∂ × a)2 µ , (3.45)and observe that the fact that A extµ couples only to v µ means that the expression(3.42) for K µ is still valid. Of course, ξ sc (x, T ) is now truly different from its s-wavecounterpart, including a possible difference in the critical exponent, since the couplingof d-wave quasiparticles to the Berry gauge field is marginal at the RG engineeringlevel and may change the quantum critical behavior of the superconductor-pseudogap(insulator) transition.3.3 Low energy effective theoryAs indicated in Fig. 3.1, the low energy quasiparticles are located at the fournodal points of the d xy gap function: k 1,¯1 = (±k F , 0) and k 2,¯2 = (0, ±k F ), hereafterdenoted as (1, ¯1) and (2, ¯2) respectively. To focus on the leading low energy behavior55

211 2Figure 3.1: Schematic representation of the Fermi surface of the cuprate superconductorswith the indicated nodal points of the d x 2 −y 2 gap.of the fermionic excitations near the nodes we follow the standard procedure [73] andlinearize the Lagrangian (3.10). To this end we write our TF spinor ˜Ψ as a sum of 4nodal fermi fields,˜Ψ = e ik 1·r ˜Ψ1 + e ik¯1·r σ 2 ˜Ψ¯1 + e ik 2·r ˜Ψ2 + e ik¯2·r σ 2 ˜Ψ¯2. (3.46)The σ 2 matrices have been inserted here for convenience: they insure that we eventuallyrecover the conventional form of the QED 3 Lagrangian. (Without the σ 2 matricesthe Dirac velocities at ¯1, ¯2 nodes would have been negative.) Inserting ˜Ψ into (3.10)and systematically neglecting the higher order derivatives [73], we obtain the nodalLagrangian of the formL D = ∑ ˜Ψ † l [D τ + iv F D x σ 3 + iv ∆ D y σ 1 ] ˜Ψ l (3.47)l=1,¯1+ ∑l=2,¯2˜Ψ † l [D τ + iv F D y σ 3 + iv ∆ D x σ 1 ] ˜Ψ l + L 0 [a µ ],where D µ = ∂ µ + ia µ denotes the covariant derivative and L 0 [a µ ] is given by (3.45):L 0 [a µ ] = 1 2 K µ(∂ × a) 2 µ ≡ 1 (∂ × a) 22e 2 µ . (3.48)µ56

v F = ∂ɛ k /∂k denotes the Fermi velocity at the node and v ∆ = ∂∆ k /∂k denotes thegap velocity. Note that v F and v ∆ already contain renormalizations coming from highenergy interactions and are effective material parameters of our theory. Similarly,K µ= 1/e 2 µ, derived in the previous Section in terms of ξ sc (x, T ), are treated asadjustable parameters which are matched to experimentally available information onthe range of superconducting correlations in the pseudogap state.The massive Doppler gauge field v µ does not appear in the above expression, sinceit merely generates short range interactions among the fermions.In contrast, Berry gauge field a µ remains massless in the pseudogap state, as itcannot acquire mass by coupling to the fermions. As seen from Eq. (3.47) a µ couplesminimally to the Dirac fermions and therefore its massless character is protected bygauge invariance. Physically, one can also argue that a µ couples to the TF spin threecurrent– in a spin-singlet d-wave superconductor SU(2) spin symmetry must remainunbroken, thereby insuring that a µ remains massless. Its massless Maxwellian dynamics(3.48) in the pseudogap state can therefore be traced back to the topologicalstate of spacetime vortex loops and directly reflect the absence of true superconductingorder or, equivalently, the presence of “vortex loop condensate” and dual order(Appendix B.1).We have also dispensed with the residual interactions represented by H res in (3.2).These interactions are generically short-ranged contributions from the p-h and amplitudefluctuations part of the p-p channel and in our new notation are exemplifiedby:The effective vertex I ll ′H res → 1 2∑l,l ′I ll ′˜Ψ†l ˜Ψ l ˜Ψ†l ′ ˜Ψ l ′ . (3.49)has a scaling dimension −1 at the engineering level. Thisfollows from the RG analysis near the massless Dirac points which sets the dimensionof ˜Ψ l to [length] −1 . The implication is that I ll ′is irrelevant for low energy physics inthe perturbative RG sense and we are therefore justified in setting I ll ′ → 0. However,the residual interactions are known to become important if stronger than some criticalvalue I c [74, 75]. In the present theory, this is bound to happen in the severelyunderdoped regime and at half-filling, as x → 0 (see Section 3.6).In this case,57

the residual interactions are becoming large and comparable in scale to the pairingpseudogap ∆, and are likely to cause chiral symmetry breaking (CSB) which leads tospontaneous mass generation for massless Dirac fermions. The CSB and its variety ofpatterns in the context of the theory (3.47,3.48) in underdoped cuprates are discussedin Section 3.6. Here, we shall focus on the chirally symmetric phase and assumeI ll ′< I c , which we expect to be the case for moderate underdoping. Therefore, wecan set I ll ′ → 0.Finally, Eq. (3.47,3.48) represents the effective low energy theory for nodal TF.This theory describes the problem of massless topological fermions interacting withmassless vortex “Beryons”, i.e. the quanta of the Berry gauge field a µ , and is formallyequivalent to the Euclidean quantum electrodynamics of massless Dirac fermions in(2+1) dimensions (QED 3 ). It, however, suffers from an intrinsic Dirac anisotropy byvirtue of v F ≠ v ∆ .3.3.1 Effective Lagrangian for the pseudogap stateLagrangian (3.47) is not in the standard from as used in quantum electrodynamicswhere the matrices associated with the components of covariant derivatives forma Dirac algebra and mutually anticommute. In (3.47) the temporal derivative isassociated with unit matrix and it therefore commutes (rather than anticommutes)with σ 1 and σ 3 matrices associated with the spatial derivatives. These nonstandardcommutation relations, however, lead to some rather unwieldy algebra. For thisreason we manipulate the Lagrangian (3.47) into a slightly different form that isconsistent with usual field-theoretic notation. First, we combine each pair of antipodal(time reversed) two-component spinors into one 4-component spinor,( ) ( )˜Ψ1˜Ψ2Υ 1 = , Υ 2 = . (3.50)Second, we define a new adjoint four-component spinor˜Ψ¯1˜Ψ¯2Ῡ l = −iΥ † l γ 0. (3.51)58

In terms of this new spinor the Lagrangian becomeswith covariant derivativesL D = ∑l=1,2Ῡ l γ µ D (l)µ Υ l + 1 2 K µ(∂ × a) 2 µ , (3.52)D (1)µ = i[(∂ τ + ia τ ), v F (∂ x + ia x ), v ∆ (∂ y + ia y )],D (2)µ = i[(∂ τ + ia τ ), v F (∂ y + ia y ), v ∆ (∂ x + ia x )].The 4 × 4 gamma matrices, defined as( )σ2 0γ 0 =, γ 1 =0 −σ 2γ 2 =( )−σ3 00 σ 3(σ1 00 −σ 1),(3.53)now form the usual Dirac algebra,{γ µ , γ ν } = 2δ µν (3.54)and furthermore satisfyTr(γ µ ) = 0 , Tr(γ µ γ ν ) = 4δ µν . (3.55)The use of the adjoint spinor Ῡ instead of conventional Υ† is a purely formal devicewhich will simplify calculations but does not alter the physical content of the theory.At the end of the calculation we have to remember to undo the transformation (3.51)by multiplying the 〈Υ(x)Ῡ(x′ )〉 correlator by iγ 0 to obtain the physical correlator〈Υ(x)Υ † (x ′ )〉.Next, to make the formalism simpler still we can eliminate the asymmetry betweenthe two pairs of nodes by performing an internal SU(2) rotation at nodes 2, ¯2:leading to the anisotropic QED 3 LagrangianL D = ∑l=1,2Υ 2 → e −i π 4 γ 0γ 1 Υ 2 , (3.56)Ῡ l v (l)µ γ µ(i∂ µ − a µ )Υ l + 1 2 K µ(∂ × a) 2 µ , (3.57)59

with v (1)µ = (1, v F , v ∆ ) and v (2)µ = (1, v ∆ , v F ).We shall start by considering the isotropic case, v F= v ∆ = 1, which althoughunphysical in the strictest sense, is computationally much simpler and provides penetratinginsights into the physics embodied by the QED 3 Lagrangian (3.57). Afterwe have understood the isotropic case we will then be ready to tackle the calculationfor the general case and will show that Dirac cone anisotropy does not modifythe essential physics discussed here. To make contact with standard literature onQED 3 , we further consider a more general problem with N pairs of nodes describedby LagrangianL D =N∑Ῡ l γ µ (i∂ µ − a µ )Υ l + 1 2 K µ(∂ × a) 2 µ . (3.58)l=1For the basic problem of a single CuO 2 layer N = 2. As we will show in the nextSection, N itself is variable and can be equal to four or six in bi- and multi-layercuprates. Our analytic results can be viewed as arising from the formal 1/N expansion,although we expect them to be qualitatively (and even quantitatively!) accurateeven for N = 2 as long as we are within the symmetric phase of QED 3 – the quantitativeaccuracy stems from a fortuitous conspiracy of small numerical prefactors[74].3.3.2 Berryon propagatorUltimately, we are interested in the properties of physical electrons. To describethose we need to understand the properties of the electron–electron interaction mediatedby the gauge field a µ . To this end we proceed to calculate the Berry gauge fieldpropagator by integrating out the fermion degrees of freedom from the Lagrangian(3.58). To one loop order this corresponds to evaluating the vacuum polarizationbubble Fig. 3.2(a). Employing the standard rules for Feynman diagrams in themomentum space [76] the vacuum polarization reads:∫d 3 kΠ µν (q) = N(2π) Tr[G 0(k)γ 3 µ G 0 (k + q)γ ν ]. (3.59)Here G 0 (k) = k α γ α /k 2is the free Dirac Green function, k = (k 0 , k) denotes theEuclidean three-momentum, and the trace is performed over the γ matrices.60

§ ¡¦£¢¡ £¥¤ ¦£ £¦©¨ £¦ ¦Figure 3.2: One loop Berryon polarization (a) and TF self energy (b).The integral in Eq. (3.59) is a standard one (see Appendix B.2 for the details ofcomputation) and the result isΠ µν (q) = N (8 |q| δ µν − q )µq ν, (3.60)q 2where |q| = √ q 2 . The one loop effective action for the Berry gauge field thereforebecomes (2π) ∫ −3 d 3 qL B with( NL B [a] =8 |q| + 1 ) (2e 2 q2 δ µν − q )µq νaq 2 µ (q)a ν (−q) . (3.61)At low energies and long wavelengths, |q|e 2 ≪ N/4, the fermion polarization completelyoverwhelms the original Maxwell bare action term and the Berryon propertiesbecome universal. In particular, the coupling constant 1/e 2 drops out at low energiesand only reappears as the ultraviolet cutoff. Physically, the medium of massless Diracfermions screens the long range interactions mediated by a µ . In QED 3 this screening61

is incomplete: the gauge field becomes stiffer by one power of q but still remainsmassless, in accordance with our general expectations. This anomalous stiffness ofa µ justifies the quadratic level expansion of L 0 (3.45) and renders higher order termsirrelevant in the RG sense. The theory therefore clears an important self-consistencycheck.At low energies the fully dressed Berryon propagator is given as an inverse of thepolarization,D µν (q) = Π −1µν (q) . (3.62)In order to perform this inversion we have to fix the gauge. To this end we implementthe usual gauge fixing procedure by replacing q µ q ν /q 2→ (1 − ξ −1 )q µ q ν /q 2 in Eq.(3.60). ξ ≥ 0 parameterizes the orbit of all covariant gauges. For example, ξ = 0corresponds to Lorentz gauge k µ a µ (k) = 0 while ξ = 1 corresponds to Feynman gauge.Upon inversion we obtain the low energy Berryon propagatorD µν (q) = 8|q|Nin agreement with previous authors [81].(δ µν − q µq νq 2 (1 − ξ) ), (3.63)3.3.3 TF self energy and propagatorTopological fermion (TF) propagator is a gauge dependent entity and one couldtherefore immediately object that as such it has no direct physical content and is of nointerest. While in the strictest sense, all observable quantities are gauge independentand the 2−point fermion function is manifestly not gauge independent, the dependenceon the choice of gauge can be confined to the field strength renormalization[82]. In this Section we will show that in the leading order in large N expansion,the TF propagator is a powerlaw. The powerlaw exponent depends on the choice ofgauge. Since we know that the theory flows to the strong coupling limit, we expectthat the powerlaw exponent describing the physical spectrum is not trivial i.e. thepoles in the spectrum are replaced by a branchcut. This means that the plane wavestates acquire finite lifetime at arbitrarily low energy.62

The lowest order (O(1/N)) self-energy diagram is depicted in Fig. 3.2(b) and reads∫d 3 qΣ(k) =(2π) D µν(q)γ 3 µ G 0 (k + q)γ ν . (3.64)Again, the computation is rather straightforward (see Appendix B.2 for details) andthe most divergent part isΣ(k) =4(2 − 3ξ)3π 2 N( ) Λ ̸k ln , (3.65)|k|where we have introduced the Feynman “slash” notation, ̸k = k µ γ µ .To the leading 1/N order, the inverse TF propagator is given by[ ( )] ΛG −1 (k) ≠k 1 + η ln|k|(3.66)with4(2 − 3ξ)η = − . (3.67)3π 2 NHigher order contributions in 1/N will necessarily affect this result. Renormalizationgroup arguments [78] and non-perturbative approaches [79] strongly suggest that Eq.(3.66) represents the start of a perturbative series that eventually resums into a powerlaw:( ) η ΛG −1 (k) ≠k|k|. (3.68)This implies real-space propagator of the formG(r) = Λ −η ̸r. (3.69)r3+η Thus, the TF propagator exhibits a Luttinger-like algebraic singularity at small momenta,characterized by an anomalous exponent η. In the Lorentz gauge (ξ = 0)we find η = −8/3π 2 N ≃ −0.13, for N = 2. This rather small numerical value forthe anomalous dimension exponent (which is even considerably smaller for N = 4or N = 6) indicates that the unraveling of the Fermi liquid pole in the original TFpropagator brought about by its interaction with the massless Berry gauge field is ina certain sense “weak”. Note also that η is negative in the Lorentz gauge while itbecomes positive, η = 4/3π 2 N ≃ 0.06 for N = 2, in the Feynman gauge (ξ = 1). Theabove results provide a strong indication that the physical, gauge-invariant fermionpropagator also has a Luttinger-like form, characterized by a small and positive anomalousdimension [58].63

