Superconductivity through Quantum Critical Fluctuations in the ...

Superconductivity through Quantum CriticalFluctuations in the CupratesVarious Collaborators but the latest workprimarily with Aji, Shekhter, Choi, Zhou.1. Introduction: Phase Diagram of the Cuprates.2. Quantum-Critical Fluctuations.3. Coupling of these fluctuations to Fermions;Derivation of vertex for d-wave pairing.4. Test of theory through comparison with laser-ARPES.5. Derivation of local quantum criticality:Topological Excitations.

The phenomena in the Cuprates has required are-examination of some of the fundamentalConcepts of Condensed Matter Physics.

Universal Phase DiagramTAntiferromagnetism?T*StrangeMetalCrossoverPseudo-GappedFermi liquidQCPx (doping)SuperconductivityLines drawn (1997) based on phenomena which occur inreasonably pure samples of all Cuprates.

TThis Region Violates Quasi-particleConcepts. Quantum CriticalFluctuations with spatially localcriticality and ω/ T scaling.↓T* CrossoverLoopordered.Fermi liquidQCPx (doping)Superconductivity

Fermi-Arc Phenomena:A New paradigm for massgeneration or gap formation.TAn anisotropic gap at chemical potentialwithout change of translational symmetry:T*StrangeMetalCrossoverFermi liquidQCPx (doping)

TT*StrangeMetalCrossoverPseudo-GappedFermi liquidQCPx (doping)This region has a scale of fluctuationsextending to ~0.4 eV, with preferred π/2scattering leading to high Tc in the d-wavechannel.

Any theoretical ideas and calculations whichdo not explain the normal state propertiesis not a candidate for understandingsuperconductivity either.All the remarkable effects should followfrom a single set of ideas.

TThis Region Violates Quasi-particleConcepts. Quantum CriticalFluctuations with spatially localcriticality and ω/ T scaling.↓T* CrossoverLoopordered.Fermi liquidQCPx (doping)Superconductivity

Effect of Fluctuations on Single-particleSpectra measured in ARPES15Im (⇤,k)(k)0d⇤ Im⇥(⇤ ) ⇥(k) =k| (k, k )| 2Predicts Linewidth proportional to for cand constant beyond. Factor of ~ 2 mom. dependence.

Recent ARPES Experiments to High Energies.oXOP Bi2201 Nodal , Meevasana et al.OP-Bi2212 Nodal, Lanzara et al.LSCO OP, Nodal , Chang et al.LSCO Nodal underdoped, Chang etThis is an experimental proof of the existence of a distinctspectrum of fluctuations with a well-defined cut-off frequency.This is what couples to fermions at just above Tc.

If there is a region of Quantum-critical Fluctuations,there may be an unusual ordered phase?Universal Phase DiagramTAntiferromagnetism?T*StrangeMetalCrossoverPseudo-GappedFermi liquidQCPx (doping)Superconductivity

Loop OrderDerived (1997) in three orbital model for cuprates a time-reversalviolating state preserving translational symmetry. More detailedcalculations by Weber et al.Model has four possible flux configurations in a cell into one of whichorder is observed.Order parameter:Cu+OCuO_CuOO CuL i = M i ˆr i L i | i = e i i| iθ= π/4, 3π/4, 5π/4, 7π/4.

s but simply characterized temperaturerties studied. Characteristic Fermi-liquidregion (II) an anisotropic gap (pseudoicalpotential is observed. Transport [?]racteristic changes below the line T p (x).erties across di erent lines also occur. Itnd the universal properties.Experiments consistent with this long-range order.This order has been observed in four distinct families ofCuprates by Polarized Neutron scattering or by DichroicARPES.YBa(2)Cu(3)O(6+x)Bi2212HgBaCuOLSCO:IIIFermi liquidCrossoverx (doping)Temperature (K)4003002001000T (K)and discussed further belowHg1201 are in good agreem400 in Figs. 1-5 demonstrate tha300netic signal belowysis method.T mag is inIt is furtherm200100crease of T ⇤ toward higher hoappearance of the pseudogapdoping. Moreover, as alread00.05 0.1 0.15 0.2p0.05 0.1 0.15 0.2Hole doping player YBCO. 3,4,9 . These releast in compounds with higpseudogap phenomenon is aa mere crossover.In Ref. 4 an attempt wascal exponent of the transitiothat the fit value depends otrinsic temperature dependenconsidered in comparisons wof 2 =0.37±0.12 is consiste5). The uncertainty in the

