The frustrated Hubbard model on the Bethe lattice - an investigation using the self-energy functional approach

Transcription

1 The frustrated Hubbard model on the Bethe lattice - an investigation using the self-energy functional approach von Martin Eckstein Diplomarbeit im Fach Physik vorgelegt der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Augsburg Februar 26 angefertigt am Lehrstuhl für Theoretische Physik III - Elektronische Korrelationen und Magnetismus - Institut für Physik der Universität Augsburg bei Prof. Dr. D. Vollhardt Zweitprüfer: Prof. Dr. T. Kopp

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3 Contents 1 Introduction 1 2 Basic theoretical concepts Hubbard model and dynamical mean-field theory The Hubbard model The limit of infinite dimensions Dynamical mean-field theory The Bethe lattice Self-energy-functional theory The Luttinger-Ward functional Derivation of the SFA Relation of SFA to DMFT Stationary points and phases Thermodynamic quantities Numerical implementation Reference system and symmetries Spin-symmetry Particle-hole and spin-symmetry The particle-hole symmetric Néel state Search for stationary points Given chemical potential Fixed density Evaluation of the self-energy functional Numerical effort Analytical calculation of the derivatives Green function and self-energy Lattice contribution to the self-energy functional Results The Mott metal-insulator transition The Mott Hubbard metal-insulator transition in the particle-hole symmetric case The doped paramagnet The paramagnetic phase for t 1 -t 2 hopping Antiferromagnetic phase I

4 II CONTENTS The strong-coupling regime The weak-coupling regime: Hartree-Fock theory Two-site approximation for the antiferromagnetic phase Summary 79 A Numerical evaluation of the trace-log term 8

5 Chapter 1 Introduction Remarkably, the properties of many solids are well explained by theories that either neglect the Coulomb interaction between the electrons or treat it as a time-independent mean field. The conventional classification of insulators and metals in terms of filled and partially filled energy bands is based on this approximation, but also the local density approximation (LDA) of density functional theory (DFT), which can make fairly precise quantitative predictions for many materials, starting from very few and experimentally easily accessible input parameters such as the crystal structure. However, shortly after Bloch and Wilson proposed their energy band theory, Mott pointed out that the electron-electron interaction can make a periodic crystal insulating even if the number of electrons is such that its energy bands would be half-filled. His proposition is now believed to explain the insulating behavior of many materials for which band theory and LDA incorrectly predict a metallic ground state, in particular transition-metal oxides like NiO, MnO, and CoO. In those systems the effect of the Coulomb interaction is more pronounced than in conventional metals like Al, since narrow d-orbitals contribute to the valence band such that electrons are more confined at the lattice sites and spacial fluctuations are suppressed. The electrons in such a Mott insulator are localized at the atoms in the lattice. The excitations that can be made, e.g. by absorption of electromagnetic radiation, create a local charge deficit or excess and require a certain minimum energy to overcome the Coulomb interaction. The typical spectrum of the Mott insulator thus consists of broad peaks well separated from the Fermi energy, which are called the Hubbard bands. Among the many materials which are now classified as Mott insulators there are some examples for which a transition from a metallic phase into the Mott insulator is observed [1, 2]. One of the experimentally and theoretically best studied examples is certainly V 2 O 3. At low temperatures this material is antiferromagnetically ordered and insulating. In the paramagnetic phase above the Néel temperature there is a first-order transition from a metallic phase at high pressure to an insulating phase at low pressure. (High pressure increases the hybridization of the d-orbitals and thus decreases the effective Coulomb interaction as the electrons become less confined in space.) Materials in which the electron-electron interaction is important often have a complicated phase diagram, that can include, e.g., superconducting and magnetically ordered phases. Up to now first-principles approach comparable to LDA do not exist for most of those highly correlated materials. Instead, it is common to study model Hamiltonians that provide a very simplified description in terms of as few as possible relevant degrees 1

6 2 CHAPTER 1. INTRODUCTION of freedom and effective interactions. The minimal model to treat the electron-electron interaction in crystals is the Hubbard model [3]. It takes only one orbital per atom into account. All other orbitals can be imagined to be inaccessibly high in energy or otherwise are completely filled and have no influence on the physics of the valence band apart from modifying the long range Coulomb interaction between the electrons to an effective short range interaction. The model is fully specified by very few parameters which are the number of electrons per lattice site, the Hubbard-U that accounts for the interaction and the hopping amplitudes that determine the overlap between the atomic orbitals and thus the band structure in the noninteracting case. Of course the interactions and degrees of freedom that are relevant for some system are not known from the beginning. In V 2 O 3, e.g., there is a small volume change across the metal-insulator transition. It is thus impossible to assume a static lattice and neglect the electron-phonon interaction if one is aiming for quantitative results. On the other hand, a qualitatively correct phase diagram for a simplified model is useful because such a model is necessarily less specific and hence can account for the properties of a whole class of materials. It is therefore interesting to see if the Hubbard model can reproduce the basic features in the phase diagram of V 2 O 3 that reappear for many other correlated materials, namely some kind of order at low temperatures and a metal-insulator transition in the high-temperature disordered phase. However in spite of all simplifications the Hubbard model can in general not be solved analytically, and numerical result require large computational effort even for very small systems. Some insights into the model can be obtained from the limits of either large or small effective Coulomb interaction. In both limits the ground state is antiferromagnetically ordered and insulating, but further properties and in particular the properties of the paramagnetic phase that is stable at higher temperatures are quite different. While the weakly correlated system is a metal, in the opposite limit a gap in the spectrum remains also above the magnetic transition. The question if the transition from the metallic-like to the insulating-like behavior is via a phase transition like in V 2 O 3 or rather via a smooth crossover cannot be answered in either of the two limits. However, there is a different limit in which the Hubbard model can be treated at least numerically, and this limit is independent of the interaction strength. It is the limit of infinite dimensions d = [4] in which the Hubbard model can be solved with dynamical mean-field theory (DMFT) [5]. In this case the Hubbard model is mapped onto a much simpler single-impurity Anderson model (SIAM) with model parameters that have to be determined self-consistently. While this mapping is only exact in d =, it can be used as an approximation for finite-dimensional systems. For threedimensional systems DMFT together with LDA has been used with success in calculating some material-specific properties [6]. At the same time DMFT is extended to real materials, open questions remain even for the Mott transition in the pure Hubbard model in infinite dimensions. DMFT in fact predicts a first-order metal-insulator transition in the paramagnetic phase, but the paramagnetic phase itself is thermodynamically not stable. More precisely the first-order transition-line ends in a second-order critical end point above which there is only a smooth crossover between metal and insulator, and this critical temperature is much lower than the Néel temperature above which the paramagnetic phase becomes thermodynamically stable. The strong tendency of the Hubbard model towards antiferromagnetism can be