3.4 Effects of Dirac anisotropy in symmetric QED 3It is natural to examine to what extend is the theory modified by the inclusion ofthe Dirac anisotropy, i.e. the finite difference in the Fermi velocity v Fand the gapvelocity v ∆ . In the actual materials the Dirac anisotropy α D = v Fv∆decreases withdecreasing doping from ∼ 15 in the optimally doped to ∼ 3 in the heavily underdopedsamples.There are two key issues: first, for a large enough number of Dirac fermion speciesN, how is the chirally symmetric infrared (IR) fixed point modified by the fact thatα D ≠ 1, and second, as we decrease N, does the chiral symmetry breaking occur atthe same value of N as in the isotropic theory. In this Section we address in detailthe first issue and defer the analysis of the second one to the future.We determine the effect of the Dirac anisotropy, marginal by power counting, bythe perturbative renormalization group (RG) to first order in the large N expansion.To the leading order δ in the small anisotropy α D = 1 + δ, we obtain the analyticvalue of the RG β αD -function and find that it is proportional to δ, i.e. in the infra redthe α D decreases when δ > 0 and the anisotropic theory flows to the isotropic fixedpoint. On the other hand, when δ < 0, α D increases in the IR and again the theoryflows into the isotropic fixed point. These results hold even when anisotropy is notsmall as shown by numerical evaluation of the β-function. Upon integration of the RGequations we find small δ acquires an anomalous dimension η δ = 32/5π 2 N. Therefore,we conclude that the isotropic fixed point is stable against small anisotropy.Furthermore, we show that in any covariant gauge renormalization of Σ due to theunphysical longitudinal degrees of freedom is exactly the same along any space-timedirection. Therefore the only contribution to the RG flow of anisotropy comes fromthe physical degrees of freedom and our results for β αDstated above are in fact gaugeinvariant.64

3.4.1 Anisotropic QED 3In the realm of condensed matter physics there is no Lorentz symmetry to safeguardthe space-time isotropy of the theory. Rather, the intrinsic Dirac anisotropyis always present since it ultimately arises from complicated microscopic interactionsin the solid which eventually renormalize to band and pairing amplitude dispersion.Thus there is nothing to protect the difference in the Fermi velocity v F = ∂ɛ kand the∂kgap velocity v ∆ = ∂∆ kfrom vanishing and, in fact, all HTS materials are anisotropic.∂kThe value of α D can be directly measured by the angle resolved photo-emissionspectroscopy (ARPES), which is ultimately a ”high” energy local probe of v Fv ∆ . Since QED 3 is free on short distances, we can take the experimental values asthe starting bare parameters of the field theory.The pairing amplitude of the HTS cuprates has d x 2 −y2 symmetry, and consequentlythere are four nodal points on the Fermi surface with Dirac dispersion around whichwe can linearize the theory. Note that the roles of x and y directions are interchangedbetween adjacent nodes. As before, we combine the four two-component Dirac spinorsfor the opposite (time reversed) nodes into two four-components spinors and labelthem as (1, ¯1) and (2, ¯2) (see Fig. 3.1).Thus, the two-point vertex function of the non-interacting theory for, say, 1¯1fermions isΓ (2)free = γ1¯1 0 k 0 + v F γ 1 k 1 + v ∆ γ 2 k 2 . (3.70)Therefore the corresponding non-interacting “nodal” Green functions areandG n 0(k) =√ gnµνγ µ k νk µ g µν k ν≡ γn µ k µk µ g µν k ν. (3.71)Here we introduced the (diagonal) “nodal” metric g (n)µν : g (1)00 =g (2)00 =1, g (1)11 =g (2)22 =v 2 F ,g (1)22 =g (2)11 =v 2 ∆ , as well as the “nodal” γ matrices γn . In what follows we assume thatboth v F and v ∆ are dimensionless and that eventually one of them can be chosen tobe unity by an appropriate choice of the ”speed of light”.Since α D ≠ 1 breaks Lorentz invariance of the theory and since it is the Lorentzinvariance that protects the spacetime isotropy, we expect the β functions for α D toacquire finite values. However, the theory still respects time-reversal and parity and65

for N large enough the system is in the chirally symmetric phase. These symmetriesforce the fermion self-energy of the interacting theory to have the formΣ 1¯1 = A(k 1¯1, k 2¯2) (γ 0 k 0 + v F ζ 1 γ 1 k 1 + v ∆ ζ 2 γ 2 k 2 ) . (3.72)where k 1¯1 ≡ k µ g µν (1) k ν and k 2¯2 ≡ k µ g µν (2) k ν . The coefficients ζ i are in general differentfrom unity. Furthermore, there is a discrete symmetry which relates flavors 1, ¯1 and2, ¯2 and the x and y directions in such a way thatΣ 2¯2 = A(k 2¯2, k 1¯1) (γ 0 k 0 + v ∆ ζ 2 γ 1 k 1 + v F ζ 1 γ 2 k 2 ) . (3.73)In the computation of the fermion self-energy, this discrete symmetry allows us toconcentrate on a particular pair of nodes without any loss of generality.3.4.2 Gauge field propagatorAs discussed above in the isotropic case, the effect of vortex-antivortex fluctuationsat T=0 on the fermions can, at large distances, be included by coupling the nodalfermions minimally to a fluctuating U(1) gauge field with a standard Maxwell action.Upon integrating out the fermions, the gauge field acquires a stiffness proportional tok, which is another way of saying that at the charged, chirally symmetric fixed pointthe gauge field has an anomalous dimension η A = 1 (for discussion of this point inthe bosonic QED see [80]).We first proceed in the transverse gauge (k µ a µ = 0) which is in some sense themost physical one considering that the ∇ × a is physically related to the vorticity, i.e.an intrinsically transverse quantity. We later extend our results to a general covariantgauge. To one-loop order the screening effects of the fermions on the gauge field aregiven by the polarization functionΠ µν (k) = N ∑∫d 3 q2 (2π) T 3 r[Gn 0 (q)γn µ Gn 0 (q + k)γn ν ] (3.74)n=1,2where the index n denotes the fermion “nodal” flavor. The above expression can beevaluated straightforwardly by noting that it reduces to the isotropic Π µν (k) once66

the integrals are properly rescaled [58]. The result can be conveniently presented bytaking advantage of the “nodal” metric g n µν asΠ µν (k) = ∑ nN√k α gαβ n 16v F v k β∆(g n µν − gn µρ k ρg n νλ k λk α g n αβ k β). (3.75)Note that this expression is explicitly transverse, k µ Π µν (k) = Π µν (k)k ν = 0, and symmetricin its space-time indices. It also properly reduces to the isotropic expressionwhen v F = v ∆ = 1.However, as opposed to the isotropic case, it is not quite as straightforward todetermine the gauge field propagator D µν . For example, as it stands the polarizationmatrix (3.75) is not invertible, which makes it necessary to introduce some gaugefixingconditions. In our case the direct inversion of the 3 × 3 matrix would obscurethe analysis and therefore, we choose to follow a more physical and notationallytransparent line of reasoning which eventually leads to the correct expression for thegauge field propagator. Upon integrating out the fermions and expanding the effectiveaction to the one-loop order, we find thatwhere the bare gauge field stiffness isL eff [a µ ] = (Π (0)µν + Π µν )a µ a ν (3.76)Π (0)µν = 1 (2e 2 k2 δ µν − k )µk ν. (3.77)k 2At this point we introduce the dual field b µ which is related to a µ asb µ = ɛ µνλ q ν a λ . (3.78)Physically, the b µ field represents vorticity and we integrate over all possible vorticityconfigurations with the restriction that b µ is transverse. We note thatL[b µ ] = χ 0 b 2 0 + χ 1 b 2 1 + χ 2 b 2 2 (3.79)where χ µ ’s are functions of k µ and upon a straightforward calculation they can befound to readχ µ = 12e 2 +N16v F v ∆67∑n=1,2g n ννg n λλ√kα g n αβ k β, (3.80)

where µ ≠ ν ≠ λ ∈ {0, 1, 2}. At low energies we can neglect the non-divergent barestiffness and thus set 1/e 2 = 0 in the above expression.The expression (3.79) is manifestly gauge invariant and has the merit of not onlybeing quadratic but also diagonal in the individual components of b µ . Thus, integrationover the vorticity (even with the restriction of transverse b µ ) is simple and wecan easily determine the b µ field correlation function〈b µ b ν 〉 = δ µνχ µ− k µk νχ µ χ ν( ∑ik 2 iχ i) −1. (3.81)The repeated indices are not summed in the above expression. Note that, in additionto being transverse, 〈b µ b ν 〉 is also symmetric in its space time indices.It is now quite simple to determine the correlation function for the a µ field and inthe transverse gauge we obtainD µν (q) = 〈a µ a ν 〉 = ɛ µij ɛ νklq i q kq 4 〈b jb l 〉. (3.82)Using the transverse character of 〈b µ b ν 〉 (which is independent of the gauge) the aboveexpression can be further reduced toD µν (q) = 1 ((δq 2 µν − q ))µq ν〈b 2 〉 − 〈bq 2 µ b ν 〉 . (3.83)It can be easily checked that in the isotropic limit the expression (3.83) properlyreduces to the results obtained in a different way.We can further extend this result to include a general gauge by writingD µν (q) = 1 ((δq 2 µν − (1 − ξ ))2 )q µq ν〈b 2 〉 − 〈bq 2 µ b ν 〉 . (3.84)where ξ is our continuous parameterization of the gauge fixing. This expression canbe justified by the Fadeev-Popov type of procedure starting from the Lagrangian(L = Π µν + 1 )2k 2 k µ k νaξ 〈b 2 〉 k 2 µ a ν , (3.85)where the stiffness for the unphysical modes was judiciously chosen to scale as k ina particular combination of the physical scalars of the theory. Note that 〈b 2 〉 can be68

determined without ever considering gauge fixing terms. In this way, the extensionof a transverse gauge ξ = 0 to a general covariant gauge is accomplished by a simplesubstitution k µ k ν → (1 − ξ )k 2 µk ν . The expression (3.84) is our final result for thegauge field propagator in a covariant gauge.3.4.3 TF self energyAs discussed above, Σ is not gauge invariant in that it has an explicit dependenceon the gauge fixing parameter. As we will show in this Section (and more generally inthe Appendix B.3), the renormalization of Σ by the unphysical longitudinal degreesof freedom does not depend on the space-time direction: the term in Σ which isproportional to γ 0 is renormalized the same way by the gauge dependent part of theaction as the terms proportional to γ 1 and γ 2 . Therefore, the only contribution tothe RG flow of α D comes from the physical degrees of freedom.We denote the topological fermion self-energy at the node n by Σ n (q). Hence, tothe leading order in large N expansion we have∫d 3 kΣ n (q) =(2π) 3 γn µ Gn 0 (q − k)γn ν D µν(k). (3.86)or explicitly∫Σ n (q) =d 3 k(2π) 3 γn µ(q − k) λ γλn γν n D µν (k), (3.87)(q − k) µ g µν (q − k) νwhere the gauge field propagator D µν is already screened by the nodal fermions (3.84).Using the fact thatγ µ γ λ γ ν = iɛ µλν γ 5 γ 3 + δ µλ γ ν − δ µν γ λ + δ λν γ µ , (3.88)where µ, ν, λ ∈ {0, 1, 2} and γ 5 ≡ −iγ 0 γ 1 γ 2 γ 3 , we can easily see thatγ n µ γn λ γn ν D µν = (2g n λµ γn ν − γn λ gn µν )D µν, (3.89)where we used the symmetry of the gauge field propagator tensor D µν . Thus,∫Σ n (q) =d 3 k (q − k) λ (2gλµ n γn ν − γn λ gn µν )D µν(k)(2π) 3 (q − k) µ gµν n (q − k) ν(3.90)69

and as shown in the Appendix B.3 at low energies this can be written asΣ n (q) = − ∑ ( )ηµ n (γn µ q Λµ) ln √qα g n µαβ q . (3.91)βHere Λ is an upper cutoff and the coefficients η are functions of the bare anisotropy,which have been reduced to a quadrature (see Appendix B.3). It is straightforward,even if somewhat tedious, to show that in case of weak anisotropy (v F = 1+δ, v ∆ = 1),to order δ 2 ,η0 1¯1 = − 83π 2 Nη 1¯11 = − 83π 2 Nη 1¯12 = − 83π 2 N(1 − 3 2 ξ − 1 35 (40 − 7ξ) δ2 )(1 − 3 2 ξ + 6 5 δ − 1 35 (43 − 7ξ) δ2 )(3.92)(3.93)(1 − 3 2 ξ − 6 5 δ − 1 35 (1 − 7ξ) δ2 ). (3.94)In the isotropic limit (v F = v ∆ = 1) we regain η n µ = −8(1 − 3 2 ξ)/3π2 N as previouslyfound by others.3.4.4 Dirac anisotropy and its β functionBefore plunging into any formal analysis, we wish to discuss some immediateobservations regarding the RG flow of the anisotropy. Examining the Eq. (3.91) it isclear that if η n 1 = ηn 2then the anisotropy does not flow and remains equal to its barevalue. That would mean that anisotropy is marginal and the theory flows into theanisotropic fixed point. In fact, such a theory would have a critical line of α D . Forthis to happen, however, there would have to be a symmetry which would protect theequality η n 1 = ηn 2 . For example, in the isotropic QED 3 the symmetry which protectsthe equality of η’s is the Lorentz invariance. In the case at hand, this symmetry isbroken and therefore we expect that η1 n will be different from ηn 2 , suggesting that theanisotropy flows away from its bare value. If we start with α D > 1 and find thatη2 1¯1 > η1 1¯1 at some scale p < Λ, we would conclude that the anisotropy is marginallyirrelevant and decreases towards 1. On the other hand if η2 1¯1 < η1 1¯1 , then anisotropycontinues increasing beyond its bare value and the theory flows into a critical pointwith (in)finite anisotropy.70

The issue is further complicated by the fact that η n µis not a gauge invariantquantity, i.e. it depends on the gauge fixing parameter ξ. The statement that, sayη n 1 > η n 2 , makes sense only if the ξ dependence of η n 1 and η n 2 is exactly the same,otherwise we could choose a gauge in which the difference η n 2 − η n 1can have eithersign. However, we see from the equations (3.92-3.94) that in fact the ξ dependenceof all η’s is indeed the same. Although it was explicitly demonstrated only to theO(δ 2 ), in the Appendix B.3 we show that it is in fact true to all orders of anisotropyfor any choice of covariant gauge fixing. This fact provides the justification for ourprocedure. Now we supply the formal analysis reflecting the above discussion.The renormalized 2-point vertex function is related to the “bare” 2-point vertexfunction via a fermion field rescaling Z ψ asΓ (2)R = Z ψΓ (2) . (3.95)It is natural to demand that for example at nodes 1 and ¯1 at some renormalizationscale p, Γ (2)R(p) will have the formΓ (2)R (p) = γ 0p 0 + v R F γ 1 p 1 + v R ∆γ 2 p 2 . (3.96)Thus, the equation (3.96) corresponds to our renormalization condition through whichwe can eliminate the cutoff dependence and calculate the RG flows.To the order of 1/N we can writeΓ (2)R (p) = Z ψγ n µ p µ(1 + ηµ n ln Λ )p(3.97)where we used the fermionic self-energy (3.91). Multiplying both sides by γ 0 andtracing the resulting expression determines the field strength renormalizationZ ψ =11 + η n 0 ln Λ p≈ 1 − η n 0 ln Λ p . (3.98)We can now determine the renormalized Fermi and gap velocitiesv R Fv F≈ (1 − η0 1¯1 ln Λ )(1 + η1¯11p ln Λ p ) ≈ 1 − (η1¯10 − η1¯11 ) ln Λ p(3.99)andv∆R ≈ (1 − η0 1¯1v ln Λ )(1 + η1¯12∆ p ln Λ p ) ≈ 1 − (η1¯10 − η1¯12 ) ln Λ p . (3.100)71

