Hubbard Model for Asymmetric Ultracold Fermionic ... - KOMET 337

Hubbard Model for AsymmetricUltracold Fermionic QuantumGasesD I P L O M A R B E I TKOMET 337Institut für PhysikJohannes Gutenberg Universität Mainzunter Anleitung von Prof. Dr. Peter van DongenvonTobias Gottwald7. Mai 2007Staudingerweg 755099 Mainz

AbstractThe experimental discovery of Bose-Einstein condensates in 1995 by W. Ketterle, C. Wiemanand E. Cornell [1] gave rise to many new experiments and to theoretical considerations.While in superfluid 4 He the condensate fraction is about 10%, in Bose-Einstein condensatesa condensate fraction of up to 99% may be reached. Furthermore, the interaction betweenthe bosons can be understood in a much better way in the dilute Bose-Einstein condensatesthan in 4 He, which is a strongly interacting system. Bose-Einstein condensates are realizedby loading bosons into a magneto-optical trap, which forms an effective harmonic potential,and by using special techniques for cooling beyond the condensation’s critical temperature [2].While, according to the Bose-Einstein statistics, in a bosonic gas there is no need ofinteraction for condensation, fermionic superfluidity occurs only if an effective attractiveinteraction takes place. We may understand the fermionic superfluidity as Bose-Einsteincondensation of Cooper-pairs, which are composed of two attracting fermions and are thereforeeffectively bosonic, when viewed from a distance much larger than the spatial extensionof the pair [3].With the help of coherent laser beams optical lattices can be superimposed [4] [5] [6] [7]. Inexperiments with superimposed optical lattices it is possible to realize Hubbard model physicsexperimentally [8] [9] [10] [11]. In fermionic systems, with the help of Feshbach-resonancesthe interaction strength of the fermions may be controlled [12] [13] [14]. Ultracold quantumgases on optical lattices are also of interest in the field of quantum transport and quantuminformation [15].In recent experiments the behavior of unbalanced Fermi-mixtures (mixtures with a majorityand a minority “spin” species) in the absence of an optical lattice was investigated [16][17]. While these experiments have been theoretically analyzed [18] [19], there seems to beneed also of investigations on unbalanced Fermi-mixtures on optical lattices.In conventional theories, whenever a Hubbard model is used to describe ultracold quantumgases, it is assumed that the hopping amplitude of the atoms is “spin” independent. InFermi-mixtures with different masses or hyperfine states this assumption needs not to be true[20], so that an investigation of Hubbard models with spin-dependent hopping amplitudes isof interest.In chapter 1 of this thesis we briefly show the connection between the fermionic Hubbardmodel and the physics of ultracold fermionic quantum gases. We will also present theHartree-Fock method of treating interactions in the weak coupling regime.In chapter 2 we diagonalize the superfluid Hartree-Fock Hamiltonian for both spindependenthopping and a magnetic field, which is needed to induce the imbalance in thefermionic system. We derive the self-consistency equations and discuss their basic properties.In chapter 3 hopping amplitudes are assumed to be spin-independent. We analyze superfluidityand phase separation occurring in an unbalanced Fermi-mixure in the attractive weakiii

coupling regime. The trapping potential is treated within the local density approximation.In chapter 4 we study the influence of an imbalance in the Fermi-mixture on superfluidityin a Hubbard model with spin-dependent hopping at weak attractive coupling. In viewof experimental applications, we focus on superfluid states, disregarding possible competingphases (like, e.g., charge density waves). In our discussions we concentrate on ground stateproperties.In chapter 5 the strong coupling limit of the asymmetric Hubbard model is presented.We derive effective Hamiltonians for translationally invariant systems for both attractive andrepulsive interactions.iv

Contents1 Introduction 11.1 Ultracold quantum gases and the Hubbard model . . . . . . . . . . . . . . . . 11.2 Unbalanced Fermi-mixtures: a brief experimental overview . . . . . . . . . . 42 Formalism at weak coupling 72.1 Weak (attractive) coupling limit at Hartree-Fock level . . . . . . . . . . . . . 72.2 Diagonalization of the grand canonical Hamiltonian . . . . . . . . . . . . . . 72.2.1 Fourier-transformation in the k-space . . . . . . . . . . . . . . . . . . 82.2.2 Bogoliubov-transformation . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Self-consistency equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Interaction-free densities of states . . . . . . . . . . . . . . . . . . . . . 112.3.2 Self consistency equations in the thermodynamic limit . . . . . . . . . 122.4 Properties of the self-consistency equations . . . . . . . . . . . . . . . . . . . 142.4.1 Range of the parameters n, m and ∆ . . . . . . . . . . . . . . . . . . . 142.4.2 Situation at half filling . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Non-polarized solutions of first type . . . . . . . . . . . . . . . . . . . 152.4.4 The grand potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Unbalanced Fermi-mixtures 193.1 Broken translational invariance and the LDA . . . . . . . . . . . . . . . . . . 193.2 Properties of the self-consistency equations at T = 0 . . . . . . . . . . . . . . 203.2.1 Uniqueness of n and m at fixed ∆ . . . . . . . . . . . . . . . . . . . . 213.2.2 Properties of the third self-consistency equation . . . . . . . . . . . . . 253.2.3 Graphical illustration of the phases . . . . . . . . . . . . . . . . . . . . 263.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Systems with finite temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.1 Infinite spatial extension . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.2 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Model with spin-dependent hopping 354.1 Charge density wave states and the repulsive model . . . . . . . . . . . . . . 354.1.1 Special particle-hole transformation . . . . . . . . . . . . . . . . . . . 354.1.2 Grand potential at half filling . . . . . . . . . . . . . . . . . . . . . . . 364.2 Superfluidity away from half filling . . . . . . . . . . . . . . . . . . . . . . . . 374.2.1 The quasiparticle energies . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.2 Hartree-Fock density of states . . . . . . . . . . . . . . . . . . . . . . . 384.3 Numerical method for the problem at fixed parameter n . . . . . . . . . . . . 394.4 Numerical results for the ground state . . . . . . . . . . . . . . . . . . . . . . 41v

4.5 Critical temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5.1 Superfluidity in non-magnetized solutions . . . . . . . . . . . . . . . . 434.5.2 Phase transition at T = T C . . . . . . . . . . . . . . . . . . . . . . . . 455 Strong coupling limit 475.1 Introduction: Exactly solvable 2-site model . . . . . . . . . . . . . . . . . . . 475.1.1 Diagonalization of the Hamiltonian . . . . . . . . . . . . . . . . . . . . 475.1.2 “Band structure” at strong coupling . . . . . . . . . . . . . . . . . . . 485.2 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.1 Second order perturbation theory . . . . . . . . . . . . . . . . . . . . . 495.2.2 Choice of the transformation operators . . . . . . . . . . . . . . . . . . 505.2.3 Structure of the second order term . . . . . . . . . . . . . . . . . . . . 515.3 Treatment of pure nearest neighbor hopping . . . . . . . . . . . . . . . . . . . 525.4 Low temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4.1 Repulsive U model at half filling . . . . . . . . . . . . . . . . . . . . . 525.4.2 Attractive U model with even number of fermions . . . . . . . . . . . 55vi

Chapter 1IntroductionAs an introduction, we show how the Hubbard model may be used to describe the physics ofultracold atoms loaded into an optical lattice. We also discuss the main motivation of thisthesis, namely the recent interest in unbalanced Fermi-mixtures.1.1 Ultracold quantum gases and the Hubbard modelAtoms loaded into optical lattices feel an effective potential, induced by the interaction oftheir dipole moment with the electric field of the coherent laser beam. In order to create anoptical lattice for an atomic species, the laser frequency must be far away from any transitionfrequency, so that the laser does not induce any transitions of the atoms’ internal state. Inthis case the effective potential arises from the AC-Stark-shift, and for hypercubic (d = 1,2,3)lattices it takes the form [2] [5] [7]:V (x) = − V 0dd∑i=1( cos 2 πx) ia, (1.1)where a = λ 2 is the lattice constant, λ is the wavelength of the laser beam and V 0 is proportionalto the laser’s intensity. As a consequence, for non-interacting particles, the correspondingSchrödinger equation can always be solved with the help of Floquet’s theorem inone dimension [21] [22]. Since the d-dimensional potential V (x) is a sum of one-dimensionalterms, the d-dimensional wave function is just a product of one-dimensional wave functions(see standard literature about quantum mechanics, [23] for example). Since we are onlyinterested in treating the tight-binding caseV 0 ≫22ma 2 , (1.2)where m is the atom’s mass, and want to derive a one-band Hubbard model, we present analternative and much shorter derivation here. We assume the validity of(1.2) and call eachpoint x i which satisfiesV (x i ) = −V 0 (1.3)a lattice site with lattice vector i. Of course we have the free choice to decide which pointon the d-dimensional lattice is identified with the origin i = 0, since we assume the systemto be translationally invariant here. The components of i are discrete (i ∈ N d ) and can beidentified with the difference vector from the lattice site to the origin. If we consider a fixed1

2 CHAPTER 1. INTRODUCTIONlattice site x i , we can expand the potential around its minimum in x = x i :V (x) ∼ − V 0d( [d − π2(x -2a 2(x - x i) 2 xi ) 4 )]+ Oa 4(1.4)∼ π2 V 02da 2 (x − x i) 2 − V 0 .With this approximation we can represent Wannier-states, localized at lattice site i, by agaussian times a Hermite polynomial, which are the well-known solutions of the harmonicoscillator problem. Since we are interested only in low temperature physics, we consider onlythe ground state of this harmonic oscillator and identify it with the Wannier-state localizedat the lattice site i. Therefore we neglect all energetically excited states. The ground state isgiven as:ψ(x) = 1l d 2)exp(− x22l 2, l =Here we present the approximation graphically:V(x), V app (x), |ψ(x)| 2 (a.u.)4√mω , ω = π √V0a md. (1.5)321-4 -2 2 4x-1Figure 1.1: Plot of periodic potential V (x) in its cosine form and its parabolically approximatedform. The probability density |ψ(x)| 2 is very small in the region where the approximationbreaks down.Hence, the harmonic oscillator approximation is good, since the probability density |ψ(x)| 2decreases rapidly with growing values of |x − x i |, so the atoms “feel” only the approximatepotential (1.4). With this tight-binding approximation we can derive a one-band Hubbardmodelas follows.Since the Wannier-functions located at different sites overlap, the atoms may tunnel fromone site to the other. In this thesis we consider only tunneling processes from one site to itsnearest neighbors and neglect all other tunneling processes. The tunneling process, whichis called “hopping” in the Hubbard model language, gives the following contribution to thecontribution to the Hamiltonian (see for example [24] [25]):H t = − ∑ (ij)σt σc † iσ c jσ, (1.6)where c † jσ and c iσdenote the creation/annihilation operators of the Wannier-states locatedat the lattice sites i/j and σ denotes the pseudo-spin, used to describe the fermion’s internal

1.1. ULTRACOLD QUANTUM GASES AND THE HUBBARD MODEL 3state or atom species. For bosons this technique works analogously, but we are interestedonly in fermions in this thesis. The hopping amplitude t σ is defined as [25]:∫)t σ := − dxψσ ∗ (x - x i)(− 2∆ + V σ (x) ψ σ (x - x j ) (1.7)2m σ(∝ ξσ d exp(−2ξ2 σ ) , ξ V0,σ m σ a 2 )14σ =2π 2 2 .Note that at this point we have built in the possibility of spin-dependent hopping, motivatedfor example in [20], which can arise from different atomic masses or from different potentials“felt” by the hyperfine states.For fermionic superfluidity there is also the need of an interaction, in contrast to bosoniccondensation. At low temperatures we have to consider only s-wave scattering [2] [25], whichleads to the interatomic potential:U(x − y) = 4πa sδ(x − y) , (1.8)mwhere a s is the s-wave scattering length, which can be positive or negative, and may becontrolled via Feshbach-resonances. In this thesis the interaction is attractive in most cases,because we want to treat fermionic s-wave superfluidity, which is not possible at repulsiveinteraction. Since the overlap of two atoms located on different sites is small compared tothe overlap of two atoms located on the same site, we consider only on-site interaction. Eachlattice site may be occupied at most once by each spin species because of the Pauli exclusionprinciple, so that on-site interaction takes place only between fermions of different pseudospinspecies. Hence the contribution to the Hubbard Hamiltonian arising from interactionreads:H U = U ∑ n i↑n i↓, n iσ= c † iσ c iσ, (1.9)iwhere the interaction strength U is given as:U := 4πa ∫sdx |ψ ↑ (x)| 2 |ψ ↓ (x)| 2 . (1.10)m(ξ ↑ ξ ↓ ) d∝(ξ↑ 2 + ξ2 ↓ ) .d2Hence, the ratio U tis highly tunable by altering the laser’s intensity, so that systems canbe tuned from the weak coupling to the strong coupling regime. In this thesis we performexclusively grand canonical calculations. Since we want to be able to control the imbalanceof Fermi-mixtures composed of two pseudo-spin species, we have to add both a chemicalpotential term and a Zeeman-term to the Hamiltonian:H µ = −µ ∑ iσn iσ, H B = B ∑ iσσn iσ, (1.11)where the bare symbol σ is to be interpreted as: “↑≡ +” and “↓≡ −”. (In this thesis we willrestrict consideration to systems with two pseudo-spin indices.) Thus, in sum, we obtain thegrand-canonical Hamiltonian:H gk = H t + H U + H µ + H B . (1.12)In chapter 2-4 we analyze this Hamiltonian in the weak coupling regime with special attentionto competing s-wave superfluid and magnetized phases. In chapter 5 we derive effectiveHamiltonians for the canonical version of (1.12) in the strong coupling regime.