7 related to the special choice of the hopping amplitudes. To simplify the calculations frequently only hopping amplitudes t 1 between nearest neighbors (NN) are kept. However already small hopping amplitudes t 2 between next-nearest neighbors (NNN) might have an appreciable effect on the antiferromagnetic phase. This is most easily understood in the strong-coupling limit, in which the electrons are localized at the atoms. The system can then gain energy by aligning their magnetic moments antiparallel on sites with overlapping orbitals. Long-range antiferromagnetic order meets this requirement on all such pairs if hopping is allowed only between NN, but it is impossible to align the moments antiparallel on all NN and NNN pairs of sites simultaneously. Hence the antiferromagnetic order is no longer energetically favorable if t 2 is large enough, i.e., it is frustrated by NNN hopping. A similar effect has been demonstrated on the hypercubic lattice for weak coupling [7] and for intermediate coupling in the infinite dimensional limit [8], using DMFT and Quantum Monte Carlo (QMC). Model calculations in DMFT are often performed on the Bethe lattice. This is not a regular crystal lattice but is recursively defined. In the limit of infinite coordination number the density of states (DOS) is semielliptic, which leads to simplifications in the DMFT self-consistency equations. Furthermore, in contrast to the DOS of a infinitedimensional hypercubic lattice the DOS on the Bethe lattice has finite bandwidth and similar van-hove singularities as in three dimensions for all coordination numbers. On the other hand the influence of the lattice topology is probably not too profound in infinite dimensions. From a practical point of view it would be desirable if frustration could be introduced in a way that preserves both the simplifications that occur for pure NN hopping, namely particle-hole symmetry at half filling which allows one to specify the chemical potential from the beginning, and those that are provided by the use of the Bethe lattice. This is indeed possible for the so-called fully frustrated two sublattice model (FFTSM) [5, 9], which involves random hopping t 2 within the two antiferromagnetic sublattices of the Bethe lattice and regular hopping t 1 between them. In this model the paramagnetic phase does not change at all by the influence of frustration. A suppression of the antiferromagnetic phase was indeed found in the FFTSM [1] but the Mott transition remains hidden inside the stable antiferromagnetic phase unless artificially large values of t 2 are assumed. It has therefore been concluded in Ref. [1] that the Hubbard model with a single band cannot even qualitatively describe the phase diagram of materials like V 2 O 3. However, in contradiction to previous propositions [5, 11] the FFTSM is not equivalent to nonrandom t 1 -t 2 hopping (NN and NNN hopping) on the Bethe lattice [12]. In fact the DOS does not remain symmetric and the DMFT self-consistency equations [13, 14] are also quite different from the equations for the FFTSM. Neither does particle-hole symmetry hold for half-filling if t 2, nor does the paramagnetic phase remain unchanged. It is thus worthwile to reinvestigate the frustrated single band Hubbard model in infinite dimensions, now using nonrandom t 1 -t 2 hopping. Quantum Monte Carlo simulations for the antiferromagnetic phase at high temperatures indicate a frustration effect [15], but the paramagnetic phase in the non particle-hole symmetric case with t 1 -t 2 hopping has not been investigated so far. Large computational effort is needed if the DMFT equations are solved by methods like numerical renormalization group (NRG) or QMC, especially in the absence of particlehole symmetry. Here we will use the self-energy functional approach (SFA) [16], a recently developed nonperturbative variational theory. In it the self-energy is parameterized by 3

8 4 CHAPTER 1. INTRODUCTION the exact self-energy of a small reference system and the parameters of this system are determined variationally. In the context of DMFT a SIAM with finitely many sites is chosen as reference system, but the SFA is formulated generally enough such that it can be extended beyond DMFT to finite dimensional systems [17], as well as to Hamiltonians with more complex interactions than the Hubbard interaction [18]. Also disordered systems have been treated within SFA [19]. For the infinite-dimensional Hubbard model a reference system with six sites is enough to achieve quantitative agreement with QMC and NRG calculations in the particle-hole symmetric paramagnetic phase [2] and already the simplest reference system with only two sites yields qualitatively correct results [21]. This two-site approximation or two-site dynamical impurity approach will be used here to study the paramagnetic phase in the non particle-hole symmetric case and also the antiferromagnetic phase, which however seems to be less successful. This work is structured as follows: In the next chapter the basic models and methods are described in detail, including a short introduction into the derivation of the Hubbard model, basic facts about the limit of infinite dimensions and DMFT, and the properties of the Bethe lattice, and the SFA. Chapter 3 explains the numerical implementation of the SFA. The results are presented in Chapter 4. All results concern the Hubbard model on the Bethe lattice and are obtained from the two-site approximation to SFA. New results are given for the paramagnetic phase in the absence of particle-hole symmetry, i.e., for NN hopping away from half filling and for t 1 -t 2 hopping. For the antiferromagnetic phase selected Hartree-Fock results are presented. Furthermore the two-site approximation is applied to the antiferromagnetic phase. It yields results with limited reliability but still provides useful insights.