Superconductivity through Quantum CriticalFluctuations in the Cuprates1. Introduction: Phase Diagram of the Cuprates.2. Quantum-Critical Fluctuations.3. Coupling of these fluctuations to Fermions;Derivation of vertex for d-wave pairing.4. Test of theory through comparison with laser-ARPES.5. Derivation of local quantum criticality:Topological Excitations.

For quantum-fluctuations in the disordered phase,one must consider a generator of rotations among them.Generator of Rotations among the four configurations.L z,i | i = e i /4 | i + ⇥/2

current operators in the link (p, p + 1). Note that this isthe lattice equivalent of transforms as (r ⇥ p), where rand p are the position and momentum vectors along thelinks of the unit-cell.Quantum-Critical Fluctuations : Aji-cmv PRL(07); PR-B(09)OCuThe critical modes are the fluctuations among the fourconfigurations of observed order, i.e., among the fourorientations of the order parameter vector L.(a)(b)FIG. 1: a: The four domains of the circulating current phaseare shown. They may be specified by the four orientationsof a vector shown in red. b: Schematic representation of thecurrent due to the operator C(j i )In the fluctuations regime the vector L has the same criticalQuantum Ashkin-Teller model: In the quantum fluctuationregime, fluctuating flux and current patterns overany region of space and time are generated by the elementaryprocess of fluctuations between the four config-spectra as a model with a continuous rotations of L, i.e. thequantum- XY model.⇧ 0 maywhereof thecell.mate fr0.1µ B pCouppling ofa collecmatrix|⇥ i ± ⇤tering mtime-rethe conin detaas theonly useratorsthe operotatiorotatesator inP (1234

Fluctuations of the pseudogap order in QC region:CuOCuOCuOOCuQuantum-critical Flucts: Topological excitations whosebasic units are local flips among these configurations leadingto loops of all sizes and shapes appearing and disappearing.

Above the QCP of the orderd Phase, (Aji and Varma, PR-B ’09, ’10)⇥(r, t; r t )=< L(r, t)L(r ,t) >= (r r )t(q, ⇥) = tanh(⇥/2T ), ⇥⇥ c c =2JI1tIm (k, ⇥)/TMarginal Fermi-Liquid finally Derived from a Microscopic ModelTc

Superconductivity through Quantum CriticalFluctuations in the Cuprates1. Introduction: Phase Diagram of the Cuprates.2. Quantum-Critical Fluctuations.3. Coupling of these fluctuations to Fermions;Derivation of vertex for d-wave pairing.4. Test of theory through comparison with laser-ARPES.5. Derivation of local quantum criticality:Topological Excitations.

Fermion pairing: An Important General Point(Miyake, Schmitt-Rink, V. (1986))k’-k’gFg: g 2 (k, k 0 )F (k, k 0 ,!)k-kSymmetry of Pairing determined by Angular distribution ofscattering from k to k’:S-wave: isotropic scattering of fermions.P-wave: preferred forward (near 0-angle) and backward(near ∏ ) scattering of fermions.D-wave: preferred scattering at +/- (π /2) .

angle independence of the Eliashberg function 2 F,,i.e., the angle averaged product of the matrix element and thespectral function of the fluctuations, as defined in Eq. 13,the possible physics of the low-energy bump around0.05 eV, the vertex corrections to the Eliashberg equation,and various other assorted issues. The finding that the Eliashbergfunction is angle independent below the cutoff c putsan important constraint on the microscopic understanding ofthe cuprates. For example, it can put a limit on the correlationlength for the commonly assumed form of the antiferromagneticAF fluctuations. A phenomenological form ofthe overdamped AF fluctuations may be written as 2,30How can a locally critical spectra give angle-dependentscattering?It can only do so due to the momentum dependenceof the vertex g(k,k’).This is in contrast with coupling through AFM fluctuationsthrough Hubbard or t-J model, (or cdw or stripes) where thevertex is proportional to AF k,k, =U 2 (k k 0 ,!) 2 / AFk − k − Q 2 2 +1 2 + / AF 2 . 18 AF , can be obtained after the integral over with bothk and k on the Fermi surface, that is,±⇡/2This also gives scattering peaked at anglein a square lattice ifQ ⇡ (⇡, ⇡)and AF , = AF k,k, ,k F ⇠>>1We will be able to distinguish between the two19where k and k have the angle and with respect to thefrom the experimental results.nodal cut, respectively. A straightforward calculation revealshigh-region orWerectionThe ek,qAs disc0.2 eV,only to8 is ontherefortex corrconclustuationfrequenwhich ithe uppbandwicorrecticlusionOne

