Towards exploiting non-abelian symmetries in the Dynamical Mean-Field Theory using the Numerical Renormalization Group

Transcription

1 master s sis chair of oretical solid state physics faculty of physics ludwig-maximilians-universität münchen Towards exploiting non-abelian symmetries in Dynamical Mean-Field Theory using Numerical Renormalization Group Katharina Maria Stadler March 8, 3 Supervisor: Prof. Dr. Jan von Delft

2

3 masterarbeit lehrstuhl für oretische festkörperphysik fakultät für physik ludwig-maximilians-universität münchen Zur Ausnutzung nicht-abelscher Symmetrien in der Dynamischen Molekularfeldorie unter Anwendung der Numerischen Renormalisierungsgruppe Katharina Maria Stadler 8. März 3 Betreuer: Prof. Dr. Jan von Delft

4 Name Studienrichtung Katharina Stadler Master Physik

5 Erklärung Hiermit erkläre ich, die vorliegende Arbeit selbstständig verfasst zu haben und keine anderen als die in der Arbeit angegebenen Quellen und Hilfsmittel benutzt zu haben. München, 8. März Katharina Stadler

6

7 Acknowledgement I would like to thank Jan von Delft for opportunity to write my master sis in his group. I could gar a great deal of experience during last year. In particular, trip to Paris was a true enrichment. In this regard, I thank Michel Ferrero and group of Antoine Georges for three great and informative days in Ecole Polytechnique and for providing QMC data. My special thanks goes to Andreas Weichselbaum, who is a deeply committed and patient supervisor. I really learned a lot in numerous intensive discussions. In particular, I would also like to thank Markus Hanl, whom I could ask so many questions and who explained all basics to me. A big thank you goes to my roommate Katharina Eissing for a really great time with a lot of coffee, tea and conversations. And last but not least, I thank all group members and our secretary, Stéphane Schoonover.

8

9 Contents Introduction Dynamical Mean-Field Theory (DMFT) 5. Basic idea of DMFT Model Hamiltonians The Hubbard model Model Hamiltonians from local density approach (LDA) Derivation of DMFT equations Local nature of self-energy in infinite dimensions On-site lattice Green s function Non-interacting lattice density of states Mapping onto an effective quantum impurity model The self-consistency procedure Numerical Renormalization Group (NRG) 3. Basic method Single impurity Anderson Hamiltonian (SIAM) Logarithmic discretization Mapping onto Wilson tight-binding chain Iterative renormalization group procedure Renormalization group flow NRG in Matrix Product State (MPS) formulation Basis states in terms of MPS Abelian and non-abelian symmetries Complete basis sets and full density matrix NRG Calculation of spectral functions Discarded weight in NRG Combining DMFT and NRG Calculation of self-energy The logarithmic discretization grid within DMFT Variation of conventional logarithmic discretization grid Zitko s adaptive logarithmic discretization grid One-band Hubbard model The Mott-Hubbard metal-insulator transition (MIT) Fermi-liquid ory V

10 CONTENTS 5.3 Fermi liquid physics and resilient quasiparticles Two-band Hubbard-Kanamori model Realistic modeling of strongly correlated materials The DMFT equations for multi-band models Hund s coupling in two-band Hubbard-Kanamori model The two-band Hubbard-Kanamori Hamiltonian in context of DMFT and NRG The effect of Hund s coupling on metal-insulator transition (MIT) Comparison of NRG and Quantum Monte Carlo results Conclusion and Outlook 9 Appendix 93 A Equation of motion for retarded Green s function 93 B G latt (ω) for a multi-band model 95 C G imp (ω) and Σ(ω) for a multi-band model 97 D G latt and non-interacting DOS for Be lattice Bibliography 5 VI

11 Chapter Introduction The realistic description of strongly correlated materials, especially in three dimensions, is a huge challenge in condensed-matter physics. As strength of electron-electron interactions is comparable to or larger than kinetic energy, perturbative methods cannot be applied to se materials. Strongly correlated materials typically exhibit partially filled d- or f-shells which are associated with narrow bandwidths of only a few electron-volts. Examples are transition metals and ir oxides (especially 3d-shell from Ti to Cu and also 4d-shell from Zr to Ag), rare earth and actinide elements (4f-shell from Ce to Yb and 5f from Th to Lr) but also certain organic compounds []. Since d- and f-orbitals are spatially more confined than s- or p-shells of simple metals or semiconductors, overlap between orbitals of neighboring atoms is smaller, which leads to narrow bands and accordingly lower kinetic energies. Thus, on average, electrons reside longer on one atomic orbital, enhancing electron-electron interactions. This implies that electron motion is dominated by strong correlation effects. These correlations give rise to intriguing quantum many-body phenomena like metalinsulator transition from a high-conductivity to an insulating phase (as in V O 3 ), itinerant (anti)ferromagnetism (like in iron and nickel), large rmoelectric response, volume changes across phase transitions (in actinides and lanthanides), colossal magnetoresistance effect (in manganites) and high temperature superconductors []. A very basic observation is Mott phenomenon which describes localization of electrons as a consequence of strong Coulomb repulsion. Even high temperature cuprate superconductors are widely assumed to be doped Mott insulators [3]. The investigation of se fascinating strong correlation effects is a challenging task for most analytical and numerical methods, as several competing mechanism dominate physics for a wide range of energy scales and lead to a nonperturbative nature of many-body problem. The Dynamical Mean-Field Theory (DMFT) [4] provides a nonperturbative many-body approach to strongly correlated systems. It has been developed over last 5 years and is one of most successful approaches for treating both local correlations and band structure information on an equal footing. Within DMFT framework, one maps a lattice model onto an effective single-site quantum impurity model in a self-consistent manner. Like in Weiss mean-field ory of classical statistical mechanics, spatial fluctuations are frozen out by this process, which leads to a purely local self-energy within DMFT. However, in contrast to classical case where all fluctuations are neglected, DMFT approach fully incorporates local quantum fluctuations in quantum

12 CHAPTER. INTRODUCTION impurity problem, thus capturing local dynamics of many-body problem. Although, this mapping is exact in limit of infinite lattice coordination number only, it may be regarded as a useful approximation for three-dimensional systems. For quantum impurity models several non-perturbative methods exist, such as Quantum Monte Carlo simulations (QMC) or Numerical Renormalization Group (NRG) approach. Quantum Monte Carlo codes are widely used and have been highly refined over last few decades. However, data is obtained on imaginary (Matsubara) frequency axis and has to be analytically continued to real axis, a procedure that is mamatically ill-defined in principle, and in practice notoriously difficult to perform accurately, especially for complex models. Furrmore, possible limitations are very low temperatures and also so called sign problem for fermionic systems. In contrast, NRG method allows to calculate physical quantities directly on real frequency axis. It is able to provide reliable data for effectively zero temperature and thus is worth exploiting as an alternative to Quantum Monte Carlo approaches. In combination with ab-initio band-structure calculations, DMFT has proved to be a successful method for investigation of realistic strongly correlated materials. Neverless, this technique has to be improved furr to deal with increasingly complex systems, requiring in turn, more efficient solvers for effective quantum impurity model. In this sis, Numerical Renormalization Group code, that has been developed by Andreas Weichselbaum in group of Jan von Delft at Ludwig-Maximilians-Universität München, is used as impurity solver within Dynamical Mean-Field Theory. It is one of most evolved codes in field of NRG. Using full density matrix approach with complete many-body basis sets, it can handle arbitrary temperatures [5]. Moreover, it is, to date, only NRG framework, that is able to exploit arbitrary abelian and non-abelian symmetries, which leads to a significant reduction of numerical effort and makes it highly suitable for investigation of multi-band models in presence of intrinsic symmetries such as channel symmetry. The goal of this work is to explore potential of our program in context of DMFT. The outline of this sis is as follows. In Chapter, concept of Dynamical Mean-Field Theory is presented. It can be regarded as an extension of classical mean-field ory to quantum many-body problems. We explain underlying physical ideas of DMFT method by taking example of one-band Hubbard model, most fundamental model for strongly correlated materials. Later on, we also refer to its applicability to more complex model Hamiltonians, as provided by local density approach (LDA), an ab-initio band structure calculation technique. The basic approximation within DMFT is to use a purely local self-energy. In limit of infinite lattice coordination, this approximation becomes exact, which can be rationalized using a scaling argument in perturbation ory. Beside local self-energy, anor important quantity within DMFT, local retarded lattice Green s function, is introduced. The significance of lattice type under consideration is discussed afterwards. Finally, we show how lattice model can be mapped onto a single-site quantum impurity model by a self-consistent procedure. At beginning of Chapter 3, we review basics of Numerical Renormalization Group method. In a second part, we concentrate on more recent developments in field of NRG and concisely present outstanding features of our NRG program, including incorporation of arbitrary abelian and non-abelian symmetries, implementation of a complete many-body basis set, full density matrix approach for finite temperatures and definition of a discarded weight within NRG. In Chapter 4, we reveal peculiarities that emerge, when combining DMFT and NRG approach. First, we review how to calculate self-energy within NRG. Thereafter, we discuss

13 variations of NRG logarithmic discretization grid, that are required when NRG is employed within DMFT. This is followed by a first consistency check for our DMFT+NRG program, using one-band Hubbard model in Chapter 5. We investigate Mott-Hubbard metal-insulator transition which occurs for an integer filling of lattice. Then we reproduce several results for a hole-doped system, given in [6], on a quantitative level. As a major application, we study two-band Hubbard-Kanamori model in Chapter 6. A short introduction to ideas of realistic modeling of strongly correlated materials reveals to what extent a two-band model is appropriate to describe transition metal compounds. Before investigating two-band Hubbard-Kanamori Hamiltonian in detail, including an analysis of possible symmetries of system, we extend DMFT equations to multi-band models. Special attention is directed to effect of Hund s coupling on metal-insulator transition in two-band model. Finally, we explicitly compare NRG to QMC results. Note that all QMC data were kindly provided by our collaborator Michel Ferrero from Centre de Physique Théoretique, Ecole Polytechnique, France (thus labeled with MF). The sis is completed by evaluating present stage of our DMFT+NRG program and giving a brief outlook to possible future directions. 3