9 Chapter 2 Basic theoretical concepts 2.1 Hubbard model and dynamical mean-field theory The Hubbard model In 1963 Hubbard, Gutzwiller, and Kanamori [3, 22, 23] introduced a highly idealized model for interacting electons on a crystal lattice which should capture important effects of the physics of transition-metal oxides. The Hamiltonian of this model, now called the Hubbard model, is given by H = iσ t ij c iσ c jσ + U i n i n i µ iσ n iσ, (2.1) where the operator c iσ creates an electron with spin σ at site i of the lattice and n iσ = c iσ c iσ is the local number operator. The first term describes noninteracting electrons hopping on a lattice, whereas the term proportional to U accounts for the local interaction between the electrons. Several approximations have to be made to derive this Hamiltonian from the basic many-particle Schrödinger equation of the solid. First of all the lattice is considered to consist of pointlike, immobile nuclei without internal degrees of freedom. Lattice distortions and other effects caused by the electron-phonon interaction are completely neglected. The next approximation concerns the electronic degrees of freedom, which are restricted to one orbital per site. This cannot be justified for most transition-metal compounds since the relevant valence orbitals are five-fold degenerate d-orbitals, and this degeneracy is usually not completely lifted by crystal-fields. A natural extension to (2.1) are thus multiband Hubbard models, which however require large computer power and could therefor only be studied in the recent years. Once these approximations are made the Hamiltonian can be written in the form H = t αβ c ασc βσ + U αβγδ c ασc βσ c δσ c γσ, (2.2) αβ,σ αβγδ σσ where α,..., δ label the single-particle states that contribute to the valence band. The corresponding wave functions will be denoted by ψ α (x),...,ψ δ (x). The hopping amplitudes t αβ and the interaction matrix elements U αβγδ are determined by overlap integrals U αβγδ = d 3 x d 3 x ψ α (x) ψ β (x ) V ee (x, x )ψ δ (x )ψ γ (x). (2.3) 5

10 6 CHAPTER 2. BASIC THEORETICAL CONCEPTS and t αβ = d 3 x ψ α (x) ( h2 2m 2 + V ion (x))ψ β (x), (2.4) where V ion (x) is the lattice potential and V ee (x, x ) the electron-electron interaction. These integrals are generally very difficult to estimate since V ion (x) and V ee (x, x ) are effective potentials that must take into account screening effects by the core-electrons that are not included in (2.2). Hubbard used Wannier states χ i (x) = N 1 2 e Rik φ k (x) (2.5) as a basis, which are constructed from the Bloch wave functions φ k (x) in such a way such that for narrow bands they are are well localized at the lattice sites R i. Under these conditions he showed [3] that the matrix element U U iiii, corresponding to the on-site Coulomb repulsion, dominates all other interaction matrix elements. The Hubbard model Eq. (2.1) is obtained by keeping only U iiii. However the remaining matrix elements that describe e.g. NN density-density interactions n iσ n jσ or correlated hopping n jσ c i σ c j σ are not smaller by orders of magnitude and thus certainly will have some influence on the physics. As a further approximation, frequently only the strongest hopping amplitudes t ij t 1 between nearest neighbors ij are kept. While hopping amplitudes decay rapidly with distance between the sites, the restriction to NN hopping on a bipartite lattice leads to particle-hole symmetry at half filling and thus favors an antiferromagnetically ordered ground state. Inclusion of small NNN hopping amplitudes (t 2 ) removes this symmetry and may thus lead to qualitative changes. The study of this effect is the main issue of this work. The Hubbard model must be considered as a minimal model for systems in which electronic correlations are important, rather than as a model for any real material. It contains both the extreme limit of noninteracting electrons and the atomic limit U =, in which the electrons are completely localized at the lattice sites. The different behavior in those limits suggests that there is a transition from a metal with delocalized electrons to an insulator with localized electrons at some intermediate U. However exact statements can only be made in one dimension [24] and for the case of infinite-range hopping [25]. In both cases the ground state is insulating for all U > and there is no Mott transition at finite interaction strength. However early approximations by Hubbard [26] and the application of the Gutzwiller variational wave function [27] suggest that such a transition exists in the paramagnetic phase. This transition is accessible in detail for the mean-field limit of infinite dimensions (cf. section 2.1.2). The Hubbard model has also various magnetic phases. In contrast to the Mott transition magnetic order can be studied in the limiting cases of large and small interaction. For large interaction and half filling the Hubbard model is equivalent to an antiferromagnetic Heisenberg-model [28] for localized spins (cf. section 4.2.1), whereas for small interaction Hartree-Fock theory can be applied which also results in an antiferromagnetic ground state at half filling for all U > (cf. section 4.2.2). Ferromagnetism exists for one electron less than half filling and U, which has been rigorously established by Nagaoka [29]. k