The corresponding renormalized Dirac anisotropy is thereforeα R D ≡ vR Fv R ∆The RG beta function can now be determined≈ α D (1 − (η2 1¯1 − η1¯11 ) ln Λ ). (3.101)pβ αD= dαR Dd ln p = α D(η 1¯12 − η1¯11). (3.102)In the case of weak anisotropy (v F = 1 + δ, v ∆ = 1) the above expression can bedetermined analytically as an expansion in δ. Using Eqs.(3.93-3.94) we obtainβ αD = 8 ( 6))3π 2 N 5 δ(1 + δ)(2 − δ) + O(δ3 . (3.103)Note that this expression is independent of the gauge fixing parameter ξ. For 0 2, β < 0 which maynaively indicate that there is a fixed point at δ = 2; this however cannot be trustedas it is outside of the range of validity of the power expansion of η µ . The numericalevaluation of the β-function shows that, apart from the isotropic fixed point and theunstable fixed point at α D = 0, β αDdoes not vanish (see Fig. 3.3). This indicatesthat to the leading order in the 1/N expansion, the theory flows into the isotropicfixed point where α D − 1 has a scaling dimension η δ = 32/(5π 2 N) > 1.3.5 Finite temperature extensions of QED 3In this Section we focus on thermodynamics and spin response of the pseudogapstate. First, in order to derive thermodynamics and spin susceptibility from theQED 3 theory [58] we generalize its form to finite T . This is a matter of some subtletysince, the moment T ≠ 0, the theory loses its fictitious “relativistic invariance”.Second, we show that the theory predicts a finite T scaling form for thermodynamicquantities in the pseudogap state. Third, we explicitly compute the leading T ≠ 072

β αD[8/3π 2 Ν]121086420−20 1 2 3 4 5α DFigure 3.3: The RG β-function for the Dirac anisotropy in units of 8/3π 2 N. Thesolid line is the numerical integration of the quadrature in the Eq. (B.52) while thedash-dotted line is the analytical expansion around the small anisotropy (see Eq.(3.92-3.94)). At α D = 1, β αD crosses zero with positive slope, and therefore at largelength-scales the anisotropic QED 3 scales to an isotropic theory.scaling and demonstrate that the deviations from “relativistic invariance” are actuallyirrelevant for T much less than the pseudogap temperature T ∗ , in the sense that theleading order T ≠ 0 scaling of thermodynamic functions remains that of the finite-Tsymmetric QED 3 . These deviations from “relativistic invariance” do, however, affecthigher order terms. We illustrate these general results by computing the leading(∼ T 2 ) and next-to-the leading (∼ T 3 ) terms in specific heat c v . Finally, we evaluatemagnetic spin susceptibility χ u and show that it is bounded by T 2 at low T butcrosses over to ∼ T at higher T , closer to T ∗ . Consequently, the Wilson ratio χ u T/c vvanishes as T → 0 implying the non-Fermi liquid nature of the pseudogap state incuprates.We start by noting that the spin susceptibility of a BCS-like d-wave superconductorvanishes linearly with temperature. The way to understand this result is to noticethat the spin part of the ground state wavefunction, being a spin singlet, remains unperturbedby the application of a weak uniform magnetic field. However, the excitedquasiparticle states are not in general spin singlets and therefore contribute to thethermal average of the spin susceptibility. Because their density of states is linear atlow energies, at finite temperature the number of quasiparticles that are excited is∼ k B T , each contributing a constant to the Pauli-like uniform spin susceptibility χ u .Thus χ u ∼ T .73

When the superconducting order is destroyed by proliferation of unbound quantumhc/2e vortices, the low-energy quasiparticle excitations are strongly interacting. Theinteraction originates from the fact that it is the spin singlet pairs that acquire aone unit of angular momentum in their center of the mass coordinate, carried bya hc/2e vortex.This translates into a topological frustration in the propagationof the “spinon” excitations. As a result, non-trivial spin correlations persist in theexcited states of the phase-disordered d-wave superconductor. At low temperature,this leads to a suppression of χ u relative to that of the non-interacting quasiparticles.We shall argue below that in the phase-disordered superconductor, χ u ∼ T 2 at lowtemperatures.Similarly, in a d-wave superconductor, linear density of the quasiparticle statestranslates into T 2 dependence of the low T specific heat. When the interactionsbetween quasiparticles are included the spectral weight is transferred to multi-particlestates. Within QED 3 theory, however, the strongly interacting IR (infra-red) fixedpoint possesses emergent “relavistivistic invariance” at long distances and low energiesand the dynamical critical exponent z = 1. Furthermore, the effective quantum actionfor vortices, deep in the phase disordered pseudogap state, introduces an additionallengthscale, the superconducting correlation length ξ τ,⊥ (labels τ and ⊥ stand fortime- and space-like, respectively). At T = 0 this scale serves as a short distancecutoff of the theory and possesses some degree of doping (x) dependence within thepseudogap state. We then argue that under rather general circumstances the low Telectronic specific heat scales as T 2 while the free energy goes as T 3 .Deep in the phase disordered pseudogap state the fluctuations in the vorticity3-vector b µ = ɛ µνλ ∂ ν a λ are described by the “bare” Lagrangian of the QED 3 theory[58]:L 0 (a µ ) = K (⊥ T2 f ⊥k , T ⊥,τ)ω , T Kb 2 0 + K (τ T2 f τk , T ⊥,τ)ω , T K ⃗ b 2 , (3.104)where f τ (0, 0, 0) = f ⊥ (0, 0, 0) = 1. f τ,⊥ are general scaling functions describing howL 0 (a µ ) is modified from its “relativistically invariant” T = 0 form as the temperatureis turned on, and K ⊥,τ is related to the superconducting correlation length as K τ ∝ξ 2 sc /ξ τ and K ⊥ ∝ ξ τ [58]. Physically, such modifications are due to changes in the74

pattern of vortex-antivortex fluctuations induced by finite T .To handle the intrinsic space-time anisotropy, it is convenient to introduce twotensorsA µν =(δ µ0 − k )µk 0 k2k 2 ⃗ k2B µν = δ µi(δ ij − k ik j⃗ k2and rewrite the gauge field action as(δ 0ν − k )0k νk 2,)δ jν , (3.105)L 0 (a µ ) = 1 2 Π0 A a µA µν a ν + 1 2 Π0 B a µB µν a ν . (3.106)For details see Appendix B.4. It is straightforward to show that( TΠ 0 A = K τ f τk , T )ω , T K ⊥,τ ( ⃗ k 2 + ω 2 ),( T = K τf τk , T ) ( Tω , T K ⊥,τ ω 2 + K ⊥ f ⊥k , T )ω , T K ⃗⊥,τ k 2 .Π 0 B(3.107)The gauge field a µ couples minimally to the Dirac fermions representing nodalBCS quasiparticles [58]. Consequently, the resulting Lagrangian readsL = ¯ψ (iγ µ ∂ µ + γ µ a µ ) ψ + L 0 (a µ ) . (3.108)The integration over Berry gauge field a µ reproduces the interaction among quasiparticlesarising from the topological frustration referred to earlier.3.5.1 Specific heat and scaling of thermodynamicsThe only lengthscales that appear in the thermodynamics are the thermal length ∼1/T , and the superconducting correlation lengths K ⊥,τ . At T = 0, the two correlationlengths enter only as short distance cutoffs for the theory since the electronic actionis controlled by the IR fixed point of QED 3 . These observations allow us to writedown the general scaling form for the free energyF(T ; x) = T 3 Φ ( K τ (x)T, K ⊥ (x)T ) . (3.109)75

In the above scaling form K ⊥ is related to the T → 0 finite superconducting correlationlength of the pseudogap state ξ sc (x). The ratio K τ /K sc describes the anisotropybetween time-like and space-like vortex fluctuations and is also a function of dopingx. The scaling expressions for other thermodynamic functions can be derived from(3.109) by taking appropriate temperature derivatives. For details see AppendicesB.5 and B.6.We are interested in the limit of the thermal length 1/T being much longer thanK τ and K ⊥ i.e. in the limit of Φ(x → 0, y → 0). This is precisely the limit in whichthe free energy approaches the free energy of the finite temperature QED 3 . Therefore,Φ(x, y) is regular at x = y = 0 (see Ref.[81]) and so in the limit of T → 0, F ∼ T 3 orc v ∼ T 2 .3.5.2 Uniform spin susceptibilityThe fermion fields ψ were defined in Ref. [58] where it is shown that the physicalspin density ψ † ↑ ψ ↑ − ψ † ↓ ψ ↓ is equal to the Dirac fermion density ¯ψγ 0 ψ.To compute the spin-spin correlation function 〈S z (−k)S z (k)〉 we introduce anauxiliary source J µ (k) and couple it to fermion three current. ThusL[ ¯ψ, ψ, a µ , J µ ] = ¯ψ (iγ µ ∂ µ + γ µ (a µ + J µ )) ψ + L 0 (a µ ) (3.110)and since it is the z-component of the spin that couples to the gauge field we have〈S z (−k)S z (k)〉 = 1 δ δZ[J µ ] δJ 0 (−k) δJ 0 (k) Z[J µ] ∣ (3.111)Jµ=0,Z being the quantum partition function. Now we let a ′ µ = a µ + J µ and integrateout both the fermions and the gauge field a ′ µ . The correlations between a′ µ fields aredescribed by the polarization matrix which to the order of 1/N can be written in theform Π µν = (Π 0 A + ΠF A )A µν + (Π 0 B + ΠF B )B µν. The resulting spin correlation functionis then readily found to be〈S z (−k)S z (k)〉 = ΠF A Π0 ⃗A k2Π F A + Π0 ⃗A k2 + ω , (3.112)276

where Π F A denotes the fermion current polarization function. Due to the scale invarianceof the (massless) fermion action, the time component of the retarded polarizationfunction has the scaling form( qΠ F Aret (q, ω, T ) = NT PAF T , ω ), (3.113)Twhere PA F (x, y) is a universal function of its arguments and N is the number of the4−component fermion species (see Appendix B.5). In the static limit, ω → 0,andwhilelim Rey→0 PF A (x, y) = 2 ln 2 + x2π 24π + O(x3 ); x ≪ 1 (3.114)lim Rey→0 PF A (x, y) = x 8 + 6ζ(3) + O(x −3 ); x ≫ 1 (3.115)πx 2lim Im 2 ln 2y→0 PF A (x, y) =π( )y y2x + O ; x ≪ 1 (3.116)x 2In addition, in the limit of ω = 0 and k → 0 we have( ) Tf τk , ∞, K τT → 1 + c T 2(3.117)k 2where c is a pure number. Combining all of these results together we haveχ(ω = 0, q → 0) =2N ln 2πK τ T 22N ln 2πc+ K τ T . (3.118)There are several sources of correction to the scaling, and it is impossible to addressall of them without an explicit model for the vortices. Instead, we will concentrate onthe particular case of the corrections to scaling due to the Dirac cone anisotropy. Itwas shown in the previous Section that the Dirac cone anisotropy α D = v F /v ∆ scalesto unity at the QED 3 IR fixed point. Alternatively, when defined as α D ≡ 1 + δ, δhas a scaling dimension η δ > 1. To the leading order in 1/N, η δ = 32/(5π 2 N) (seethe previous Section). For simplicity, we assume that the bare action for the gaugefield is e −2 Fµν 2 . Then the T dependence of the free energy scales asF = T ( 3 TΦvF2 e , v )F2 v ∆= T ( )3 TΦvF2 e , 1 + δ(T ) 277(3.119)

Thus, in the limit of T → 0 we haveF = T 3v 2 FΦ (0, 1) + T 3+η δδvF2 0 Φ ′ (0, 1) . (3.120)Therefore, while the leading order scaling of the free energy is analytic T 3 , the correctionto scaling is non-analytic T 3+η δ.3.6 Chiral symmetry breakingIn this Section we shall try to understand the physical nature of the instabilitiesof the symmetric phase of QED 3 or Algebraic Fermi Liquid.We show that a phase disordered d-wave superconductor, as introduced in previoussections, becomes an antiferromagnet [83]. The antiferromagnetism emergesvia a phenomenon of spontaneous chiral symmetry breaking (CSB)[59, 60]. Awayfrom half filling, the broken symmetry phase typically takes the form of an incommensuratespin-density-wave (SDW), whose periodicity is tied to the Fermi surface.Furthermore, we show that numerous other states, most notably a d+ip and a d+isphase-incoherent superconductors (dipSC, disSC) and “stripes”, i.e. superpositionsof 1D charge-density-waves (CDW) and phase-incoherent superconducting-densitywaves(SCDW), as well as continuous chiral rotations among them, are all energeticallyclose and competitive with antiferromagnetism due to their equal membershipin the chiral manifold of two-flavor (N = 2) QED 3 . This large chiral manifold ofnearly degenerate states plays the key role in the QED 3 theory as the culprit behindthe complexity of the HTS phase diagram.The above results place tight restrictions on this phase diagram and provide meansto unify the phenomenology of cuprates within a single, systematically calculable“QED 3 Unified Theory” (QUT). Any microscopic description of cuprates, as longas it leads to the large d-wave pairing pseudogap with T ∗ ≫ T c → 0, will conformto the general results of QUT. In particular, all the physical states in natural energeticproximity to a d-wave superconductor are the ones inhabiting the above chiralmanifold.78

For the sake of concreteness we restate the starting Lagrangian 3.47L QED = ¯ψ n c µ,n γ µ D µ ψ n + L 0 [a µ ] + (· · · ). (3.121)Here ψ † α = ¯ψ α γ 0 = (η † α , η† ᾱ) are the four-component Dirac spinors with η † α = 1 √2Ψ † α (1+iσ 1 ), η † ᾱ = 1 √2Ψ † ᾱσ 2 (1 + iσ 1 ), and Ψ † α = (ψ † ↑α , ψ ↓α). Fermion fields ψ σα (r, τ) describe‘topological fermions’ of the theory and are related to the original nodal fermionsc σα (r, τ) via the singular gauge transformation (3.4). Index n labels (1, ¯1) and (2, ¯2)pairs of nodes while α labels individual nodes, µ = τ, x, y(≡ 0, 1, 2). D µ = ∂ µ + ia µis a covariant derivative, c τ,n = 1, c x,1 = c y,2 = v F , c x,2 = c y,1 = v ∆ . The gammamatrices are defined as γ 0 = σ 3 ⊗ σ 3 , γ 1 = −σ 3 ⊗ σ 1 , γ 2 = −σ 3 ⊗ σ 2 , and satisfy{γ µ , γ ν } = 2δ µν . The Berry gauge field a µ encodes the topological frustration ofnodal fermions generated by fluctuating quantum vortex-antivortex pairs and L 0 isits bare action. The loss of superconducting phase coherence caused by unbinding ofvortex pairs is heralded in (3.121) by a µ becoming massless:L 0 → 1 (∂ × a) 22e 2 τ + ∑τi12e 2 i(∂ × a) 2 i ; (3.122)here e 2 τ, e 2 i (i = x, y), as well as the velocities v F (∆) , are functions of doping x and T .Along with residual interactions between nodal fermions, denoted by the ellipsis in(3.121), these parameters of QUT arise from some more microscopic description andwill be discussed shortly.L QED (3.121) possesses the following peculiar continuous symmetry: borrowingfrom ordinary quantum electrodynamics in (3+1) dimensions (QED 4 ), we know thatthere exist two additional gamma matrices, γ 3 = σ 1 ⊗ 1 and γ 5 = iσ 2 ⊗ 1 that anticommutewith all γ µ . We can define a global U(2) symmetry for each pair of nodes,with generators 1 ⊗ 1, γ 3 , −iγ 5 and 1 2 [γ 3, γ 5 ], which leaves L QED invariant. In QED 3this symmetry can be broken by two “mass” terms, m ch ¯ψn ψ n and m PT ¯ψn12 [γ 3, γ 5 ]ψ n .Spontaneous symmetry breaking in QED 3 as a mechanism for dynamical mass generationhas been extensively studied in the field theory literature [59]. It has beenestablished that while m PT is never spontaneously generated [84], the chiral mass m chis generated if number of fermion species N is less than a critical value N c . While79