14 CHAPTER. INTRODUCTION 4

15 Chapter Dynamical Mean-Field Theory (DMFT) The first step towards Dynamical Mean-Field Theory (DMFT) was given by W. Metzner and D. Vollhardt in 989. In ir pioneering work [7] y studied Hubbard model of correlated lattice fermions in limit of infinite dimensions. With properly applied scaling of Hamiltonian, diagrammatic treatments are drastically simplified compared to finite dimensions, while competing physical mechanism between kinetic energy and electron-electron interaction is maintained. This new limit of infinite lattice coordination gave better insight to different approximation schemes for strongly correlated problems. The basic DMFT framework was established in 99 by A. Georges and G. Kotliar [8]. They extended idea of classical mean-field ory to quantum many-body problems and mapped complex quantum lattice problem onto a single-site effective problem - quantum impurity (Anderson) model, that consists of a correlated local impurity system with a small number of degrees of freedom, coupled to a macroscopic non-interacting bath, whose structure has to be determined in a self-consistent manner. The quantum impurity model can be solved with previously developed methods (such as Monte Carlo or NRG). In limit of infinite dimensions, mapping can be carried out exactly. That construction led to an intensive investigation of various model Hamiltonians and opened a new direction of research, including study of realistic models of strongly correlated materials. Here material specific informations, like several orbitals, actual lattice type and corresponding density of states, are taken into account. A recent extension of DMFT method, called Cluster-DMFT, accounts for short-range spatial correlations and is able to treat inhomogeneous materials, such as disordered alloys, thin films or multi-layered nanostructures [9],[, Chapter ]. A detailed review of Dynamical Mean-Field Theory and its applications is provided by A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg in [4]. An introduction to DMFT in context of realistic modeling of strongly correlated materials can be found in []. 5

16 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT). Basic idea of DMFT To understand basic ideas of Dynamical Mean-Field Theory for quantum lattice problems, it is advisable to revisit underlying physical concept of mean-field ory for familiar classical Ising model H = J ij S i S j h S i (.) ij i with ferromagnetic coupling J ij > between spins of different neighboring lattice sites i and j and an external magnetic field h. The number of nearest neighbors defines coordination number z. The main notion of mean-field approach is to reduce a complex lattice model to a single site problem with effective parameters, that exhibits less degrees of freedom. The dynamics of whole lattice is n represented by interaction of degrees of freedom at single site with an external field, Weiss effective field, which is comprised of all or degrees of freedom at or sites. The effective single site Hamiltonian can be written in terms of Weiss field h eff and spin degree of freedom at a given site i = : H eff = J ij S j + h S = (zjm + h) S = h eff S. (.) j The effective bath h eff is obtained by neglecting correlated spin fluctuations [S i S ][S j S ] =, (.3) which leads to rmal average of magnetization S j. For translationally invariant systems we get S j = m and J ij = J. The sum over lattice sites j can n be simplified to a product with lattice coordination number z. With this, one arrives from Eq. (.) at well-known mean-field equation for magnetization at finite temperature T (β = k B T ), m = tanh(βh + zβjm), (.4) which can be reformulated in terms of Weiss effective field, h eff = zj tanh(βh eff ) + h. (.5) Eq. (.4) and Eq. (.5) have form of a self-consistency condition for magnetization and Weiss field, respectively. The approximation of neglecting correlated spin fluctuations becomes exact in limit of an infinite coordination number z. For lower dimensions, z serves as a control parameter for validity of mean-field approximation. Yet, to obtain a sensible limit z, where physical quantities like effective magnetic field h eff remain finite, we have to rescale coupling constant J as J = J z, J = const. (.6) The basic steps of Weiss mean-field ory can be directly extended to quantum many-body problems [4] (see Fig.. for one-band Hubbard model): 6

17 .. BASIC IDEA OF DMFT exact for z! Weiss effective field t! t p z, DMFT t = const. k (!) = (!) (!) =Im G latt (!) + (!) scaling J! J z, approximation self-consistency equation classical MFT J = const. h[s i hsi][s j hsi]i = h e = zj tanh( h e )+h Figure.: Basic steps of DMFT compared to classical Weiss mean-field ory. In both methods a complex lattice problem is mapped onto an effective single-site problem. The dynamics of whole lattice is n represented by interaction of degrees of freedom at single site with an external field, Weiss effective field, which is comprised of all or degrees of freedom at or sites. The Weiss field ( static magnetic field h eff for classical Ising model and dynamical hybridization function Γ(ω) for quantum case) is determined by a self-consistency condition. The mapping is based on an approximation (neglecting correlated spin fluctuations in classical case and a momentum independence in self-energy in DMFT), that is exact in limit of infinite lattice coordination z. In this limit, physical quantities (like ferromagntic coupling J for Ising model and hopping amplitude t for Hubbard model (see Sec...)) have to be rescaled. A complex quantum lattice problem is mapped onto a single-site quantum impurity problem (Anderson model), that has to fulfill a self-consistency condition (see Sec..3.5). The whole lattice dynamics is n captured by local single-particle retarded lattice Green s function (see Sec..3.) G R µν(t) = iθ(t) {ĉ µ (t), ĉ ν} T (.7) with fermionic annihilation (creation) operator ĉ ( ) µ,ν, acting locally on one specific lattice site with possible quantum labels µ, ν (e.g. for different spins or several orbitals) and (real) time evolution in Heisenberg picture ĉ µ (t) = e iĥt ĉ µ e iĥt. As we will use retarded Green s function throughout this sis we will skip superscript R from now on. The basic approximation within DMFT is to freeze out spatial fluctuations by introducing a purely local (site-diagonal) self-energy Σ ij (ω) z Σ(ω)δ ij, (.8a) or expressed in Fourier space, a momentum-independent self-energy Σ k (ω) z Σ(ω). (.8b) 7

18 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT) Yet, in contrast to classical case, where magnetization is static, DMFT approximation fully incorporates local temporal quantum fluctuations. Hence, we achieve a dynamical (frequency dependent) mean-field ory for quantum case. As for Ising model, approximation becomes exact in limit of infinite lattice coordination also for quantum lattice problems (which was shown by W. Metzner and D. Vollhardt in 989 [7]). In this limit appropriate scaling of investigated Hamiltonian leads to a local nature of perturbation ory and refore to a momentum-independent self-energy. Again coordination number constitutes a control parameter for validity of approach. As z is reasonably large in three dimensions (e.g. z = 6 for a cubic lattice and z = for a face centered cubic lattice), we can apply DMFT method in good approximation to three dimensional lattice problems.. Model Hamiltonians To study properties of strongly correlated materials, we need model Hamiltonians that can be solved numerically (with DMFT method). The most fundamental model for strongly correlated materials is one-band spin- Hubbard model. For more realistic descriptions, model Hamiltonians can be constructed via local density approach (LDA)... The Hubbard model The one-band spin- Hubbard model was introduced by J. Hubbard in its seminal work of 963 [] and is a tight-binding description for solids including interactions: Ĥ = µ ˆn iσ + t ij ĉ iσĉjσ + U ˆn i ˆn i, (.9) iσ i ij σ where ˆn i σ ĉ iσĉiσ is particle number operator of site i. ĉ( ) iσ denote corresponding annihilation (creation) operators for conduction electrons with spin σ =,. ij is a sum over nearest neighbors i and j only. As common for tight binding approximation, solid is modeled by various sites i, representing constituent atoms (or molecules) of material, that are arranged in a specific type of lattice. The atoms exhibit orbitals occupied by a certain number of electrons, some of which can contribute to transport by hopping between orbitals centered around different sites. In case of one-band Hubbard model, only one orbital per site with maximally two electrons (one for each spin) contributes to transport. The first term of model Hamiltonian gives local single-particle energy level in terms of chemical potential µ. The second term yields kinetic energy of conduction electrons, which is determined by hopping amplitude t ij for an electron of spin σ to be annihilated at site j and again created at site i. This transfer amplitude (between nearest-neighbor sites i and j in case of Eq. (.9) or between different orbitals in general) can be calculated by considering an overlap integral between Wannier functions, representing corresponding localized orbitals. To simplify our model, we assume hopping amplitude to be site-independent in following, i. e. t ij = t. The Fourier transform of kinetic energy part of Eq. (.9) results in band dispersion ε k and momentum distribution operator ˆn kσ : tĉ iσĉjσ = ε kˆn kσ. (.) kσ ij σ 8