11 2.1. HUBBARD MODEL AND DYNAMICAL MEAN-FIELD THEORY The limit of infinite dimensions The first successful attempt to explain an ordering phenomenon from a microscopic theory was perhaps the mean-field theory for magnetic systems, put forward by Weiss in 197 (cf. section 4.2.1). While this theory gives incorrect answers to many questions of quantitative nature especially close to the critical point, and predicts a phase transition in dimensions d = 1, 2 where this is excluded by the Mermin-Wagner theorem, it leads to a thermodynamically consistent picture for the phase transition in d = 3 that qualitatively resembles many features of magnetic transitions in real materials. Later it turned out that this theory becomes exact in the limit of infinite dimensions d = (or infinite lattice coordination Z) [3]. The approximation is thus controlled by the parameter (1/Z) in the sense that it becomes more reliable as 1/Z approaches zero, and the success of the theory in d = 3 might indicate that 1/Z is already small in this case. The limit of infinite dimension has been carried over to interacting fermionic lattice systems by Metzner and Vollhardt in 1989 [4], where it again leads to a controlled approximation: the dynamical mean-field theory (DMFT) [5]. In performing this limit for the Hubbard model the hopping matrix elements have to be rescaled in order to preserve the relative strength of kinetic and potential energy. The amplitude t d for hopping between d-th nearest neighbors has to be of the form t d = t d Z d/2 t d = const., (2.6) where it is assumed that the number of d-th nearest neighbors is proportional to Z d in leading order in 1/Z. The rescaled matrix element t d is independent of Z. If this scaling is applied to the hopping Hamiltonian on the hypercubic (hc) lattice (only NN hopping t t 1, Z = 2d) the local DOS approaches a Gaussian, ρ hc (ɛ) d 1 t 2π e (ɛ/t ) 2 /2 (2.7) which has a finite second moment ɛ 2 = t 2. The most important simplification in d = is that the self-energy Σ becomes local in real space or, equivalently, k-independent, i.e., Σ ij,σ (ω) Z δ ij Σ σ (ω) (2.8) Σ σ (k, ω) Σ σ (ω). (2.9) This can be concluded from the diagrammatic perturbation expansion of the self-energy either in real-space [4] or momentum-space [31]. Further simplifications in d = concern, e.g., disordered systems for which the coherent potential approximation becomes exact, or the exact evaluation for the expectation values of the Gutzwiller variational wave function [32]. A review is given in Ref. [33] Dynamical mean-field theory DMFT (for a review see Ref. [5]) reduces the interacting lattice problem to a local problem in which the on-site interaction 1 is kept but the lattice environment is modeled by 1 The discussion in this section applies only to local interactions. For a possible extension of DMFT for nonlocal interactions such as NN density-density interactions (extended DMFT) see Ref. [18].

12 8 CHAPTER 2. BASIC THEORETICAL CONCEPTS the coupling of the local degrees of freedom to a bath of noninteracting electrons. The properties of the bath and the coupling must be determined self-consistently. This local problem, which is known as the single-impurity Anderson model (SIAM), can be modelled by the Hamiltonian H (i) SIAM = Unc n c µ(n c + n c ) + k,σ V (i) k (c σa kσ + h.c.) + (ɛ (i) kσ µ)a kσ a kσ. (2.1) k,σ Here the site index i indicates that for phases that are not translationally invariant the effective local model must depend on the lattice site. In (2.1) the impurity orbital (c σ,c σ) hybridizes with a bath of noninteracting electrons (a kσ,a kσ ) which is described by the last term in the Hamiltonian. In the second term V (i) k gives the hybridization strength between the bath state k and the impurity orbital. On the other, since this is only an auxiliary model, one can integrate out the bath degrees of freedom in the Grassmann variable path integral 2 for the partition function to obtain an effective action A[c σm, c σm ; G iσm ]. Local correlation functions can then be calculated from it by a path integral, G iσ (iω n ) = 1 D[c Z σnc σn ]c σnc σn e A[G iσm,c σm,cσm]. (2.11) The effective action resulting from (2.1) is obtained as A[c σm, c σm ; G iσm ] = nσ β c σ(iω n )Gσnc 1 σ (iω n ) U c (τ)c (τ)c (τ)c (τ)dτ, (2.12) where G 1 iσn = iω n + µ kσ iω n + µ ɛ (i) kσ is the noninteracting Green function of the bath. This second formulation of the local ( V (i) k ) 2 (2.13) problem illustrates the mean-field nature of the theory, since G 1 iσ (iω n) plays precisely the role of am undetermined mean field at site i in Weiss theory. However in the present case this so-called Weiss-field is frequency-dependent, hence dynamic. The same effective action as Eq. (2.12) is obtained when all electronic degrees of freedom (apart from those connected to a single site) are integrated out in the Hubbard model in infinite dimensions [35], which proves the equivalence of the two models. To make use of this equivalence one has to provide a second equation to determine the Weiss field G 1 iσ (iω n) or equivalently the parameters of the effective SIAM. This equation can be obtained by noting that together with functional form (2.12) of the action also the (local) self-energy Σ iσ of the two models must coincide if the Green functions G iσ coincide. Thus the impurity self-energy, which is obtained from the Green function via the Dyson equation G iσ (iω n ) 1 = G 1 iσn Σ iσ(iω n ) (2.14) of the SIAM must at the same time obey the lattice Dyson equation G 1 (iω n ) = G 1 (iω n) Σ(iω n ) = iω n + µ H Σ(iω n ). (2.15) 2 For Grassmann variables and path integrals in many-body physics see, e.g., Ref. [34].