Coupling of Quantum-critical fluctations to fermions.Aji, Shekhter, cmv (PRB-2010)In continuum limit, L z (r) is the angular momentumoperator for collective modes.It couples to the angular momentum operator forfermions:+ (r)(r p) (r)H coupl / L z (kk 0 ) + k 0 (ik ⇥ k 0 ) kk-k’k k’Orbital Moment Analog of the familiar coupling:J⇥ +k⇥ k · S

Pairing VertexScattering through π/2 favoredk’-k’ik k ik k: g 2 (k ⇥ k 0 ) 2 hLLi(k, k 0 ,!)k-kProject to different angular momentum channels|k × k ′ | 2 =1/2(k 2 x + k 2 y)(k → k ′ ) − 1/2(k 2 x − k 2 y)(k → k ′ ) − 1/2(2k x k y )(k → k ′ ).Repulsive in s-wave channel, equally attractive ind(x 2 y 2 ) and d(xy) channels.

Superconductivity through Quantum CriticalFluctuations in the Cuprates1. Introduction: Phase Diagram of the Cuprates.2. Quantum-Critical Fluctuations.3. Coupling of these fluctuations to Fermions;Derivation of vertex for d-wave pairing.4. Test of theory through comparison with laser-ARPES.5. Derivation of local quantum criticality:Topological Excitations.

Deducing the Fluctuation Spectra Coupling to Fermions andLeading to Pairing in Cuprates from High-Resolution ARPES DataGeneralization of the MacMillan-Rowell procedureto d-wave Superconductors.Data from the Group of Xingjiang Zhou, BeijingAnalysis done in collaboration with the group ofHan-Yong Choi, Seoul, Korea.This work is ongoing.

MacMillan-Rowell ProcedurePairing dynamicsBased on Eliashberg formulation of Pairing withretarded interactions and Schrieffer-Scalapino-Wilkins! For LTS: McMillan-Rowell inversion ofconductance of S/I/N or S/I/S (1965).use of that theory for Tunneling spectra.Pairing dynamics! For LTS: McMillan-Rowell inversion ofconductance of S/I/N or S/I/S (1965).! Fit by invertingthe Eliashberg eq.to extract and .! Fit by invertingthe Eliashberg I eq.to extract and .V

dI/dV ⇥ SdI/dV ⇥ N( )2 F (⇥)

Ordinary Tunneling measurements: not useful forfinite angular momentum pairing.But ARPES can be used in this case.Procedure suggested : (Vekhter and cmv (2002))Extract from the data using Eliashberg Equations:XF`(k, !) ⌘ g 2 (k, k 0 )F (k, k 0 ,!)P`(k 0 )k 0Need to deduce bothF 0 (k, !) and F 2 (k, !)Requirements on the data are very severe.The requirements are beginning to be met.Must also check the validity of the Eliashberg Equations

High Resolution Laser ARPES data. Zhou et al., Beijing.Analysis: Choi, CV, Zhoubending back dispersion. (a1)-(f1). Original photoemissionnodal and anti-nodal. (a2)-(f2). The corresponding imageat 107K.