TaCSBdSC¡¡¢¡¢¡¢¡¢¡¡¢¡¢¡¢¡¢¢¡¢¡¢¡¢ ¡ ¡ T*bCSBdSC¡¡¢¡¢¡¢¡¢QED3SDW/AF(CSB)£¡£¡£ ¤¡¤¡¤£¡£¡£ ¤¡¤¡¤¤¡¤¡¤ £¡£¡£T cdSCxFigure 3.4: Schematic phase diagram of a cuprate superconductor in QUT. Dependingon the value of N c (see text), either the superconductor is followed by a symmetricphase of QED 3 which then undergoes a quantum CSB transition at some lower doping(panel a), or there is a direct transition from the superconducting phase to the m ch ≠0 phase of QED 3 (panel b). The label SDW/AF indicates the dominance of theantiferromagnetic ground state as x → 0.still a matter of some controversy, standard non-perturbative methods give N c ∼ 3for isotropic QED 3 [59, 60], but as we shall discuss shortly, anisotropy and irrelevantcouplings present in Lagrangian (3.121) can change the value of N c .Let us now assume that CSB occurs and the mass term m ch ¯ψn ψ n is generated. Wewish to determine what is the nature of this chiral instability in terms of the originalelectron operators. To make this apparent, let us consider a general chiral rotationψ n → U (n)ch ψ n with U (n)ch= exp(iθ 3n γ 3 + θ 5n γ 5 ). Within our representation of Diracspinors (3.121), the m ch ¯ψn ψ n mass term takes the following form:m ch cos(2Ω n ) [ η † α σ 3η α − η † ᾱσ 3 ηᾱ]++m ch sin(2Ω n ) θ 5n + iθ 3nΩ nη † ασ 3 ηᾱ + h.c. , (3.123)where Ω n = √ θ 2 3n + θ 2 5n. m ch acts as an order parameter for the bilinear combinationsof topological fermions appearing in (3.123). In the symmetric phase of QED 3 (m ch =80

21Q Q2211Q121 2Figure 3.5: The “Fermi surface” of cuprates, with the positions of nodes in the d-wavepseudogap. The wavectors Q 1¯1, Q 2¯2, Q¯1¯2, etc. are discussed in the text.0) the expectation values of such bilinears vanish, while they become finite, 〈 ¯ψ n ψ n 〉 ≠0, in the broken symmetry phase.The chiral manifold (3.123) is spanned by the “basis” of three symmetry breakingstates. When re-expressed in terms of the original nodal fermions c σα (r, τ), two ofthese involve pairing in the particle-hole (p-h) channel – a cosine and a sine spindensity-wave(SDW):〈c † ↑α c ↑ᾱ − c † ↓α c ↓ᾱ〉 + h.c. (cos SDW)i〈c † ↑α c ↑ᾱ − c † ↑ᾱ c ↑α〉 + (↑→↓) (sin SDW) (3.124)and are obtained from Eq. (3.123) by setting Ω n equal to π/4 or 3π/4. Rotationswithin the chiral manifold (3.123) at fixed Ω n correspond to the sliding modes ofSDW.A simple physical picture emerges here: we started from a d-wave superconductingphase, our parent state. As one moves closer to half-filling and true phase coherenceis lost, strong vortex-antivortex pair fluctuations, acting under the protective umbrellaof a d-wave particle-particle (p-p) pseudogap, spontaneously induce formationof particle-hole “pairs” at finite wavevectors ±Q 1¯1 and ±Q 2¯2, spanning the Fermi81

surface from node α to ᾱ (Fig. 3.5). The glue that binds these p-h “pairs” and playsthe role of “phonons” in this pairing analogy is provided by the Berry gauge field a µ .Such “fermion duality” is a natural consequence of the QED 3 theory (3.121). Remarkably,we find the antiferromagnetic insulator being spontaneously generated in formof the incommensurate SDW. As we get very near half-filling and Q 1¯1, Q 2¯2 approach(±π, ±π), SDW acquires the most favored state status within the chiral manifold –this is the consequence of umklapp processes which increase its condensation energywithout it being offset by either the anisotropy or a poorly screened Coulomb interactionwhich plagues its CDW competitors to be introduced shortly. It seems thereforereasonable to argue that this SDW must be considered the progenitor of the Neel-Mott-Hubbard insulating antiferromagnet at half-filling. Thus, QED 3 theory (3.121)explains the origin of antiferromagnetic order in terms of strong vortex-antivortexfluctuations in the parent d-wave superconductor. It does so naturally, through itsinherent and well-established chiral symmetry breaking instability [59].The chiral manifold (3.123) contains also a third state, a p-p pairing state correspondingto Ω n = 0 or π/2 and best characterized as a d+ip phase-incoherentsuperconductor:i〈ψ ↑α ψ ↓α − ψ ↑ᾱ ψ ↓ᾱ 〉 + h.c. (dipSC) . (3.125)We have written dipSC in terms of topological fermions ψ σα (r, τ) since use of theoriginal fermions leads to more complicated expression which involves the backflow ofvortex-antivortex excitations described by gauge fields a µ and v µ (such backflow termsdo not arise in the p-h channel). This state breaks parity but preserves time reversal,translational invariance and superconducting U(1) symmetries. To our knowledge,such state has not been proposed as a part of any of the major theories of HTS. It isan intriguing question whether this d+ip phase-incoherent superconductor can be theactual ground state at some dopings in some of the cuprates. Its energetics does notsuffer from long range Coulomb problems but it is clearly inferior to the SDW veryclose to half-filling since, being spatially uniform, it receives no help from umklapp.Until now, we have discussed the CSB pattern only within individual pairs ofnodes, (1,¯1) and (2,¯2). What happens if we allow for chiral rotations that mix nodes82

1 and ¯2 or 1 and 2? A whole new plethora of states becomes possible, with chiralmanifold enlarged to include a superposition of one-dimensional p-h and p-p states,an incommensurate CDW accompanied by a non-uniform phase-incoherent superconductor(SCDW) at wavevectors ±Q 12 and ±Q¯2¯1 (Fig. 3.5):1√2〈c † ↑1 c ↑2 + c † ↑¯2 c ↑¯1 + h.c.〉 + (↑→↓)(CDW)1√2〈ψ ↑1 ψ ↓2 + ψ ↑¯2ψ ↓¯1 + h.c.〉 + (↑↔↓) (SCDW) (3.126)These same states, rotated by π/2, are replicated at wavevectors ±Q 1¯2 and ±Q 2¯1(Fig. 2). In a fluctuating d x 2 −y 2 superconductor these CDWs and SCDWs run alongthe x and y axes and are naturally identified as the “stripes” of QUT. Note, however,these are not the only one-dimensional states in QUT – among the states in the chiralmanifold (3.123) are also “diagonal stripes”, the combination of a SDW (3.124) along±Q 1¯1 and a dipSC (3.125) which opens the mass gap only at nodes (2, ¯2), or viceversa. Furthermore, a phase-incoherent d+is superconductor (disSC) is also presentwithin the chiral enlarged manifold, since it results in alternating signs for differentnodes with equal number of positive and negative “masses” for the two-componentnodal fermions:i〈ψ ↑1 ψ ↓1 + ψ ↑¯1ψ ↓¯1 + h.c.〉 + (1 → 2) (disSC) . (3.127)In contrast, in a d+id phase incoherent superconductor these “masses” have thesame sign for all the nodes producing a maximal breaking of the PT symmetry [84].Consequently, a d+id phase-incoherent superconductor is not spontaneously inducedwithin the QED 3 theory.In the isotropic (v F = v ∆ ) N = 2 QED 3 all these additional states plus arbitrarychiral rotations among them are completely equivalent to those discussed previously.It is here where we confront the problem of intrinsic anisotropy in Eq. (3.121). Suchanisotropy cannot be rescaled out and manifestly breaks the U(2)×U(2) degeneracyof the full N = 2 chiral manifold down to two separate U(1)×U(1) (3.123) chiralgroups discussed previously. This is reflected in the general increase in energy ofthe states from the enlarged chiral manifold. For example, the anisotropy raises the83

energy of our “stripe” states (3.126) relative to those of SDW, dipSC or “diagonalstripes”. However, when the long range Coulomb interactions and coupling to thelattice are included in the problem, as they are in real materials, it is conceivablethat the “stripes” would return in some form, either as a ground state or a long-livedmetastable state at some intermediate doping. disSC is also adversely affected byanisotropy but to a lesser extent and might remain competitive with SDW, dipSCand “diagonal stripes”. This state breaks time reversal symmetry but preserves parityand the discussion concerning dipSC below Eq. (3.125) applies to disSC equally well.How do we use these general results on CSB in QUT to address the specifics ofcuprate phase diagram? To this end, we need some effective combination of phenomenologyand more microscopic descriptions to determine the parameters v F , v ∆ , e τ ,e i and residual interactions (· · · ) appearing in L QED (3.121). The main task is todetermine what is the sequence of states within QUT that form stable phases as thedoping decreases toward half-filling under T ∗ in Fig. 3.4. While this is an extensiveproject for the future, we outline here some of the general features. First, withinthe superconducting state e τ , e i → 0 and a µ becomes massive thus denying the CSBmechanism its main dynamical agent. We therefore expect that the superconductoris in the symmetric phase and its nodal fermions form well-defined excitations[58].As we move to the left in Fig. 3.4, the phase order is suppressed and e τ , e i becomefinite, reflecting the unbinding of vortex-antivortex excitations[58]. For all practicalpurposes, this is precisely what the experiments imply. Now, the key question iswhether the QED 3 (3.121) remains in its symmetric phase or whether it immediatelyundergoes the CSB transition and generates finite gap (m ch ≠ 0).This question is difficult to answer in detail, because N c can depend non-universallyon higher order coupling constants. For instance, short range interactions, while perturbativelyirrelevant, effectively increase N c if stronger than some critical value[74].Such interactions, typically in the form of short range three-current terms[21], arise inmore microscopic models used to derive L QED and are prominent among the residualterms denoted by ellipsis in (3.121). Their strength generically increases as x → 0.These residual interactions play a dual role in QUT. First, they can conspire with theanisotropy to produce the situation depicted in panel (b) of Fig. 3.4, where the CSB84

takes place as soon as the phase coherence is lost. Second, once the chiral symmetryhas been broken, the residual interactions further break the symmetry within thechiral manifold (3.123) and play a role in selecting the true ground state.3.7 Summary and conclusionsWe considered the effect of fluctuations in the d-wave superconductor, concentratingon the role of the phase fluctuations and their effect on the nodal fermions.The pairing pseudogap provided a reference point of our analysis which allowed us toclassify various quasiparticle interactions according to their fate under the renormalizationgroup. By carefully treating the interactions of the quasiparticles with thevortex phase defects, we find that the low energy effective theory for the quasiparticlesinside the pairing protectorate is the (2+1) dimensional quantum electrodynamics(QED 3 ) with inherent spatial anisotropy, described by Lagrangian L D specified byEqs. (3.47,3.57).Within the superconducting state the gauge fields of the theory are massive byvirtue of vortex defects being bound into finite loops or vortex-antivortex pairs. Suchmassive gauge fields produce only short ranged interactions between our BdG quasiparticlesand are therefore irrelevant: in the superconductor quasiparticles remainsharp in agreement with prevailing experimental data [85]. Loss of the long rangesuperconducting order is brought about by unbinding the topological defects – vortexloops or vortex-antivortex pairs – via Kosterlitz-Thouless type transition and itsquantum analogue. Remarkably, this is accompanied by the Berry gauge field becomingmassless. Such massless gauge field mediates long range interactions between thefermions and becomes a relevant perturbation. Exactly what is the consequence ofthis relevant perturbation depends on the number of fermion species N in the problem.For cuprates we argued that N = 2n CuO where n CuO is the number of CuO 2layers per unit cell. If N < N c ≃ 3 [59], the interactions cause spontaneous openingof a gap for the fermionic excitations at T = 0, via the mechanism of chiral symmetrybreaking in QED 3 [59]. Formation of the gap corresponds to the onset of AF SDWinstability [58, 83] which must be considered as a progenitor of the Mott-Hubbard-85

Neel antiferromagnet at half-filling. If, on the other hand, N > N c as will be thecase in bilayer or trilayer materials, the theory remains in its chirally symmetric nonsuperconductingphase even as T → 0 and AF order arises from within such stateonly upon further underdoping (Fig. 1). We call this symmetric state of QED 3 analgebraic Fermi liquid (AFL). In both cases AFL controls the low temperature, lowenergy behavior of the pseudogap state and in this sense assumes the role played byFermi liquid theory in conventional metals and superconductors. In AFL the quasiparticlepole is replaced by a branch cut – the quasiparticle is no longer sharp – andthe fermion propagator acquires a Luttinger-like form Eq. (3.68). To our knowledgethis is one of the very few cases where a non-Fermi liquid nature of the excitationshas been demonstrated in dimension greater than one in the absence of disorder ormagnetic field. This Luttinger-like behavior of AFL will manifest itself in anomalouspower law functional form of many physical properties of the system.Dirac anisotropy, i.e. the fact that v F ≠ v ∆ , plays important role in the cuprateswhere the ratio α D = v F /v ∆ in most materials ranges between 3 and 15−20. Suchanisotropy is nontrivial as it cannot be rescaled and it significantly complicates anycalculation within the theory. Using the perturbative renormalization group theorywe have shown that anisotropic QED 3 flows back into isotropic stable fixed point.This means that for weak anisotropy at long lengthscales the universal properties ofthe theory are identical to those of the simple isotropic case. It remains to be seenwhat are the properties of the theory at intermediate lengthscales when anisotropy isstrong.The QED 3 theory of the pairing pseudogap, as presented in here, starts from aremarkably simple set of assumptions, and via manipulations that are controlled inthe sense of 1/N expansion, arrives at nontrivial consequences, including the algebraicFermi liquid and the antiferromagnet.86

Appendix AQuasiparticle correlation functionsin the mixed stateA.1 Green’s functions: definitions and identitiesThe one-particle Green’s function matrix [52] is defined asĜ αβ (r 1 τ 1 ; r 2 τ 2 ) ≡ −〈T τ Ψ α (1)Ψ † β (2)〉(A.1)where T τ denotes imaginary time ordering operator and α = 1, 2 denotes componentsof a Nambu spinor Ψ † (1) which is a shorthand for Ψ † (r 1 , τ 1 ) = ( ψ † ↑ (r 1, τ 1 ), ψ ↓ (r 1 , τ 1 ) ) .Due to the time independence of the Hamiltonian (2.2), the Green’s function (A.1)depends only on the imaginary time difference τ = τ 1 − τ 2 . Therefore, its Fouriertransform is given byĜ(r 1 , r 2 ; iω) = ∫ β0 eiωτ Ĝ(r 1 , r 2 , τ)dτ∑Ĝ(r 1 , r 2 ; τ) = 1 e −iωτ Ĝ(rβ1 , r 2 ; iω).iω(A.2)Here β = 1/(k B T ), T is temperature, and the fermionic frequency ω = (2l + 1)π/β, l ∈ Z.Using the above relations, it is straightforward to derive the spectral representationof the following correlation functions between Ψ and its imaginary time derivatives87