13 2.1. HUBBARD MODEL AND DYNAMICAL MEAN-FIELD THEORY 9 In this expression all boldfont quantities are matrices in single-particle indices i and σ and H is the noninteracting part of the Hubbard Hamiltonian. Eqs. (2.11), (2.14), and (2.15) provide a self-consistent set of equations for the determination of the Green function and the self-energy of the Hubbard model. It is exact only in infinite dimensions because only then is the effective local action of the Hubbard model of the form (2.12), but it can be used as an approximation for finite-dimensional systems. If the self-energy is not only local but also site-independent it just leads to a shift in the chemical potential and (2.15) can be simplified to ρ(ɛ) G iσ (iω n ) = dɛ iω n + µ Σ σ (iω n ) ɛ, (2.16) where ρ(ɛ) is the noninteracting local DOS of the Hubbard model. However if some kind of symmetry breaking is present such that the self-energy does not have full symmetry of the Hamiltonian, the evaluation of the Dyson equation is more involved. The solution of the mean-field equations still involves an interacting many-body problem, namely the SIAM. This can be done by numerical methods like Quantum Monte Carlo simulations (QMC) or numerical renormalization group (NRG) techniques, or by approximate analytical methods like iterated perturbation theory (IPT). Another possible method involves the exact diagonalization of a finite SIAM and is described later (section 2.2.3) in more detail The Bethe lattice Basic properties and application to DMFT The underlying lattice structure for all calculations in this work is the Bethe lattice. This is not a regular Bravais lattice but an infinite tree in the sense of graph theory [36, 37]. Any two of its vertices are connected by a single path and each vertex has the same number of branches Z, as shown in Fig The number n d of sites that can be reached from a given center in d NN steps is then given by Z(Z 1) d 1 and thus increases exponentially with d. As a consequence for any finite portion of the Bethe lattice such as displayed in Fig. 2.1 the surface contains a macroscopic fraction of sites. Hence average physical properties are always dominated by surface effects and a regular thermodynamic limit does not exist. Nevertheless the Bethe lattice is sometimes used in theoretical physics, as some problems involving disorder and/or interactions can be treated exactly when defined on this lattice [38, 39]. Simplifications also occur in the context of DMFT, which becomes exact on the Bethe lattice for Z if the hopping amplitudes are scaled according to Eq. (2.6). We will use the equivalent scaling t d = t d K d/2, t d = const., (2.17) where K = Z 1 is called the connectivity. In DMFT the Bethe lattice is widely used in model calculations because the local DOS of a hopping Hamiltonian with only NN hopping has the shape of a semiellipse 4 ɛ 2 ρ(ɛ) = (2.18) 2π

14 1 CHAPTER 2. BASIC THEORETICAL CONCEPTS Figure 2.1: One way to represent the Bethe lattice for Z = 4. The sizes of the discs at the vertices and the bondlength have no physical meaning. All sites are equivalent. Note that the Bethe lattice is bipartite, i.e., it can be subdivided into two sublattices A and B (marked red and blue) such that all sites adjacent to a site in A are in B and vice versa.

15 2.1. HUBBARD MODEL AND DYNAMICAL MEAN-FIELD THEORY 11 (t 1 has been set to one). This special form allows an explicit solution and inversion of the integrated Dyson equation (2.16) such that together with (2.14) the selfconsistency equation can be written in closed form: G iσ (iω n ) 1 = iω n + µ G iσ (iω n ) 1. (2.19) Furthermore the DOS at the band-edges ɛ = ±2t 1 resembles the three-dimensional case and one is not confronted with the peculiarity of an unbounded spectrum as for the hypercubic DOS [Eq. (2.7)]. A closed form of the self-consistency equations can also be derived for the more general case of NN (t 1 ) and NNN (t 2 ) hopping and a self-energy that depends on spin σ and sublattice γ = A, B of the bipartite Bethe lattice [13, 14]. It is then by γσn = iω n + µ + h γσ t 2 1 G γσn(1 t 2G γσn) (1 t 2 γ G γ σn) 2 t 2 2 G γσn 1 t, (2.2) 2 G γσn G 1 where h γσ is an additional sublattice and spin dependend external field in the Hamiltonian. The proof of this equation and also of Eq. (2.18) must proceed by different methods than it would on regular lattices, since a k-space cannot be defined on the Bethe lattice. This is because the treelike structure of the Bethe lattice does not allow an embedding as a regular lattice in finite dimensional Euclidean space. 3 Various methods such as the renormalized perturbation expansion (RPA) [41] or a mean-field like approach [39] have been developed to solve the problem for pure NN hopping. These methods can make use of the absence of any closed loops on the Bethe lattice if only NN hopping is allowed, but their application becomes more involved for t 2 [14]. However a special topological property of the Bethe lattice [13] makes it particularly suitable for the study of quite general hopping Hamiltonians among which t 1 -t 2 hopping is only one example. This property and its consequences will be explained in the following paragraph, which closely follows Ref. [13]. Hopping on the Bethe lattice On the Bethe lattice all hopping Hamiltonians of the general form H hopp = d t d H d, (2.21) where H d = ij,d i,j =d i j (2.22) describes hopping only between sites that are separated by a distance d i,j = d, can be treated on the same footing. 4 The topological basis for the following derivation is the so-called distance-regularity of the Bethe lattice which means that the number of path that connect two sites i and j by a sequence of n NN steps depends only on the distance 3 The Bethe lattice can be mapped onto a regular tiling of two-dimensional hyperbolic space [4]. However so far not much use has been made of this observation. 4 Distance denotes the topological distance, defined as the minimum number of NN steps between two sites.