of Figs. 7 and 8 where the position of the band bottom withrespect to the Fluctuation Fermi energy spectra is given. just Theabove latter isTc.about 1 eVin the nodal Slighty direction underdoped and about 0.25 Sample eV at 25°. Bi2212. As wasshown in aInversion simple calculationusing Elaishberg 27 and as ismethod:natural, the bandbottom serves as an effective cutoff in the fluctuation spectrumwhenno it isotherlowertheoretical than intrinsicimput.cutoff. So one im-GUIDES TOIn this section, wewhich point to futureA peak carrying10% of spec. wt.BOK et al.A featureless angleind. (MFL) spectrawith rest of weight.Cut-off is smaller ofabout 0.4eVor bottom of bandPHYSICAL REVIEW B 81, 174516 2010FIG. 7. Color online The Eliashberg function extracted fromthe real part of the self-energy at T=107 K. Notice the remarkablecollapse of the Eliashberg functions below 0.2 eV for differenttilt angles.FIG. 6. Color online Comparison between the real part of theself-energy and the MEM fitting of Eq. 11 for the tilt angles =0° and 15°. The squares are the extracted real part self-energy of174516-6FIG. 9. Color onlithe imaginary parts of thgiven for representativeoverall as expected.FIG. 8. Color online The angle dependence of the Fermi velocityand band bottom calculated from Eq. 4.

The Eliashberg functionAnalysis on an overdoped sample with Tc = 64 KBoth sample data are consistent with a fluctuation spectrawhich is q independent and matrix elements which areq-dependent.Consistency of analysis;Tc calculated backwards from spectra in the first caseis 105 K \pm 10K, Second case 88 \pm 10K,Expt. 91 K and 64 K.

erg function is angle independent below the cutoff c putsan important constraint on the microscopic understanding ofthe cuprates. For example, it can put a limit on the correlationlength for the commonly assumed form of the antiferromagneticAF fluctuations. A phenomenological form ofthe overdamped AF fluctuations by the form: may be written as 2,30AFM fluct. spectra or any other spectra specified by acorrelation lengthis ruled out. AF k,k, = 2 / AFk − k − Q 2 2 +1 2 + / AF 2 . 18 AF , can be obtained after the integral over with bothMOMENTUM DEPENDENCE OF THE SINGLE-PARTICLE…k and k on the Fermi surface, that is,1.00.8 AF , = AF k,k,⇥ = a, , =0, 5, .., 2519where k and0.6k have the angle and with respect to thenodal cut, respectively. A straightforward = a/⇥ calculation revealsΧAFΘ,Ω0.4that a weak dependence of AF , means that /a10.2for AF , where AF is the characteristic AF energy scale.0.0This is shown 0in Fig. 1 10 for 2 /a=1/ 3 4 with5the blue and for/a=1 with the red lines. ΩΩFor AF each , the angles are =0° ,5° ,10° ,15° ,20° ,25° from above. As expected,FIG. 10. Color online The model Eliashberg function calculated, from becomes the overdamped angle independent as is decreased. The AFcollapse of 2 AF fluctuations of Eq. 18. The red andblue lines are forF, the AF correlation implieslength that/a=1 if it and is due 1/, to respec- the AF fluc-As discuss0.2 eV, thatonly to the8 is only atherefore fotex correcticonclusionstuation spefrequencywhich PHYS in dthe upper cbandwidthcorrectionsclusions haOne finaduced specfor the supparticle selthen followof the fluctof the supestudy of thAn angle-independentquantum-critical spectra hyliquid description ofmicroscopically 31,32 to beregion of the phase diagratum melting of the loopunderdopedregion of the cdeduced 2 F, whichthe low-energy bump atpresence of this bump maIm , which is notT. The linearity maytheory and earlier ARPESnot have the high resolutTc < 10 K for correlation length of a lattice constant.

How could one imagine superconductivity from AFMfluctuations when AFM corr. length goes down whileTc is increasing?VOLUME 82, NUMBER 26 P H Y S I C A L R E V I E W L E T T EFIG. 1. q width (closed circles) and T C (open squares) versusoxygen content. FIG. 2. Supercofull square corres

SummaryMotivation: Attempt to understand all Anomalies in theCuprates through a single set of ideas.Loop Order at low carrier density.Collective modes of Fluctuations of loop orderhave local quantum-criticality to give Marginal Fermi-liquid.Vertex coupling these to fermions promotes d-wavepairing with some unique signatures.Verification through quantitative comparison with highresolution ARPES data.Expect results on other samples with systematic variationof Tc and deduced spectra.