∂ τ Ψ ≡ ˙Ψ: ⎧⎪ ⎨⎪ ⎩〈T τ Ψ α (1)Ψ † β (2)〉〈T τ ˙Ψα (1)Ψ † β (2)〉〈T τ Ψ α (1) ˙Ψ † β (2)〉= − 1 β〈T τ ˙Ψα (1) ˙Ψ † β (2)〉∑∫e −iwτiωdɛÂ(r 1, r 2 ; ɛ)iω − ɛ⎧+1⎪⎨−ɛ+ɛ⎪⎩−ɛ 2(A.3)The spectral function Â(r 1, r 2 ; ɛ) can be written in terms of eigenfunctions Φ n (r) =(u n (r), v n (r)) T of the Bogoliubov-deGennes Hamiltonian Ĥ0 in the form:Â αβ (r 1 , r 2 ; ɛ) = ∑ nδ(ɛ − ɛ n )Φ nα (r 1 )Φ † nβ (r 2),(A.4)where ɛ n is the eigen-energy associated with an eigenstate labeled by the quantumnumber n. Substituting (A.4) into (A.3) we can write the above correlation functionssolely in terms of the eigenfunctions Φ n (r):⎧〈T τ Ψ α (1)Ψ † β⎪⎨(2)〉+1⎧⎪〈T τ ˙Ψα (1)Ψ † β (2)〉〈T τ Ψ α (1) ˙Ψ † β⎪⎩(2)〉=− 1 ∑e Φ nα(r −iwτ 1 )Φ † nβ (r ⎨2) −ɛ nβiω − ɛiω,nn +ɛ n〈T τ ˙Ψα (1) ˙Ψ † β (2)〉 ⎪ ⎩−ɛ 2 n(A.5)In the calculations that follow we will also encounter Matsubara summations overthe fermionic frequencies ω = (2l + 1)π/β, l ∈ Z, of the form:S nm (iΩ) = 1 ∑1β (iω − ɛ n )(iΩ + iω − ɛ m )where Ω = 2πk/β, k ∈ Z, is an outside bosonic frequency.evaluated using standard techniques (see e.g. [52]) and yields:iωS nm (iΩ) =f n − f mɛ n − ɛ m + iΩ .(A.6)The sum (A.6) can be(A.7)Here f n is a short hand for the Fermi-Dirac distribution function f(ɛ n ) = ( 1 +exp(βɛ n ) ) −1.In order to derive spin and thermal currents we will need the explicit form of thegeneralized velocity operator V introduced in (2.33):( )πiˆvm ∆V =iˆv ∆∗88π ∗m(A.8)

where the gap velocity operator is given byiˆv ∆ = i∆ 0 η δ δe iφ(r)/2( e iδ·p − e −iδ·p) e iφ(r)/2(A.9)and the canonical momentum equalsπ =− i 2 δe i r+δr (p− e c A)·dl + h.c. (A.10)The following identities for operators ˆv ∆ and π will be used in the next section:{π ∗ Ψ † · πΨ = i∇ · (Ψ † πΨ) + Ψ † π 2 Ψ(A.11)π ∗ Ψ † · πΨ = −i∇ · (π ∗ Ψ † Ψ) + (π ∗ ) 2 Ψ † ΨΨ † ∆Ψ − ∆Ψ † Ψ = 1 2 ∇ · (Ψ † ˆv ∆ Ψ − ˆv ∆ Ψ † Ψ ) (A.12)The above equations are straightforward to derive in continuum, while on the tightbindinglattice Eqs. (A.11) and (A.12) imply a symmetric definition of the latticedivergence operator.The identities (A.11,A.12) explicitly ensure that the generalized velocity operatorV µ is Hermitian i.e it satisfies the following identity:∫∫drV ∗ Ψ † (r) Ψ(r) ≡ drVαβ ∗ Ψ† β (r) Ψ α(r)∫= drΨ † (r) VΨ(r).(A.13)A.2 Spin current and spin conductivity tensorThe time derivative of spin density ρ s = 2 (ψ† ↑ ψ ↑ − ψ † ↓ ψ ↓) can be written in Nambuformalism as˙ρ s = i [H, ρs ] = 2 ( ˙Ψ † Ψ + Ψ † ˙Ψ)(A.14)Using equations of motion (2.10) together with the explicit form of Ĥ0 operator (2.8)we obtain˙ρ s = i ( (π ∗ ) 22 2m Ψ† 1 Ψ 1 − Ψ † π 212m Ψ 1 + ˆ∆Ψ † 1 Ψ 2 − Ψ † ˆ∆Ψ 1 2)+ ˆ∆ ∗ Ψ † 2 Ψ 1 − Ψ † 2ˆ∆ ∗ Ψ 1 − π22m Ψ† 2 Ψ 2 + Ψ † (π ∗ ) 222m Ψ 2 . (A.15)89

Using the identities (A.11,A.12) it is easy to show that˙ρ s = − 4 ∇ µ(Ψ † V µ Ψ + V ∗µ Ψ† Ψ) = −∇ · j s (A.16)where the generalized velocity operator V µ is defined in Eq. (A.8), and the lastequality follows from the continuity equation (2.31) relating the spin density ρ s andthe spin current j s . Upon spatial averaging and utilizing Eq. (A.13) we find the q → 0limit of the spin currentj s µ = 2 Ψ† V µ Ψ.(A.17)The evaluation of the spin current-current correlation function (2.29) is straightforwardand yields:D µν (iΩ) = − 24∫ β0e iτΩ[ V µ (2)V ν (4)×〈T τ Ψ † (1)Ψ(2)Ψ † (3)Ψ(4)〉 1, 2 → (x, τ)3, 4 → (y, 0)]dτ (A.18)Here Ψ(1) ≡ Ψ(r 1 , τ 1 ) and similarly the operator V µ (2) acts only on functions of r 2 .Using Wick’s theorem, identity (A.5), and upon spatial averaging over x and y weobtainD µν (iΩ) = 24∑〈n|V µ |m〉〈m|V ν |n〉S nm (iΩ).mn(A.19)The double summation extends over the eigenstates |m〉 and |n〉 of the Hamiltonian(2.8), and S mn (iΩ) is given in Eq. (A.7). Analytically continuing iΩ → Ω + i0 wefinally obtain the expression for the retarded correlation function:D R µν(Ω) = 24∑mnVµ nm Vνmnɛ n − ɛ m + Ω + i0 (f n − f m ).(A.20)where V µ mn is defined in (2.35) and f n is a short hand for the Fermi-Dirac distributionfunction f(ɛ n ) = ( 1+exp(βɛ n ) ) −1 . Finally, we substitute the last equation into (2.28)to obtainσµν s = 2 ∑ Vµ nm Vνmn4i (ɛmn n − ɛ m + i0) (f 2 n − f m ) (A.21)90

A.3 Thermal current and thermal conductivity tensorIn order to calculate thermal currents and thermal conductivity in the magneticfield we introduce a pseudo-gravitational potential χ = r · g/c 2 [48, 49, 50, 51]. Thisformal procedure is useful because it illustrates that the transverse thermal responseis not given just by Kubo formula, but in addition it includes corrections related tomagnetization. Throughout this section = 1. The pseudo-gravitational potentialenters the Hamiltonian, up to linear order in χ, as∫H T = dx (1 + χ 2 )Ψ† (x) Ĥ0 (1 + χ 2 )Ψ(x),(A.22)where H 0 is the Bogoliubov-deGennes Hamiltonian (2.8). The equations of motionfor the fields Ψ thus becomei ˙Ψ = [Ψ, H T ] = (1 + χ 2 )Ĥ0(1 + χ 2 )Ψ = (1 + χ)Ĥ0Ψ − i∇ µ χV µ Ψ. (A.23)The last equality follows from the commutation relation (2.33). (Note that for χ ≠ 0the Eq. (A.23) differs from Eq. (2.10). Throughout this section ˙Ψ will refer to Eq.(A.23) unless explicitly stated otherwise).To find thermal current j Q we start with the continuity equationḣ T + ∇ · j Q = 0.The Hamiltonian density h T follows from Eq. (A.22) and readsh T = 12m ∗ (π ∗ µ ˜Ψ † 1π µ ˜Ψ1 − π µ ˜Ψ† 2 πµ ∗ ˜Ψ)2 − µ ˜Ψ † ασαβ 3 ˜Ψ β+ 1 ( )˜Ψ†2 α ∆ αβ ˜Ψβ + ∆ αβ ˜Ψ† α ˜Ψβ(A.24)(A.25)where ˜Ψ = (1 + χ 2 )Ψ. Taking the time derivative of the Hamiltonian density h T and91

using Eqs.(A.11) and equations of motion (A.23) we obtainḣ T = i[H T , h T ] =i (2m ∇ ˙˜Ψ† µ Π µ ˜Ψ − Π∗ ˙˜Ψ)µ ˜Ψ†+ 1 ( )˜Ψα ∆ αβ˙˜Ψβ − ˙˜Ψ† 2α ∆ αβ ˜Ψβ+ 1 )(∆ αβ˙˜Ψ†2α ˜Ψβ − ∆ αβ ˜Ψ† α˙˜Ψβ(A.26)where we introduced matrix operators( )(0 ˆ∆∆ αβ =, Π µˆ∆ ∗ αβ0= πµ 00 πµ∗). (A.27)Here ˙˜Ψ = (1 +χ2 ) ˙Ψ and π µ is defined in Eq. (A.10). Finally we use (A.12) to extractj Q from (A.26). Upon spatial averaging and using the Hermiticity of V µ (A.13) thethermal current j Q readsNote that the expression for j Q contains two termsjµ Q = i ( )˜Ψ† V µ˙˜Ψ − ˙˜Ψ† V µ ˜Ψ . (A.28)2j Q = j Q 0 + jQ 1(A.29)where j Q 0 is independent of χ and jQ 1is linear in χ. Explicitly:j Q 0 (r) = 1 2 Ψ† {V, Ĥ0}Ψ(A.30)andjµ,1(r) Q = − i 4 ∂ νχΨ † (V µ V ν − V ν V µ ) Ψ+ ∂ ()νχ4 Ψ† (x ν V µ + 3V µ x ν )Ĥ0 + Ĥ0(3x ν V µ + V µ x ν )Ψ(A.31)where {a, b} = ab+ba. Analogously to the situation in the normal metal, the thermalaverage of j Q 1bydoes not in general vanish in the presence of the magnetic field [54].The linear response of the system to the external perturbation can be described〈j Q µ 〉 = 〈jQ 0µ〉 + 〈j Q 1µ〉 = −(K µν + M µν )∂ ν χ ≡ −L Q µν ∂ νχ (A.32)92

whereK µν = − δ〈jQ 0µ〉δ ∂ ν χ = − lim Pµν(Ω) R − Pµν(0)RΩ→0 iΩ(A.33)is the standard Kubo formula for a dc response [52], P R µν(Ω) being the retardedcurrent-current correlation function, andM µν = − δ〈jQ 1µ〉δ ∂ ν χ= − ∑ nɛ n f n 〈n| {V µ , x ν } |n〉 + ∑ ni4 f n〈n|[V µ , V ν ]|n〉 (A.34)is a contribution from “diathermal” currents [54]. Note that the latter vanishes for thelongitudinal response while it remains finite for the transverse response. As we willshow later in this section, at T = 0 there is an important cancellation between (A.33)and (A.34) which renders the thermal conductivity κ µνthe singularity from the temperature denominator in Eq. (2.48).well-behaved and preventsThe retarded thermal current-current correlation function Pµν R (Ω) can be expressedin terms of the Matsubara finite temperature correlation function∫ β〉P µν (iΩ) = − dτe〈T iΩτ τ j µ (r, τ)j ν (r ′ , 0)as0P R µν(Ω) = P µν (iΩ → Ω + i0).The current j µ (r, τ) in Eq. (A.35) is given by(A.35)(A.36)j µ (τ) = 1 2 (Ψ† (τ)V µ ∂ τ Ψ(τ) − ∂ τ Ψ † (τ) V µ Ψ(τ))(A.37)As pointed out by Ambegaokar and Griffin [56] time ordering operator T τ and timederivative operators ∂ τ do not commute and neglecting this subtlety can lead to formallydivergent frequency summations[38]. Taking heed of this subtlety, substitutionof (A.37) into (A.35) amounts to∫ βP µν (iΩ) = − 1 dτe iΩτ V µ αβ4(2)V γδ〈T ν (4) τ ˙Ψ† α (1)Ψ β (2) ˙Ψ † γ (3)Ψ δ(4)0〉− ˙Ψ † α(1)Ψ β (2)Ψ † γ(3) ˙Ψ δ (4) − Ψ † α(1) ˙Ψ β (2) ˙Ψ † γ(3)Ψ δ (4) + Ψ † α(1) ˙Ψ β (2)Ψ † γ(3) ˙Ψ δ (4) ,(A.38)93

where the notation follows Eq. (A.18) and Eq. (2.10). Upon utilizing the identities(A.5) and (A.13) and performing the standard Matsubara summation we haveP µν (iΩ) = 1 ∑〈n|V µ |m〉〈m|V ν |n〉(ɛ n + ɛ m ) 2 S nm (iΩ) (A.39)4nmwhere S mn is given by Eq. (A.7). Analytically continuing iΩ → Ω + i0 we obtainP R µν (Ω) = 1 4∑nm(ɛ n + ɛ m ) 2 Vµ nm Vνmnɛ n − ɛ m + Ω + i0 (f n − f m ).(A.40)Note that the only difference between the Kubo contribution to the thermal response(A.40) and the spin response (A.20) is the value of the coupling constant. In thecase of thermal response (A.40), the coupling constant is (ɛ n + ɛ m )/2 which is eigenstatedependent, while in the case of spin response (A.20) it is /2 and eigenstateindependent.Using Eq. (A.33) we find that the Kubo contribution to the thermal transportcoefficient is given byThis can be written aswhereandK µν = − i 4∑nm(ɛ n + ɛ m ) 2(ɛ n − ɛ m + i0) 2 V nmµ V mnν (f n − f m ). (A.41)K µν = K (1)µν + K(2) µν (A.42)K µν (1) = − i ∑ 4ɛ n ɛ m4 (ɛnm n − ɛ m + i0) V nm2 µ Vν mn (f n − f m ) (A.43)K (2)µν = − i 4∑nmV nmµ V mnν (f n − f m ) (A.44)Similarly, we can separate the “diathermal” contribution (A.34) asM µν = M (1)µν + M (2)µν (A.45)where M (1)µν and M (2)µν refer to the first and second term in (A.34) respectively. Usingthe completeness relation ∑ m|m〉〈m| = 1 it is easy to show thatM µν (2) = i ∑(f n − f m )Vµ nm Vν mn . (A.46)4mn94