16 12 CHAPTER 2. BASIC THEORETICAL CONCEPTS between those points. This property is not shared by regular lattices. For example, on the square lattice two sites that are two NN steps apart are connected via two paths if they lie on opposite vertices of a plaquette but only by one if they lie on the same axis. The distance-regularity implies that all powers of the NN hopping Hamiltonian H 1 can be expressed as a linear combination of the H d (H 1 ) n = n a (d) n H d. (2.23) This is most easily proved by comparing matrix elements j... i on both sides of the equation. The matrix element on the right side gives a (d i,j) n by definition, while on the left side one has (H 1 ) n ij. It is well known in graph theory (and can be proven directly by writing out the matrix multiplication) that for any lattice this quantity is equal to the number of paths of length n connecting sites i and j. But this number depends only on the distance d i,j for the Bethe lattice, and thus (2.23) is correct if a (d) n is taken to be this number. One can now go one step further and view (2.23) as a set of equations for the hopping Hamiltonians H d. The definition of the coefficients a n (d) implies that this is a triangular system of equations (a n (d) = for d > n, a (n) n = 1) which can be solved iteratively to express each H d as polynomials of degree d in H 1. The first few coefficients a (1) 1 = 1, a (2) 2 = 1, a (1) 2 =, a () 2 = Z,... can be figured out by simply counting the paths, leading to H 2 = H 2 1 Z (2.24) H 3 = H 3 1 (2Z 1)H 1. (2.25) The full solution of Eq. (2.23) can be obtained by first deriving a recursive relation for the coefficients a n (d), which can then be solved [13]. The final result is most conveniently expressed in terms of the generating function for the hopping Hamiltonians H d, given by H d x d = d= 1 x 2 1 xh 1 + Kx2. (2.26) As a consequence all types of isotropic hopping (2.21) on the Bethe lattice can be written as a function of the NN hopping Hamiltonian H 1, H hopp = d t d H d ɛ(h 1 ). (2.27) In analogy with regular lattices where any translational invariant hopping Hamiltonian is a function of momentum, the function ɛ(h 1 ) will be referred to as the dispersion belonging to a given set of hopping parameters. Two dispersions are important in this work, namely the cases of t 1 -t 2 hopping H hopp = t 1 H 1 + t 2 H 2 = t 2 (H 1 ) 2 + t 1 H 1 Zt 2 (2.28) K t 2 ( H1 ) 2 + t 1 H1 t 2 (2.29)

17 2.1. HUBBARD MODEL AND DYNAMICAL MEAN-FIELD THEORY 13.8 t ρ t 1,t 2 (ω) t 2 = t 2 =1/9 t 1 t 2 =3/7 t 1 t 1 = Figure 2.2: The DOS ρ t (ω) for t 1-t 1,t 2 hopping on the Bethe lattice (cf. Ref. [13]) (t = 2 t t 2 2 ). For t 2 /t 1 > 1/4 there is a square-root singularity at the lower band edge. In the plot its position is indicated by an arrow. The DOS for t 2 /t 1 < can be obtained by employing the symmetry ρ t (ω) = 1, t ρ 2 t ( ω), that follows from the dispersion (2.29) 1,t 2 and Eq. (2.32). ω/t and of hopping with exponentially decreasing hopping matrix elements [13] H hopp = t w d 1 Hd = d=1 K t ( 1 w 2 ) /K w 1 w H 1 + w 1 2 ( ) t 1 w 1 w H 1 + w 1 2 (2.3) (2.31) Here H d = H d /K d/2 such that the spectrum of H 1 is given by Eq. (2.18) for K. In the latter case the parameter w < 1 determines the decay of the hopping matrix elements. (Pure t 1 hopping corresponds to w =.) The noninteracting DOS is obtained from the dispersion ɛ(λ) and the DOS (2.18) for H 1 in the usual way by evaluating ρ(ɛ) = λ 2 dλ δ[ɛ ɛ(λ)]. (2.32) 2π This is done in Ref. [13] for various dispersions. In Fig. 2.2 we plot the DOS ρ t (ɛ) for 1,t 2 t 1 -t 2 hopping for various values of t 1 and t 2 (in the limit Z ). It is given explicitly by [13] ρ Θ[t 2 t (ɛ) = 1 + 4t 2 (t 2 + ɛ)] 2 4 λ i (ɛ) 2 (2.33) 1,t 2 t t 2 (t 2 + ɛ) 2π i=1

18 14 CHAPTER 2. BASIC THEORETICAL CONCEPTS where λ 1,2(ɛ) = t 1 ± t t 2 (t 2 + ɛ) 2t (2.34) 2 are the solutions of the quadratic equation ɛ(λ) = ɛ where the dispersion (2.29) is used for ɛ(λ). 2.2 Self-energy-functional theory The formulation of a theory in terms of some variational principle often allows to construct approximations in a systematic way either by a simplification of the variational functional or by a restriction of the variational space. Many approaches in solid state physics, among them the Hartree-Fock approximation, DMFT, and density-functional theory, fit into this scheme [42]. In this section the variational principle due to Luttinger and Ward [43] and a derived approximation, the self-energy functional approach (SFA) [16] will be discussed The Luttinger-Ward functional The Luttinger-Ward functional 5 ˆΦ[G] was first constructed for a many-body Hamiltonian with general two-particle interactions of the form H int = αβγδ U αβγδ c αc β c δc γ (2.35) in 196 by Luttinger and Ward [43] in order to derive a diagrammatic expansion of the thermodynamic potential in terms of the full interacting Green function G. By definition it is a functional of G only and it can be characterized by the following three properties [44]. (i) The functional derivative of ˆΦ[G] yields another functional ˆΣ[G] = δˆφ[g] δg, (2.36) which takes the value of the physical self-energy Σ phys = G 1 G 1 phys when the physical Green function G phys is inserted. Here G is the noninteracting Green function. (ii) The thermodynamic potential Ω is obtained from ˆΦ[G] by evaluating the expression βω = ˆΦ[G] + Tr log[ G] Tr(ΣG) (2.37) at the physical values of G and Σ. With help of the Dyson equation Σ = G 1 G 1 (2.38) expression (2.37) can be viewed as a functional ˆΩ[G] of G only (G is then considered as a fixed parameter), which is stationary at the physical Green function, i.e. δˆω[g] =. (2.39) δg Gphys 5 Throughout this section functionals will be denoted by a hat and quantities printed in bold font are matrices in single particle states and Matsubara frequencies (or imaginary time slices).