4 CHAPTER 1. INTRODUCTION1.2 Unbalanced Fermi-mixtures: a brief experimental overviewIn this section we present the experimental motivation for analyzing unbalanced Fermimixtureson optical lattices. In recent experiments [16] [17] the superfluidity of two-componentunbalanced Fermi-mixtures has been examined experimentally. Here we show brieflythe results obtained in the experiment done at Rice university [16]. In this experiment twocomponentFermi-mixtures with different proportions of the components were cooled downbelow the critical temperature of the superfluid phase transition. With the help of in-situimages, the spatial distribution of the majority and of the minority species were imaged. Theresults are shown graphically here:Figure 1.2: In-situ images of the spatial particle distributions are represented here. In the leftimage the population of the minority cloud is decreased from (a) to (f), while the temperatureis constant. In each part the upper picture shows the distribution of the majority species,the middle picture shows the distribution of the minority species and the lower picture showstheir difference. In the right image the parts (a)/(b) and (c)/(d) are measured at differenttemperatures. The parts (a) and (b) correspond to the lower temperature, which is the sameas in the left image. The parts (c) and (d) are measured at a higher temperature, but stillbelow the critical temperature of the superfluid phase transition.From Figure 1.2 we conclude that:- There is superfluidity in unbalanced Fermi-mixtures at low temperatures.- Phase separation takes place.- A superfluid core with balanced occupation is in the middle of the trap.- In the outer region we find the unpaired part of the majority cloud.- There is a second critical temperature beyond the superfluid-to-normal critical temperature.Between those critical temperatures we find a polarized superfluid phase.Hence, unbalanced Fermi-mixtures differ from balanced ones, where we find only one criticaltemperature and no phase separation, since the complete cloud is superfluid below the critical

1.2. UNBALANCED FERMI-MIXTURES: A BRIEF EXPERIMENTAL OVERVIEW 5temperature and it is normal above of it.Since these experiments were performed without the use of optical lattices and until todaytheory has looked mainly at these experiments [18] [19], we decided in this thesis to use theHubbard model to investigate unbalanced Fermi-mixtures loaded into optical lattices. Theresults we have obtained are presented in chapter 3.

6 CHAPTER 1. INTRODUCTION

Chapter 2Formalism at weak couplingIn this chapter we will derive the self-consistency equations for the problem of an attractiveon-site interaction. The grand-canonical Hamiltonian is given by:H gk = − ∑ (ij)σt σc † iσ c jσ + U ∑ in i↑ n i↓ − µ ∑ iσn iσ + B ∑ iσσn iσ . (2.1)In this context we identify the spin indices with ↑≡ + and ↓≡ −. We have to discuss the termsappearing in H gk . The first term describes as usual the (in general spin-dependent) hoppingof the fermions. The second one describes the on-site interaction, where U is negative. Thethird term is as usual the chemical potential term which mainly controls the average totalnumber of particles in the system. The fourth one is the Zeeman term which is mainly usedto control the unbalance between particles with spin ↑ and spin ↓. Of course controling thetotal number of particles and their unbalance cannot be done independently, as we will seelater. We use the grand canonical ensemble especially at attractive interaction, since we wantto treat superfluidity, which is described by pair creation/annihilation terms at Hartree-Focklevel, and therefore it is more comfortable to use a formulation in Fock space with a variablenumber of particles.2.1 Weak (attractive) coupling limit at Hartree-Fock levelIn order to treat the problem at Hartree-Fock level, taking into account superfluidity, we haveto decouple the interaction term as follows:n i↑ n i↓ → 〈n i↑ 〉n i↓ + 〈n i↓ 〉n i↑ − 〈n i↑ 〉〈n i↓ 〉 + ∆ ∗ i c i↓ c i↑ + ∆ i c† i↑ c† i↓ − |∆ i |2 . (2.2)The superfluid order parameter ∆ iis defined as the average of the pair annihilation operator:∆ i = 〈c i↓c i↑〉 ; ∆ ∗ i = 〈(c i↓ c i↑ )† 〉 = 〈c † i↑ c† i↓ 〉 . (2.3)Analogously, its complex conjugate is defined as the average of the pair creation operator.With these tools we are able to derive self-consistency equations. A derivation of the Hartree-Fock approximation by using first order perturbation theory with the help of Lagrangianfactors can be found in the appendix of [26].2.2 Diagonalization of the grand canonical HamiltonianIn the following we will assume that translational invariance is not broken, so that all averagescan be taken as i-independent:〈n iσ 〉 → n σ ; ∆ i → ∆ . (2.4)7

8 CHAPTER 2. FORMALISM AT WEAK COUPLINGStarting from the spin-dependent average particle numbers per site, n ↑ and n ↓ , we introducethe total particle number per site and the magnetization per site:Inverting those relations we have:n = n ↑ + n ↓ ; m = n ↑ − n ↓n ↑ + n ↓. (2.5)n ↑ =n(1 + m)2; n ↓ =n(1 − m). (2.6)2Thus we have three parameters, namely n, m and ∆, which have to be determined selfconsistently.2.2.1 Fourier-transformation in the k-spaceAs in the interaction-free case we will diagonalize the Hamiltonian via a Fourier-transformationin k-space. To do so, we introduce the Fourier-transformed creation and annihilationoperators:c † kσ = √ 1 ∑exp(−i k · i)c † iσ; c kσ= 1 ∑√ exp(i k · i)c iσ. (2.7)N NiOf course, N means the total number of lattice sites. Sums over all lattice vectors of numberoperators remain formally unchanged by the Fourier-transformation, the pair creation andannihilation terms are transformed into momentum-free pair creation and annihilation terms,so that we can write the Hartree-Fock Hamiltonian as:H gk → H HFgk = H C + H N + H P , (2.8)iwith the constant-term H C (includes no operators):{H C = −NU |∆| 2 + n2 (1 − m 2 })4, (2.9)the counting term H N :H N = ∑ kσ{−2t σ ε(k) − µ + Bσ + U}(1 − mσ)n kσ (2.10)2and the pair creation/annihilation term H P :H P = U|∆| ∑ k{exp(−iδ)c−k↓ c k↑+ exp(iδ)c † }k↑ c† −k↓ . (2.11)Here n kσ= c † kσ c kσis the k-space number operator and ∆ = |∆|exp(iδ) has been decomposedinto its absolute value and its phase. The one-particle kinetic energy ε(k) is given by:ε(k) =d∑cos(k l ) , (2.12)l=1where d is the dimension of the system and the lattice is assumed to be hypercubic. Inthe following we show how the Hamiltonian (2.8) can be diagonalized with the help of aBogoliubov-transformation.

10 CHAPTER 2. FORMALISM AT WEAK COUPLINGIn order to complete the diagonalization we have to determine the relation between thetransformation parameters α kσand ϕ kσ. At this point it is useful to define the non-superfluidgrand potential per particle:and especially the average:E kσ=Un(1 − mσ)2− µ + Bσ − 2t σε(k) (2.23)E k:= E k↑ + E k↓2= Un2 − µ − (t ↑ + t ↓)ε(k) . (2.24)Note that all these energies are only implicitly dependent on k (via ε(k)). Hence they aresymmetric under the transformation k → −k.We will now write down the full Hamiltonian in the new language of the b operators anddirectly symmetrize it. One finds for the various termsH N → 1 ∑ ()() ()][E2 kσ−vkσ ∗ b −k−σ +u −k−σ b† kσu −k−σb kσ−v kσb † −k−σ+ (k,σ) ↔ (−k, −σ)andH P → U 2kσ∑kσ[ ()() ]σ ∆ − vkσ ∗ b −k−σ + u −k−σ b† kσ− v−k−σ ∗ b kσ + u kσ b† −k−σ+ h.c.(2.25), (2.26)while H C remains unchanged, of course. Now we have to determine α kσand ϕ kσso that paircreation and annihilation terms vanish in the sum. Writing down u kσand u kσin terms ofα kσand ϕ kσ, and demanding that pair creations vanish, leads to equations determining ourBogoliubov parameters. For the phase-factor we may choose:ϕ kσ= δ , (2.27)where δ is the superfluid order parameter’s phase-factor, defined in (2.11). For the angleparameter α kσwe get:tan(2α kσ) = U|∆|σE k, (2.28)so that the angle parameter is defined by the ratio of interaction times the absolute value ofthe superfluid order parameter and the “average” grand canonical energy. In order to satisfy(2.28) we choose the cosine and the sine of the double Bogoliubov-angle as:cos(2α kσ) =|E k|√ ; sin(2α kσ) = sign(E k )U|∆|σ . (2.29)Ek 2 + U2 |∆|√E 2 k 2 + U2 |∆| 2This choice also guarantees that at U = 0 all cosine terms reduce to unity, as they should. If weEhad chosen alternatively cos(2α kσ) = √ kand sin(2α E 2k +U 2 |∆| 2 kσ ) = U|∆|σ√E 2k +U 2 |∆| 2, for example,the Boguliubov-matrix in (2.13) would not be equal to the identity matrix at U = 0. Finally,the two non-constant terms in the grand canonical Hamiltonian can be rewritten as:H P + H N = ∑ k{Ek − sign(E k)√E 2 k + U2 |∆| 2} + ∑ kσE kσ¯n kσ, (2.30)

2.3. SELF-CONSISTENCY EQUATIONS 11withE kσ= −Unmσ√+ Bσ − (t2σ− t −σ)ε(k) + sign(E k) Ek 2 + U2 |∆| 2 . (2.31)Of course the new number operator is given by the product of the Bogoliubov quasiparticlecreation and annihilation operators as ¯n kσ= b † kσ b kσ .Obviously, in the new quasiparticle language the Hamiltonian is completely independent ofthe superfluid phase-factor δ, so that from now on we may treat ∆ as a real non-negativenumber (∆ ∈ R + 0 ).2.3 Self-consistency equationsIn this section we will restrict consideration to the thermodynamic limit (N → ∞) and derivethe self-consistency equations which determine (n,m,∆) for given parameters (U,t σ ,B,µ) intranslationally invariant systems. We will first introduce the Hubbard model’s interactionfreedensity of states in order to obtain the equations in the form of one-dimensional integralequations, which are better to handle than equations involving d-dimensional k-sums.2.3.1 Interaction-free densities of statesWe will here introduce the density of states (DOS) of the hypercubic lattices in variousspatial dimensions to be able to replace d-dimensional sums by one-dimensional integrals inthe thermodynamic limit as follows:1N∑where the thermodynamic limit meanskf ( ε(k) ) ∫ ∞TD−−→ dεν d (ε)f(ε) , (2.32)lim −∞N → ∞ µ,B = const . (2.33)The replacement of k-sums by energy integrals, as in (2.32), is useful, since in our Hamiltonianthe complete k-dependence is given via ε(k). Here the d-dimensional density of states ν d (ε)is given by:ν d (ε) = 1 ∑δ ( ε − ε(k) ) . (2.34)NkNote that this definition differs slightly from the one which is usually found in literature,where hopping is not (pseudo-)spin-dependent and ¯ε ≡ 2tε is taken as a variable instead ofε. With the help of a Fourier-transformation the density of states in d-dimensions can beexpressed via Bessel-functions as follows (see [27], [28]):ν d (ε) = 1 π∫ ∞This means for the DOS in one- and two-dimensional systems:0dx [ J 0 (x) ] d cos(εx) , (2.35)1ν 1 (ε) =π √ − |ε|) and (2.36)1 − ε2Θ(1 ν 2 (ε) = 1 ( ) ε 2(1π 2K − Θ(2 − |ε|) , (2.37)2)where K(x) is a complete elliptic integral of the first kind and Θ(x) is the unit step function.For the three-dimensional DOS such a simple representation in terms of special functionsdoes not exist and we will calculate it only numerically. The graphs of the DOS in 1 - 3dimensions are sketched graphically in Figures 2.1-2.3.

12 CHAPTER 2. FORMALISM AT WEAK COUPLING1.41.210.80.60.40.2-1 -0.5 0.5 1Figure 2.1: One-dimensional free DOS ν 1 (ε)0.80.60.40.2-3 -2 -1 1 2 3Figure 2.2: Two dimensional free DOS ν 2 (ε)0.250.20.150.10.05-4 -2 2 4Figure 2.3: Three dimensional free DOS ν 3 (ε)For our purposes in this thesis it is important that the interaction-free DOS is symmetricwith respect to ε: ν d (ε) = ν d (−ε). In numerical calculations the interaction-free DOS ind = 3 will always be approximated by a piecewise linear fit, consisting of 288 line segments,in order to reduce the numerical effort. While in d = 1,2 the interaction-free DOS diverges atthe van-Hove-singularities, in d ≥ 3 the piecewise linear approximation becomes acceptable,since the van-Hove-singularities do not cause the function ν 3 (ε) itself but only its derivativesto diverge. The normalization of the 3-dimensional interaction-free DOS remains nearlyunchanged through piecewise linearization, instead of the exact normalization∫ ∞−∞dεν d (ε) = 1 we get∫ ∞−∞dεν 3,lin (ε) ≈ 1,00017 , (2.38)so that the relative error of the normalization is of the order ∼ 10 −4 and, hence, we canassume that the piecewise linear version of the DOS in d = 3 is sufficiently good to be usedfor integrations.2.3.2 Self consistency equations in the thermodynamic limitWe are now able to derive the self-consistency equations in k-space. With the help of thethermodynamic averages we may calculate the parameters (n,m,∆) in dependency of theBogoliubov quasiparticle-eigenenergies E kσ. The averages of the Bogliubov operators are

2.3. SELF-CONSISTENCY EQUATIONS 13well-known, since the grand canonical Hamiltonian commutes with the Bogoliubov numberoperator ¯n kσ . The thermodynamic averages in the grand-canonical ensemble can be writtenas:〈b † kσ b k ′ σ ′ 〉 = δ kk ′δ σσ ′f β(E kσ) ; 〈b kσb k ′ σ ′ 〉 = 〈b † kσ b† k ′ σ ′ 〉 = 0 , (2.39)since, at Hartree-Fock level, the Bogoliubov-quasiparticles do not interact. Here f β (x) is theFermi function at inverse temperature β:f β (x) =1exp(βx) + 1. (2.40)The averages of the occupation numbers are of course given by: 〈n kσ〉 = 〈c † kσ c kσ〉. Thesemay be expressed with the help of the inverse Bogoliubov-transformation as follows:〈c † kσ c kσ 〉 = cos2 (α kσ)〈b † kσ b kσ 〉 + sin2 (α kσ)〈b −k−σb † −k−σ 〉 (2.41)〈c † k↑ c† −k↓ 〉 = cos(α k↑ )sin(α k↑ )e−iδ 〈b † k↑ b k↑ − b −k↓ b† −k↓ 〉 (2.42)〈c −k↓c k↑〉 = 〈c † k↑ c† −k↓ 〉∗ , (2.43)since from (2.39) we know that pair creation/annihilation terms of Bogoliubov-quasiparticlesdo not contribute. Using that our parameters are supposed to be translationally invariant, wedirectly obtain our self-consistency equations by performing a k-summation and using thatdouble products of sine and cosine functions of an angle α may be expressed as sine or cosinefunctions of the double angle 2α. With the help of (2.29) we finally get:n = 1 + 1 Nm = 1nN∆ =U∆2N∑kσ∑ |E k|(√ f β (E kσ) − 1 ), (2.44)Ek 2 + U2 |∆| 2 2kσ|E k|σ√E 2 k + U2 |∆| 2 f β (E kσ) and (2.45)∑ sign(E k) (√ f β (E kσ) − 1Ek 2 + U2 |∆| 2 2kσ). (2.46)These equations are only valid in the thermodynamic limit, so that we replace the d-dimensionalk-summation by a one-dimensional ε-integration as introduced in (2.32). Theself-consistency equations can now be rewritten as:n = 1 +m =1 n∆ = U∆ 2∫ ∞−∞∫ ∞−∞∫ ∞|E(ε)| [ (dεν d (ε) √ fβ E↑ (ε) ) (+ f β E↓ (ε) ) − 1 ] , (2.47)E 2 (ε) + U 2 |∆| 2|E(ε)| [ (dεν d (ε) √ fβ E↑ (ε) ) (− f β E↓ (ε) )] and (2.48)E 2 (ε) + U 2 |∆| 2−∞sign ( E(ε) ) [ (dεν d (ε) √ fβ E↑ (ε) ) (+ f β E↓ (ε) ) − 1 ] . (2.49)E 2 (ε) + U 2 |∆| 2Obviously (2.49) is solved automatically if the superfluid parameter vanishes (∆ = 0), otherwiseit can be rewritten as:1 = U ∫ ∞sign ( E(ε) ) [ (dεν d (ε) √ fβ E↑ (ε) ) (+ f β E↓ (ε) ) − 1 ] . (2.50)2E 2 (ε) + U 2 |∆| 2−∞The phase of the superfluid order parameter δ does not play a role, evidently.