TQuantum-critical fluctuationsof the Loop Ordered Stateand a Marginal fermi-liquidAntiferromagnetism?T*CrossoverA line ofphase transitions.Fermi liquidLoop-ordered Stateending at a QCP.Anisotropic gap atchem. potential.x (doping)QCPSuperconductivityHigh Tc d-wave superconductivitydue to large freq. scale of qtm.crit.fluctuations and π/2 scattering due tothe 4 configs. in loop-ord. state.

From New Yorker - April 25, 2011

Quantum Critical Fluctuations(Vivek Aji, cmv : PRL 07, PRB-09, PRB-10)Classical model:is equivalent in critical properties to a generalized xy model orInteracting Rotors Model with anisotropy:The anisotropy is marginally irrelevant in the fluctuation regime but strongly relevant in theordered phase. The model has no diverging specfic heat at the transition.+Quantum Generalization of the Model:Dissipative xy or Quantum-Rotor Modelexp(i i ) L + i :L is the angular momentum operator for the Rotors.H = L z i 2 /2I + J(L + i L j+ h.c.)+ Dissipative terms ( ).This model has been suggested to have a Quantum Critical Point at(Chakravarty, Kivelson, Luther, Ingold, Zimanyi, M. Fisher, .... )α = α c .

Theory of Criticality2D XY model with dissipationS = ⇥ d⇤ij⇥ J cos (⇥ i ⇥ j )+ ⇥ d⇤i˙2C + ⇥ dkd⌅ |⌅| k 2 |⇥ k,⇥ | 2Exact Transformation in terms of two sets of orthogonal variables:S= d⇤drdr J⇥ v (r, ⇤)⇥ v (r , t)ln |r r |+ drd⇤d⇤ ⇥ w (r, ⇤)⇥ w (r, ⇤ )ln |⇤ ⇤ |Singularities decouple in space and timeFluctuation spectrumIm (k, ⇥)/TTc

t finite temperature where the correlation funcder-parameterphase. Nevertheless e ı changes a from phaseexponential transitiontoPrincipal Steps:temperature where the correlation funcarametere ı changes from exponential totransition. The quantum dissipative generalizatiomodel includes two dynamical terms and is given bpower law. This is the Kosterlitz-Thoulesstransition. 25,26 The quantum dissipative generalizmodel includes two dynamical terms and is giveZ = D i exp − 0= Z D i exp − 0S diss =−d0did ij,klC2 2 ij − J cos ij − klij,kl ij − kl ij,kl − ij − kl − + S diss ,e capacitance andS=R Q /R, where R Q =h/4e 2 .cs of this phase transition diss d ,=−isd0better understoodij,klin terms of the − topological defects of the system. To dndard procedure of using the Villain transform and integrating out the phase degrees of freedom. 20 Thacitance olves expanding Villain =R Transformation:the Q periodic /R, where function R Q =h/4e in terms 2 . of a periodic Gaussianthis phase transitionexp − J is better understood 1 − cos ij − kl in termsij,klm ij;klexp − of J the topological defects of ij − kl − 2m ij;kl /2 2 ,the system. Tprocedure of using the Villain transform and integrating out the phase degrees of freedom. 20ij,klre integers that live on the links of the original. We can combine the two link variables m i,j;i+1,jto one integrated two-component over. vector m i,j that lives onof the lattice see Fig. 1. We expand the quadtransform to Fourier space. Keeping the lead-diC2 2 ij − J cos ij − kl xy − x+1,y − 2m xx,y+ S diss ,2 ij − kl − ij − kl 2expanding the periodic function in terms of a periodic Gaussianexp − J ij,kl1 − cos ij − kl m ij;klexp − J ij − kl − 2m ij;kl /2 2 ,ij,klNow the model is Gaussian in theθ’swhich can be, 2 a 2 2 xx xy +4a x xy m x,yxwhere m x,y is the x component of the vector fieldthe integer m . In the absence of dissipation,egers that live on the links of the original − − 2m x 2 a 2 2 +4a m x+