19 2.2. SELF-ENERGY-FUNCTIONAL THEORY 15 (iii) The functional ˆΦ[G] is universal, i.e. its functional form is the same for all Hamiltonians with have the same (local or nonlocal) two-particle interaction (2.35) and does not depend on the hopping part of the Hamiltonian. Note that this implies that also the functional Σ[G] ˆ is universal. Figure 2.3: The first terms of a diagrammatic expansion of ˆΦ[G]. Lines represent the full Green function G. The derivation of self-energy functional theory depends only on those properties. However a mathematical proof for the existence of the functional ˆΦ[G] is highly nontrivial [44]. The first construction was given in terms of a diagrammatic expansion [43], where ˆΦ[G] is defined as the sum over all skeleton diagrams of the grand potential, i.e. all diagrams without self-energy insertions, in which the noninteracting Green function is replaced by the full propagator (Fig. 2.3). Properties (i) to (iii) are then readily verified: The universality (iii) holds by definition since the integral expressions for the diagrams depend only on the interaction part of the Hamiltonian. The derivative of each diagram with respect to one of the propagator lines corresponds to opening up this line. Keeping all symmetry factors for the diagrams, one finds that derivative of the series yields the skeleton expansion for the self-energy (Fig. 2.4). The last point (ii) requires some calculation, which will not be presented here (see e.g. [43, 45]). A proof that the skeleton expansion for the self-energy or the grand potential converges is not available in most cases. However, there exists a different, nonperturbative derivation of the Luttinger-Ward functional [44] using the path-integral formulation, which can also be used as starting point for generalizations of the functional [18]. Up to now the Luttinger-Ward functional has been used mostly in formal derivations rather than for explicit calculations, since its functional form is generally not known. The Luttinger theorem [43], which states that for a translationally invariant Fermi liquid the volume inside the Fermi surface is unchanged by interactions, can be derived by assuming the existence of this functional. Furthermore, approximations that preserve the conservation laws for energy, momentum, angular momentum and particle number, so-called conserving approximations [46, 47], can be derived from the Luttinger-Ward functional. Figure 2.4: Skeleton expansion ˆΣ[G] for the self-energy. The diagrams are obtained by cutting in the series for ˆΦ[G] (Fig. 2.3). Not all diagrams corresponding to the derivatives of the terms in Fig. 2.3 are shown.

20 16 CHAPTER 2. BASIC THEORETICAL CONCEPTS Derivation of the SFA To construct the self-energy functional approach (SFA) [16] from the variational principle Eq. (2.39) one starts with the Legendre transform of the Luttinger-Ward functional to obtain a functional of Σ only, ˆF[Σ] = ˆΦ ] ] [Ĝ[Σ] Tr [Ĝ[Σ]Σ. (2.4) The functional Ĝ[Σ] is the inverse of ˆΣ[G], defined by Eq. (2.36). As ˆΣ[G] and ˆΦ[G] are universal, the same also holds for ˆF[Σ]. Furthermore, one has the property δˆf[σ] δσ = Ĝ[Σ] (2.41) in analogy to Eq. (2.36). Starting from Eq. (2.37) one can now define a functional βˆω[σ] = ˆF[Σ] [ + Tr log (G 1 Σ) 1], (2.42) which has then the stationary property δˆω[σ] δσ =. (2.43) Σphys and ˆΩ[Σ = Σ phys ] = Ω phys. (2.44) The SFA is a direct application of this variational principle. No approximation is made to the functional ˆΩ[Σ] defined by Eq. (2.42) but instead its stationary points are determined only in a restricted set of self-energies. This is very similar to the standard implementation of the Ritz variational principle, where the approximate ground state is determined by minimizing the exact energy expectation value E(α) = ψ α H ψ α / ψ α ψ α not in full Hilbert space but only among a certain class of variational wave functions ψ α (that are parameterized α). Two problems have to be overcome to make use of the variational principle (2.43). First of all it is not intuitively clear how to to write down a good ansatz for the self-energy directly, and secondly the functional form of ˆF[Σ] is generally not known and thus it cannot be evaluated for arbitrary self-energies. In the SFA both problems are solved simultaneously. The variational self-energy is chosen to be the exact self-energy of a so-called reference Hamiltonian, which is much simpler then the original Hamiltonian but has the same functional ˆF[Σ]. Due to the universality of this functional one has considerable freedom in choosing the reference system. This allows to leave some parameters as free and optimize them according to Eq. (2.43). If the reference Hamiltonian is simple enough to allow a full diagonalization the functional ˆF[Σ] can then be evaluated by taking the difference [ ˆF[Σ ] = Ω Tr log (G 1 Σ ) 1], (2.45) where Σ and Ω are the exact (physical) quantities of the reference system and G is its noninteracting Green function. The value of the functional ˆΩ[Σ] for the original system at this self energy Σ = Σ is then given by [ ˆΩ[Σ] = Ω Tr log (G 1 Σ ) 1] [ + Tr log (G 1 Σ ) 1] (2.46)

21 2.2. SELF-ENERGY-FUNCTIONAL THEORY 17 and can be calculated exactly. Note that the properties of the original lattice enter only via the third term, [ ˆΩ latt β 1 Tr log (G 1 Σ) 1] (2.47) which will be referred to as the lattice contribution to the self-energy functional. The reference system The choice of the reference system determines the variational ansatz for the self-energy. In order to have the same functional ˆF[Σ] its Hamiltonian must have the same interaction part as the original system but the hopping matrix elements and the on-site energies are not fixed. One is only restricted by the fact that the reference system has to be simple enough to allow a full diagonalization. Since it is known that the self-energy of the infinite-dimensional Hubbard model can be identified with the self-energy of a SIAM (cf. section 2.1.3) the natural choice for a reference system in this case is a SIAM with finitely many bath sites (cf. Fig. 2.5). Note that the addition of uncorrelated bath sites leaves the interaction part of the Hamiltonian unchanged and thus does not change the Luttinger-Ward functional even though the Hilbert space is increased. The surprising fact is that already one bath site is enough to reproduce the basic phase diagram of the Mott transition in the paramagnetic phase [21], and three are sufficient to achieve a good quantitative agreement with the results of NRG and QMC [2]. Figure 2.5: Schematic representation of possible reference systems for the the Hubbard model on a cubic lattice. Disconnected finite SIAMs correspond to an approximation to DMFT equations (left), finite clusters of the original lattice system are used in VCA (right). Finite dimensional systems can be studied by using finite clusters of the original system and optionally additional bath sites as reference system [17]. This variational cluster approach (VCA) has the advantage that important relations of the self-energy like the causality are fulfilled by construction. Although the functional (2.46) was derived for general two-particle interactions of the form (2.35), it is of little use in the case of nonlocal interactions such as density-density