19 .3. DERIVATION OF THE DMFT EQUATIONS The bandwidth of one-band Hubbard model is determined by hopping amplitude t (see Sec..3.3). The third term of Hubbard Hamiltonian includes on-site Coulomb repulsion U between electrons sitting at same site i. The Coulomb interaction can be approximated as purely local because it is in general heavily screened for strongly correlated materials. Its implication leads to a competition between itinerancy and localization of electrons and can give rise to intriguing many-body phenomena like Mott-Hubbard metal-insulator transition (MIT), which will be discussed in detail in Chapter 5... Model Hamiltonians from local density approach (LDA) A very recent development in field of DMFT is realistic modeling of materials, considering actual electronic and lattice structure - LDA+DMFT approach. Here art of modeling is to incorporate material specific details, but to keep model as simple as possible at same time in order to allow for its numerical treatment. In that sense, Hubbard Hamiltonian is minimal Hamiltonian for a system with a narrow band at Fermi level, where all or degrees of freedom are projected out [, Chapter 6]. To describe complexity of real materials, Hubbard model typically has to be extended (see Chapter 6). While DMFT method is based on model Hamiltonians with empirically determined physical parameters, density functional ory (DFT) and its local density approach (LDA) are ab initio methods for calculation of realistic band structures (yet without considering strong correlation effects) [, Chapter ]. In LDA method an ab-initio basis of localized Wannier functions is constructed, that carries relevant details about structure and chemistry of a given material (like number and type of orbitals contributing to transport) [, Chapter 6]. These functions are n used to determine hopping integrals (or analogously dispersion relation) and local electronic energy levels of orbitals. The interaction parameters can, in principle, also be calculated from LDA Wannier functions. However, it is still a major challenge to obtain screened Coulomb integrals ab initio [, Chapter 6]. Note that model Hamiltonians based on LDA contain an additional term ( double counting correction ) that cancels electron-electron interaction contained in hopping term. Normally this correction is absorbed in local electronic energy levels of orbitals. In Sec. 6 we give a more detailed introduction in realistic modeling of strongly correlated materials and investigate a Hamiltonian with material specific details (two degenerate orbitals)..3 Derivation of DMFT equations In this section we want to present basic ideas of DMFT in more detail and derive DMFT equations which will be later solved with NRG method. For simplicity, we concentrate on one-band Hubbard model. Yet, method can be adapted to more complex models in a straightforward manner (see Sec. 6.)..3. Local nature of self-energy in infinite dimensions We now have a closer look how simplification of a momentum-independent self-energy arises for limit z. As for classical case we have to ensure a non-trivial limit for physical 9

20 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT) quantities (like internal energy per site), which leads to a scaling condition for Hubbard Hamiltonian. While interaction term, as well as term containing chemical potential, are purely local and refore not affected by number of nearest neighbors z, kinetic energy per site diverges with growing coordination number and has to be properly rescaled to retain competition between kinetic and interaction energy. This can be easily understood from corresponding non-interacting density of states (DOS) ρ (ε) = δ(ε ε k ) (.) N B with ρ (ε)dε =. Following argumentation of W. Metzner and D. Vollhardt [7], we consider a simple d-dimensional hypercubic lattice with dispersion relation k d ε k = t cos k i (.) i= and coordination number z = d. The DOS can be interpreted as probability density to find an one-particle excitation of energy ε = ε k for randomly chosen k i. In limit d, dispersion relation is n a sum over essentially independent random numbers t cos k i with only constraint that d i= cos k i = ε t. As a consequence, one can apply central limit orem, leading to a gaussian DOS [, Chapter 3], ρ H (ε) d t πd exp that retains a finite width only if hopping amplitude scales like t = t z = [ ( ) ] ε t, (.3) d t d, t = const. (.4) The rescaled Hamiltonian Ĥ = µ iσ ˆn iσ + ij σ t z ĉ iσĉjσ + U i ˆn i ˆn i (.5) has now a well-defined limit z and kinetic energy per site remains finite by construction. The kinetic energy can be written in terms of full interacting fermion propagator G ij : E kin = ij t ĉ z i ĉj = lim t + ij t z dω πi G ij(ω)e iωt. (.6) (Note that we have skipped spin index for convenience, as it will not affect following derivations. It can be simply reintroduced afterwards.) As kinetic energy per site is of order for z, we can derive from above expression that G ij has to scale like ( ) G ij O z for i j =, (.7a) G ii O(), (.7b) since sum over all nearest-neighbors yields an order z while scaling of hopping amplitude goes like / z [, Chapter 5].

21 .3. DERIVATION OF THE DMFT EQUATIONS This scaling behavior leads to strong simplifications in perturbation ory in z, including a momentum-independent self-energy [4]. This may be best demonstrated diagrammatically using skeleton expansion for self-energy of Hubbard model in real space (see Fig..), which solely consists of full interacting propagator lines G ij (double lines) and local interaction vertices (dashed lines). (a) (!) = (higher orders) full interacting propagator i interaction vertex (local interaction ) (b) first diagram with two different internal vertices i j : O( p z ) i j O( p z ) O( p ) z O(z 3 ) Figure.: Panel (a) shows skeleton expansion for self-energy of Hubbard model as in [, Chapter 5] with arrows and labels omitted. The full interacting propagator lines G ij scale like / z for i j. For second order diagram (depicted in panel (b)) we see that it vanishes as / z. The three independent propagator lines give a factor of z 3. The sum over nearest neighbors yields anor factor of order z. So in limit z this diagram scales like / z and refore does not contribute to self-energy. It can be easily shown that two different internal vertices i j are at least connected by three independent propagator lines, each of which scale like / z (see second order diagram in Fig.. (b) as example). So a factor of at least z 3 is obtained for each diagram (of order and with different internal vertices i j). The sum over nearest neighbors yields anor factor of order z. Thereby, any non-local contribution of skeleton perturbation expansion vanishes at least as / z and all diagrams fully collapse to a single site in limit of infinite coordination number z [4]. Thus self-energy becomes a purely local quantity in real space, Σ ij (ω) z Σ(ω)δ ij, (.8a) and its Fourier transform is momentum independent, Σ k (ω) z Σ(ω). (.8b) Yet, self-energy is still frequency dependent. So DMFT approximation Eq. (.8b) only freezes out spatial fluctuations but completely accounts for temporal quantum fluctuations, which are at bottom of interesting features of strongly correlated materials [4].

22 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT) Although this derivation of DMFT approximation is based on a perturbative expansion of self-energy with a finite radius of convergence [], central statement of a momentumindependent self-energy and all subsequent DMFT equations are proven to be non-perturbative (see for example cavity method in [4]). The fact that DMFT approximation becomes exact in limit of infinite coordination ensures that physical constraints, such as causality of self-energy, are retained within DMFT approach []. In complete analogy, DMFT approximation can be adapted to more complex models than one-band Hubbard Hamiltonian discussed above..3. On-site lattice Green s function In previous section, it was shown that all diagrams in skeleton expansion of self-energy collapse to a single site. So, self-energy reduces to a functional that only depends on local Green s function. As a consequence, two central quantities in DMFT are (retarded) on-site lattice Green s function and local self-energy, which are both dynamical and able to capture genuine correlation effects [, Chapter ]. For translationally invariant lattice Hamiltonians (such as one-band Hubbard Hamiltonian) and skipped spin index, one can write retarded lattice Green s function in momentum space as G latt,k (ω) ĉ k ĉ k ω = (.9) ω + µ ε k Σ latt,k (ω) with all interaction effects considered by self-energy Σ latt,k (ω). Note that vectors, as well as matrices, are printed in bold. Using DMFT approximation, we arrive at simplified form G latt,k = ω + µ ε k Σ latt (ω) = G latt,k(ω Σ latt (ω)) (.) with non-interacting expression for retarded lattice Green s function G latt,k(ω) = ω + µ ε k. (.) A general derivation of above equations, based on an equation of motion ansatz (see Appendix A), is given in Appendix B. ω ω + = ω + iδ with positive infinitesimal δ accounts for causality of retarded Green s function. The local or on-site lattice Green s function (which is chosen to be at site r =, for convenience) can n be calculated as Fourier transform G latt (ω) = G latt,k (ω)e ik(r=) N B k = N B ω + µ ε k Σ latt (ω) = k ρ (ɛ) dε ω + µ ε Σ latt (ω), (.a) (.b) (.c) where N B denotes number of k-points of Brillouin zone and ρ (ε) is non-interacting density of states (DOS) of specific lattice (see Sec..3.3 for details). Note that although self-energy is purely local within DMFT, Hubbard model is not reduced to a purely local model as hopping between different lattice sites still enters via momentum-dependent

23 .3. DERIVATION OF THE DMFT EQUATIONS dispersion relation ε k. As we mainly want to investigate transport properties, we are interested in local density of states (spectral function) It obeys sum rule A(ω) π Im G latt(ω). (.3) dωa(ω) =, (.4) which is especially important for DMFT approach, and can generally be expressed in Lehman representation as a sum over δ-functions. From Fourier transform G µµ (ω) of expression in Eq. (.7), we obtain A µ (ω) = ab e βea + e βe b a ĉ µ b δ(ω (E b E a )) A µ (ω) + A µ (ω) (.5) Z with partition function Z, eigenvalues E a and corresponding many-body eigenstates a of full Hamiltonian. The index µ can denote any degrees of freedom, such as spin σ and orbital m, depending on system under consideration. The local density of states can be used to calculate average occupation number of a specific impurity level. ˆn µ ˆd ˆd µ µ = dωa µ (ω)n F (ω, T ) (.6a) with Fermi-Dirac distribution function n F (ω, T ), or equivalently, ˆn µ = dωa µ (ω). (.6b) For SIAM, we have µ σ. In essence, goal of DMFT procedure is to obtain shape of spectral function to high accuracy in order to deduce transport properties..3.3 Non-interacting lattice density of states As can be observed in Eq. (.c), specific lattice type only enters via non-interacting density of states, or equivalently over dispersion relation in case of Eq. (.b). Neverless, material specific configuration of constituent atoms is also reflected in physical parameters, such as Coulomb interaction strength and number of orbitals considered in model Hamiltonian. For model calculations, it is common to use infinite-dimensional limit of non-interacting DOS as an approximation to finite-dimensional lattices [, Chapter 5]. A rar artificial, but important lattice type, that is often used for model calculations within DMFT, is infinite Caley tree, which is also called Be lattice. The corresponding lattice DOS is semi-elliptic for infinite coordination z, ρ B (ε) = ( ) ε, ε [ D, D], (.7) πd D 3