14 CHAPTER 2. FORMALISM AT WEAK COUPLING2.4 Properties of the self-consistency equationsIn this section we discuss some general properties and special cases of the self-consistencyequations. First we determine the range of the parameters at Hartree-Fock level which isuseful for performing the numerical evaluations. Then we will treat the special cases of halffilling and balanced mixtures, to show how the self-consistency equations can be simplifiedthere.2.4.1 Range of the parameters n, m and ∆We will now briefly derive the range of the parameters n, m and ∆. Since the parameters nand m derive from the averages of the fermionic occupation operator 〈n iσ〉, it is trivial thatthey are in the range:0 ≤ n ≤ 2 ; −1 ≤ m ≤ 1 . (2.51)For the superfluid order parameter ∆ which is an average of the pair annihilation operator, weare also able to determine a range for its absolute value. Since the pair creation/annihilationtakes place at only one site (s-wave superfluidity) we may perform a calculation in the one-siteFock space in the base:|0〉 , | ↑〉 , | ↓〉 , | ↑↓〉 , (2.52)since this base is complete for one site. The singly occupied states | ↑〉 and | ↓〉 do notcontribute to pair creation/annihilation averages, so that the absolute value of ∆ can bemaximized only in states which are free from single occupations:|max〉 = α|0〉 + β| ↑↓〉 with |α| 2 + |β| 2 = 1 . (2.53)The averages of pair annihilations are now given as:∆ = 〈max|c i↓c i↑|max〉 = βα ∗ , (2.54)∆ ∗ = 〈max|c † i↑ c† i↓ |max〉 = αβ∗ .Since the absolute values of α and β can be written as a sine and a cosine of an angle ϕ, wecan write:|∆| = sinϕcos ϕ = 1 2 sin(2ϕ) ≤ 1 , (2.55)2so the absolute value of ∆ is always between 0 and 1 2. The phase-factor δ may have any valuebetween 0 and 2π, of course.It is from now on clear that in numerical calculations, where we are looking for solutionsof the self-consistency equations, the region of interest is the volume (0 ≤ n ≤ 2) × (−1 ≤m ≤ 1) × (0 ≤ |∆| ≤ 1 2). The phase-factor δ is not accessible through numerical calculations,it is a random value the system “chooses” by spontaneous symmetry-breaking.Furthermore note that the fermionic superfluid has an upper limit in contrast to thebosonic superfluid order parameter required for the description of BEC.2.4.2 Situation at half fillingWe will now show that half filling is realized with the chemical potential µ = U 2, if either theZeeman term is 0 (B = 0) or the hopping is spin-independent (t ↑ = t ↓ ). In the spin-symmetricHubbard model, where both conditions are fulfilled, there is particle-hole symmetry which

2.4. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS 15directly leads to a chemical µ = U 2a particle-hole transformationpotential at half-filling. It is well-known that performingc † iσ→ exp(iQ · i)c iσ, (2.56)c iσ→ exp(iQ · i)c † iσ,Q = (π,π,... ,π) T ,leads to the desired result, if the lattice is bipartite. Furthermore, the model is symmetricunder a spin exchange transformation:c † iσ → c† i−σ; c iσ→ c i−σ. (2.57)If we take the Hamiltonian (2.1) at B = 0, particle-hole symmetry still leads to µ = U 2 athalf filling. Spin-exchange symmetry is however broken, the transformation (2.57) leads to amodified hopping termH T → − ∑ t −σc † iσ c jσ, (2.58)(ij)σso that particles hop with the hopping-amplitude of the opposite spin species.The Hamiltonian (2.1) at t ↑ = t ↓ ≡ t is neither symmetric under the transformation (2.56)nor under (2.57). But if we combine both transformations at µ = U 2, we get our symmetryback:HS.E. −−→P.H.− ∑ t c † iσ c jσ + U ∑ (1 − n i↑− n i↓+ n i↑n i↓) (2.59)(ij)i− U ∑(1 − n2 iσ) − B ∑ σ(1 − n iσ)iσiσ= − ∑ t c † iσ c jσ + U ∑(ij)i= H ,so that we still have half filling at µ = U 2 .n i↑ n i↓ − µ ∑ iσn iσ + B ∑ iσσn iσIf both B ≠ 0 and t ↑ ≠ t ↓ , these arguments do not work and therefore the calculation ofthe chemical potential at half filling becomes more complicated. These symmetry relationswere shown for the exact model, but they are still valid in the Hartree-Fock approximation,where some of the operators have been replaced by averages.These symmetry relations at half filling can be used to reduce the numerical problem ofsolving the self-consistency equations by one parameter, since at µ = U 2, under the conditionsdiscussed above, (2.47) reduces exactly to n = 1 at all temperatures and needs not to besolved numerically.2.4.3 Non-polarized solutions of first typeWe will distinguish two types of non-polarized solutions (m = 0). The first type of nonpolarizedsolutions, which we will discuss in this section, arises from the choice of the hoppingamplitudes t σ , the chemical potential µ and the magnetic field B occurring in the Zeemanterm. This type of solutions is non-magnetic even at U = 0, in contrast to the second type

16 CHAPTER 2. FORMALISM AT WEAK COUPLINGsolutions. At Hartree-Fock level with U < 0 the particles with spin σ “feel” attracted by themean-field of the particles with spin −σ, which causes a part the of majority spin speciesto switch to the minority species. Additionally, superfluidity may occur at U < 0, so thatthe superfluid energy gap is so large that the magnetization is suppressed as a result of theinteraction. We call this phenomenon non-magnetized solution of second type.First we observe that there is trivially no magnetization if the hopping is spin-independentand the Zeeman term is switched off (B = 0). Putting m = 0 into (2.48) at random (n, |∆|)leads directly to the solution of (2.48), since E σ(ε) becomes independent of σ. As soon aswe have B ≠ 0 in the spin-independent hopping model, it is not possible to obtain nonmagnetizedsolutions of the first type.An additional special case is the spin-dependent hopping model in absence of the Zeemanterm at half filling. Since we have n = 1 and µ = U 2, the quasiparticle energies become oddas a function of ε, if we assume m = 0:E σ(ε) → (t −σ− t σ)ε + sign ( E(ε) ) √E 2 (ε) + U 2 |∆| 2 (2.60)E(ε) → −(t ↑ + t ↓ )ε .Putting (2.60) into (2.48) and using the symmetry of the d-dimensional DOS leads to:m =∫ ∞0dεν d (ε)|E(ε)|√E 2 (ε) + U 2 |∆| 2 [fβ(E↑ (ε) ) + f β(− E↑ (ε) ) (2.61)−f β(E↓ (ε) ) − f β(− E↓ (ε) )] .Since the Fermi-function has the property f β (x) + f β (−x) = 1 we directly obtain that (2.48)is solved for m = 0 in this special case, so that the problem reduces to a one-dimensionalone, namely solving (2.49) at n = 1 and m = 0. Away from half filling we cannot expect themagnetization to vanish in this case, since the bands of the different spin species differ fromeach other.Non-magnetized solutions of the first type are of course also possible away from half fillingby appropriate tuning of the magnetic field B in the Zeeman term, which, in general, becomestemperature-dependent.2.4.4 The grand potentialSince we are treating the model in the grand-canonical ensemble, it is useful to determine thegrand potential in dependence of the parameters (n,m,∆). As we will see later, it is possiblein this system that first order phase transitions occur. That means that the self-consistencyequations (2.47) - (2.49) have more than one solution, and so we have to compare the grandpotentials at those parameters to be able to decide which solution has lowest grand potentialand is therefore thermodynamically stable. From the grand-canonical partition function the

2.4. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS 17grand potential per site follows as:ω = Ω N= − 1 N ln Z gk{= −U |∆| 2 + n2 (1 − m 2 )}+ 1 ∑{}E4 N k− sign(E k)√Ek 2 + U2 |∆| 2k− 1 ∑ln ( 1 + exp(−β E kσ ) )NβTD−−→limkσ{−U |∆| 2 + n2 (1 − m 2 )}∫4∞+ dεν d (ε){E(ε) − sign ( E(ε) )√ }E 2 (ε) + U 2 |∆| 2− 1 β−∞∑σ∫ ∞−∞(dεν d (ε) ln 1 + exp ( − β E σ (ε) )) . (2.62)At T = 0 (β → ∞) the last term in (2.62) reduces to:− 1 β∑σ∫ ∞−∞(dεν d (ε) ln 1 + exp ( − β E σ (ε) )) → ∑ σ∫ ∞−∞dεν d (ε) E σ (ε)Θ ( − E σ (ε) ) , (2.63)where Θ(x) is the unit step function. With (2.62) we are now able to decide numericallywhich solution has lowest grand potential.

18 CHAPTER 2. FORMALISM AT WEAK COUPLING

Chapter 3Unbalanced Fermi-mixturesIn this chapter we will discuss a Fermi-mixture with unequal occupation numbers for thedifferent spin species. We will assume that the system is in the ground state T = 0 andthat the hopping amplitudes are the same for both spin-species. The parabolic potentialcaused by the magneto-optical trap will be introduced and treated within the local densityapproximation (LDA). We will discuss the phases which can occur in such a system and shownumerical results for given trap parameters.3.1 Broken translational invariance and the LDAIn this section we will introduce the concept of the LDA for treating the trapping potential.Ultracold quantum gases are usually trapped in a magneto-optical trap, which can be describedby a quadratic potential for the atoms. The trapping potential and the superimposedoptical lattice form the effective potential in space which the atoms feel [5]. We illustrate thepotential here in one dimension (arbitrary units):2121-2 -1 1 2-1-2 -1 1 2-1-2-2Figure 3.1: Potentials felt by the atomsRed curve: Lattice potentialGreen curve: Trapping potentialFigure 3.2: Sum of the lattice and trapping potentialsIn our one-band tight-binding approximation we assumed that the atoms can be describedas Wannier-states, which are localized around specific lattice sites. Here the position of alattice site is defined, e.g., by a minimum of the translationally invariant lattice potential.We will now assume that the trapping potential varies slowly enough in space, so that thedescription via the Wannier-states is still acceptable and the hopping of atoms from one site19

20 CHAPTER 3. UNBALANCED FERMI-MIXTURESto its nearest neighbor can still be described by the overlap-integral, as in the translationallyinvariant case. This means that the ground state Wannier function is still approximativelygaussian, as described in section 1.1.In Wannier state language the trapping potential can now be described as:H Ω = ∑ iσ(i · Ω · i)n iσ, (3.1)where Ω is a symmetric and positive matrix. We have assumed that the trapping potentialis the same for both spin-species.In addition we will assume that the trapping potential varies slowly enough for the LDAto be valid, i.e., implying that we may combine the chemical potential term H µ with thetrapping term H Ω :µ → µ(i) = µ 0 − i · Ω · i . (3.2)Since we treat our model in the grand-canonical ensemble, µ 0 is used to control the total numberof fermions in the system. Applying the LDA now means that each lattice site is treatedas an infinite translationally invariant system, so that we have to solve a self-consistencyequation for each lattice site. We may assume that the LDA is a good approximation for thistype of systems, since the size of a trapped atom cloud is of the order of some millimetersand the laser beams have wavelengths less than one micrometer, so that fermions at a chosenlattice site“see”neighboring lattice sites with nearly the same local chemical potential. So thewhole system is treated as a composition of “quasi translationally invariant” parts in LDA.3.2 Properties of the self-consistency equations at T = 0We are now particularly interested in showing, which phases may occur in an unbalancedFermi-mixture at low temperatures. We restrict consideration to the case T = 0, sincethe phase diagram in an unbalanced mixture is much more complicated than in a balancedsituation. At T = 0 we have to replace all Fermi-functions by unit step functions, so that(2.47) - (2.49) become (∆ ∈ R + 0 in this chapter, since the phase δ is unimportant, as we haveseen in chapter 2):∫ ∞n = 1 +m =1 n∆ = U∆ 2−∞∫ ∞−∞∫ ∞|E(ε)| [ (dεν d (ε) √ Θ − E↑ (ε) ) + Θ ( − E ↓ (ε) ) − 1 ] , (3.3)E 2 (ε) + U 2 ∆ 2|E(ε)| [ (dεν d (ε) √ Θ − E↑ (ε) ) − Θ ( − E ↓ (ε) )] and (3.4)E 2 (ε) + U 2 ∆ 2−∞sign ( E(ε) ) [ (dεν d (ε) √ Θ − E↑ (ε) ) + Θ ( − E ↓ (ε) ) − 1 ] . (3.5)E 2 (ε) + U 2 ∆ 2A non-trivial question is now, which solutions the Equations (3.3) - (3.5) may have. We willnow present analytical arguments which will lead us to a numerical procedure for solvingEquations (3.3) - (3.5).