2 2where m x,y is the x component of thethe integer m x,y;x+1,y . In the absence ofcompetition between the kinetic-energenergy terms, the former minimized bydisordered stabilizing an insulating phminimized by a fixed value of ij staductor. Since ’s are bosonic degrees ofimpose the boundary condition that ij odicity in the imaginary-time directionof the field ij implies that there is anfreedom that has to be accounted for wnumber. At T=0, the imaginary- timinfinite in extent, and the nondissipatthree-dimensional 3D XY universalitycretize the imaginary-time direction inon a three-dimensional lattice. Introduwhich only live on the spatial links, thedisordered stabilizing and m i,j;ij+1 into onean two-component insulating vector m i,j phase that lives on while the latterthe site i, j of the lattice see Fig. 1. We expand the quadraticatermfixed and transform value to Fourier ofspace. Keeping the lead-xminimized bying quadratic term i,j − i+1,j −a x xy , where ij stabilizing a superconductor.m is Since a discretea is the latticeconstant, ’s are x=aiinteger bosonic and y=aj, we getfield degrees which of freedom, lives on we links. need toimpose the boundary condition that ij = ij 0. The periodicityin the imaginary-time i, j+y direction and the compactnessof the field ij implies that there is an additional degree ofm i,j;i,j+ym mfreedom that has to bei-x,accountedj; i, ji, j; i+x, jfor which is the windingi-x, ji, j i+x, jnumber. At T=0, the imaginary- time direction becomesm i ,j-y; i, jinfinite in extent, and the nondissipative model is in thethree-dimensional 3D XY i-x, j-y universality class. First we discretizethe imaginary-time direction in units of and workZ = exp(a)on a three-dimensional lattice. Introducing the variables m k, mwhich only live on the spatial links, the action is 19,20After integrating over the sZ = mm i,j;i,j+yexpk,m i, j; i+x, j− 4 2 J(b)FIG. 1. Color online The directed link variables are labeled asshown in the 2 a. b We define a two-component vector living onthe sites of them original · mC/c lattice whose + components k 2 /are the two directedlinks variables: m=m i,j;i+x,j ,m i,j;i,j+x . n m ijJck m 2C/c n 2 + Jck 2 + n k 2− 4 2 J nC/c 2 n + Jck 2 + n k 2,174501-2− 4 2 JckJC/c 2 n +− 4 2 J 2 nm · mC/c + k 2 / nC/c 2 n + Jck 2 + n where c=a/, J→Ja 2 , C→Ca 2 /have also redefined →a 3 . The last tetor is unimportant and may be dropped.has two possible phase transitions, depecapacitance C or the dissipation term the second term in Eq. 5 dominates inlow-frequency limit. The former correswith the dynamic critical z=1, i.e., we5

ppose we separate m into the usual transverse parms of a vortex field v k, n s of a vortex field v k, n Vortex field:ik m t k, n = v k, n ik m t k, n = v k, n d a longitudinal part through introducing the wak,Split m into its transverse and longitudinal parts:a longitudinal part through introducing the w,Warp field: n m k, n = ckˆ w k, n . n m k, n = ckˆ w k, n .uite miraculously the partition function in Eq. 5 canritten exactly asThen one finds that the singular part of the action decouples as:e miraculously the partition function in Eq. 5 caexpn Z = J v , w k, k 2 vk n 2 −en exactly asS= d⇤drdr J⇥ v (r, ⇤)⇥ v (r , t)ln |r r |+ drd⇤d⇤ ⇥ w (r, ⇤)⇥ w (r, ⇤ )ln |⇤ ⇤ |Z = exp J 2 vk n 2 −J 2 k 24 wk n 2w k n 2