24 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT) with half bandwidth D = t and rescaled nearest-neighbor hopping t = t / z [4] (see Appendix D). Anor example of an infinite DOS, gaussian hypercubic lattice DOS, has already been introduced in Sec..3.. In analogy to Be lattice for infinite coordination z, an effective half bandwidth D = t of twice standard deviation of gaussian DOS is introduced for hypercubic case: ρ H (ε) = [ exp π(d/) ( ) ] ε, ε [, ]. (.8) D/ As can be seen in Fig..3 (c) DMFT results (here spectral function for symmetric one-band Hubbard model with U/D =, µ/d = and T/D ) are qualitatively equal for eir lattice type. ρ(ε)*d ρ( (a) (b) d=3 cubic lattice d=3 z =3 Be lattice hypercubic lattice A(ω)*D U/D=, µ/d=, T/D=.6e (c) spectral function Be lattice hypercubic lattice ε/d 4 4 ω/d Figure.3: Panel (a) contains an illustration for cubic lattice in three dimensions (red) and Be lattice with coordination z = 3 (blue). In panel (b) Be and hypercubic lattice DOS are plotted for infinite coordination z. Additionally cubic lattice DOS for three dimensions is sketched in black (curve adapted from [3]) to show that red curve is a good approximation for finite dimensional case d = 3. In panel (c) we plotted corresponding spectral functions for symmetric one-band Hubbard model with U/D =, µ/d = and T/D, which are qualitatively equal for eir lattice type. Besides, re are also or DOS considered in literature, like DOS for infinite version of three-dimensional diamond lattice or Lorentzian DOS involving long-range hopping (see [4] for details). On a model level, Be DOS for infinite coordination z is often favored, as expression for local lattice Green s function Eq. (.c) can n be obtained analytically and leads to a quite simple form of self-consistency condition (see Sec..3.5). The local lattice Green s function is formally Hilbert transform of non-interacting density of states G latt (ξ) = dε ρ (ε) ξ ε (.9) 4

25 .3. DERIVATION OF THE DMFT EQUATIONS with ξ = ω +µ Σ(ω) and Im ξ = Im Σ(ω) > (as we consider retarded Green s function). For semi-elliptic DOS we obtain G latt (ω) = ) (ξ D ξ D, (.3) using a continued fraction expansion for local retarded Green s function (see Appendix D for details). For LDA Hamiltonians local lattice Green s function is directly computed from sum over Brillouin zone in Eq. (.b), that explicitly contains LDA band structure..3.4 Mapping onto an effective quantum impurity model In this section, we discuss non-trivial part of how to compute self-energy in expression of local lattice Green s function, which represents central quantity in DMFT approach. The big simplification was given in 99 by A. Georges and G. Kotliar [8]: lattice model can be mapped onto a quantum (Anderson) impurity model with effective parameters, that exhibits same local interaction term as original model. Impurity Anderson models and associated Kondo physics of strongly correlated electrons have been studied intensively since 98 s and are often well-understood due to a variety of efficient numerical codes [, Chapter ]. Since interaction effects only enter locally through momentum-independent self-energy in DMFT, self-energy of a single-site quantum impurity Hamiltonian with equal local interaction resembles that of lattice model: Σ latt (ω) = Σ imp (ω) Σ(ω). (.3) As a consequence, dynamics of lattice model, which is encoded in its local lattice Green s function, can be determined via Green s function of impurity model - a single site coupled to an effective bath - analogously to classical mean-field ory, where local magnetization is described as a spin on a single site, that is coupled to an effective magnetic field ( Weiss mean-field) []. So mapping of lattice model onto effective quantum impurity can be written as G latt (ω)! = G imp (ω). (.3) For single-band Hubbard Hamiltonian, effective quantum impurity model is single impurity Anderson model (SIAM): Ĥ SIAM = Ĥimp + Ĥbath + Ĥcpl, (.33) Ĥ imp = σ Ĥ bath = kσ Ĥ cpl = kσ ε d ˆd σ ˆdσ + U ˆd ˆd ˆd ˆd, ε k ĉ kσĉkσ, V k ( ˆd σĉ kσ + h.c.). (.33a) (.33b) (.33c) For more complex lattice problems, SIAM will be generalized to multi-band (or multi-orbital) Anderson models (see Sec. 6.). 5

26 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT) 4.3 Key points of Wilson chain mapping (a) (b) tu -- --! z! k - (!) - (c) time t t Figure 4.: Sketch of key points of mapping to W bridization V and, thus, constant hybridization function (!) model where impurity (red) is coupled by V (light blue) with constant hybridization function. In panel (b) energy Figure.4: In DMFT approach a quantum lattice problem Hubbard is mapped self- is represented cally(here discretized with model) = and each interval consistently onto a single-site effective problem ( single impurity Anderson Left,same we illustrate circles) which give model). rise to hybridization function d Hubbard model where electrons can hop between different lattice sites chain with ais hopping The final Wilson depicted amplitude in panel (c) where cou t and feel a Coulomb repulsion of strength U, when y and meet onfirst one Wilson site. On we show a. The thinning site is right, again given by V n sketch of SIAM (adapted from [4]), where a single impurity coupled via amplitude Vk to an pict is exponentially decreasing couplings of order. In p effective bath. The effect of P environment on impurity site isofdescribed Weiss field - to Wilson structure waveby functions corresponding hybridization function Γ(ω) = π k Vk δ(ω εk ). In essence, impurity represents a given lattice site index n respective wave functionssite have support only (red), hybridization describes influence of remaining lattice (blue). Thus exponentially growing size.spatial correlations in bath are frozen out while local quantum fluctuations (indicated in red box) are completely retained by mapping onto local impurity model. solved iteratively as will be explained in next section. Fo typical energy resolution introduced through this site is g ( ) n n A given site of lattice (at r = ) is represented by impurity. dˆσ denotes' annihilation tn = ( + ) n (creation) operator for an electron of spin σ =, on impurity. εd is local energy level of characterizing few lowest-lyin impurity site and has to be equal to local single-particle level µtypical of spacing lattice.between As in finite length n with n sufficiently large. lattice case, local site has four different possible quantum states. It can be unoccupied ( i), Foror a non-interacting Fermi with Mofparticles (and siz singly occupied with an electron of spin up or down ( σi), doubly occupied by sea electrons particle level spacing at Fermi energy scales like /M. Acco spin up and down ( i), respectively. In last case, electrons feel a Coulomb repulsion of (see Sec..) this also holds for interacting case. Thus, we m strength U. The interaction term in SIAM is equal to local interaction of Hubbard and so model. The rest of lattice is described by an effective bath of non-interacting fermions n n ˆ bath ), that hybridizes with impurity via hopping amplitude given bymv /. cˆ( ) are, L /, (H k kσ corresponding annihilation (creation) operators in 3 For bath. Thus, we local impurity model historical reasons are using exact formula in Eq. (4.7) w as it is only an estimate anyway. allows for charge and spin fluctuations. As impurity is embedded in an effective environment, it may undergo transitions between different quantum phases. The effect of external bath onto impurity is fully contained in hybridization function, (ω) = X k Vk. ω εk (.34) With ω ω +, (ω) is analytic in upper complex plane. Hence, it is sufficient to concentrate 6

27 .3. DERIVATION OF THE DMFT EQUATIONS on imaginary part, Γ(ω) Im (ω) = π k V k δ(ω ε k ), (.35) which plays role of effective Weiss field in DMFT. It is frequency dependent and hence dynamical. The hybridization, however, is not known a priori, but has to be determined in a self-consistent manner. The effective impurity model n reproduces actual mean-field solution of original many-body problem. An illustration of self-consistent mapping onto an effective quantum impurity model is shown in Fig..4. The local interacting Green s function of effective impurity model is completely given in terms of hybridization function (ω) and self-energy Σ(ω): G imp (ω) ˆd ˆd ω = ω ε d (ω) Σ(ω), ε d = µ. (.36) This can be derived from equation of motion (see Appendix C for a detailed calculation), ω ˆd ˆd ω = { ˆd, ˆd } T [Ĥ, ˆd] ˆd ω. (.37) In following, we want to derive DMFT self-consistency condition for calculation of Weiss effective field Γ(ω) and present a possible solution process to obtain finally local lattice Green s function and corresponding spectral function..3.5 The self-consistency procedure The mapping of an original lattice onto an effective impurity model is performed by equating local lattice Green s function with impurity Green s function in Eq. (.3), G latt (ω)! = G imp (ω), which imposes a self-consistent condition for Weiss effective field Γ(ω). Inserting impurity Green s function Eq. (.36) into Eq. (.3) yields G latt (ω) + Σ(ω) = ω ε d (ω) = G imp(ω) (.38) with ε d = µ, and a simple relation for imaginary part, Γ(ω) = Im (ω) = Im(G latt (ω) + Σ(ω)). (.39) This is DMFT self-consistency condition, relating for each frequency Weiss effective field and local lattice Green s function. Thus, it leads to a closed set of DMFT equations, that can be solved iteratively and which finally fully determines hybridization function Γ(ω), local self-energy Σ(ω) and accordingly local Green s function. The DMFT equations are summarized in iterative DMFT procedure in Fig..5. In principle, self-consistency equations can be formulated differently, depending on how impurity model is solved. Since quantum impurity models have been investigated intensively over last decades, a variety of (partly numerically extensive) techniques are available. Quantum Monte Carlo (QMC) methods are widely used and a lot of different algorithms (like Hirsch-Fye QMC or continuous-time (CT) QMC) have been successfully developed. Possible limitations are very low temperatures and so called sign problem. Furrmore, QMC methods compute physical quantities in dependence of imaginary Matsubara frequencies. Therefore (Pade) continuation to real frequency axis has to be performed, which may become difficult for complex models (see Chapter 6). Or quantum impurity solvers include Numerical Renormalization Group 7