3.2. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS AT T = 0 213.2.1 Uniqueness of n and m at fixed ∆We will now show that at fixed ∆ there is only one solution for (3.3) and (3.4). At first werewrite (3.3) and (3.4) in terms of n σ := 〈n iσ 〉:n σ = 1 2 + 1 2∫ ∞−∞|E(ε)| [ (dεν d (ε) √ sign − Eσ (ε) )] . (3.6)E 2 (ε) + U 2 ∆ 2We will use Banach’s fixed point theorem in order to show that there is only one uniquesolution for the coupled Equations (3.6). If (3.6) has only one solution, it is trivial that thereis only one point (n,m) that solves (3.3) and (3.4). In order to be able to use Banach’stheorem we must first define a metric on the space n ↑ × n ↓ := [0,1] × [0,1]. Here we chooseto take the maximum-metric:d(n 1 ,n 2 ) := ||n 1 − n 2 || ∞ = max{|n 1,σ − n 2,σ |} . (3.7)Banach’s theorem states that a mappinq I(·) is a contraction, if there is a constant q so that:d(I(n 1 ), I(n 2 )) ≤ q d(n 1 ,n 2 ) (∀n 1 ,n 2 ; 0 ≤ q < 1) (3.8)and that a contraction has one definite fixed point:I(n 0 ) = n 0 . (3.9)The mapping I will be identified with the integrations of (3.6), and so, if we have proved thatour integrals are contractions in the sense of (3.8), we know that there is only one solutionfor (3.6), namely the fixed point.We will not be able to show that our mapping I is a contraction on the whole region[0,1] × [0,1]. But we will show that, when two points n 1 and n 2 are close enough to eachother, the mapping I is a contraction with a well defined constant q. We will first show thatthe local property of being a contraction can be continued to a global property. We will thenillustrate this graphically and afterwards prove the local property of I being a contraction.n 3n 2 n 1In 3In 2In 1Figure 3.3: Overlap of two regions,where I is a contraction.Figure 3.4: Points after mapping I.In 2 is still in the overlapping region.

22 CHAPTER 3. UNBALANCED FERMI-MIXTURESAs we see in Figures 3.3 and 3.4, points n 1 and n 3 which share a point n 2 in their regions,where I is a contraction, are mapped onto the same fixed point by I, since applying I atimes to them givesd ( I a (n 1 ), I a (n 2 ) ) ≤ q a d ( ) a→∞n 1 ,n 2 −−−→ 0 , (3.10)d ( I a (n 3 ), I a (n 2 ) ) ≤ q a d ( n 2 ,n 2)a→∞−−−→ 0 , (3.11)so that n 1 , n 2 and n 3 must converge to the same limit, namely the fixed point. Note thatI can map any point onto a different one, far away from its original, in general. For ourpurpose it is only important that two points close to each other are mapped onto pointswhich are then closer to each other in the sense of d( ·, ·) and that we are able to representthe whole parameter space by unions of regions where I is a contraction. Note also that wehave used a maximum norm to define our metric d( ·, ·), and in the illustration we have usedan euklidean norm to define the regions, where I is locally a contraction. This has been donefor convenience only, but it is unimportant, since in finite dimensions all norms are equivalent.We are now going to show the local validity of (3.8). We will show that, under certaincircumstances, there is a constant q so that (3.8) is satisfied in a small region of a point n 1 .So we will write n 2 = n 1 + δn, and treat δn as a small parameter. We will use a Taylorexpansionto first order in δn. Having shown the validity of (3.8) for small δn, (3.8) becomesvalid for the whole parameter region with the arguments of the Figures 3.3 and 3.4. We mustfirst distinguish between two cases which can occur while doing the proof:E ↑/↓ (ε) , E(ε)E ↑/↓ (ε) , E(ε)8866442-3 -2 -1 1 2 3-2ε2-3 -2 -1 1 2 3-2ε-4-4Figure 3.5: E ↑/↓ (ε) and E(ε) change theirsign at the same value of ε.Figure 3.6: E ↑/↓ (ε) and E(ε) change theirsign at different values of ε.1. For the parameter value n 1 , E(ε) and E σ change their sign at the same value of ε for afixed choice of σ as shown in Figure 3.52. For the parameter value n 1 , E(ε) and E σ change their sign at different values of ε for afixed choice of σ as shown in Figure 3.6Of course we have to choose δn small enough, so that both n 1 and n 2 belong to the samecategory. We will now treat the first case. We choose one of both spin species σ and performan estimate for (3.8):In 1σ − In 2σ = 1 ∫ []∞E 2 (ε)dεν d (ε) √2E22 (ε) + U 2 ∆ − E 1 (ε)√ . (3.12)2 E21 (ε) + U 2 ∆ 2−∞

3.2. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS AT T = 0 23Since δn is small, both E 1 (ε) and E 2 (ε) have the same zeroes as E 1σ (ε) and E 2σ (ε). Hencewe can do a Taylor-expansion in powers of δn, which leads to:∫ ∞|In 1σ − In 2σ | = 1 |U 3 ∆ 2 |dε ν d (ε)4 −∞ } {{ } (E1 2(ε) + ≤ν d (ε U2 ∆ 2 ) 3 2max)[≤|U|4t ν E 1 (ε)d(ε max ) ||δn|| ∞≤|U|2t ν d(ε max ) ||δn|| ∞ ,√E21 (ε) + U 2 ∆ 2 ] d−d|δn ↑ + δn ↓ |} {{ }≤ 2||δn|| ∞(3.13)since E(ε) = U(n ↑+n ↓ )2− µ − 2tε and ν d (ε) = 0 for |ε| ≥ d. Note that this procedure worksonly for d ≥ 3, since in one or two dimensions the interaction-free DOS diverges at thevan-Hove-singularities. As (3.13) is valid for both spin species, we have:so as we need q < 1, we must have||In 2 − In 1 || ∞ ≤ |U|2t ν d(ε max ) ||n 2 − n 1 || ∞} {{ }, (3.14):=q|U|t

24 CHAPTER 3. UNBALANCED FERMI-MIXTURESE ↑/↓ (ε) , E(ε)8642-3 -2 -1 1 2 3-2ε-4Figure 3.7: The functions E ↑/↓ (ε) and E(ε) are represented at different parameters here. Thebrighter curves correspond to the parameter n 1 , the darker ones correspond to the parametern 2 . The variation δn is small.As we can see from Figure 3.7 the first terms in (3.16) contribute only between the twozeroes of the blue curves and the two zeroes of the red curves. We will derive an estimate foreach of the contributing regions:1. Contribution of E(ε)-term- The variation δn causes a shift of U(δn ↑+δn ↑ )2in the y-position of E(ε).- This causes a shift of U(δn ↑+δn ↑ )4tin the y-position of E(ε).- This shift has to be multiplied by 2, because the sum of the sign functions changesby 2.- Then we can estimate the absolute value of this contribution by |U|2t ν d(ε max ) ||δn|| ∞ .2. Contribution of the E σ (ε)-term- The variation δn causes a shift, which can be estimated to be less than U|(δn ↑ +δn ↑ )| in the y-position of E σ (ε).- This causes a shift of maximum |U(δn ↑+δn ↑ )|2tin the y-position of E σ (ε).- This shift has to be multiplied by 2, because the sum of the sign functions changesby 2.- Then we can estimate the absolute value of this contribution by |U|tν d (ε max ) ||δn|| ∞ .Hence the sum of all these contributions gives (the third one is well known from the first casewe have presented):so that instead of (3.15) we get||In 2 − In 1 || ∞ ≤ |2U|tν d (ε max ) ||n 2 − n 1 || ∞ , (3.17)|U|t

3.2. PROPERTIES OF THE SELF-CONSISTENCY EQUATIONS AT T = 0 25In the following we assume the more restrictive condition, namely (3.18), to guarantee thatour mapping I is a contraction. Note that when we are neither in the first case nor in thesecond one, but we are on the boundary between them, we can combine our arguments ofboth cases by dividing δn in two regions, which always permits one of both arguments. sothat we come to the same conclusions. With the condition (3.18) it is guaranteed that (3.3)and (3.4) have a unique solution at fixed ∆, at least at weak coupling, where the Hartree-Fockapproximation is valid.3.2.2 Properties of the third self-consistency equationWe will present the possible solutions of (3.5) here. In order to do this, we will discuss thederivative with respect to ∆ of the integral in (3.5). As we have shown in 3.2.1, there is onlyone unique pair of values of n and m which is able to solve the self-consistency equations.Hence it is quite easy to see that (3.4) is solved with m = 0 for |B| < |U∆|, so that Equation(3.4) is solved analytically for |B| < |U∆|. At |B| > |U∆|, m = 0 does not solve (3.4), sothat there is one solution with m ≠ 0 for fixed ∆ in this region. As in section 3.2.1 we willdistinguish two different cases.In the first case, |B| < |U∆|, Equation (3.5) can for ∆ > 0 be written as:1 = |U|2∫ ∞−∞1dεν d (ε) √ , (3.19)E 2 (ε) + U 2 ∆ 2since E(ε) and E σ (ε) change their sign at the same value of ε. As a consequence, increasing∆ causes the zero of E(ε) to be shifted by U δn4t, where δn is the shift in n caused by theincrease of ∆ (n is a function of ∆ because of (3.3).). This shift causes a very small changein the value of the integral in (3.19), since it does not change the height of the square rootterm, it only shifts its position relatively to the interaction-free DOS. Hence differences inthe value of the integral arise exclusively from contributions near the band’s edge, where theDOS goes to zero (d ≥ 3). Thus an increase ∆ causes the integral in (3.19) to decrease, sincethe square root term becomes smaller at all values of ε.In the second case, |B| > |U∆|, Equation (3.5) can for ∆ > 0 be written as:1 = |U|2∫ ∞−∞dεν d (ε) Θ( E ↑ (ε) · E ↓ (ε) )√E 2 (ε) + U 2 ∆ 2 , (3.20)since the distance between the root of E(ε) to the points, where E σ (ε) changes its sign, isthe same for both spin species, the point belonging to one spin species being greater thanthe root of E(ε) and the one belonging to the other spin species being smaller. Hence anincrease of the value of ∆ tends to increase the value of the integral in (3.20), since the unitstep function vanishes in the region, where the square root term’s value is greatest. In themajority of cases, numerical calculations have shown that the integral’s value is enhanced byincreasing ∆. A decrease was observed very rarely. We will now discuss the possible solutionsof (3.5) with the help of a graphical demonstration.

26 CHAPTER 3. UNBALANCED FERMI-MIXTURES3.2.3 Graphical illustration of the phasesHere we will discuss the phases which can occur. In order to do this, we will plot the solutionsof (3.3) and (3.4) as a function of ∆. In addition we will plot the residual of (3.5):Res(∆) := U ∫ ∞sign ( E(ε) ) [ (dεν d (ε) √ Θ − E↑ (ε) ) + Θ ( − E ↓ (ε) ) − 1 ] − 1 , (3.21)2E 2 (ε) + U 2 ∆ 2−∞where n and m are determined self-consistently in order to illustrate solutions of (3.5) graphically.Note that Equation (3.5) is solved automatically for ∆ = 0 or for Res(∆) = 0.n0.63R [. . .] − 10.6250.620.20.6150.1 0.2 0.3 0.4∆0.61-0.20.6050.60.1 0.2 0.3 0.4∆-0.4Figure 3.8: Occupation number and residual of (3.5) in dependence of ∆ at B = 0nmR [. . .] − 10.6350.630.6250.0350.030.025-0.10.1 0.2 0.3 0.4 0.5∆0.620.6150.610.020.0150.01-0.2-0.30.6050.60.1 0.2 0.3 0.4 0.5∆0.0050.1 0.2 0.3 0.4 0.5∆-0.4Figure 3.9: Occupation number, magnetization and residual of (3.5) in dependence of ∆ atweak magnetic fieldn0.6350.630.625m0.050.04R[. . .] − 1-0.10.1 0.2 0.3 0.4 0.5∆0.620.03-0.20.6150.610.6050.020.01-0.3-0.40.60.1 0.2 0.3 0.4 0.5∆0.1 0.2 0.3 0.4 0.5∆Figure 3.10: Occupation number, magnetization and residual of (3.5) in dependence of ∆ atstrong magnetic filedNote that the lines in Figures 3.8 - 3.10 are just a guide for the eye. In these Figures we seeplots of n, m and the residual of (3.5) in dependence of ∆. With the discussions of section3.2.2 and the help of Figures 3.8 - 3.10 we are now able to determine the phases occurring inthe various parameter regions:- In Figure 3.8 there is no magnetization, since B = 0. So the residual is decreasing withincreasing ∆. There is exactly one solution of (3.5).