22 18 CHAPTER 2. BASIC THEORETICAL CONCEPTS interactions V = i,j V ij n i n j, (2.48) because in this case no reference system with the same interaction as the lattice system can be separated into disconnected small clusters by a any choice of the hopping. To treat those interactions a generalization of the variational principle (2.39) has been proposed [18], which uses some two-particle Π correlation function in addition to the Green function G and as variables. In the case of density-density interactions this would be Π ij (τ, τ ) = T τ n i (τ)n j (τ ). Starting from this generalized variational principle an approximation is constructed in analogy to SFA, in which again the relevant variables, which are the selfenergy and the irreducible part of Π, can be parameterized by simple reference systems containing clusters of the original system and fermionic or bosonic bath sites Relation of SFA to DMFT As only the infinitely coordinated Bethe lattice is studied in this work, the only relevant reference system is the SIAM. The notation for the SIAM reference system used in the following is H = Un 1 n 1 + σ n s l=1(ɛ lσ µ)n lσ + σ n s V lσ (c lσ c 1σ + h.c.), (2.49) l=2 where the correlated impurity site is labelled by 1 and sites l = 2...n s are bath sites. In analogy to to Eq. (2.13) the noninteracting Green function for this model is with the hybridization function G 1,1σ (ω) = ω + µ ɛ 1σ σ (ω), (2.5) σ (ω) = N l=2 V 2 lσ ω + µ ɛ lσ. (2.51) We now show that in the limit of infinitely many bath sites n s the SFA with this reference system becomes indeed equivalent to DMFT [21]. We start by writing down the Euler equation ˆΩ[Σ(x)]/ x i = for the stationary points of ˆΩ[Σ], where x = (x 1...x N ) are the parameters of the reference system. In doing this one can use Eq. (2.41) to arrive at the expression ˆΩ[Σ(x)] x i = Tr [ δ ˆF[Σ] Σ x i ] [ + Tr (G 1 Σ) δσ [ ( = Tr (G 1 Σ) 1 G[Σ] ) Σ x i ] 1 Σ x i ]. (2.52) This is zero if Σ is a stationary point. The DMFT self-consistency equation however requires that the local matrix elements of the expression under the trace vanishes as ( ) (G 1 Σ) 1 G[Σ] =. (2.53) iσ,iσ

23 2.2. SELF-ENERGY-FUNCTIONAL THEORY 19 Eq. (2.52) is clearly a projection of this condition into some subspace, and only if the parameterization of Σ by x spans all possible local self-energies Eq. (2.53) can be concluded from the vanishing of (2.52). In its application to DMFT the SFA is very similar to the ED method [5] that has been used previously to solve the DMFT equations. Both methods parameterize the self-energy by a finite SIAM such that the self-consistency is fullfilled approximately. However in ED the parameters of the finite SIAM are optimized by minimizing some ad hoc error η that can be defined, e.g., by comparing noninteracting bath Green function G 1,1σ with the Weiss field G1σ 1 obtained from the self-consistency equation (using the full bath Green function) on some Matsubara frequencies as [5] η = n max n= G 1,1σ (iω n) G 1 1σ (iω n) 2. (2.54) Compared to SFA this has the practical advantage that only a minimization has to be done instead of finding the stationary points and furthermore the evaluation the error η is much simpler that the evaluation of the trace-log terms in Eq. (2.42) (cf. section 3.3.4). However the SFA has proven to converge faster with the number of bath sites [2] and the value of its error functional ˆΩ has a physical meaning at the stationary point that makes it more appropriate to address thermodynamic questions Stationary points and phases The stationary points of the functional (2.42) correspond to physical self-energies and the value of the functional at these points is the grand potential in the corresponding phases. After restricting the variational space, there arise two obvious questions: (i) Is there more than one stationary point of the functional, and (ii), if this is the case, how can one decide which of them corresponds to the best approximation to the thermodynamic potential? These questions are addressed in Ref. [48] and the answers are shortly summarized in this section. Trivial stationary points In SFA one ideally attempts to approach the exact solution by considering a sequence of reference systems with increasing complexity that in the end recovers the full solution. In the context of DMFT one tries to improve the solution by adding more and more uncorrelated bath sites to the reference system, whereas in VCA one observes how the solution changes as the cluster size n c is increased. So one generally is dealing with reference systems that consist of at least two subsystems A and B and a reference Hamiltonian H = H A (x A) + H B (x B) + H AB (v), where H A(B) act on the subsystem A(B) only and depend on variational parameters x A(B) whereas H AB couples the two subsystems and depends on some parameters v. A main result of Ref. [48] is that whenever x A and x B are stationary points if only subsystem A ar B is used as reference system ( x A, x B, v = ) is a stationary point for the more complex reference system. In short, no stationary point is lost when going to a more complex reference system. Stationary points in parts of the variational space in which the reference system decouples into simpler systems are termed trivial stationary points. Assuming that for each