28 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT) (NRG), exact diagonalization, density matrix renormalization group (DMRG) or analytical yet approximate schemes like iterated perturbation ory or noncrossing approximation (for a list with references see [4] and []). We will use Numerical Renormalization Group (NRG) to solve quantum impurity model, a method that is well-established in realm of quantum impurity problems. It offers a clear advantage in that it calculates data directly on real frequency axis. Therefore self-consistency procedure generates (frequency-dependent) hybridization, which is a natural input for NRG method. (!) (!) = F (!) G(!) (!) =Im G latt (!) + (!) NRG G(!) =h ˆdk ˆd i! F (!) =h[ ˆd, ĤU ]k ˆd i! G latt (!) Z G latt (!) = (") d"! + µ " (!) (!) (!) Figure.5: DMFT self-consistency procedure with NRG method as impurity solver. Starting from an arbitrary hybridization function Γ(ω), we obtain self-energy from NRG approach, solving effective quantum impurity problem for first DMFT iteration. The self-energy is n inserted into expression for lattice Green s function. The self-consistency condition closes DMFT loop, yielding a new hybridization function. The iterative procedure is repeated until convergence is reached. At start of calculation, we know neir hybridization nor local Green s function. One refore starts self-consistency procedure with an arbitrary input hybridization Γ(ω). This completely defines quantum impurity model that has to be solved in first iteration with NRG method. The NRG impurity solver, depicted as a black box in Fig..5, returns self-energy of SIAM, which is equal to lattice self-energy and thus can be inserted into integral expression Eq. (.c) for lattice Green s function. The calculation of selfenergy is not performed via common Dyson equation Σ(ω) = G imp (ω) G imp (ω), but given as ratio of a two-particle and a one-particle Green s function (see box in Fig..5). The details of this NRG specific approach are presented in Sec. 4.. With self-consistency condition Eq. (.39), we arrive at a new hybridization function, that serves as input for next iteration. After performing several DMFT iterations, iterative procedure converges to 8

29 .3. DERIVATION OF THE DMFT EQUATIONS a stable and usually unique solution, independently of initial input hybridization. Only in some special cases (e.g. close to a Mott transition) different input regimes lead to more than one stable solution [] (see Sec. 5.). From converged version of lattice (or impurity) Green s function, finally, we also have local spectral function of strongly correlated many-body system, and hence can investigate its dynamics. For model calculations, it is convenient to use semi-elliptic Be lattice DOS, since DMFT self-consistency equations reduce to a compact form, i.e. without numerical integration. With ( analytic expression for lattice Green s function Eq. (.3), G latt (ω) = /D ξ ξ D ), and ξ = ω + µ Σ(ω) (see Sec..3.3), we can be simply derive ( ) D ω ε d (ω) = ω + µ G latt, (.4) by inserting Eq. (.3) into Eq. (.38). From this equation we explicitly see that ε d = µ in impurity model. Furr, it leads directly to compact expressions ( D (ω) = ) G latt(ω) and Γ(ω) π Eq. (.3) = ( D ) A(ω). (.4) The thick line in illustration of DMFT loop in Fig..5 depicts that non-trivial and numerically most time-consuming part of DMFT self-consistency procedure is computation of self-energy via impurity model. The accuracy of impurity solver completely determines overall success of DMFT procedure. Especially for more evolved models with additional degrees of freedom (like multi-band Hubbard models) high numerical efficiency and flexibility of algorithm is required. In this sis, we introduce powerful NRG program of our group (developed by A. Weichselbaum) in framework of DMFT. It is a highly refined and accurate numerical code that is able to deal efficiently with complex systems in context of symmetries, as arbitrary abelian and non-abelian symmetries can be exploited. In next chapter we will first give an introduction to general concept of Numerical Renormalization Group and n present outstanding features of our NRG code that make it, also in DMFT framework, highly competitive. In Chapter 4, we will discuss problems that can arise when NRG is used within DMFT. Special attention is directed at issue of a frequency-dependent hybridization function within NRG. 9

30 CHAPTER. DYNAMICAL MEAN-FIELD THEORY (DMFT)

31 Chapter 3 Numerical Renormalization Group (NRG) The Numerical Renormalization Group (NRG) was developed in early 97 s by K. G. Wilson to solve Kondo model in a fully nonperturbative way. In this model, a single magnetic impurity interacts with conduction band electrons of a nonmagnetic metal describing dilute magnetic alloys. NRG has subsequently evolved to become most reliable numerical method available for treating quantum impurity models. A quantum impurity model consists in general of a discrete quantum system with a small number of degrees of freedom that is coupled to a macroscopic bath of non-interacting fermions (or bosons). Each system on its own can in principle be solved exactly, coupling of both, however, leads to strongly correlated quantum many-body phenomena. As we do not consider any bosonic problems in this work, we concentrate on Hamiltonian for a fermionic quantum impurity system, which can be written in terms of three different parts: Ĥ = Ĥimp + Ĥbath + Ĥcpl. (3.) The impurity Hamiltonian Ĥimp may contain arbitrary interactions, like Coulomb repulsion or Hund s coupling terms. Since by assumption impurity has just a small number of degrees of freedom, Ĥ imp can be diagonalized exactly. The non-interacting bath term Ĥbath usually describes a (quasi)continuous excitation spectrum covering a broad range of energies: n c Ĥ bath = ε kµ ĉ kµĉkµ (3.) µ= k with µ =,..., n c labeling different kinds of electrons like electrons from different baths (e.g. µ =, for two baths) or just electrons with different spin (µ =, ). ĉ kµ and ĉ kµ create and annihilate a bath electron of type µ and energy ε kµ, respectively. The energy support of bath is defined by bandwidth of specific problem. For a standard NRG calculation one normally chooses interval [ D, D] with D = and Fermi energy situated in middle of band at ε F =. Within DMFT, however, band structure changes after every iteration of self-consistency procedure. Hence it is more convenient to define all energies in units of D= using bandwidth of non-interacting density of states (see Sec..3.3), which remains

32 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) fixed during entire calculation. Neverless, for simplicity, we will adhere to standard NRG convention with a half bandwidth of D = for subsequent derivations. The third part of Hamiltonian Ĥcpl couples two subsystems. Such an interacting quantum-mechanical system can display a variety of interesting physics depending on system parameters and energy scale under consideration. For Kondo Hamiltonian re is a crossover from a high temperature regime with free impurity spin to a strong coupling regime with completely screened spin below Kondo temperature. As a renormalization group approach, NRG method allows to solve all relevant energy scales by performing successive diagonalization from high to low energy scales (see Sec. 3..4). The special about Numerical Renormalization Group is that this can be done completely nonperturbatively in all system parameters. Thus NRG method is highly suitable as an impurity solver within DMFT, in which coupling of impurity to bath can have arbitrary strength and structure, since it is determined self-consistently by mapping a lattice model of strongly correlated electrons onto quantum impurity model. Since K.G. Wilson s first application of NRG method to Kondo problem in 975 [5], it was continuously improved with a wide range of applications, including description of quantum dots and also DMFT (see e.g. R. Bulla, A. C. Hewson and Th. Pruschke 998 [6], R. Bulla 999 [7], R. Zitko and Th. Pruschke 9 [8] for one-band Hubbard model and T. Pruschke and R. Bulla 5 [9], R. Peters and T. Pruschke [], R. Peters and N. Kawakami [] for two-band Hubbard model). In first part of this chapter we give an overview over basic NRG technique following paper of H. R. Krishna-murthy, J. W. Wilkins and K. G. Wilson [] and review of R. Bulla, T. A. Costi and T. Pruschke [3]. The second part concentrates on more recent developments in field of NRG based on a matrix product state (MPS) formulation as used in this work (see A. Weichselbaum and J. von Delft 7 [5], A. Weichselbaum [4] and [5] and dissertation of W. Münder [4] for an overview). 3. Basic method The starting point of Wilson s NRG approach is to coarse-grain continuous bath spectral function in energy space. This leads to a model with a discrete set of states that can be mapped onto a semi-infinite tight-binding chain with exponentially decaying couplings, called Wilson chain. The Wilson chain Hamiltonian is solved numerically by iterative diagonalization. (See Fig. 3. and Fig. 3. for a sketch of NRG steps.) It is absolutely crucial to choose a logarithmical discretization scheme to rewrite Hamiltonian, since this yields an exponentially enhanced low-energy resolution, as compared to a linear discretization, and allows to resolve even lowest relevant energy scale of system. 3.. Single impurity Anderson Hamiltonian (SIAM) For simplicity we restrict our discussion of NRG method to single impurity Anderson Hamiltonian (SIAM) in Eq. (.33). A pictorial representation of model is given in panel (a) of Fig. 3.. The effective bath of non-interacting fermions is described by annihilation (creation) operators ĉ ( ) kσ, which fulfill standard fermionic anticommutation relation {ĉkσĉ k σ } = δ kk δ σσ. The imaginary part of hybridization function, Γ(ω) = π k V k δ(ω ε k), completely

33 3.. BASIC METHOD encodes influence of bath onto impurity. So any reformulation of Hamiltonian, which leaves hybridization Γ(ω) unchanged, leads to same physics with respect to impurity of SIAM. As demonstrated for a constant hybridization in [], a possible one-dimensional energy representation of SIAM Hamiltonian Eq. (.33) is achieved by replacing discrete bath operators ĉ ( ) kσ by continuum annihilation (and creation) bath operators â( ) εσ via transformation ĉ ( ) kσ â( ) εσ ρ(ε). (3.3) ρ(ε) denotes non-interacting density of states for continuous reformulation dερ(ε). (3.4) k The transformation in Eq. (3.3) is defined such that standard fermionic anticommutation relation {â εσ, â ε σ } = δ(ε ε )δ σσ applies. Note that with this, operators â ( ) εσ acquire unit (energy), and thus only combination in Eq. (3.3) is dimensionless. The bath n covers a continuous excitation spectrum and Hamiltonian reads Ĥ = Ĥimp + dε ε â εσâ εσ + Γ(ɛ) dε σ σ π ( ˆd σâ εσ + h.c.), (3.5) with energy representation of hybridization function Γ(ε) πρ(ε)v (ε). (3.6) It is this continuous form of Hamiltonian Eq. (3.5) that needs to be discretized. 3.. Logarithmic discretization For logarithmic discretization of conduction band, K. G. Wilson [5] introduced a discretization parameter Λ >, which defines discretization points as with ɛ n > ɛ n+, leading to intervals ɛ n = Λ n, n =,,,..., (3.7) I n = { [ ɛ n, ɛ n+ ] for n < [ɛ n+, ɛ n ] for n > (3.8) with logarithmically decreasing width d n ɛ n ɛ n+ = Λ (n+) (Λ ). Typical values for Λ are.5 up to 4, depending on complexity of impurity site. In principle, continuum limit is recovered by taking Λ. The continuous energy support of bath can now be coarse-grained in terms of logarithmic intervals dε dε, (3.9) n I n 3