3.3. NUMERICAL METHOD 27- In Figure 3.9 two solutions of (3.5) can be seen, since there are two roots of the residual,one for m ≠ 0 at smaller ∆ and one with m = 0 for larger ∆.- In Figure 3.10 one finds that there is no solution of (3.5) with ∆ ≠ 0.- There is always a trivial solution of (3.5), namely ∆ = 0.If there is more than one solution of the self-consistency equations, the grand potentials haveto be compared and the solution which minimizes this potential is thermodynamically stableat Hartree-Fock level. Hence, we may summarize the results of this section as follows:1. There are three competing phases:- Non-superfluid magnetized (B ≠ 0) phase- Superfluid magnetized phase- Superfluid non-magnetized phase2. All phase transitions are in general of first order, since they are characterized onlyby the grand potentials of the two phases being equal, not by the divergence of anysusceptibility.We will call the superfluid non-magnetized phase at B ≠ 0 a non-magnetized phase of secondtype, since the absence of magnetization is a consequence of the superfluidity and thereforecaused by interaction.3.3 Numerical methodIn this section we will present the numerical method used to solve the self-consistency equations(3.3) - (3.5). We have proved analytically in section 3.2 that at fixed ∆ the coupledEquations (3.3) and (3.4) can be treated as a contraction in the sense of Banach’s fixed pointtheorem if the relative interaction strength |U|tis small enough. In addition to that we haveseen with the help of (2.55) that 0 ≤ ∆ ≤ 1 2, and we know that ∆ = 0 always solves (3.5).So we are able to do a scan through the values of ∆ between 0 and 1 2and thereby find allpossible solutions of the self-consistency equations. We will solve (3.3) and (3.4) directly byusing Banach’s theorem:

28 CHAPTER 3. UNBALANCED FERMI-MIXTURESStart with random values n 1 and m 1Calculation of n i+1 and m i+1by integration using n i and m i ; i → i + 1Check if equations approximatively solved|n i − n i−1 | < ǫ n ∧ |m i − m i−1 | < ǫ mnoyesEnd of iteration scheme, output of n i and m iFigure 3.11: Iteration scheme for solving (3.3) and (3.4) at fixed ∆. The abort-criteriaare chosen as: ǫ n = 10 −4 , ǫ m = 10 −3 .In Figure 3.11 we see the numerical solution scheme for (3.3) and (3.4) at fixed ∆. It hasbeen used that (3.3) and (3.4) are a contraction in the sense of (3.7), so that this approximationscheme is always able to solve the equations numerically with the method of successiveapproximations. We have observed that the scheme 3.11 finishes within less than 20 iterationsteps for interaction values up to |U|t= 4, so that (3.18) is a very restrictive convergencecriterion (many rigorous estimations were done). Since with increasing order parameter ∆,the parameter n approaches half filling and the magnetization vanishes, we can conclude thatconvergence improves with increasing values of ∆.

3.3. NUMERICAL METHOD 29Input of ∆ min and ∆ maxChecking product of Res(∆ min ) · Res(∆ max )Usage of scheme 3.11 and definition (3.21)(+1)Output: No solution(−1)20 secant iteration steps to find root|∆ min − ∆ max | < ǫ ∆Adaptation of ∆ min and ∆ max as bounds for ∆|∆ min − ∆ max | > ǫ ∆Bisection method until |∆ min − ∆ max | < ǫ ∆Output: ∆ = ∆ min+∆ max2Figure 3.12: Iteration scheme for solving (3.5) between two values ∆ min and ∆ max ,where∣ does not change the sign.s.c.∂Res(∆)∂∆The scheme represented in Figure 3.12 is used for solving (3.5) numerically. It is based on theassumption that the derivative ∂Res(∆)∂∆ ∣ does not change its sign between ∆ min and ∆ max ,s.c.where “s.c.” means that in this derivative n and m are also changed self-consistently. Withthis assumption there is only the possibility of maximally one solution between ∆ min and∆ max . The sign-check tells us if there is one solution or none at all. Since the self-consistentderivative is not accessible easily, we have tried apply the a secant method in order to solve(3.5). Although this method works very well in most cases, in exceptions it converges tooslowly, so that we abort this method if it takes more than 20 iteration steps and switch toa bisection method, which is a numerical brute-force method that is always able to find theroot.

30 CHAPTER 3. UNBALANCED FERMI-MIXTURESSolve (3.3) and (3.4) at ∆ = 0 with scheme 3.11Try to find solution for ∆ min = 0 and ∆ max = min { 1, ∣ ∣B∣ }2 UIf ∣ ∣B∣ U 0 it is possible that no solution exists. If ∣ ∣B∣ U< 1 2,we have divided therange of ∆-values (see (2.55)) in two parts, namely ∆ < ∣ ∣ ∣B∣ U and ∆ > B∣ U , since there isstrong evidence, that in each part the self-consistent derivative ∂Res(∆)∂∆∣ does not changes.c.its sign (section 3.2.2). As a consequence there is no more than one solution per part. Havingfound all possible solutions, the one which corresponds to the absolute minimum of the grandpotential is taken as the correct (i.e. thermodynamically stable) one. With this procedurewe are able to solve (3.3)- (3.5) for any values of {U,t,µ,B} if the relative interaction ∣ ∣U ∣t isnot too strong.3.4 Numerical resultsWe will now present results for the space-dependent parameters in a trap, calculated withthe scheme, represented in Figure 3.13, within the LDA. We have used a hypercubic latticewith different matrices Ω. In our Figures we present the x-z planes of the three dimensionallattices and show the different phases occurring in such a system.

3.4. NUMERICAL RESULTS 31x-position10521.5246z-position810.510 0nDiff.0.010.00750.0050.00250546810z-positionx-position102∆0.20.150.10.050546810z-positionx-position102Figure 3.14: Spatial distribution of the parameters n, (n ↑ −n ↓ ) = nm and ∆ in the x-z-planeat trap parameters: {U = −3, t = 1, µ 0 = 2, B = −0.1, Ω = Diag(0.05,0.3,0.1)}.

32 CHAPTER 3. UNBALANCED FERMI-MIXTURESx-position107.52.551.5246z-position10.58 0nDiff.0.0040.0030.0020.00102.55x-position7.5102468z-position∆0.20.150.10.0502.55x-position7.5102468z-positionFigure 3.15: Spatial distribution of the parameters n, (n ↑ −n ↓ ) = nm and ∆ in the x-z-planeat trap parameters: {U = −3, t = 1, µ 0 = 0, B = −0.05, Ω = Diag(0.05,0.3,0.1)}.

3.4. NUMERICAL RESULTS 33x-position6421.5246z-position810.510 0nDiff.0.0020.00150.0010.000502x-position46246810z-positionFigure 3.16: Spatial distribution of the parameters n and (n ↑ − n ↓ ) = nm in the x-z-planeat trap parameters: {U = −2, t = 1, µ 0 = 1, B = −0.01, Ω = Diag(0.2,0.3,0.1)}. The orderparameter ∆ vanishes in the whole system and is not shown.We will now discuss the numerical results presented in Figures 3.14 - 3.16. These Figuresdemonstrate that:1. Phase separation occurs: at different lattice sites different phases are found.2. The superfluid polarized phase never occurs.3. We see a shell structure of non-superfluid magnetized and superfluid non-magnetizedrings.4. In Figure 3.14 we have the magnetized phase in the center of the trap, the superfluidring sorrounding it and a magnetized phase in the outer region.5. In Figure 3.15 the magnetized phase in the center is missing, we have a superfluid phasein the center sorrounded by a magnetized phase.6. In Figure 3.16 superfluidity breaks down in the whole system. There is only the magnetizedphase left.

34 CHAPTER 3. UNBALANCED FERMI-MIXTURES7. The occupation number n depends very weakly on the phase occurring.We also see that superfluidity favors half filling, which is not surprising, since the superfluidorder parameter can only be maximized at half filling (see section 2.4.1). In further investigationswe have never been able to find the magnetized superfluid phase, so that we mayassume that this phase does not occur at least at T = 0.These results could be directly observed in experiments with the help of in-situ imaging.It is not possible to image the superfluid order parameter directly, but the difference betweenthe occupation numbers of the different spin-species can be directly observed [16] [17].3.5 Systems with finite temperaturesWe will now briefly discuss some topics about systems at low temperatures and mention somedifferences and similarities to systems in the ground state.3.5.1 Infinite spatial extensionMathematically the difference arises from the fact that at finite temperatures the systemsare described with the help of a Fermi-function, which at T = 0 is replaced by a unit stepfunction. In our LDA calculation we have used that for some critical value of µ < µ C theoccupation numbers vanish exactly:n = 0 ∧ m = 0 ∧ ∆ = 0 (∀ µ < µ C ) . (3.22)This fact arises from the sharp edge at the Fermi-level, which causes all states to be empty.At T > 0 the Fermi-function has no sharp edge, so that there is no µ C in that sense. As aconsequence in our LDA-calculation we get:n ∝ e −βµ ∝ e −β(i·Ω·i) , ( |i | → ∞) , (3.23)so that we have to define an abortion criterion, n < n C = 10 −4 for example to avoid aninfinite loop.3.5.2 Phase transitionsThe phase transition in systems with B = 0 is of second order, since∂Ω∂ 2 ∣Ω ∣∣∣∆=0∂∆∣ = 0 and∆=0∂∆ 2 < 0 (∀ T < T C ) , (3.24)so that ∆ = 0 is not a minimum of the grand potential. That means for T = 0, that wefind a superfluid order parameter ∆ ≠ 0 as thermodynamically stable solution of the selfconsistencyequations. Increasing T causes ∆ to decrease until a critical temperature T C isreached, where ∆ → 0. At T > T C there is no other solution than ∆ = 0 for (3.5). In systemswith B ≠ 0, phase transitions are of first order, so that there is no similar argumentationas in non-magnetized systems. With B ≠ 0 we have to take into account that transitions offirst order between a non-magnetized superfluid phase, a magnetized superfluid phase and anormal phase may occur. More detailed information is found in section 4.5.

Chapter 4Model with spin-dependent hoppingIn this chapter we will treat effects arising from spin-dependent hopping in translationallyinvariant systems. The magnetic field is switched off (B = 0) and hopping is spin-dependent(t ↑ ≠ t ↓ ) during this chapter. First we explain our focus on superfluid solutions. We thendiscuss superfluidity in the attractive U model away from half filling with the resulting magnetizationat T = 0 and critical temperatures of the phase transition at half filling. Allnumerical calculations have been done for three-dimensional systems.4.1 Charge density wave states and the repulsive modelIn this section we compare the attractive U model (2.1) at B = 0 with its repulsive version athalf filling. We show their equivalence and compose a“dictionary”of corresponding propertiesin order to motivate our focus on superfluid phases in this thesis neglecting possible competingphases, like, e.g., charge density wave phases (CDW).4.1.1 Special particle-hole transformationHere we present the equivalence of the model (2.1) at attractive interaction and B = 0 withthe repulsive model at half filling in a magnetic field. The proof is done by performing aspecial particle-hole transformation:c † i↑→ c † i↑c † i↓→ exp(iQ · i)c i↓(4.1)Q = (π,π,... ,π) T .As was the case for the general particle-hole transformation, studies in (2.56), the specialparticle-hole transformation (4.1) leaves the hopping-term H t unchanged, but the interactionterm H U changes its sign. The Zeeman-term in the repulsive model is transformed into anadditional chemical potential term in the attractive model. We show the transformation from35

36 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPINGthe repulsive to the attractive model here:H tP.H.−−−→ −H t (4.2)H UH µ ′H B ′P.H.−−−→P.H.−−−→ − U 2P.H.−−−→−U ∑ n i↑ n i↓ + U ∑ n i↑(4.3)ii( ∑n i↑ − ∑ )n i↓ + N(4.4)i i)B ′ ( ∑iσn iσ + N. (4.5)Hence, the repulsive model (2.1) at half filling with the chemical potential µ ′ = U 2and themagnetic field B ′ is mapped onto the attractive model away from half filling with the chemicalpotential µ = − |U|2 − B′ and B = 0, since the constant terms (proportional to N) arephysically inactive. Note that the variables µ ′ and B ′ refer to the repulsive model, while thevariables µ and B refer to the attractive model.Furthermore, with the use of the transformation (4.1), we may identify the spatial componentsof the staggered magnetization ˜S in the repulsive model with staggered occupationand superfluidity in the attractive model. The corresponding quantities are presented here:Spin comp. U > 0 U < 0 Quantity1 ˜S 1 2 〈exp(iQ · i)(c† i↑ c i↓ + c† i↓ c i↑ )〉 12 〈c i↑ c i↓ + c† i↓ c† i↑ 〉 Re(∆)i ˜S 2 2 〈exp(iQ · i)(c† i↑ c i↓ − c† i↓ c i↑ )〉 − i 2 〈c i↑ c i↓ − c† i↓ c† i↑ 〉 Im(∆)1 ˜S 3 2 〈exp(iQ · i)(n i↑ − n i↓ )〉 12 〈exp(iQ · i)(n i↑ + n i↓ − 1)〉 sTable 4.1: Spatial components of the staggered magnetization in the repulsive model andtheir correspondences in the attractive model, where s is the staggered density parameter ofthe CDW phase.With these correspondences, we are able to compare the model at attractive interaction withthe model at repulsive interaction.4.1.2 Grand potential at half fillingIn order to understand the ground state properties of the attractive model (2.1) with B = 0at half filling we must first discuss some symmetry properties. In the spin-independenthopping case t ↑ = t ↓ the system is rotationally invariant in each spatial direction, so we havea SU(2)-symmetry. Thus in the repulsive model there is no distinguished spatial directionfor the staggered magnetization. Hence the superfluid and the CDW-phase in the attractivesystem are energetically degenerate according to Table 4.1. In the spin-dependent hoppingcase t ↑ ≠ t ↓ , the SU(2)-symmetry is broken and there is only a U(1)-symmetry left, whichdescribes rotation around the z-axis of the repulsive model and a rotation in the complexplane of the superfluid order parameter in the attractive model. Hence the z-direction isa distinguished spatial direction in the repulsive model’s staggered magnetization and theCDW-phase is energetically distinguishable from the superfluid phase. At T = 0 the CDW

4.2. SUPERFLUIDITY AWAY FROM HALF FILLING 37phase is energetically favored. The grand potential of the CDW phase is given as:{ω CDW = −U s 2 + 1 }− ∑ ∫ ∞dεν d (ε) √ 4t42 σ ε2 + U 2 s 2 , (4.6)σ 0where 0 ≤ s ≤ 1 2is the staggered density parameter (see Table 4.1). The self-consistencyequation for s is given as:1 = |U|2∑σ∫ ∞0dεν d (ε)and the self-consistency equation for the superfluid phase writes:1 = |U|∫ ∞0dεν d (ε)1√4t2 σ ε 2 + U 2 s 2 (4.7)1√(t↑ + t ↓ ) 2 ε 2 + U 2 ∆ 2 . (4.8)If we switch on the asymmetry around the symmetric case by replacing:t σ → t + σλ , λ ≪ t , (4.9)the right hand sides of the self-consistency equations (4.7) and (4.8) do not depend on λin first order, so that neither the staggered density parameter s nor the superfluid orderparameter ∆ change their values, implying s = ∆+O(λ 2 ). The difference between the grandpotentials of the phases at s = ∆ has a definite sign:ω Sup − ω CDW > 0 , (s = ∆, T = 0) , (4.10)so that at small hopping asymmetry the CDW phase is preferred at half filling. But from theliterature [29] it is well known that, in the symmetric hopping model at repulsive interaction,the staggered magnetization in x- or y-direction is energetically lower in a magnetic field.This implies (after a special particle-hole transformation) the thermodynamic stability ofthe superfluid phase in the attractive model away from half filling (see Table 4.1). In thisthesis we analyze the asymmetric hopping model away from half filling, hence we assumethat superfluidity occurs. Furthermore, in recent developments both experimentalists andtheoreticians investigate only superfluid phases, while CDW phases seem to be experimentallyirrelevant. (See for example: [9], [30], [11], [31], [32], [10] and [33]). This leads us to restrictconsideration to superfluid phases in this thesis, while possible CDW phases are ignored.4.2 Superfluidity away from half fillingWe will now discuss our model (2.1) at B = 0. We derived the Hartree-Fock self-consistencyequations in chapter 2. As we have seen in section 2.4.3 we cannot expect the system to benon-magnetized away from half filling even if B = 0.4.2.1 The quasiparticle energiesThe quasiparticle energies (B = 0) are for asymmetric hopping amplitudes defined as:E σ (ε) =E(ε) =−Unmσ − (t σ − t −σ )ε + sign ( E(ε) ) √E22 (ε) + (U∆) 2 , (4.11)Un2 − µ − (t ↑ + t ↓ )ε .