Comparison of (A.44) and (A.46) yields M (2)µν + K (2)µν = 0. Therefore the thermalresponse coefficient is given byL Q µν = K(1) µν + M (1)µν . (A.47)Utilizing commutation relationships (2.33), M µν (1) can be expressed in the form:∫ ()M µν (1) = ηf(η)Tr δ(η − Ĥ0)(x µ V ν − x ν V µ ) dη (A.48)where the integral extends over the entire real line. The rest of the section followsclosely Smrčka and Středa [50]. We define the resolvents G ± :G ± −1≡ (η ± i0 − Ĥ0) (A.49)and operators()dGA(η) = iTr V +µ V dη νδ(η − Ĥ0) − V µ δ(η − Ĥ0)V dG −ν dη) B(η) = iTr(V µ G + V ν δ(η − Ĥ0) − V µ δ(η − Ĥ0)V(A.50)ν G −To facilitate Sommerfeld expansion we note that response coefficients K (1)µν (T ),M µν (1) (T ), σµν s (T ) have generic form∫L(T ) =f(η)l(η)dη,(A.51)and after integration by parts∫L(T ) = −dfdη ˜L(η)dη.(A.52)Here ˜L(ξ) is defined as˜L(ξ) ≡∫ ξ−∞l(η)dη.(A.53)Note that L(T = 0) = ˜L(ξ = 0) and in particular the spin conductivity at T = 0satisfies σ s µν(T =0) = ˜σ s µν(ξ =0). Identities (A.52) and (A.53) will enable us to expressthe coefficients at finite temperature through the coefficients at zero temperature. Forexample, it follows from Eq. (A.48) that˜M (1) (ξ) =∫ ξ−∞()ηTr δ(η − Ĥ0)(x µ V ν − x ν V µ )95(A.54)

and from Eqs.(A.21, A.50)˜σ s µν (ξ)= 1 4∫ ξ−∞A(η)dηSimilarly, coefficient K µν (1) from (A.43) can be written as∫K µν (1) = −i dηf(η)η ∑ ( Vnmµ Vνmnδ(η − ɛ n )ɛ m(η − ɛ m + i0) − V µ mn V nm )ν2 (η − ɛ m − i0) 2so that˜K (1)µν (ξ) ≡ −i ∫ ξ−∞dη η ∑ δ(η − ɛ n )ɛ m(Vnmµ Vνmn(η − ɛ m + i0) − V mn2µ Vνnm )(η − ɛ m − i0) 2(A.55)(A.56)(A.57)Using definitions (A.50), ˜K(1) (ξ) can be expressed as˜K (1)µν (ξ) = ∫ ξ−∞After integration by parts one obtains∫ξµν (ξ) = ξ2 A(η)dη +˜K (1)−∞η 2 A(η)dη +∫ ξ−∞∫ ξ−∞ηB(η)dη.((η 2 − ξ 2 ) A(η) − 1 )dBdη.2 dη(A.58)(A.59)As shown in Ref. ([50]) the last term in this expression is exactly compensated by˜M (1) (ξ): This becomes evident after noting that definitions in Eq. (A.50) imply[A − 1 dB(η)= 1 2 dη 2 Tr]dδ(η − Ĥ0) (x µ V ν − x ν V µ ) . (A.60)dηSubstituting the last identity into the Eq.(A.59) and integrating the second term byparts we obtain∫ξµν (ξ) = ξ2˜K (1)−∞∫ ξA(η)dη −−∞()ηTr δ(η − Ĥ0)(x µ V ν − x ν V µ ) dη(A.61)The second term here is equal to −M (1)µν from (A.54). After the cancellation the resultsimply reads:˜L Q µν(ξ) =˜K(1)µν (ξ) +˜M(1)µν (ξ) = ξ 2 ∫ ξ96−∞A(η)dη.(A.62)

Or, using (A.55)Finally, from Eq.(A.52) we find( ) 2ξ˜L Q µν2˜σ (ξ) = µν s (ξ). (A.63)L Q µν(T ) = − 4 2 ∫ df(η)dη η2˜σ s µν(η)dη.(A.64)97

Appendix BField theory of quasiparticles andvorticesB.1 Jacobian L 0Here we derive the explicit form of the “Jacobian” L 0 for two cases of interest: i)the thermal vortex-antivortex fluctuations in 2D layers and ii) the spacetime vortexloop excitations relevant for low temperatures (T ≪ T ∗ ) deep in the underdopedregime.B.1.12D thermal vortex-antivortex fluctuationsIn order to perform specific computations we have to adopt a model for vortexantivortexexcitations. We will use a 2D Coulomb gas picture of vortex-antivortexplasma. In this model (anti)vortices are either point-like objects or are assumedto have a small hard-disk radius of size of the coherence length ξ 0 which emulatesthe core region. As long as ξ 0 ≪ n − 1 2 , where n = n v + n a is the average densityof vortex defects, the two models lead to very similar results and both undergo avortex-antivortex pair unbinding transition of the Kosterlitz-Thouless variety.98

Above the transition we have∫∑∫exp [−β d 2 rL 0 ] = 2 −N lDϕ(r)A,B×δ[∇ × v − 1 2 ∇ × (∇ϕ A + ∇ϕ B )](B.1)×δ[∇ × a − 1 2 ∇ × (∇ϕ A − ∇ϕ B )] .The phase ϕ(r) is due solely to vortices and we can rewrite (B.1) as:∑ 2 −N l ∑ ∏N v ∫∏N a ∫d 2 r i d 2 r j e −βEc(Nv+Na)N v !N a !N v,N aA,Bi×δ [ ∑N vρ v (r) − δ(r − r i ) ] δ [ ∑N aρ a (r) − δ(r − r j ) ]i×δ [ N A N v∑v∑Bb(r) − π δ(r − r A i ) + π δ(r − r B i )ijNaA N∑a∑B+π δ(r − r A j ) − π δ(r − r B j ) ] .jiji(B.2)Here N v (N a ) is the number of free vortices (antivortices), N l = N v + N a , r i (r j )are vortex (antivortex) coordinates and ρ v (r) (ρ a (r)) are the corresponding densities.b(r) = (∇ × a(r)) z = π(ρ A v − ρB v − ρA a + ρB a ) and E c is the core energy which we haveabsorbed into L 0 for convenience. We now express the above δ-functions as functionalintegrals over three new fields: d v (r), d a (r) and κ(r):∑ 2 −N l ∑ ∏N v∫∏N a∫d 2 r i d 2 r j e −βEc(Nv+Na)N v !N a !∫×N v,N aA,BDd v Dd a Dκ exp { ∫i∫+ii∫+ij∑N vd 2 rd v (ρ v (r) − δ(r − r i ))∑N ad 2 rd a (ρ a (r) − δ(r − r j ))d 2 rκ [ N A N v∑v∑Bb(r) − π δ(r − r A i ) + π δ(r − r B i )iNaA N∑a∑B+π δ(r − r A j ) − π δ(r − r B j ) ]} .j99iiji(B.3)

The integration over δ-functions in the exponential is easily performed and the summationover A(B) labels can be carried out explicitly to obtain:∫[ ∫Dd v Dd a Dκ exp i d 2 r ( d v ρ v + d a ρ a + κb )]e −βEcNv N vN v !N v,N a i× ∑× e−βEcNaN a !∏∫∏N aj∫d 2 r i exp(−id v (r i )) cos(πκ(r i ))d 2 r j exp(−id a (r j )) cos(πκ(r j )).(B.4)In the thermodynamic limit the sum (B.4) is dominated by N v(a) = 〈N v(a) 〉, where〈N v(a) 〉 is the average number of free (anti)vortices determined by solving the fullproblem. Furthermore, as 〈N v(a) 〉 → ∞ in the thermodynamic limit the integrationover d v (r), d a (r) and κ(r) can be performed in the saddle-point approximation leadingto the following saddle-point equations:−ρ v (r) + 〈N v 〉Ω v (r) = 0−ρ a (r) + 〈N a 〉Ω a (r) = 0−b(r) + [〈N v 〉Ω v (r) + 〈N a 〉Ω a (r)](B.5)(B.6)(B.7)×π tanh(πκ(r)) = 0withΩ m (r) =e dm(r) cosh(πκ(r))∫d2 r ′ e dm(r′ ), m = a, v.cosh(πκ(r ′ ))Eqs. (B.5-B.7) follow from functional derivatives of (B.4) with respect to d v (r), d a (r)and κ(r), respectively. We have also built in the fact that the saddle-point solutionsoccur at d v → id v , d a → id a , κ → iκ.The saddle-point equations (B.5-B.7) can be solved exactly leading to:d v (r) = ln ρ v (r) − ln cosh(πκ(r))d a (r) = ln ρ a (r) − ln cosh(πκ(r))(B.8)(B.9)whereκ(r) = 1 []b(r)π tanh−1 . (B.10)π(ρ v (r) + ρ a (r))100

Inserting (B.8-B.10) back into (B.4) finally gives the entropic part of L 0 /T :ρ v ln ρ v + ρ a ln ρ a − 1 [ ] (∇ ×π (∇ × a) z tanh −1 a)zπ(ρ v + ρ a )+ (ρ v + ρ a ) ln cosh tanh −1 [ (∇ × a)zπ(ρ v + ρ a )], (B.11)where ρ v(a) (r) are densities of free (anti)vortices. We display L 0 in this form tomake contact with familiar physics: the first two terms in (B.11) are the entropiccontribution of free (anti)vortices and the Doppler gauge field ∇ × v → π(ρ v − ρ a )(〈∇ × v〉 = 0). The last two terms encode the “Berry phase” physics of topologicalfrustration. Note that the “Berry” magnetic field b = (∇ × a) z couples directly onlyto the total density of vortex defects ρ v + ρ a and is insensitive to the vortex charge.This is a reflection of the Z 2 symmetry of the original problem defined on discrete(i.e., not coarse-grained) vortices. We write ρ v(a) (r) = 〈ρ v(a) 〉 + δρ v(a) (r) and expand(B.11) to leading order in δρ v(a) and ∇ × a:L 0 /T → (∇ × v) 2 /(2π 2 n l ) + (∇ × a) 2 /(2π 2 n l ),(B.12)where n l = 〈ρ v 〉+〈ρ a 〉 is the average density of free vortex defects. Both v and a havea Maxwellian bare stiffness and are massless in the normal state. As one approachesT c , n l ∼ ξ −2sc→ 0, where ξ sc (x, T ) is the superconducting correlation length, and vand a become massive (see the main text).B.1.2Quantum fluctuations of (2+1)D vortex loopsThe expression for L 0 [j µ ] given by Eq. (3.39) follows directly once the systemcontains unbound vortex loops in its ground state and thus can respond to the externalperturbation A extµ over arbitrary large distances in (2+1)-dimensional spacetime.This is already clear at intuitive level if we just think of the geometry of infiniteversus finite loops and the fact that only unbound loops allow vorticity fluctuations,described by 〈j µ (q)j ν (−q)〉, to proceed unhindered. Still, it is useful to derive (3.39)and its consequences belabored in Section II within an explicit model for vortex loopfluctuations.101

Here we consider fluctuating vortex loops in continuous (2+1)D spacetime andcompute L 0 [j µ ] using duality map to the relativistic Bose superfluid [86]. We againstart by using our model of a large gap s-wave superconductor which enables us toneglect a µ in the fermion action and integrate over it in the expression for L 0 [v µ , a µ ](3.11). This gives:whereande − d3 xL 0=˜S =N∑∮S 0l=1(∂ × ∂ϕ(x))∞∑N=01N!ds l + 1 2N∏∮l=1N∑∮l,l ′ =1µ = 2πn µ(x) = 2πDx l [s l ]δ ( j µ (x) − n µ (x) ) e − ˜Sds l∮N∑∮lds l ′g(|x l [s l ] − x l ′[s l ′]|)Ldx lµ δ(x − x l [s l ]) .(B.13)(B.14)(B.15)In the above equations N is the number of loops, s l is the Schwinger proper time (or“proper length”) of loop l, S 0 is the action per unit length associated with motion ofvortex cores in (2+1)-dimensional spacetime (in an analogous 3D model this wouldbe ε c /T , where ε c is the core line energy), g(|x l [s l ]−x l ′[s l ′]|) is the short range penaltyfor core overlap and L denotes a line integral. We kept our practice of including coreterms independent of vorticity into L 0 . Note that vortex loops must be periodic alongτ reflecting the original periodicity of ϕ(r, τ).We can think of vortex loops as worldline trajectories of some relativistic charged(complex) bosons (charged since the loops have two orientations) in two spatial dimensions.L 0 without the δ-function describes the vacuum Lagrangian of such atheory, with vortex loops representing particle-antiparticle virtual creation and annihilationprocess. The duality map is based on the following relation between theGreen function of free charged bosons and a gas of free oriented loops:∫G(x) = 〈φ(0)φ ∗ d 3 k e ik·x ∫ ∞(x)〉 == dse −sm2(2π) 3 k 2 + m 2 dd 0∫×d 3 k(2π) 3 eik·x−sk2 =∫ ∞dse −sm2 d0( ) 312e− x24s ,4πs(B.16)102

where m d is the mass of the complex dual field φ(x).integral representation we can write:( ) 3/2 ∫ 1x(s)=x[e − 1 4 x2 /s = Dx(s ′ ) exp − 1 ∫ s]ds ′ ẋ 2 (s ′ )4πsx(0)=04 0Within the Feynman path, (B.17)where ẋ ≡ dx/ds. Furthermore, by simple integration (B.16) can be manipulatedintoTr [ ∫ ∞ln(−∂ 2 + m 2 d )] ds= −0= −s e−sm2 d∫ ∞0∫dss e−sm2 dd 3 k(2π) 3 e−sk2∮Dx(s ′ ) exp[− 1 ∫ s]ds ′ ẋ 2 (s ′ )4 0(B.18)where the path integral now runs over closed loops (x(s) = x(0)). In the dual theorythis can be reexpressed asTr [ ∫ln(−∂ 2 + m 2 d )] =Combining (B.18) and (B.19) and usingd 3 k(2π) 3 ln(k2 + m 2 d ) .(B.19)Tr [ ln(−∂ 2 + m 2 d) ] = ln Det(−∂ 2 + m 2 d) ≡ W(B.20)finally leads to:e −W =∞∑N=01N!N∏[∫ ∞l=10∮ds le −s lm 2 ds lDx(s ′ l ) ]×exp[− 1 4N∑∫ sll=10ds ′ lẋ2 (s ′ l ) ]. (B.21)This is nothing else but the partition function of the free loop gas. The size of loopsis regulated by m d . As m d → 0 the average loop size diverges. On the other hand,through W (B.20), we can also think of (B.21) as the partition function of the freebosonic theory.To exploit this equivalence further we write∫Z d = Dφ ∗ Dφe − d3 xL d, (B.22)whereL d = |∂φ| 2 + m 2 d |φ|2 + g 2 |φ|4 , (B.23)103