34 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) resulting in exact representation of coupling term Ĥcpl, Γ(ɛ) dε σ π ( ˆd σâ εσ + h.c.) = σ ( ˆd σ γ n â nσ + h.c.). (3.) n Here we have introduced â nσ γ n I n dε Γ(ε) π âεσ dεφ no(ε)â εσ, (3.) I n where γ n = I n Γ(ε) dε π (3.) represents a proper normalization, such that â ( ) nσ are standard fermionic operators, that destroy (create) particles of spin σ in bath interval I n, respectively. In principle, y can be completed into a basis set of orthonormal polynomials (labeled by integer p) â npσ = dε Φ np(ε)â εσ, (3.3) such that continuous bath operators can be written as In â εσ = np Φ np (ε)â npσ. (3.4) By construction, impurity directly couples to p = components of conduction band states only. Thus hybridization term is still represented exactly, neglecting p states for discretization of conduction band term. In contrast, bath Hamiltonian is approximated by dε ε â εσâ εσ ξ n â nσânσ. (3.5) σ nσ While in general npσ Ĥbath np σ δ pp, we only keep terms p = p = for representative energies ξ n in intervals I n : ξ n nσ Ĥbath nσ = I dεγ(ε)ɛ n I n dεγ(ε), (3.6) where npσ â npσ and denotes vacuum state. Altoger discretization procedure approximates bath Hamiltonian by only one single state per interval (see Fig. 3. (b)), which replace originally infinite many continuous states in that interval, while keeping overall coupling of impurity to bath exact, Ĥ = Ĥimp + ξ n a nσânσ + ( ˆd σ γ n â nσ + h.c.). (3.7) nσ σ n The coarse graining influences calculation of physical quantities. Firstly, it has shortcoming of (slightly) misrepresenting effective hybridization of bath. This, however, 4

35 3.. BASIC METHOD can be systematically improved upon ([6] and [7]). Second and more importantly, Wilson chain lost its character as a true rmal bath by eliminating a large degree of fermionic levels. A general method to increase resolution of physical quantities and to reduce artificial oscillations due to discretization is to average over obtained data of various (N z ) discretization grids for fixed Λ ( z-averaging ) [8]. To this end, one introduces a parameter z, uniformly distributed in (, ], that shifts original discretization points: ɛ z = and ɛ z n = Λ n z for n =, 3,.... (3.8) However, z-averaging procedure cannot replace true continuum limit Λ by taking limit N z. In this work, we will mainly use a slight variation of grid given in Eq. (3.8) and Eq. (3.7) to discretize Hamiltonian (see Sec. 4..). Moreover, we exploit fact that within DMFT we only have to calculate self-energy with NRG. As R. Bulla pointed out in 998 [6], this quantity can be determined via ratio of a two-particle and a one-particle retarded Green s function (see Sec 4.). This self-energy-trick leads to a reduction of systematic errors for calculation of self-energy within NRG and significantly reduces artificial oscillations, dispensing with need for a discretization correction. As we will discuss in detail in Sec. 4., an appropriate choice of discretization intervals is absolutely essential within DMFT. As one has to handle hybridization functions of arbitrary shape, concrete way of discretizing Hamiltonian can have strong impact on overall efficiency and accuracy of NRG calculations. Therefore we will also investigate an alternative discretization scheme in Sec. 4.., which was suggested by R. Zitko in 9 [7] and takes actual shape of hybridization function into account. It is based on requirement to exactly reproduce conduction band density of states after z-averaging of bath alone Mapping onto Wilson tight-binding chain The next step of NRG procedure is an exact mapping of discrete Hamiltonian onto a semi-infinite tight-binding chain. The coupling term of Hamiltonian directly leads to introduction of a new conduction band state, with normalization constant ˆf σ = γ n â nσ, (3.9) ξ ξ = n γ n = n dε Γ(ε) π, (3.) representing all bath states that are directly coupled to impurity. This motivates a complete unitary transformation from set of bath operators â ( ) nσ to a new set of mutually orthogonal ( ) operators ˆf nσ that exhibit only nearest-neighbour coupling []. The transformation can be achieved by a standard tridiagonalization procedure (e.g. Lanczos algorithm, see Sec. 4..). As a result, we obtain Wilson chain Hamiltonian: Ĥ = Ĥimp + ξ ( ˆd ˆf σ σ + h.c.) + σ [ε n ˆf nσ ˆfnσ + t n ( ˆf ˆf nσ n+σ + h.c.)]. (3.) σ,n= 5

36 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) D : ; < =>? Figure 3.: Sketch of main steps for mapping an impurity model onto a Wilson chain. Here we consider SIAM with a constant hybridization function Γ (or k-independent V, respectively). Panel (a) shows impurity model, where an impurity (red) is coupled to a non-interacting fermionic bath via hybridization V. The first step of Wilson s NRG approach is to coarse-grain energy band by logarithmic discretization and to integrate out bath degrees of freedom in every interval. This leads to a model with a discrete set of states (blue circles) as depicted in (b), that still features same coupling to impurity as in original model. The discretized Hamiltonian is n mapped onto a semi-infinite tight-binding chain shown in panel (c). The exponentially decaying couplings of Wilson chain are illustrated by thinning bonds between different sites. The coupling of impurity to first Wilson chain site is again given by V. Panel (d) presents physical interpretation of Wilson chain sites as a sequence of shells with exponentially growing size, centered around impurity. Figure adapted from [4]. ( ) The very first site of this semi-infinite tight-binding chain represents impurity ( ˆd σ ), which ( ) is coupled to first site of conduction band ( ˆf σ ) with strength ξ. All or sites of chain belong to bath and couple to ir next neighbor via hopping matrix elements t n. The on-site energies are given by ε n, where ε n = for all n if Γ(ω) = Γ( ω). For a constant hybridization Γ, one can derive an analytical expression for t n [5]: t n = ( ( + Λ )( Λ n ) Λ n Λ n 3 ) Λ n n ( ) + Λ Λ n. (3.) For a non-constant hybridization Γ(ω), t n is calculated numerically. Eir way, due to underlying logarithmic coarse graining, hopping matrix elements fall off exponentially for large n: n t n Λ n. (3.3) Note that we will later refer to t n Λ n+ for large n, which is convention. This behavior is a direct consequence of logarithmic discretization Eq. (3.7) and is of major conceptual importance for whole method to work. The tight-binding chain with exponentially decreasing couplings is depicted in Fig. 3.(c). While operator ˆd ( ) σ of impurity part of SIAM Hamiltonian Eq. (.33a) acts on 6

37 3.. BASIC METHOD impurity site, operator f nσ ( ) annihilates (or creates) a particle at site n of Wilson chain. The physical interpretation of se Wilson chain sites is a sequence of shells in bath, centered around impurity in position space (see Fig. 3. (d)). The wave function corresponding to Wilson chain operator ˆf ( ) σ shows a maximum closest to impurity. Therefore ˆf σ ( ) annihilates (or creates) electrons at location of impurity. Analogously, operators ˆf nσ (n > ) act on shells furr away from impurity. The size of spherical shells is growing exponentially, which is related to exponential decrease of hopping matrix elements [5]. In case of SIAM Hamiltonian with only n c = different types of electrons (spin up and spin down), every site of Wilson chain can display four different states σ n = {,,, }, an unoccupied, a singly occupied with spin up or spin down and a doubly occupied site n. Therefore Hilbert space for one site has dimension d = (nc=) = 4. If we consider a more complex model with two impurity levels, where each couples to a separate conduction electron band, site dimension increases to d = (nc=4) = 6. Altoger, size of Hilbert space for whole chain Hamiltonian grows like d imp d n+, when considering Wilson chain up to and including site n. Here d imp represents local dimension of impurity site. Therefore this many-body system can generally not be solved exactly. Yet, re is a controlled numerical procedure to tackle such problems, presented in following Iterative renormalization group procedure The Wilson chain Hamiltonian Eq. (3.) can be solved using an iterative renormalization group (RG) procedure invented by K.G. Wilson [5]. This introduces a series of Hamiltonians ĤN: Ĥ N = Λ N Ĥ imp + ξ ( ˆd ˆf σ σ + h.c.) + σ N σ,n= ε n ˆf nσ ˆfnσ + N σ,n= t n ( ˆf nσ ˆf n+σ + h.c.) (3.4) with a scaling factor Λ N, canceling N-dependence of t N, which will be useful for investigation of flow diagrams in Sec. (3..5). The original Hamiltonian is n obtained as From Eq. (3.4) one arrives at recursion relation Ĥ N = ΛĤN + Λ N σ Ĥ = lim N Λ N ĤN. (3.5) ε N ˆf Nσ ˆf Nσ + Λ N which can be interpreted as a renormalization group transformation R: t N ( ˆf ˆf N σ Nσ + h.c.), (3.6) σ Ĥ N = R(ĤN ) (3.7) (see Sec. (3..5) for a discussion of RG aspect). Using Eq. (3.6) and Ĥ = Λ [ Ĥ imp + σ ε ˆf σ ˆf σ + ] ξ ( ˆd ˆf σ σ + h.c.) σ (3.8) as starting point, we can solve Wilson chain Hamiltonian Eq. (3.5) by iterative diagonalization. 7