38 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPINGWhile the quasiparticle energy for symmetric hopping is decreasing with increasing parameterε, for asymmetric hopping the quasiparticle energies are not monotonic for both spinspecies. So there is the possiblility of additional zeroes in the quasiparticle-energy functionof the spin species with the lower hopping amplitude. This causes the proof of section 3.2 tobe not valid anymore, since in the proof it has been assumed that the quasiparticle energiesE σ change their sign at most once. In the spin-dependent hopping case this is only valid, ifthe asymmetry does not cause additional zero points in the quasiparticle energy. However,even if the proof of section 3.2 breaks down for t ↑ ≠ t ↓ , there still seems to be uniquenessof the solution of the self-consistency equations: In numerical studies of the model we neverobserved multiple solutions of n and m at fixed ∆.E ↓ (ε) , E ↑ (ε)7.552.5-3 -2 -1 1 2 3-2.5ε-5-7.5-10Figure 4.1: Quasiparticle energies for asymmetric hopping. The curve corresponding to thespin species with the lower hopping amplitude is not monotonic (red curve).4.2.2 Hartree-Fock density of statesThe absence of the symmetry in the quasiparticle energies influences the (Hartree-Fock) DOSof the up and down particles, which is defined as:ν d,σ (E) :=∫ ∞−∞dεν d (ε)δ ( E − E σ (ε) ) . (4.12)The asymmetric hopping now causes the spin-dependent DOS to have additional singularitiesof 0 th order (i.e., jumps). If we assume t ↓ < t ↑ , the following changes can be established:- The square-root singularity at the superfluid gap in ν d,↑ is replaced by a 0 th ordersingularity.- The DOS ν d,↓ gets an additional singularity of 0 th order.- The superfluid gap in ν d,↓ becomes smaller than 2|U∆|.The normalization of ν d,σ is not affected, of course. Note that the following Figures are madefor the parameters: U = −5, t ↑ = 3, t ↓ = 0.5, µ = −2, n = 1, m = 0.4, ∆ = 0.3. Theseparameters are not meant to be useful in the sense of solving the self-consistency equations,they are just for illustration of the DOS.

4.3. NUMERICAL METHOD FOR THE PROBLEM AT FIXED PARAMETER N 39ν 3,↑ (E)0.120.10.080.060.040.02-15 -10 -5 5 10 15 20EFigure 4.2: Hartree-Fock DOS of the spin species with the greater hopping amplitude. Insteadof a square-root singularity there is a 0 th order singularity at the border of the superfluidgap.ν 3,↓ (E)10.80.60.40.2-6 -5 -4 -3 -2 -1 1EFigure 4.3: Hartree-Fock DOS of the spin-species with the lower hopping-amplitude. Anadditional 0 th order singularity appears and the superfluid gap is smaller than 2|U∆|.So there are more effects arising from spin-dependent hopping away from half filling thanjust the possibility of magnetization. The different widths of the superfluid gaps could beobserved experimentally, for example.4.3 Numerical method for the problem at fixed parameter nIn this section we will present a numerical method for solving the self-consistency equationsat fixed n. Since we fix the occupation number, µ becomes a function of U, µ → µ(U). Weanalyze the magnetization m and the superfluid order parameter ∆ in the ground state.

40 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPINGThe self-consistency equations are formally the same as (3.3) - (3.5), of course, since we havediagonalized the Hamiltonian (2.1) including both spin-dependent hopping and a Zeemanterm.In contrast to the case treated in chapter 3 we have never found more than one solutionof (3.5) with ∆ ≠ 0 in absence of the Zeeman-term, so that we have to compare at most twosolutions with the help of the Helmholtz free energy per site, which is defined as:f(T, N,n ↑ ,n ↓ ) = 1 N F(T, N,n ↑,n ↓ ) (4.13)= ω(T, N,µ ↑ ,µ ↓ ) + µ ↑ n ↑ + µ ↓ n ↓= ω(T, N,µ ↑ ,µ ↓ ) + µ n .Here µ is spin independent, since we have chosen B = 0. However, for fixed n the chemicalpotential µ depends on the fixed parameters {U,t σ ,n}. Note that µ may be spin-dependent,since σ is a pseudospin and can therefore be used to describe different atom species, forexample. The numerical evaluation has been done with the use of the following algorithm:Begin with µ(U = 0) and m(U = 0)Determine µ min and µ max as relevant regionBisection at fixed m until |n(µ) − n fix | < ǫ nDetermine m by successive approximationUse method to solve (3.4), where µ is fixedCheck if |n(µ) − n fix | < ǫ n and |m i − m i−1 | < ǫ myesnoOutput of µ(U) and m(U)Figure 4.4: Solution algorithm for µ and m at fixed (U,t σ ,n,∆). The parameters ǫ n and ǫ mare chosen as in algorithm 3.11.

4.4. NUMERICAL RESULTS FOR THE GROUND STATE 41Thus the algorithm is very similar to the algorithm presented in Figure 3.11, where we presentthe way of solving (3.3) and (3.4) at fixed µ and ∆. Since µ appears only implicitly in theself-consistency equations, we decouple the Equations (3.3) and (3.4) at fixed n and use abisection method to solve (3.3) at fixed m and a successive iteration as in Figure 3.11 to solve(3.4). Afterwards we check if both equations are solved within the desired accurancy ǫ n andǫ m .The algorithm for solving the self-consistency equations uses scheme 4.4 and is presentedhere:Find µ and m at ∆ = 0Scan for sign changes in Res(∆)Sign change found?noTake ∆ = 0 as solutionyesUse combination of secant and bisection methodin order to solve (3.5) (as in algorithm 3.11)Compare Helmholtz free energy of the solutionsOutput of µ, m and ∆Figure 4.5: Solution algorithm for the self-consistency equations (3.3)-(3.5). algorithm 4.4 isused implicitly to obtain µ and m.With this procedure we are able to find the ground state of our spin-dependent hoppingHamiltonian at fixed occupation number n. We will now present results obtained with theuse of this method.4.4 Numerical results for the ground stateWe will now present the results of numerical calculations done according to algorithm 4.5.As hopping amplitude we have chosen t ↑ = 1 and t ↓ = 0.5. We calculate the parameters µ,m and ∆ as a function of U for different occupation numbers n.

42 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPING-0.5µ-2.5µ-1-2.6-2.7-1.5-2.8-5 -4 -3 -2 -1-2.5U-5 -4 -3 -2 -1-2.9-3.1-3.2U0.08m0.8m0.060.60.040.40.020.2-5 -4 -3 -2 -1U-5 -4 -3 -2 -1U0.4∆0.14∆0.30.120.10.20.080.060.10.040.02-5 -4 -3 -2 -1U-5 -4 -3 -2 -1UFigure 4.6: µ, m and ∆ as a function of U near half filling on the left, away from half fillingon the right.As parameters for these calculations we have used µ(U = 0) = −0.5, approximatively correspondingto n ≈ 0.79, for the solutions near half filling and µ(U = 0) = −2.5, approximativelycorresponding to n ≈ 0.18, for the solutions far away from half filling. We have alsodone a calculation exactly at half filling, where we have m = 0 and µ = U 2(see section 2.4.3).The order parameter ∆ as function of U behaves as follows:

4.5. CRITICAL TEMPERATURES 430.4∆0.30.20.1-5 -4 -3 -2 -1UFigure 4.7: Superfluid order parameter ∆ as a function of U. For small values of U, the orderparameter becomes exponentially small.The numerical results can now be summarized as follows:- At half filling we are able to show analytically that ∆ > 0 is the correct solution.- Numerically we are not able to detect ∆ for small values of U (|U| ≪ t ↑ + t ↓ ), since ∆becomes exponentially small.- Near half filling (n ≈ 0.79) we need U to be greater than a critical value in order to getsuperfluidity.- Superfluid solutions are non-magnetized solutions of the second type near half filling.- Far away from half filling (n ≈ 0.18) we need U to be even greater than in the situationnear half filling.- Superfluid solutions are magnetized far away from half filling.With these calculations we have been able to see the basic effects arising from spin-dependenthopping. It is not surprising that, the further we are away from half filling the more oursystem tends to be magnetized and non-superfluid, since the symmetric pairing is suppressedby magnetization and the non-interacting system is rigorously magnetized.4.5 Critical temperaturesIn this section we present the behavior of the critical temperature of the superfluid orderparameter ∆ in the superfluid phase and the critical temperature of the order parameter s ofthe CDW phase at half filling. We first show that at 0 ≤ T < T C,∆/s the superfluid and theCDW solution always have a lower free energy then the non-symmetry-broken one. Then wepresent the self-consistency equation at T = T C,∆/s in order to calculate T C,∆/s numerically.4.5.1 Superfluidity in non-magnetized solutionsWe now show that non-magnetized solutions of first type (see section 2.4.3) are always superfluidbelow a certain critical temperature T C,∆/s . This will be demonstrated for the modelwith spin-dependent hopping, half filling and B = 0. The proof at spin-independent hoppingand B = 0 is analogous for the superfluid case. (The filling is unimportant here, since awayfrom half filling we do not expect magnetization to occur at B = 0.) In order to show this

44 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPINGwe have to consider the grand potential or the free energy. Which one we consider is unimportant,since at half filling µ · n is constant as a function of ∆. We will formulate our proofwith the help of the grand potential for the superfluid case:{ω = −U ∆ 2 + 1 }− 1 4 β∑σ∫ ∞−∞(dεν d (ε) ln 1 + exp ( − β E σ (ε) )) (4.14)E(ε) = −(t ↑ + t ↓ )ε (4.15)E σ(ε) = (t −σ− t σ)ε + sign ( E(ε) ) √E 2 (ε) + U 2 ∆ 2 (4.16)and demonstrate for derivatives:∂ω∂∆∣ = 0 and∆=0∂ 2 ω∂∆ 2 ∣∣∣∣∆=0< 0 for β > β C . (4.17)Hence the solution ∆ = 0 is not a minimum of the grand potential, and the correct solutionmust have ∆ ≠ 0. The first derivative is given as:∂ω∣∂∆∣∆=0= ∆[= 0 ,|2U| + ∑ σ∫ ∞−∞dεν d (ε)11 + exp ( β E σ (ε) ) · sign ( E(ε) ) ]· U 2√E 2 (ε) + U 2 ∆ 2∆=0(4.18)and the second derivative follows as:∂ 2 ∣ [ω ∣∣∣∆=0∂∆ 2 = |2U| + ∑ ∫ ∞1dεν d (ε)σ −∞ 1 + exp ( β E σ (ε) ) · sign ( E(ε) ) ]· U 2√E 2 (ε) + U 2 ∆ + 2 O(∆2 )= |2U| + U ∑ ∫ ∞2 1P dεν d (ε)σ −∞ 1 + exp ( β E σ (ε) ) · 1E(ε)= |2U| − U 2 ∑ σβ→∞−−−→∫ ∞0∫ ∞(β Eσ(ε)2tanhdεν d (ε)E(ε))) tanh(Eσ(ε)2= |2U| − U ∑ ( ε 2 dεν dσ 0 β E(ε)( )|2U| − U 2 ν d (0) ∑ ∫ ∞ tanh Eσ(ε)2dεσ 0 E(ε)= −∞ .The symbol P denotes the integral’s principal value, of course. On the other hand we obtainin the limit of infinite temperature:)∂ 2 ω∂∆ 2 ∣∣∣∣∆=0 β→0−−−→ |2U| > 0 . (4.20)Hence there is a critical temperature T C,∆ such that at low temperatures T < T C,∆ the nonsuperfluidsolution is thermodynamically unstable, while at high temperatures T > T C,∆ itminimizes the grand potential. The proof for the CDW case is completely analogous andleads to the same conclusion.∆=0(4.19)

4.5. CRITICAL TEMPERATURES 454.5.2 Phase transition at T = T CWe will now derive the self-consistency equation for the critical temperature at half fillingfor the superfluid case. We could use Equation (2.50) at µ = U 2, n = 1 and m = 0 or (4.18)as a starting point. The critical temperature T C is defined as the temperature where thesolution of (2.50) is uniquely given by ∆ = 0. Equation (2.50) has a unique solution ∆ ≠ 0for T < T C , while it has no solution for T > T C . For T < T C we obtain ∂∆∂T< 0, so thatlim ∆(T) = 0 (T < T C ) . (4.21)T →T CHence the phase transition is of second order, since there is no jump in the order parameterat T = T C . The self-consistency equation for T C is now given by:0 = U ∫ ∞2 P 1 [ (dεν d (ε) fβC,∆ E↑ (ε) ) (+ f βC,∆ E↓ (ε) ) − 1 ] − 1 (4.22)−∞ E(ε)( )= |U| ∑∫ ∞ tanh Eσ(ε)2T C,∆dεν d (ε)− 1 .2E(ε)Analogously we obtain for the CDW phase:σ00 = |U|2∑σ∫ ∞0dεν d (ε)( )tanh Eσ(ε)2T C,sE σ (ε)− 1 . (4.23)Since the right hand sides of (4.22) and (4.23) are monotonic functions on T C,∆/s and have arange of values between -1 and +∞, there is always a solution for fixed {t σ ,U}. Hence theseare simple root-finding problems in one variable, namely T C,∆ and T C,s . We have solved theseproblems numerically for fixed t ↑ + t ↓ = 2 and fixed U for values of t ↓ between 0 and 1. Thesolutions are presented graphically in Figures 4.8 and 4.9:T C,∆ , T C,s (a.u.)0.020.01750.0150.01250.010.00750.0050.00250.2 0.4 0.6 0.8 1t ↓T C,∆ , T C,s (a.u.)0.160.140.120.10.080.060.040.020.2 0.4 0.6 0.8 1Figure 4.8: Critical temperatures T C,∆ for the superfluid order parameter and T C,s for thestaggered density order parameter in dependence of t ↓ at fixed t ↑ + t ↓ = 2 and U = −1.t ↓