and argue that Z d describes vortex loops with short range interactions in (B.13).This can be easily demonstrated by decoupling |φ| 4 through Hubbard-Stratonovichtransformation and retracing the above steps. The reader is referred to the book byKleinert for further details of the above duality mapping [86].We can now rewrite the δ-function in (B.13) asδ ( j µ (x) − n µ (x) ) ∫→ Dκ µ exp ( ∫i d 3 xκ µ (j µ − n µ ) )(B.24)and observe that in the above language of Feynman path integrals in proper timei ∫ d 3 xκ µ n µ (B.15) assumes the meaning of a particle current three-vector n µ coupledto a three-vector potential κ µ . Employing the same arguments that led to (B.23) wenow have:L d [κ µ ] = |(∂ − iκ)φ| 2 + m 2 d |φ|2 + g 2 |φ|4 , (B.25)which leads toe −d3 xL 0 [j µ] →∫Dφ ∗ DφDκ µ e − d3 x(−iκ µj µ+L d [κ µ]). (B.26)In the pseudogap state vortex loop unbinding causes loss of superconducting phasecoherence. In the dual language, the appearance of such infinite loops as m d → 0implies superfluidity in the system of charged bosons described by L d (B.23). Thedual off-diagonal long range order (ODLRO) in 〈φ(x)φ ∗ (x ′ )〉 means that there areworldline trajectories that connect points x and x ′ over infinite spacetime distances– these infinite worldlines are nothing else but unbound vortex paths in this “vortexloop condensate”. Thus, the dual and the true superconducting ODLRO are twoopposite sides of the same coin.L d [κ µ ] is the Lagrangian of this dual superfluid in presence of the external vectorpotential κ µ . In the ordered phase, the response of the system is just the dual versionof the Meissner effect. Consequently, upon functional integration over φ we areallowed to write:∫e − d3 xL 0 [j µ] = Dκ µ e − d3 x(−iκ µj µ+Md 2κµκµ) , (B.27)104

where Md2 = |〈φ〉|2 , with 〈φ〉 being the dual order parameter. The remaining functionalintegration over κ µ finally results in:L 0 [j µ ] → j µj µ4|〈φ〉| 2 . (B.28)We have tacitly assumed that the system of loops is isotropic. The intrinsic anisotropyof the (2+1)D theory is easily reinstated and (B.28) becomes Eq. (3.39) of the maintext.It is now time to recall that we are interested in a d-wave superconductor. Thismeans we must restore a µ to the problem. To accomplish this we engage in a bit ofthievery: imagine now that it was v µ whose coupling to fermions was negligible andwe could integrate over it in (3.11). We would then be left with only the δ-functioncontaining a µ . Actually, we can compute such L 0 [b µ /π], where ∂ × ∂a = b, withoutany additional work. Note that L 0 contains only vorticity independent terms. We canequally well proclaim that it is the A(B) labels that determine the true orientationof our loops while the actual vorticity is simply a gauge label – in essence, v µ and a µtrade places. After the same algebra as before we obtain:which is just the Maxwell action for a µ .L 0 [b µ /π] → b µb µ4π 2 |〈φ〉| 2 , (B.29)Of course, this simple argument that led to (B.29) is illegal. We cannot just forgetv µ . If we did we would have no right to coarse-grain a µ to begin with and would haveto face up to its purely Z 2 character (see Section II). Still, the above reasoning doesillustrate that the Maxwellian stiffness of a µ follows the same pattern as that of v µ :both are determined by the order parameter 〈φ〉 of condensed dual superfluid. Thus,we can write the correct form of L 0 , with both v µ and a µ fully included into ouraccounting, asL 0 [v µ , a µ ] → (∂ × v) µ(∂ × v) µ4π 2 |〈φ〉| 2 + (∂ × a) µ(∂ × a) µ4π 2 |〈φ〉| 2 , (B.30)where our ignorance is now stored in computing the actual value of 〈φ〉 from theoriginal parameters of the d-wave superconductor problem. With anisotropy restoredthis is precisely Eq. (3.45) of Section II.105

B.2 Feynman integrals in QED 3Many of the integrals encountered here are considered standard in particle physics.Since the techniques involved are not as common in the condensed matter physics weprovide some of the technical details in this Appendix. A more in-depth discussioncan be found in many field theory textbooks [76].B.2.1Vacuum polarization bubbleThe vacuum polarization Eq. (3.59) can be written more explicitly asΠ µν (q) = 2NTr[γ α γ µ γ β γ ν ]I αβ (q)(B.31)with∫I αβ (q) =d 3 k(2π) 3 k α (k β + q β )k 2 (k + q) 2 .(B.32)The integrals of this type are most easily evaluated by employing the Feynman parametrization[76]. This consists in combining the denominators using the formula1 Γ(a + b)=A a Bb Γ(a)Γ(b)∫ 10dxxa−1 (1 − x) b−1[xA + (1 − x)B] , a+b(B.33)valid for any positive real numbers a, b, A, B. Setting A = k 2 and B = (k + q) 2allows us to rewrite (B.32) as∫ 1 ∫I αβ (q) = dx0d 3 k k α (k β + q β )(2π) 3 [(k + (1 − x)q) 2 + x(1 − x)q 2 ] . 2(B.34)We now shift the integration variable k → k − (1 − x)q to obtain∫ 1∫d 3 k k α k β − x(1 − x)q α q βI αβ (q) = dx(2π) 3 [k 2 + x(1 − x)q 2 ] , (B.35)20where we have dropped terms odd in k which vanish by symmetry upon integration.We now notice that k α k β term will only contribute if α = β and we can thereforereplace it in Eq. (B.35) by 1 3 δ αβk 2 . With this replacement the angular integrals aretrivial and we haveI αβ (q) = 12π 2 ∫ 10dx∫ ∞0dkk 2 13 δ αβk 2 − x(1 − x)q α q β[k 2 + x(1 − x)q 2 ] 2 . (B.36)106

The only remaining difficulty stems from the fact that the integral formally divergesat the upper bound. This divergence is an artifact of our linearization of the fermionicspectrum which breaks down for energies approaching the superconducting gap value.It is therefore completely legitimate to introduce an ultraviolet cutoff. Such UVcutoffs however tend to interfere with gauge invariance preservation of which is crucialin this computation. A more physical way of regularizing the integral Eq. (B.36) isto recall that the gauge field must remain massless, i.e. Π µν (q → 0) = 0. To enforcethis property we write Eq. (B.31) asΠ µν (q) = 2NTr[γ α γ µ γ β γ ν ][I αβ (q) − I αβ (0)],(B.37)and we see that proper regularization of Eq. (B.36) involves subtracting the value ofthe integral at q = 0. The remaining integral is convergent and elementary; explicitevaluation givesI αβ (q) − I αβ (0) = − |q| (δ αβ + q )αq β. (B.38)64 q 2Inserting this in (B.37) and working out the trace using Eqs. (3.54) and (3.55) we findthe result (3.60). Identical result can be obtained using dimensional regularization.B.2.2TF self energy: Lorentz gaugeFor simplicity we evaluate the self energy (3.64) in the Lorentz gauge (ξ = 0).Extension to arbitrary covariant gauge is trivial. Eq. (3.64) can be written aswithwhere we used an identityΣ L (k) = − 2π 3 N γ µI µ (k)∫I µ (k) =d 3 qq µq · (k + q)|q| 3 (k + q) 2 ,(γ µ γ α γ ν δ µν − q )µq ν= −2 ̸q q αq 2 q . 2(B.39)(B.40)Since the only 3-vector available is k, clearly the vector integral I µ (k) can only havecomponents in the k µ direction, I µ (k) = C(k)k µ /k 2 . By forming a scalar product107

k µ I µ (k) we obtain∫C(k) = k µ I µ (k) =d 3 q (q · k)(q · k + q2 )|q| 3 (k + q) 2 . (B.41)Combining the denominators using Eq. (B.33) and following the same steps as in thecomputation of polarization bubble above we obtainC(k) = 3 2∫ 10dx √ ∫xd 3 q× (2x − 1)(k · q)2 − (1 − x)k 2 q 2 + x(1 − x) 2 k 4[q 2 + x(1 − x)k 2 ] 5/2 .(B.42)The angular integrals are trivial and after introducing an ultraviolet cutoff Λ andrescaling the integration variable by |k| the integral becomesC(k) = 2πk 2 ∫ 10dx √ ∫ Λ|k|x dq (5x − 4)q4 + 3x(1 − x) 2 q 2. (B.43)0 [q 2 + x(1 − x)] 5/2The remaining integrals are elementary. Isolating the leading infrared divergent termwe obtainC(k) → − 4π 3 k2 ln Λ|k| .Substituting this to Eq. (B.39) we get the self energy (3.65).(B.44)B.3 Feynman integrals for anisotropic caseB.3.1Self energyAs shown in the Section V, to first order in 1/N expansion, the topological fermionself energy can be reduced to∫d 3 k (q − k) λ (2gλµ n Σ n (q) =γn ν − γn λ gn µν )D µν(k)(2π) 3 (q − k) µ gµν n (q − k) . (B.45)νwhere D µν (q) is the screened gauge field propagator evaluated in the Section 3. Inorder to perform the radial integral, we rescale the momenta asK µ = √ g n µν k ν; Q µ = √ g n µν q ν (B.46)108

and obtain∫ d 3 KΣ n (q)=)√gK νn µν(Q−K) λ (2 √ (g n λµ γn ν −γ λ gµν)D n µν. (B.47)(2π) 3 v F v ∆ (Q−K) 2At low energies we can neglect the contribution from the bare field stiffness ρ in thegauge propagator (see Section 3) and the resulting D µν (q) exhibits 1 q scalingD µν()K ν√ gnµν= F µν(θ, φ)|K|(B.48)Now we can explicitly integrate over the magnitude of the rescaled momentum K byintroducing an upper cut-off Λ and in the leading order we find that∫ Λ( )dK K2 (Q−K) λK(Q−K) =−Λ ˆK Λ ( 2 λ + ln Q λ −2QˆK)λ ˆK · Q0(B.49)where ˆK = (cos θ, sin θ cos φ, sin θ sin φ). Since ˆK µ is odd under inversion while F µνis even, it is not difficult to see that the term proportional to Λ vanishes upon theangular integration. Thus∫Σ n (q) =dΩ(Q(2π) 3 λ − 2v F v ˆK)λ ˆK · Q (×)∆(×)(2 √ g n λµ γn ν − γ λg n µν )F µν(θ, φ) ln( ΛQ). (B.50)Using the fact that the diagonal elements of F µν (θ, φ) are even under parity, whilethe off-diagonal elements are odd under parity, the above expression can be furthersimplified towhere the coefficients η n µΣ n (q) = − ∑ µthereby reduced to a quadrature∫ηµ n =( )(γµq n µ ηµ) n Λln √qα gαβ n q β(B.51)are pure numbers depending on the bare anisotropy and are(dΩ(2v F v ∆ (2π) ˆK 3 µ ˆKµ −1)(2gµµF n µµ − ∑ νg n ννF νν )+ ∑ ν≠µ4 ˆK µ ˆKν√ gnµµ√ gnνν F µν). (B.52)109

Repeated indices are not summed in the above expression, unless explicitly indicated.In the case of weak anisotropy (v F = 1 + ɛ, v ∆ = 1) we can show that the Eq.(B.52)reduces to Eqs.(3.92-3.94).Consider now the effect of the (covariant) gauge fixing term on η µ . Let us definethe part of F µν which depends on the gauge fixing parameter ξ as F (ξ)µν . The generalform of this term is F (ξ)µν = ξ k µ k ν f(k) where f(k) is a scalar function of all threecomponents of k µ ; f(k) does in general depend on the anisotropy. Upon rescaling withthe nodal metric, see Eq.(B.46), we have F (ξ)µν = ξ 1 √ gµµK µ1√ gννK ν ˜f(K), where ˜f(K)is the corresponding scalar function of K µ . Substituting F (ξ)µν into the Eq. (B.52) wefind∫ ( dΩηµ ξ =ξ ˜f(K)(2v F v ∆ (2π) ˆK 3 µ ˆKµ −1)(2 ˆK µ ˆKµ − ∑ νˆK ν ˆKν )+ ∑ ν≠µ4 ˆK µ ˆKν ˆKµ ˆKν), (B.53)where ηµ ξ is the part of η µ which comes entirely from the gauge fixing term. Usingthe fact that ∑ ˆK ν ν ˆKν = 1, it is a matter of simple algebra to show thatη ξ µ =ξ ∫ dΩ ˜f(K)v F v ∆ (2π) 3(B.54)i.e., the dependence on the index µ drops out! That means that the renormalizationof η µ due to the unphysical longitudinal modes is exactly the same for all of itscomponents. Therefore, the difference in η 1 and η 2 , which is related to the RGflow of the Dirac anisotropy comes entirely from the physical modes and is a gaugeindependent quantity. Note that this statement does not depend on the choice of thecovariant gauge, i.e. on the exact form of the function f, only on the fact that thegauge is covariant.B.4 Physical modes of a finite T QED 3Free massless Abelian gauge theory in d + 1− space-time dimensions has d − 1physical degrees of freedom. The familiar example is the world we live in: light hasonly two (transverse) physical degrees of freedom. Using the modern terminology,photon is a spin 1 massless gauge particle, which restricts its spin projections to110

S = ±1. These two degrees of freedom contribute to the specific heat of a photongas.When the gauge field interacts with matter, the situation is a little more involved.In 3 + 1D QED, matter effectively decouples from light and so the leading ordercontribution to the thermodynamics is just a sum of the contributions from the freefermions and free light. However, exchange of a massless particle introduces long rangeinteractions between the fermions, thereby affecting the spectrum of the quantumtheory. While there are no collective modes in a free theory, the coupled theory hasmulti-particle collective states in the spectrum, and these states will contribute tothe thermodynamics of the coupled system.One way of computing such contributions, is to integrate out the fermions. Thepartition function can be written as Z = Z 0 fermi Zeff gauge , where Z0 fermifermion contribution to the free energy, while Z effgaugegives the freegives the contribution from thedressed gauge field. If this gauge field action is expanded to quadratic order, then thecontribution from the gauge field can be computed exactly. It can be understood asfermion dressing or renormalization of the free gauge field. In general, there are twodistinct processes: (1) the renormalization of the transverse components that contributedto thermodynamics even without any fermions, and (2) the renormalizationof the longitudinal components, which did not contribute to thermodynamics in theabsence of matter. The latter can be understood as arising solely from the collectivemodes of the fermions.To see how all of this comes about in detail, consider the free gauge theory inEuclidean space-time:L 0 (a µ ) = 12e 2 a µΠ µν a ν ; Π µν = (k 2 δ µν − k µ k ν ). (B.55)For future benefit, we shall rewrite the polarization tensor using(A µν = δ µ0 − k ) (µk 0 k2δk 2 k 2 0ν − k )0k ν,k(2B µν = δ µi δ ij − k )ik jδk 2 jν ,(B.56)where k µ = (ω n , k) and ω n = 2πnT is the bosonic Matsubara frequency. It is straightforwardto see that δ µν − k µ k ν /k 2 = A µν + B µν , and that the gauge field action is111