What is a “Warp”:A warp is an arrangement of the Divergence of a Field at a point is surrounded symmetricallyFor a general vector field oneby does expectoppositetwo kinds of“charges” whose summed value is equal to the “charge”vortices, the additional topological entity to describe theat the center.FIG. 3: A phase slip between nearest neighbors in time producesa vortex and an antivortex on neighbouring plaqiettes.In time one bond has acquired a nonzero value, representedby the red bond. If the value on the bond is -1, the vorticityat site 1 is negative and on site 2 is positive.ThereforeFIG. 4: A phase slipWarpsevent in rime resultshavein a changenoin theinteractionssurrounded at theby four annearest equalneighbor butwithopposite sites. Incharge eachgeneraldistributedthe magnitudeotherlink variables. For a change of 2π at site {i, j}, thefourlinksofthe divergence at the site will equal to the number of nearestconnected to it acquire the values shown in the figure.timeThusis a phase slip1/(τ-τ’).on a link is equivalentTheirto a localproliferationin are most one latticebelowspacing apart; theainteractioncrit. valueisoflocal in space. Although this is physically obvious fromdissipation gives the “local” quantum-critical fluctuationnext section.ρ i ps Θ (τ − τ i) [(̂x + ŷ) δ (r − r ij )spectra.Field configuration of a Warpspace and time vortex current.Local phase slips and WarpsThe periodic in time boundary condition allows forδτ− ŷδ (r − r ij − aŷ)] Θ (τ − τ i )Such a vector field distribution has no curl and hence doesnot effect the vorticity. On the other hand the divergencei.e. an arrangement ofδτmonopoles in which the “charge”is nonzero and phase slip events generate field configurationsthat are orthogonal to those created by vortices.sources to generate an arbitrary distribution. For the2+1 dimensional quantum model we have, besides thewinding number sector in time. Events that change thewinding number sector, i.e. local phase slips, acts assources for a divergence in the vector field. Just as avortex is equivalent to an electric charge in the dual language,the sources created by phase slips can be shownto be a local distribution of monopoles (ρ m ). Given the“Charge” configuration of a Warpdistribution in eqn.15, the corresponding configuration ofmonopoles, which we term the charge of the phaseslip isρ m (r, τ) = ∇ · m(r, τ) (16)= [4δ (r − r ij ) − δ (r − r ij + âx)− δ (r − r ij − âx) − δ (r − r ij + aŷ)− δ (r − r ij − aŷ)] Θ (τ − τ i )The monopole distribution equivalent to a phaseslip isshown in fig.5. The total monopole charge of the configurationis zero. Since the distribution has azimuthalFIG. 5: Warps are configurations of the field m with finitesymmetry all harmonics are zero. This is the two dimensionallattice {i, j}, the realization divergenceof has thea configuration magnitude of 4 of at athe charge site anddivergence but no curl. For a warp of unit strength at site-1over a spherical shell of radius a in three dimensions. Themagnetic neighbors field due on the to the lattice. charges is confined within onein space. They come and go in time and their interaction inunit cell around the site of the phase slip and is zero outside.Thus two phase slip events can interact only if theythis discussion, ∇×m t we (r, will τ) = demonstrate ρ v (r, τ) this explicitly in the (18)m l (r, τ) = ∑Phase slip events generateia local vortex current whichis divergenceless but has − âxδ finite (r − rcurl.ij − âx)−ŷδ (r − r ij − aŷ)]5In Fourier space we getm t (k, ω n ) = ıẑ × kk 2 ρ v (k, ω n )( ) 1 − e−ık x a ̂x +m l (k, ω n ) =ıωρ ps (k, ω n ) = ∑ iρ i pse −ık·r i−ıThe locality in space is reflfactors (1 − exp (−ık x a)) andk x a, k y a ≪ 1, to leading orderm l (k, ω n ) ≈ a (k x̂x +ω nThe two components of the fieonal to each other. Comparingnent of m l generated by a phalength limit and a warp (eqn.portional to each other. In papartition function can be recas

Correlation function in the Quantum critical RegionVortices are confined; they do not contribute to the dynamics.Above the critical value , the “warps” proliferate and disorder theVelocity field. The correlation functions can be calculated exactly.Correlation function of the Order Parameter atPhysical Reason: Interactions of topological defects (warps)is spatially local and logarithmic in time.

Theory of the Loop OrderPhysics beyond the Hubbard ModelThree-Orbital Model with on-site and nearest neighbor Interactions.oocuUse Operator identity:Vn i n j = V/2 |J ij | 2 + n i + n j ).J ij = ic + i c j + H.C. ⇥ current operator in the link (i j)SupposeV2 < J ij >= x ij .Then effective kinetic energy parameter on the link (i j) ist ij + ix ij = |˜t ij |e ia ij.This is equivalent to a vector potentiala ijon the link (i j).

Definitive Evidence that there is a phase transition at T*Fanelli et al. (APS March, 2011)Ultrasonic atten. and velocity:Singularities at T*(x)