46 CHAPTER 4. MODEL WITH SPIN-DEPENDENT HOPPINGT C,∆ , T C,s (a.u.)0.50.40.30.20.10.2 0.4 0.6 0.8 1t ↓Figure 4.9: Critical temperatures T C,∆ for the superfluid order parameter and T C,s for thestaggered density order parameter in dependence of t ↓ at fixed t ↑ + t ↓ = 2 and U = −3.In Figures 4.8 and 4.9 we tune the hopping parameters between the Falicov-Kimball limit andthe isotropic Hubbard limit at fixed U and t ↑ +t ↓ . As we can see, while the critical temperaturefor the superfluid phase T C,∆ increases with the symmetry of the hopping amplitudes, forthe CDW phase the critical temperature T C,s is higher for asymmetric hopping amplitudes.Obviously superfluidity favors a spin-symmetric Hamiltonian, while the CDW phase favorsasymmetry. The critical temperatures are quasi-identical if the asymmetry is small: t ↑ ≈ t ↓ .This is an additional argument for neglegting the CDW phase away from half filling, sincethe difference of the critical temperatures is a measure for the energetical difference of thecompeting phases, and we know that the CDW phase becomes thermodynamically unstableaway from half filling in the symmetric model t ↑ = t ↓ [29], where the critical temperatures areidentical. Interestingly the critical temperature T C,s does not have its maximum at t ↓ = 0:T C,s (a.u.)0.860.850.840.830.820.810.80.2 0.4 0.6 0.8 1t ↓Figure 4.10: Critical temperature T C,s for the staggered density order parameter in dependenceof t ↓ at fixed t ↑ + t ↓ = 2 and U = −4.5. We have chosen a large value of |U| in orderto improve the visibility of the maximum of T C,s .The critical temperature of the superfluid phase at half filling can be seen as an upperbound for the critical temperature away from half filling, since at half filling superfluidity ismaximal. In the isotropic Hubbard model limit it can also be seen as an upper bound forthe critical temperature of a system with a magnetic field, since a magnetic field also tendsto suppress superfluidity.

Chapter 5Strong coupling limitIn this chapter we will treat a Hubbard model with spin-dependent hopping in the strongcoupling limit. We will first investigate the “band structure” of the exactly solvable 2-sitemodel at strong coupling in order to get a first impression of strong coupling behavior andintroduce the concept of effective double occupancy. Then we will explain perturbationtheory for the generalized model (5.6) at strong coupling and derive the first and second orderperturbative contributions. The perturbation theory is based on a concept presented in [34].With these results we will analyze the strong coupling limit of the Hamiltonian (5.6), restrictedto nearest neighbor interaction. Finally we will show that under these conditions the U ≫max{t σ } case can be mapped onto a H XXZ model (at half filling) and the U ≪ −max{t σ }case can be mapped onto a hard-core boson model with nearest neighbor interaction (at aneven number of fermions).5.1 Introduction: Exactly solvable 2-site modelIn this section we discuss the exactly solvable 2-site Hubbard model with spin-dependenthopping in order to understand its behavior at strong coupling:H = U ∑ in i↑n i↓− ∑ i≠jt σc † iσ c jσ; i,j = 1,2 (5.1)at half filling. The Hilbert space at half filling is ( 42)= 6 dimensional. We introduce thefollowing notation for the fully antisymmetric basis states:and analogously for the other states.|1 ↑,2 ↓〉 = c † 1↑ c† 2↓|vacuum〉 , (5.2)5.1.1 Diagonalization of the HamiltonianObviously the spin polarized states |1 ↑,2 ↑〉 and |1 ↓,2 ↓〉 are eigenstates with energiesE 1/2 = 0. This reduces our problem to a 4 × 4 matrix problem:The basis is chosen as:H =⎛⎜⎝U −t ↓ −t ↑ 0−t ↓ 0 0 −t ↑−t ↑ 0 0 −t ↓0 −t ↑ −t ↓ U⎞⎟⎠ . (5.3)47

48 CHAPTER 5. STRONG COUPLING LIMIT⎛⎜⎝1000⎞⎛⎟⎠ = |1 ↑,1 ↓〉;⎜⎝0100⎞⎛⎟⎠ = |1 ↑,2 ↓〉;The eigenvalues of H can easily be calculated:√E 3/4 = U 2 ±⎜⎝0010⎞⎛⎟⎠ = |2 ↑,1 ↓〉;√2U+ (t ↑ − t ↓ )42 ; E 5/6 = U 2 ±These eigenvalues may now be analyzed at strong coupling.5.1.2 “Band structure” at strong coupling⎜⎝0001⎞⎟⎠ = |2 ↑,2 ↓〉 .(5.4)2U+ (t ↑ + t ↓ )42 . (5.5)In Figures 5.1 and 5.2 we demonstrate the energy levels at strong coupling graphically (parameters:U = ±15; t ↑ = 3; t ↓ = 1.5; half filling):Energy levels a.u.1512.5107.552.50Figure 5.1: E 1−6 atEnergy levels a.u.-2.5-7.5-10-12.5-15Umax{t ≫ 1 σ}0-5Figure 5.2: E 1−6 atUmax{t σ} ≪ −1Obviously the energy levels are close to the values 0 and U. A Taylor expansion ( of E)1−6with respect to the hopping amplitude shows that each level behaves like 0 + O t 2 σUor( )U + O t 2 σU. From now on, at strong coupling, we call levels close to an integer n multipleof U, levels with effective double occupancy n. Note that effective double occupancy differsfrom real double occupancy in the language of the c operators, since real double occupancyis not a good quantum number at t ↑ or t ↓ different from 0 in this language. Levels withn = 0, for example, may have 〈n i↑n i↓〉 ≠ 0. In the next section we will perform a unitarytransformation, which transforms the c operators into d operators, so that effective doubleoccupancy in the c language is mapped onto real double occupancy in the new d language.5.2 General procedureIn this section we treat the general modelH = U ∑ in i↑n i↓− ∑ ijσt ijσc † iσ c jσ; t ijσ≥ 0 , (5.6)which we rewrite as H(c) = H U (c)+H t (c) in the strong coupling limit (U/max{t ijσ } → ±∞).We perform an unitary transformationc † iσ = exp[S(d)]d† iσexp[−S(d)] , (5.7)

5.2. GENERAL PROCEDURE 49with an antihermitian operator S:S † = −S . (5.8)Thus we transform the set of fermionic creation and annihilation operators c into new“dressed”fermionic operators d so that the Hamiltonian can be rewritten as H(d) = H U (d)+H t (d). The aim of this transformation is that with the new operators the number of doubleoccupancies remains unchanged by the new hopping term H t (d). Double occupancy in the“new language” of the d-operators is a good quantum number. More formally we have:[H U (d), H t (d)] = 0 , (5.9)so that the application of H t to a state of the Fock-space does not influence the number ofdouble occupancies counted by H U . Equation (5.9) determines how S(d) has to be chosen. Inorder to do this we treat the hopping term perturbatively by multiplying it with a bookkeepingparameter λ and performing an expansion in powers of λ:H t (c) → −λ ∑ ijσt ijσc † iσ c jσ; λ ≥ 0 . (5.10)Putting (5.7) into (5.6) gives us H in terms of d:H = exp[S(d)]{U ∑ i¯n i↑¯n i↓− λ ∑ ijσt ijσd † iσ d jσ}exp[−S(d)] , (5.11)where ¯n iσ= d † iσ d iσis the number operator for dressed fermions. Obviously, at λ = 0, theHamiltonian is unperturbed, so that we choose d = c and therefore S(d) = 0. With thischoice we get H U (d) = H U (c) = H, so that the transformation is the identity transformationat λ = 0. Therefore we expand S(d) and H t (d) in powers of λ:andS(d) =H t (d) =∞∑S n (d)λ n (5.12)n=1∞∑H t,n (d)λ n (5.13)n=1in order to get H(d) in powers of λ. Generally n th order perturbation theory is performed bydetermining S 1 -S n−1 so that (5.9) is satisfied in the n lowest powers of λ:[H U (d), H t (d)] = O(λ n+1 ) . (5.14)If we also want to express the old operators c correctly in n th order of λ, it is necessary tocalculate S n , too, but, for determining the energy, knowledge of S n is unnecessary.5.2.1 Second order perturbation theoryWe will now neglect all terms of O(λ 3 ) and higher in order to satisfy (5.14) for n = 2:exp[±S(d)] = 11 ± λS 1 (d) + λ 2 { 1 2 S2 1 (d) ± S 2(d)} + O(λ 3 ) , (5.15)so that H(d) can also be expressed in powers of λ:H(d) = H U (d) + λH t,1 (d) + λ 2 H t,2 (d) + O(λ 3 ) . (5.16)

50 CHAPTER 5. STRONG COUPLING LIMITBy expanding (5.11) we obtain the coefficients:H U (·) = H U (·) (5.17)H t,1 = [S 1 (d),H U (d)] + H t (d) (5.18)H t,2 = [S 1 (d),[S 1 (d),H U (d)]] + [S 1 (d),H t (d)] + [S 2 (d),H U (d)] . (5.19)We must now arrange S 1 and S 2 such that they satisfy:S † 1/2 = −S 1/2 (5.20)in order to fulfill (5.8). We also have to satisfy (5.14) so that the number of double occupanciesis preserved by the first and second order contributions of H t :[H t,1/2 ,H U ] = 0 . (5.21)Therefore it is useful to decompose the original hopping term as follows:where H +/0/− are given by:H t (d) = −λ[H + + H 0 + H − ] , (5.22)H += ∑ ijσt ijσ¯n i−σ(1 − ¯n j−σ)d † iσ d jσ(5.23)H 0= ∑ ijσt ijσ(1 − ¯n j−σ− ¯n i−σ+ 2¯n i−σ¯n j−σ)d † iσ d jσ(5.24)H −= ∑ ijσt ijσ¯n j−σ(1 − ¯n i−σ)d † iσ d jσ. (5.25)Here H + (H − ) is the part of the hopping term that increases (decreases) the number of doubleoccupancies and H 0 is the part that leaves the number of double occupancies unchanged. Thisis expressed more formally by the commutation relation:In addition they fulfill:[H U (d),H p ] = UpH p p ∈ {+,0, −} . (5.26)H † 0 = H 0 ; H † + = H − ; H † − = H + . (5.27)5.2.2 Choice of the transformation operatorsWe will now choose the transformation operators S 1/2 according to (5.14) in second order ofλ and according to (5.20). Equation (5.14) is obviously satisfied in first order of λ by thechoice:S 1 (d) = 1 U (H + − H − ) . (5.28)With this choice we obtain the first order correction to H U :H t,1 = H 0 . (5.29)This means that first order perturbation theory at strong coupling can be performed byneglecting all terms in (5.6) which change the number of double occupancies. In first order

= ∑ ijlσ= − ∑ ijlσ= ∑ ijlσ= ∑ ijlσ= − ∑ ijlσ= − ∑ ijlσ5.2. GENERAL PROCEDURE 51(resetting the bookkeeping parameter λ to 1) we get a Hubbard model with“reduced”hopping.In second order in λ we may choose S 2 (d). Equations (5.14) and (5.20) are fullfilled by:The second order correction to the Hamiltonian is given by:S 2 (d) = 1U 2[H + + H − ,H 0 ] . (5.30)H t,2 = 1 U [H +,H − ] . (5.31)We are now able to calculate (5.31) in terms of d † and d in order to analyze the structure ofsecond order perturbation theory at strong coupling.5.2.3 Structure of the second order termNow we focus attention on Hubbard models of the type (5.6) with the constraint:t iiσ = 0 ; ∀i,σ . (5.32)That means, that each site has the same local energy, which is chosen equal to zero. We thenobtain an expression for H t,2 in terms of d and d † . Therefore it is necessary to calculate thecommutation relations appearing in (5.31):The second order contribution can be written as:[d † iσ d jσ ,d† kη d lη ] , (5.33)[d † iσ d jσ , ¯n l−η (1 − ¯n k−η)] . (5.34)H t,2 = 1 U6∑H t2,i (5.35)i=1with the components:H t2,1t ijσt liσ(1 − ¯n i−σ)¯n j−σ¯n l−σd † lσ d jσ(5.36)H t2,2t ijσt jlσ(1 − ¯n i−σ)¯n j−σ(1 − ¯n l−σ)d † iσ d lσ(5.37)H t2,3t ijσt il−σ(1 − ¯n j−σ)¯n lσd † iσ d jσ d† i−σ d l−σ(5.38)H t2,4t ijσt lj−σ¯n i−σd † iσ d jσ (1 − ¯n lσ )d† l−σ d jσ(5.39)H t2,5t ij−σt jlσ¯n l−σ¯n iσd † jσ d lσ d† i−σ d j−σ(5.40)H t2,6t ij−σt liσ(1 − ¯n l−σ)d † lσ d iσ (1 − ¯n jσ )d† i−σ d j−σ. (5.41)At this point we recognize that the second order contributions can be described as pairwisehopping of fermions. The terms H t2,1 and H t2,2 describe double hopping of one fermionto a different or to the same site and the other terms describe pairwise hopping of twofermions with different pseudospin indices. None of the terms changes the number of doubleoccupancies in the d-language.