nowL 0 (a µ ) = 1 2 Π0 Aa µ A µν a ν + 1 2 Π0 Ba µ B µν a ν , (B.57)where Π 0 A = Π0 B = (ω2 n + k 2 )/e 2 . We shall see later that to order 1/N the matterfields modify Π A and Π B , but not the generic form of Eq. (B.57).Finite temperature breaks Lorentz invariance in that it introduces a fixed length inthe temporal direction. Therefore, it is not convenient to use covariant gauge fixing(as at T = 0); instead we shall work in Coulomb’s gauge ∇ · A = 0. This gaugeeliminates one of the d + 1 degrees of freedom. In the free gauge theory, there is onemore non-physical degree of freedom. To see that this is indeed so, integrate out thetime component of the gauge field a 0 . Schematically, this proceeds as follows:L = Π 00 a 2 0 + 2a i Π i0 a 0 + a i Π ij a j ; i ≠ 0; (B.58)(= Π 00 a 0 + 1 ) 2Π 0i a i − 1 (Π i0 a i ) 2 + a i Π ij a j (B.59)Π 00 Π 00=k 2ω 2 n + k2 Π Aã 2 0 + Π B a i B ij a j ,(B.60)where we shifted the time component of the gauge field in the functional integral,ã = a 0 + 1Π 00Π 0i a i , and used the Eq. (B.56).In the free field theory Π A = Π 0 A = (ω2 n + k2 ) and Π B = Π 0 B = (ω2 n + k2 ). Thus,(L 0 = k 2 ã 2 0 + (ωn 2 + k 2 ) δ ij − k )ik jak 2 i a j .(B.61)Since the coefficient in front of ã 2 0 is independent of ω n and therefore independentof temperature, this mode does not contribute to thermodynamics.In addition,δ ij − k i k j /k 2 projects out the spatial longitudinal mode, and only d − 1 transversemassless modes contribute to thermodynamics.When the gauge field interacts with the fermions, the effective quadratic actionfor the gauge field is modified by the fermion renormalization of the polarizationfunctions: Π A,B = Π 0 A,B + ΠF A,B . While it is still true that the longitudinal spatialcomponent of a µ is projected out, it is no longer true that the coefficient in frontof ã 2 0 is independent of ω n .We must therefore include this ”mode” in computingthe thermodynamical free energy. For instance, in the problem of electrodynamics of112

metal with a Fermi surface, this mode is sharp at the plasma frequency. In QED 3 ,inclusion of this mode is essential to preserve the ”hyperscaling” of the (z = 1) theory,in that c v ∼ T 2 .B.5 Finite Temperature Vacuum PolarizationIn this section we shall compute the one loop polarization matrix of the finitetemperature QED 3 .Π µν (q) = N β∑∫ω nΠ µν (q) = N βd 2 k(2π) 2 T r[G 0(ω n , k)γ µ G 0 (Ω + ω n , k + q)γ ν ],∑∫ω n(B.62)d 2 k(2π) 2 T r[γ µ γ α γ ν γ β ]k α (k β + q β )k 2 (k + q) 2 , (B.63)where k = (ω n , k) = ((2n + 1)πT, k). In the Euclidean space we haveand sowhereT r[γ µ γ α γ ν γ β ] = 4(δ µα δ νβ − δ µν δ αβ + δ µβ δ να )Π µν (q) = 4N β∑∫ω nd 2 k L µν (k; q)(2π) 2 k 2 (k + q) , 2L µν (k; q) = k µ (k ν + q ν ) + k ν (k µ + q µ ) − δ µν k α (k α + q α )(B.64)(B.65)(B.66)where repeated indices (α) are summed.At T = 0, the natural way to proceed is to use Feynman parameters. However,there is no advantage in doing so at finite T . Therefore, we shall first sum over thefermionic Matsubara frequencies ω n ; then, we perform the integral over k which wesplit into an integral over the angle and the integral over the magnitude of k. Theintegral over the angle can be performed analytically. However, the last integral overthe magnitude of k cannot be performed in the closed form. This happens quite frequentlyin theories dealing with degenerate fermions. However, our results will differfrom such theories in an important qualitative way: there is no lengthscale associatedwith the Fermi energy and this prohibits the use of Sommerfeld expansion. Instead,113

the theory is quantum critical and the only scales are given by the external frequencyand momentum, and by the temperature. We shall therefore use the expression forthe polarization matrix Π µν without performing the final integral for general (Ω, q, T )and evaluate Π µν only for some of its limiting forms.To proceed, defineS µν (k, T ; iΩ, q) = 1 ∑ L µν (iω n , k; iΩ, q)(B.67)β (k 2 − (iωω n ) 2 )((k + q) 2 − (iω n + iΩ) 2 )n∮dz L µν (z, k; iΩ, q) 1= −2πi (k 2 − z 2 )((k + q) 2 − (z + iΩ) 2 ) e βz + 1 , (B.68)Γwhere the contour Γ includes the poles of the function n F (z) = 1/(e βz + 1) in thepositive (counterclockwise) sense and the negative sign comes from the residue ofn F (z) being −1. By deforming the contour around the remaining four poles it is easyto see thatS µν (k, T ; iΩ, q) = n F (|k|) L µν (|k|, k; iΩ, q)2|k| ((iΩ + |k|) 2 − (k + q) 2 ) −n F (−|k|) L µν (−|k|, k; iΩ, q)2|k| ((iΩ − |k|) 2 − (k + q) 2 ) +n F (|k + q|) L µν (|k + q| − iΩ, k; iΩ, q)− n F (−|k + q|) L µν (−|k + q| − iΩ, k; iΩ, q).2|k + q| ((iΩ − |k + q|) 2 − k 2 ) 2|k + q| ((iΩ + |k + q|) 2 − k 2 )So∫Π µν (Ω, q) = 4Nd 2 k(2π) 2 S µν(k, T ; iΩ, q).(B.69)(B.70)We shall now concentrate on Π ij where i, j denote the spatial indices. From herewe shall be able to extract both Π F A and ΠF B . Since we eventually integrate over allk, we can shift k + q → k in the last two terms in the Eq. (B.69) as well as reversethe direction of k to obtain∫( )d 2 k 2n F (|k|) − 1 2ki k j + k i q j + k j q i − δ ij (k · q − iΩ|k|)Π ij (iΩ, q) = 4N+ (iΩ → −iΩ) .(2π) 2 2|k|(iΩ) 2 + 2iΩ|k| − 2k · q − q 2(B.71)Note that all the T dependence resides in the Fermi occupation factor n F . Since itsargument in the Eq.(B.71) is positive definite, at T = 0, n F (|k|) = 0 and the resultis Π µν (q; T = 0) = Nq/8(δ µν − q µ q ν /q 2 ) as obtained by dimensional regularization.114

Therefore, at finite T we can writewhere∫δΠ ij (iΩ, q) = 4NΠ µν (q; T ) = N q 8(δ µν − q )µq ν+ δΠq 2 µν (iΩ, q) (B.72)( )d 2 k n F (|k|) 2ki k j + k i q j + k j q i − δ ij (k · q − iΩ|k|)+ (iΩ → −iΩ) .(2π) 2 |k| (iΩ) 2 + 2iΩ|k| − 2k · q − q 2(B.73)We next rotate the coordinate axis so that the vector q is aligned with the x-axis.Since the denominator is even under reflection by the (new) x-axis we can simplifythe above equality to readδΠ ij (iΩ, q) = N π 2 (δ ij − q iq jwhereq 2 ) ∫ ∞∫N q i q ∞jdknπ 2 q 2F (k)00∫ 2π0∫ 2πdkn F (k)0dθ J B(iΩ, |q|, k; cos θ)K(iΩ, |q|, k) − cos θ +dθ J A(iΩ, |q|, k; cos θ)+ (iΩ → −iΩ),K(iΩ, |q|, k) − cos θ (B.74)J A (iΩ, |q|, k; cos θ) = iΩ2|q| + 1 2 cos θ + k|q| cos2 θ,J B (iΩ, |q|, k; cos θ) =iΩ + 2k2|q|− 1 2 cos θ − k|q| cos2 θ,K(iΩ, |q|, k) = (iΩ)2 + 2iΩk − q 2. (B.75)2k|q|Now, Π F µν = Π F A A µν + Π F B B µν (see Eq. B.56). Once written in this form, we can readoff the finite T corrections to Π F A and ΠF B :δΠ F A = q2 + Ω 2 ∫N ∞dknΩ 2 π 2 F (k)δΠ F B = N ∫ ∞dknπ 2 F (k)00∫ 2π0∫ 2π0dθ J A(iΩ, |q|, k; cos θ)K(iΩ, |q|, k) − cos θdθ J B(iΩ, |q|, k; cos θ)+ (iΩ → −iΩ)K(iΩ, |q|, k) − cos θ (B.77)+ (iΩ → −iΩ)(B.76)The above integrals can be computed analytically using contour integration since∫ 2π∮ ( )dz z + z−1dθ f(cos θ) =iz f , (B.78)20where the contour of integration is a unit circle around the origin encircled counterclockwise.CSince eventually we will need retarded polarization function in orderto compute physical quantities, we analytically continue the outside frequency115

iΩ → Ω + i0 + . Putting everything together we finally arrive atΠ F A,B (iΩ → Ω + i0+ , |q|, T ) = Π F A,B ret (Ω, |q|, T )(B.79)where(Ω, |q|, T ) =√ [√Θ(q 2 −Ω 2 ) |q| 1 − Ω2 16 ln 2 T1 + 1 − Ω28 q 2 π |q| q − 4 2 π∫4 1dx √ ( ( |q|1 − xπ2 n F x − Ω ))− 4 ∫ 1Ω2T |q| π 0|q|ReΠ F retAΘ(Ω 2 −q 2 ) |q|84πand∫ ∞Ω|q|√Ω 2q 2 − 1 [−dx √ x 2 − 1 n F( |q|2T√16 ln 2 T Ω 2π |q| q − 1 + 4 2 π(x − Ω ))− 4 ∫ ∞|q| πImΠ F Aret (Ω, |q|, T ) = Θ(q 2 − Ω 2 ) |q|8[ ∫ 4 ∞dx √ { ( |q|xπ2 − 1 n F2T1sgnΩ Θ(Ω 2 − q 2 ) |q|8√Ω 2q 2 − 1 [ 4π√11 − Ω2q × 2 ))(x − Ω|q|∫ 1−1∫ Ω|q|0dx √ ( ( |q| Ω1 − x 2 n F2T(x + Ω )) ] +|q|dx √ 1 − x 2 n F( |q|2T∫ Ω|q|1dx √ ( |q|x 2 − 1 n F2Tdx √ x 2 − 1 n F( |q|2T− n F( |q|2Tdx √ 1 − x 2 {n F( |q|2T(x + Ω|q|(x + Ω|q|(x + Ω|q|))}]+|q| − x ))−( )) Ω|q| − x +)) ](B.80)))− 1 }]2(B.81)116

where Θ(x) is a Heaviside step function. Similarly,ReΠ F retB (Ω, |q|, T ) =√ ⎡1 − Ω2 ⎣1 +q 2Θ(q 2 −Ω 2 ) |q|8∫4 1π Ω|q|x 2dx√ n F1 − x2Θ(Ω 2 −q 2 ) |q|84πand∫ ∞Ω|q|√Ω 2x 2( |q|2T⎡q − 1 ⎣ 2dx√ x2 − 1 n F( |q|2T16 ln 2πT√|q|Ω 2q 21 − Ω2q 2(x − Ω ))− 4 |q| π16 ln 2πT|q|(x − Ω|q|Ω 2q 2√Ω 2ImΠ F Bret (Ω, |q|, T ) = Θ(q2 − Ω 2 ) |q|8[ ∫ 4 ∞{ (x 2 |q|dx n Fπ2T1√x2 − 1sgnΩ Θ(Ω 2 − q 2 ) |q|8√Ω 2q 2 − 1 [ 4π∫ 10− 4− 1πq 2))+ 4 π√∫ ∞1− 4 π∫ Ω|q|0x 2dx√ n F1 − x2∫ Ω|q|1x 2dx√ n F1 − x2x 2( |q|2Tdx√ x2 − 1 n Fx 2dx√ x2 − 1 n F1 − Ω2q × 2 )) ( |q|− n F|q| 2T(x + Ω∫ 1−1dx√ 1 − x2The expressions for the imaginary parts of Π F retAx 2{n F( |q|2Tand Π F retB( |q|2T(x − Ω|q|( |q|2T( Ω|q| − x ))−(x + Ω )) ] +|q|( |q|2T(x + Ω|q|( Ω|q| − x ))−(x + Ω )) ]|q|))}]+(B.82)))− 1 }]2(B.83)(B.81) and (B.83) can befurther simplified using an infinite series of Bessel functions, but there is no furthersimplification of the real parts of Π F retAand Π F Bret . Nevertheless, there is an importantfact that ought to be stressed: the expressions (B.80-B.83) reflect the quantum criticalscaling of the theory near its T = 0 fixed point. As can be seen from above, we canwrite the following scaling relations:Π F retA,B (Ω, |q|, T ) = NT P F A,B( |q|T , Ω )T(B.84)In the limiting cases of small or large argument, both real and imaginary part of the(function PA,BF q , ) ΩT T can be evaluated analytically.117

B.6 Free energy scaling of QED 3We shall now use the results of the previous section to compute the T dependenceof the thermodynamic free energy from which we can find the electronic contributionto the specific heat. Since to order 1/N the effective action for the gauge field isquadratic, we can perform the functional integral over its two ”modes” exactly. Thus,where Π A =F = 1 β ln(Det(Π A)) + 1 β ln(Det(Π B)),q2(Π 0 q 2 +Ω 2 A + ΠF A ) and Π B = Π 0 B + ΠF B . In addition we have,F = 1 ∑∫d 2 q (ln [ΠA (iΩβ (2π) 2 n , q)] + ln [Π B (iΩ n , q)] ) .Ω n(B.85)(B.86)Now, ln(z) is analytic in the upper complex plane with a branch-cut along the xaxis. Recognizing this fact, we can trade the summation over the bosonic Matsubarafrequencies Ω n = 2πnT for an integral along the branch cut:1 ∑∮dz ln [Π A,B (z, q)]ln [Π A,B (iΩ n , q)] =βΩΓ 2πi e βz − 1n== 2= 2∫ ∞−∞∫ ∞−∞∫ ∞dΩ ln [Π A,B (Ω + i0 + , q)] − ln [Π A,B (Ω − i0 + , q)]2πie βΩ − 1dΩ Im ln [Π A,B (Ω + i0 + , q)]2π e βΩ − 1−∞dΩ2π( )1ImΠrete βΩ − 1 tan−1 A,B (Ω, q)ReΠ retA,B (Ω, q)(B.87)Now, the crucial observation about the polarization functions is that they can berescaled as( |q|Π F A,B ret (Ω, |q|, T ) = NT P A,BFΠ 0ret2TA,B (Ω, |q|, T ) =e 2 P0 A,BT , Ω T( |q|T , Ω T)). (B.88)This can be used to argue that the temperature dependence of free energy readsF = −NT 3 3ζ(3) ( ) T4π+ T 3 Φ , (B.89)Ne 2118

where the fist term is the contribution from N free 4−component Dirac fermionsand the second term comes from the interaction with the gauge field. By numericalintegrations we have found out that the function Φ(x) is regular at x=0 (there aretwo log singularities coming from Π A and Π B which cancel). This result shows thatthere is no anomalous dimension in the free energy i.e. that the z = 1 hyperscaling atthe QED 3 quantum critical point holds. Therefore, the leading scaling of the specificheat in finite T QED 3 is c v ∼ T 2 .119

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VitaOskar Vafek was born on April 18, 1976. He received a Bachelor of Arts degreein Chemistry from Goucher College in 1998. He will be a Postdoctoral Scholar in theStanford Institute for Theoretical Physics (SITP) and the Department of Physics atStanford University beginning September 1, 2003.127