38 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) First we diagonalize Hamiltonian Ĥ exactly and obtain a set of eigenstates and eigenenergies. Then we continue with Ĥ. For a general RG step from site N to N, one applies strategy in Fig. 3.: (a) (b) (c) (d) (e) E N s E N Es N kept s Figure 3.: General RG step from site N to N: The many-particle spectrum Es N of iteration N-, shown in (a), is rescaled by Λ, as depicted in (b). Then a new site N of Wilson chain is added, lifting degeneracy of eigenenergies Es N by diagonalization of new system. This leads to eigenenergies Es N with respect to site N in (c), which are shifted in (d) such that new ground state has zero energy. In (e) state space of shell N is truncated with respect to a certain energy E keep, indicated by dashed red line. (a) In preceding iteration ĤN was diagonalized as ĤN s N eigenenergies Es N and eigenstates s N. = Es N s N with (b) The first term in recursion relation Eq. (3.6) rescales eigenenergies Es N spacing of order, by Λ. with level (c) Then a new site of Wilson chain is added via second and third term of Eq. (3.6), which can be regarded as a perturbation of order Λ <, lifting degeneracy of eigenenergies Es N. As new site is represented by a basis set σ N of dimension d, Hilbert space for ĤN grows by a factor of d, leading to an enlarged basis set s, σ N = σ N s N for matrix representation of Ĥ N. Diagonalization of this matrix yields new eigenvalues Es N with corresponding eigenstates s N, which are connected to old states s N via an unitary transformation A [N] : s N = σ N s[a [σ N ] ] ss σ N s N. (3.9) A [N] stands for all of d matrices [A [σ N ] ] ss = ( s N σ N ) s N. This specific formulation of unitary transformation exhibits structure of so called matrix product states (MPS). It will turn out in Sec. 3. that an implementation of whole NRG procedure in terms of MPS is quite powerful. (d) In a next step, ground state energy of iteration N is set to zero, for convenience. 8

39 3.. BASIC METHOD (e) As already pointed out, it is numerically not feasible to keep all eigenstates and eigenenergies during complete iterative procedure as Hilbert space grows exponentially with added sites of Wilson chain. Hence, we have to truncate state space from a certain iteration N onwards (typically N 5 for one-band model) and fix size of Hilbert space, which can be done by keeping only D lowest lying eigenstates. This truncation scheme is motivated by fact that an additional site can be seen as a perturbation of relative strength Λ. In perturbation ory, influence of high-energy states on low-energy spectrum is small, if perturbation is weak compared to energy of high lying levels. Thus, basic NRG assumption of energy scale separation can be justified for sufficiently large Λ (typically Λ ). Instead of keeping a fixed number of states, we can also truncate with respect to a fixed rescaled energy E keep, which corresponds more to idea of energy scale separation. Each site of Wilson chain features a characteristic energy scale or resolution ω N = aλ N (3.3) with a chosen( such that rescaled couplings approach unity for sufficiently large Wilson tn ) chains: lim N ω =. So E keep is kept fixed and is given in units of ω N N for different sites N. Indeed, re is a priori no guarantee for a specific truncation parameter D or E keep to give good results. One way to test validity of truncation scheme is to vary truncation parameter and observe its influence on outcome. In this work however, we use a quantitative criterion (called discarded weight) to analyze convergence for a given NRG calculation, which can be extracted from same NRG run (see Sec and A. Weichselbaum [4] for more details). We stop our iterative procedure by discarding all states at Wilson chain site N max, which is determined through required energy resolution for actual physical problem. E N s 3... N Figure 3.3: Sketch of unscaled energy levels of different Wilson chain sites with ground state energies set to zero. While kept states are successively refined with every step of iterative procedure discarded states remain degenerate. Adapted from [4]. 9

40 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) Remarkably, it is possible to construct a complete basis from all discarded states of all iterations as will be presented in Sec. 3..3, which can n be used to calculate physical quantities. Altoger iterative procedure, described here, leads successively to an increasing resolution of low energy spectrum (described by kept states), while discarded high-energy states remain degenerate (see Fig. 3.3) Renormalization group flow As indicated by Eq. (3.7) whole iterative procedure can be interpreted as a RG transformation. In general, a RG transformation R mapps a certain Hamiltonian Ĥ( K), depending on a set of parameters K, on a Hamiltonian of same form with renormalized parameters K : R(Ĥ( K)) = H( K ) [3]. The parameters normally display a RG flow to certain fixpoints. However in case of NRG, Hamiltonian does not have same form before and after RG transformation, except for case that system reached a stable fixed point (up to even-odd alternations). Neverless it is possible to extract information about physics of impurity model by analyzing flow of (rescaled) eigenergies Es N along Wilson chain sites N. Depending on specific model energy flow diagram can show different fixed points regimes, which are connected to different physical behavior of system. For SIAM with constant hybridization function, three different regimes can be found: free orbital regime for very small N, local moment regime for intermediate N and strong coupling regime below T K, which is also called Fermi-liquid fixed point (see [] for a detailed discussion and Fig. 3.4 for a plot). Within DMFT flow diagram also gives a hint to physics of system. For one-band Hubbard model, for example, a Fermi-liquid fixed point is reached in metallic regime (see Sec. 5 for more details on one-band Hubbard model). 4 E s N (even spectrum) 3 N(T K ) U=.8, µ=.4, Γ=., Λ= N Figure 3.4: Energy flow diagram for symmetric SIAM with constant hybridization Γ =., U =.8, ε d = U/, Λ = and E keep = 7. Here only even NRG iterations are plotted as fermionic finite-size systems show even-odd oscillations in ir many-particle spectrum (a similar picture is obtained for odd iterations). For first few sites we find free orbital regime where impurity is hardly affected by coupling to bath. Between sites N and N 35 impurity still features a definite spin but exhibits spin fluctuations. The Kondo temperature (T K = ) is associated with crossover (at around N = 5) to strong coupling regime where local impurity moment is completely screened by conduction band electrons. U=.8, ε d =.4, Γ=., B=, Λ=, N=, A Λ, D=86 3

41 3.. NRG IN MATRIX PRODUCT STATE (MPS) FORMULATION 3. NRG in Matrix Product State (MPS) formulation During whole iterative RG procedure, we have to keep track of all calculated kept and discarded states and corresponding eigenenergies. Eq. (3.9) suggests to use unified framework of matrix product states (MPS), as each step of iterative diagonalization can be expressed as a basis transformation in terms of A-matrices. For a general introduction to MPS we refer reader to [9]. In following, we want to denote operators with a hat and ir matrix representation without hat, respectively. 3.. Basis states in terms of MPS Reinserting definition of basis states Eq. (3.9) for all iterations up to N, this expresses an arbitrary eigenstate s X N in terms of a sum over a product of matrices: s X N = [A [σ N ] KX ] ss σ N s K N (3.3) σ N s = σ N...σ σ imp [A [σ ] KK... A[σ N ] KK A[σ N ] KX ] σ imp s σ N σ N σ σ imp (3.3) with X = K or D for kept or discarded states, respectively. σ imp represents impurity basis. The A-matrices, written as [A [σ N ] KX ] ss, are three dimensional tensors in space spanned by s K N, σ N and s X N. As shown in Fig. 3.5, MPS formulation has an intuitive diagrammatical representation. impi A [] KK [N ] A KK A [N] KX s i X N i N i N i si K N Figure 3.5: Diagrammatical representation of Eq. (3.3). The red part depicts basis transformation Eq. (3.3). The indices of corresponding A-matrix appear as legs: s to left, σ N at bottom and s to right. In red box s K N is represented in terms of A-matrices as in Eq. (3.3). Connected legs symbolize contractions over related indices. Figure adapted from [4]. The MPS structure of Eq. (3.3) also makes evaluation of operators more transparent. Consider Wilson chain operators in Eq. (3.6). We only store matrix representation [f ( ) Nσ,loc ] σ N σ N in terms of local state space σ N, and transform it into required basis by means of A-matrices: [f ( ) Nσ ]KK ss = K ( ) N s ˆf Nσ s K N = [A [σn ] KK ] s s[f ( ) Nσ,loc ] σ N σ σ N σ N N [A[σ N ] KK ] s s. (3.33) s 3

42 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) In addition, re are several or numerical issues, like handling of fermion signs, emanating from ir anti-commutation relations (see [4] for more details), and implementation of symmetries (see next section). All in all MPS language allows a very elegant and transparent numerical treatment of NRG method. 3.. Abelian and non-abelian symmetries For numerically expensive models, like multi-band models, where required dimension of kept state space can get extremely large, it is absolutely crucial to exploit as many symmetries of model Hamiltonian as possible. Only recently, a general framework for implementation of non-abelian and abelian symmetries in context of matrix-product and tensor network states was established by A. Weichselbaum [5]. It uses a computationally straightforward unified tensor representation for quantum symmetry spaces, called QSpace. Basis states labeled in terms of abelian symmetries, like particle (or charge) conservation, lead to a block-diagonal structure in matrix representation of Hamiltonian and subdivide general operators into symmetry sectors due to well-defined selection rules. As a result, all nonzero matrix elements are organized within a few dense data blocks, reducing numerical effort. In presence of non-abelian symmetries (like SU() spin ) matrix elements of se data blocks are not independent of each or and it is possible to furr compress m. To this end, one introduces qn; q z to label state space in terms of symmetry eigenbasis. q (q, q,..., q ns ) are quantum (q-) labels for irreducible representation of symmetries S λ (λ =,,..., n s ) of Hamiltonian Ĥ, with [Ĥ, Sλ α] = and [Sα, λ Sα λ ] = for λ λ, resulting in symmetry blocks q. Sα λ is a specific generator for symmetry S λ. The multiplet index n stands for a particular multiplet within a specific set of conserved quantum numbers, q. The internal structure of each multiplet in q is given by additional quantum labels qz λ (called z-labels), which are completely determined by symmetries under consideration. Using this state space label structure, basis transformation Eq. (3.9) can be written as Qñ; Q z = Qn;Q z ql;q z ( A [q] ) [l] Q Q nñ C[qz] Qn; Q Q z Qz z ql; q z, (3.34) where Qn; Q z represents basis states of Wilson chain and ql; q z indicates local state space of one specific site. The basic observation is now that A-tensors are split into ( two parts. A [q] ) [l] Q Q acts only on multiplet level, which leads to a strong size reduction of nñ A-tensors. The internal multiplet structure, defined by z-labels, is fully taken care of by Clebsch Gordan coefficients C [qz]. Q z Qz In description of operators, Clebsch Gordan coefficient space also factorizes. Based on Wigner Eckart orem, matrix representation of a specific irreducible operator set ˆF q reads Q n ; Q z ˆF ( q q z Qn; Q z = F [q] ) [] QQ nn C[qz] Q zq (3.35) z ( with reduced matrix elements F [q] ) [] QQ = Q n ˆF q Qn on multiplet level. The superscript [] indicates that we consider a single irreducible operator set. nn So all tensor objects, which are relevant for NRG calculations can be split into two (rank-3) structures, operating in reduced multiplet space and Clebsch Gordan coefficient space, 3