52 CHAPTER 5. STRONG COUPLING LIMIT5.3 Treatment of pure nearest neighbor hoppingIn this section we treat a simplified model with the special choice of t ijσ :{tσ if i,j nearest neighborst ijσ =0 else .(5.42)This will reduce our second order contribution to:H t2,1 = ∑t 2 σ¯n i+α,−σ (1 − ¯n i,−σ )¯n i+β,−σ d† i+α,σ d i+β,σ(5.43)H t2,2H t2,3H t2,4H t2,5H t2,6iαβσ= − ∑t 2 σ¯n i,−σ(1 − ¯n i+α,−σ)(1 − ¯n i+β,−σ)d † i+α,σ d i+β,σ(5.44)iαβσ∑= t ↑ t ↓ (1 − ¯n i+α,−σ)¯n i+β,σd † i,σ d i+α,σ d† i,−σ d i+β,−σ(5.45)iαβσ= t ↑ t ↓∑iαβσ= −t ↑ t ↓∑iαβσ¯n i+α,−σd † i+α,σ d i,σ (1 − ¯n i+β,σ )d† i+β,−σ d i,−σ(5.46)¯n i+α,−σ¯n i+β,σd † i,σ d i+α,σ d† i+β,−σ d i,−σ(5.47)∑= −t ↑ t ↓ (1 − ¯n i+α,−σ)d † i+α,σ d i,σ (1 − ¯n i+β,σ )d† i,−σ d i+β,−σ. (5.48)iαβσIn this notation α and β are summed over all lattice vectors with a length of one, so thatthey connect the lattice site i to its nearest neighbor. We have used that t σ t −σ = t ↑ t ↓ forboth σ =↑ and for σ =↓.5.4 Low temperature limitWe will show that under the condition (5.42) the model (5.6) can be mapped onto a H XXZmodel for U ≫ max{t σ } and can be mapped onto a hard-core-boson model with nearestneighbor interaction for U ≪ −max{t σ } , by restricting consideration to low temperatures:k B T ≪ |U| . (5.49)With the restriction (5.49) we can make the assumption that only states with the lowestenergy in 0 th order are occupied. This means that for a fixed number of particles in theU ≫ max{t σ } case the number of double occupancies is minimal and in the U ≪ −max{t σ }case it is maximal. We will first treat the repulsive U model and then we will treat theattractive U model. In the special cases we analyze, all contributions either vanish or can bedrastically simplified. We will focus on the repulsive U model at half filling and the attractiveU model with an even number of fermions.5.4.1 Repulsive U model at half fillingWe will restrict consideration to half filling in the repulsive case. In this case, if we minimizethe number of double occupancies, in 0 th order we have a highly degenerate ground state.Each site is singly occupied, there is no doubly occupied or empty site left. The system iscompletely degenerate with respect to the spin configuration. We will show that second orderperturbation theory will lead to an effective H XXZ model.

5.4. LOW TEMPERATURE LIMIT 53At first we look at the first order contribution:H t,1 = H 0 = ∑ (ij)σt σ (1 − ¯n j−σ− ¯n i−σ+ 2¯n i−σ¯n j−σ)d † iσ d jσ, (5.50)where the notation (ij) signifies that i and j are nearest neighbors, so that in the sum over(ij) each bond occurs twice. Because each site is singly occupied, the first order contributionvanishes: Either (for fixed i,j) both spins are parallel, and then hopping is forbidden becauseof the Pauli exclusion principle, or they are antiparallel, and then the hopping is suppressedby the (1 − ¯n j−σ− ¯n i−σ+ 2¯n i−σ¯n j−σ) term, since hopping would cause an increase of thedouble occupancies. Every term of the sum is therefore equal to 0.Also most of the second order contributions vanish, because they have combinations ofoperators, which are only able to create or annihilate doubly occupied sites:- H t2,1 = 0 because of the occurrence of ¯n i+α,−σd † i+α,σ- H t2,3 = 0 because of the occurrence of d † i,σ d† i,−σ- H t2,4 = 0 because of the occurrence of d i,σd i,−σ- H t2,5 = 0 because of the occurrence of ¯n i+β,σd † i+β,−σ.Hence the only second order contributions arise from H t2,2 and H t2,6 . These terms can bedrastically simplified, since each site is assumed to be singly occupied. Terms with α ≠ βare nonzero only if double occupancies exist. Additionally using that in our subspace eachsite is singly occupied with one fermion we have the identity:¯n iσ= 1 − ¯n i−σ. (5.51)With the help of (5.51) we can simplify the remaining contributions as:H t2,2 = − ∑ iασt 2 σ¯n i,−σ¯n i+α,σ(5.52)and (by again using the single occupancy of each site):∑H t2,6 = −t ↑ t ↓ d † i+α,σ d i,σ d† i,−σ d i+α,−σ. (5.53)iασThus H t2,2 describes a nearest neighbor spin interaction, while H t2,6 describes nearest neighborspin exchange.We will now show that the second order contribution can be rewritten as an effectiveanisotropic Heisenberg Hamiltonian. Because the number operators ¯n i,σcommute, we canrewrite H t2,2 by symmetrizing the sum as:H t2,2 = − 1 2 (t2 ↑ + t2 ↓ )∑ iασ¯n i,−σ¯n i+α,σ. (5.54)We introduce:( d† )d † i = i↑d † i↓; d i =( di↑d i↓)(5.55)

54 CHAPTER 5. STRONG COUPLING LIMITand the Pauli matrices:( ) 0 1σ 1 =1 0; σ 2 =( 0 i−i 0); σ 3 =( 1 00 −1), (5.56)so that we can write the single-site spin-operators ( = 1) as:S i,ν = 1 2 d† i · σ ν · d i; ν ∈ {1,2,3} , (5.57)where ν denotes the spatial component of S i . With the help of these tools we can rewrite oursecond order contributions as:H t2,2 = (t 2 ↑ + t2 ↓{S )∑ i,3 S j,3 − 1 }4 (d† i · d i )(d† j · d j ) (5.58)(ij)H t2,6∑}= 2t ↑ t ↓{S i,1 S j,1 + S i,2 S j,2(ij)(5.59)by using that spin-flip terms may be expressed by linear combinations of S i,1 and S i,2 :where S i,± are the spin-raising/lowering operators.S i,± = S i,1 ± iS i,2 , (5.60)The term (d † i · d i )(d† j · d j) can be neglegted, since at one fermion per site it is a constantand hence physically inactive. In hypercubic lattices it reduces to:∑(d † i · d i )(d† j · d j) = 2dN , (5.61)(ij)where d is the dimensionality and N is the total number of lattice sites. Finally we are ableto write the effective Hamiltonian as:H = H U + 1 U{H t2,2 + H t2,6}= 4t ↑t ↓U∑〈ij〉S i · S j + 2(t ↑ − t ↓ ) 2U∑S i3 S j3 , (5.62)since H U gives no contribution because of the absence of doubly occupied states. Note thatin this notation each bond contributes only once in contrast to the“(ij)”-notation. As we see,we have derived a antiferromagnetic H XXZ model with the constraint J Z ≥ J X . The model isable to reproduce the correct isotropic Hubbard model limit (t ↑ = t ↓ ) and the Falicov-Kimballlimit (t ↑ = 0, t ↓ > 0 or t ↑ > 0, t ↓ = 0). In the first case we obtain an isotropic Heisenbergmodel:H Hub → 4t2 ∑S i · S j (5.63)Uand in the second case we obtain an Ising model:H Fal → 2t2U〈ij〉〈ij〉∑S i,3 S j,3 . (5.64)In one dimension these effective models correspond to the results presented in [35].〈ij〉

5.4. LOW TEMPERATURE LIMIT 555.4.2 Attractive U model with even number of fermionsWe will now show that at negative U our model can be mapped onto a hard-core boson model.At attractive U the ground state is the state with the maximal number of double occupancies.With an even number of fermions, that means all fermions are paired. In 0 th order the groundstate is highly degenerated, the position of the paired fermions has no influence on the energy.Again the first order contribution vanishes, since hopping of one fermion immediatelydecreases the number of doubly occupied states.In second order we again have to consider the role of H t2,1 - H t2,6 . Most of these termsvanish again:- H t2,2 = 0 because of the occurrence of (1 − ¯n i+β,−σ)d † i+β,σ- H t2,3 = 0 because of the occurrence of ¯n i+β,σd i+β,−σ- H t2,5 = 0 because of the occurrence of ¯n i+α,−σd i+α,σ- H t2,6 = 0 because of the occurrence of (1 − ¯n i+α,−σ)d † i+α,σ.These combinations of operators in these terms have an output that involves only singlyoccupied states. Again we have only two terms which are nonzero and these terms alsocontribute only for α = β. At first we will introduce bosonic creation and annihilationoperators:b † i = d† i↑ d† i↓; b i= d i↓d i↑(5.65)with the standard bosonic commutation realations:and the “hard-core” properties:[b † i ,b† j ] = 0 ; [b i ,b j ] = 0 (5.66)(b † i )2 = 0 ; (b i) 2 = 0 . (5.67)In the completely doubly occupied subspace we can identify the fermionic counting operatorswith the bosonic counting operators:With these definitions we obtain:¯n i↑≡ ¯n i↓≡ ñ i, ñ i= b † i b i. (5.68)H t2,1 = (t 2 ↑ + t2 ↓ )∑ (ij)(1 − ñ i)ñ j(5.69)∑H t2,4 = 2t ↑ t ↓ b † j b i. (5.70)We can additionally simplify H t2,1 by using:∑ñ i= Z n tot , (5.71)(ij)(ij)

56 CHAPTER 5. STRONG COUPLING LIMITwhere n tot is the total number of hard-core bosons and Z is the number of nearest neighborsper site (Z = 2d for hypercubic lattices). Putting these results into (5.35), and neglectingthe constant contributions H U and the one arising from (5.71), we obtain:H = (t2 ↑ + t2 ↓ )|U|∑(ij)ñ i ñ j− 2t ↑t ↓|U|∑b † j b i, (5.72)so that we have mapped our model onto a hard-core boson model with repulsive nearestneighbor interaction with the strength Ũ = (t2 ↑ +t2 ↓ )|U|and nearest neighbor hopping with thestrength ˜t = 2t ↑t ↓|U|. The parameters always satisfy Ũ ≥ ˜t. In the isotropic Hubbard limit(t ↑ = t ↓ ) we have Ũ = ˜t, and in the Falicov-Kimball limit (t ↑ = 0 or t ↓ = 0) hopping iscompletely suppressed (˜t = 0).(ij)

Summary and OutlookIn this thesis we have analyzed different forms of asymmetric Hubbard models whichcan be used to describe ultracold asymmetric Fermi-mixtures. We have diagonalized theHartree-Fock Hamiltonian (U < 0) for superfluid phases for both asymmetric hopping andspin-dependent chemical potentials and we have derived the self-consistency equations.At symmetric hopping we have rigorously shown, which solutions the self-consistencyequations may have, and we have implemented an algorithm, which respects these properties,in order to solve the self-consistency equations. Within the local density approximation(LDA) we have shown that, at T = 0 in unbalanced Fermi-mixtures in a parabolic potential,phase separation takes place. A superfluid non-magnetized phase and a non-superfluidmagnetized phase were found. The third phase predicted as possible by our analytical calculationsnever minimized the grand potential and was therefore not thermodynamically stableat T = 0.For asymmetric hopping we presented arguments, why CDW phases in translationallyinvariant systems can be neglected, and we restricted consideration to superfluid phases. AtT = 0 and away from half filling we found both magnetized and non-magnetized superfluidphases. The critical temperatures at half filling were calculated numerically as a function ofthe hopping asymmetry.Finally we analyzed the Hubbard Hamiltonian with spin-dependent hopping at strong couplingfor both attractive and repulsive interaction. As a result of second order perturbationtheory we obtained effective models for the strong coupling limit. We derived ab anisotropicH XXZ -Heisenberg model for repulsive interaction at half filling, which reduces to the isotropicHeisenberg model in the Hubbard model limit and to an Ising model in the Falicov-Kimballmodel limit. For the attractive model at strong coupling with an even number of fermions,we derived a hard-core boson model with nearest-neighbor hopping and nearest-neighbor repulsiveinteraction for the bosons.In this thesis we performed numerical calculations in the weak coupling limit at Hartree-Focklevel, and we derived effective models in the strong coupling limit. Since, forultracold quantum gases, experimentalists are able to tune the relative interaction strength,weak coupling limit calculations may be very useful. Yet, on the other hand, it is importantto understand the physics beyond the weak coupling limit, so that the application ofnon-perturbative methods, for example DMFT or QMC, to asymmetrical Hubbard models isindispensible for reaching the intermediate interaction regime.In order to treat the trapping potential we used the LDA. Instead of using the LDAformalism,it is possible to treat the parabolic potential exactly by restricting to finite-sizelattices, where the size of the lattices may be estimated by the LDA-results. This methodis especially useful for situations in which the LDA breaks down, for small particle numbersor for a parabolic potential that varies fast in space. The exact treatment of the parabolicpotential in combination with non-perturbative treatment of the interaction is also of greatinterest, of course.57

In the case of spin-dependent hopping we have presented arguments for the neglegt ofCDW phases. In order to complete this argument, a proof that this phase is thermodynamicallyunstable away from half filling could be added.In the future, multiband Hubbard models and dissipative multiband Hubbard modelscould become realizable in ultracold quantum gases, and therefore methods like DMFT+QMC+ 1 ω[27] could be used to analyze these systems.58

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DanksagungAn erster Stelle möchte ich mich bei Herrn Prof. Peter van Dongen für die Betreuung meinerDiplomarbeit im Laufe des letzten Jahres bedanken. Er war jederzeit ausgesprochen hilfsbereitund gewährte mir stets die Freiheit meine eigenen Ideen zu verwirklichen.Bei Herrn Prof. Martin Reuter bedanke ich mich für die zahlreichen Unterhaltungen undinsbesondere für die Ratschläge während der Diplomprüfungszeit.Ich bedanke mich bei Prof. Nils Blümer, der mir bei IT-Problemen stets weitergeholfen hat.Außerdem bedanke ich mich bei Eberhard Jakobi vor allem für die interessanten und inspirierendenDiskussionen, die wir im Laufe der Zeit geführt haben.Frau Helf danke ich dafür, dass sie keine Mühe davor gescheut hat mir bei organisatorischenAngelegenheiten weiterzuhelfen.Ich möchte mich bei meinen Eltern, Elisabeth und Thomas bedanken, die mir ein sorgenfreiesStudium ermöglichten und nicht zuletzt bei meiner Freundin Regina.

SelbstständigkeitserklärungIch versichere hiemit, dass ich die Arbeit selbstständig verfaßt und außer den in der Arbeitangegebenen keine anderen Hilfsmittel und Quellen benutzt habe.Mainz, den 7. Mai 2007Tobias Gottwald