43 3.. NRG IN MATRIX PRODUCT STATE (MPS) FORMULATION respectively. As final data structure is same for both spaces, a general tensor representation (of arbitrary rank), QSpace, is introduced, which allows a well-organized, unified treatment of all NRG operations in terms of symmetries (see [5] for all details). The QSpace formulation is valid for general non-abelian symmetries (such as SU(N)), symplectic symmetry group, point symmetries and also abelian symmetries. Thus, a maximum number of symmetries can be exploited for a certain model which leads to an enormous gain in numerical efficiency Complete basis sets and full density matrix NRG As already mentioned before, matrix product states, obtained from iterative diagonalization, do not span whole Hilbert space of chain Hamiltonian as states have been discarded on way. However, F. B. Anders and A. Schiller pointed out that complete many-body basis sets can be constructed out of discarded states of all iterations, which, importantly, also represent approximate eigenstates of full Hamiltonian (see [3] and [3]). Therefore, it is possible to calculate physical quantities like spectral functions very accurately (see [5] for a detailed discussion). These complete basis sets are also easily implemented in QSpace formalism introduced above. Up to and including site N, we have a complete set of eigenstates { s K N } for ĤN if N N, as for se iterations no states have been truncated. If we want to span complete Fock space F Nmax of full chain, we have to include all subsequent sites N + to N max by a set of d (Nmax N) degenerate environmental states e N σ Nmax σ N+, build out of ir local state spaces. Then we can write complete Fock space as (d impd Nmax+ ) = se se K N K N se (3.36) with se K N = e N s K N for N N. Analogously, we can define states se X N = e N s X N (3.37) for N < N < N max with X = K, D for sites of Wilson chain, where states have already been discarded. For a general site N, se states can be regarded as approximate eigenstates of full chain Hamiltonian ĤN max, Ĥ Nmax se X N EN s se X N, (3.38) with eigenenergies Es N still d (Nmax N) -fold degenerate. Eq. (3.38) is referred to as NRGapproximation, emphasizing basic NRG argument of energy scale separation in this context. The accuracy in energy is determined by characteristic energy scale ω N of shell N. By construction, discarded states of different shells are orthogonal to each or, D M se s e D N = δ MNδe N e Nδ ss, (3.39) and also to kept states of same shell and shells still to follow: { K M se s e D N = M N δ en e N [A[σ M+] KK... A [σ N ] KD ] ss M < N. (3.4) 33

44 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) The overlap of kept states with discarded states of later shells is determined by A-matrices that map kept states on discarded ones. Using se considerations, we can now introduce complete basis sets for full Fock space F Nmax, given by discarded states of all iterations, (d impd Nmax+ ) = N max N>N se se D D N N se, (3.4) where we have defined all states of last iteration N max as discarded states, for convenience. For calculation of physical quantities, like spectral functions, we need to perform rmal averages = Tr[ˆρ...] with full rmal density matrix ˆρ = e βĥ/z, which can be expressed in terms of complete basis sets using NRG-approximation Eq. (3.38) [5]: N max ˆρ se D e βen s N N>N se Z D N se = w N ˆρ [N] DD (3.4) N>N with β = k B T. The eigenenergies EN s represent non-rescaled energies relative to a common energy reference. So full rmal density matrix can be written in terms of properly normalized rmal density matrices for discarded states of shells N > N, with Z N = D s e βen s relative weight ˆρ [N] DD = Z N s e βen s se D N D N se, (3.43) such that Tr[ρ [N] DD ] =, that enter a sum over all shells N > N with w N = Z Nd (Nmax N) Z and N max N>N w N =. (3.44) The factor d (Nmax N) in w N takes account of degeneracy of environmental states. In contrast to widely used single-shell approximation w N = δ NNT (with N T defined by Wilson chain sites with characteristic energy scales of order of temperature), relative weights w N in full (rmal) density matrix (FDM) approach [5] are peaked around shell, corresponding to energy scale of temperature, with a finite width of several (5 to ) iterations. Therefore, given a long enough Wilson chain, all relevant shells for a given temperature are automatically included. Thereafter, Wilson chain effectively terminates on its own. Calculating physical quantities, complete basis sets avoid double counting of basis states as occurred in previous methods. General local operators ˆB (acting on sites up to N ) can be simply expressed in terms of complete basis sets by F. B. Anders and A. Schiller (see [5, Eq. (6)] and [4, Sec. 4.5.]). Eq. (3.4) can be fully taken care of within NRG. When tracing out partial environments from earlier shells, se density matrices become reduced density matrices, similar to ones introduced by W. Hofstetter in his density-matrix (DM)-NRG [3]. The latter, however, was still based on heuristic patching schemes. Moreover, FDM-NRG also allows to fully take care of fermionic signs [3] Calculation of spectral functions We can now use complete basis set and FDM-NRG approach for calculation of spectral functions to describe impurity dynamics in rmal equilibrium. The general definition 34

45 3.. NRG IN MATRIX PRODUCT STATE (MPS) FORMULATION of a spectral function reads A BC (ω) = and corresponding Lehman representation A BC (ω) = a,b dt π eiωt ˆB(t)Ĉ T (3.45) e βea b Ĉ a Z a ˆB b δ(ω E ba ) (3.46) with Z = a e βea and E ba = E b E a. In FDM formulation we get A BC (ω) = N w N A BC N (ω). (3.47) Expressed in terms of discarded states only, it is possible to calculate A BC (ω) by performing a single backward run from last Wilson chain site to first one, while all eigenstates and eigenenergies were calculated before in a forward run (see [4] for a detailed explanation). This way, sum rules, as dωa BC (ω) = ˆBĈ T, are fulfilled to high accuracy (.% after broadening of discrete data) [4]. After a NRG run we obtain Eq. (3.47) as a sum over discrete data points (ω j, a j ): A raw (ω) = j a j δ(ω ω j ). (3.48) To broaden this sequence of δ-peaks we define a broadening kernel K(ω, ω ) representing a log- Gaussian function with frequency-dependent width (dealing with fact that we have less dense data at large frequencies due to logarithmic discretization) above a certain frequency ω. Below ω, a smooth transition from log-gaussian to a regular Gaussian of width ω sets in to get finite values for ω. So smooned spectral function is defined by A(ω) dω K(ω, ω )A raw (ω ) (3.49) with ( K(ω, ω ) = Θ(ω, ω ) e log ω /ω πσ ω σ σ 4 ) for ω ω (3.5) and a broadening parameter σ. A detailed discussion of broadening kernel is given in appendix of [5]. For DMFT application it is important to choose σ as small as possible to carve out complete structure of hybridization function during DMFT self-consistency procedure. However very strong artificial oscillations should be avoided as well, because se are possibly not completely corrected by self-energy-trick and can grow during DMFT iterations. See Fig 3.6 for an example of broadened raw data for spectral function of one-band Hubbard model. For calculations within this work, we mainly apply following strategy. First, we roughly reach convergence within DMFT self-consistency procedure by applying no z-averaging and larger broadening. Then, we refine structure of hybridization function by performing some more iterations using z-averaging and an appropriate smaller value for σ until convergence is also reached for se parameters. 35

46 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) spectral function discrete data for A(ω) A(ω) A(ω)*D ω/d Figure 3.6: Smooned spectral function (local density of states) for particle-hole-symmetric oneband Hubbard model with U/D =, µ/d = U/, T/D, N z = 4 and Λ =. In grey raw data is plotted as Araw(ω) 8 ω. The blue curve represents smooned spectral function A(ω). With a broadening of σ =. spectral function still shows artificial oscillations around ω =, as can be observed in inset. These artefacts can be reduced by applying self-energy-trick (see Fig. 4.) Discarded weight in NRG To end of chapter about NRG method, we want to introduce discarded weight mentioned during description of iterative diagonalization procedure, which can be regarded as a control parameter that indicates, if whole NRG run was successfull. The numerical treatment of iterative RG procedure is based on truncation of high-energy states, rationalized by energy-scale separation along Wilson chain. However, it is not clear a priori how many low-energy states one should keep for an appropriate description of lowenergy spectrum still to follow. Motivated by density matrix renormalization group, where an a priori truncation of state space can be defined through discarded weight in its reduced density matrix, a similar criterion within NRG was introduced by A. Weichselbaum in [4]. It is based on question how important states kept some iterations earlier have de facto been for specification of ground state at iteration N. To analyze this influence quantitatively, we consider fully mixed density matrix of g N -fold degenerate ground state space G at iteration N : ˆρ,N s g N N N s (3.5) s G with a Schmidt rank equal to g N and trace out its N smallest energy shells to arrive at reduced density matrix ˆρ [N;N ] (N = N N ), which can be expressed in terms of kept states of iteration N (see Eq. (3.53)). To trace out one site, a so called backward update is performed (see Fig. 3.7): ˆρ [K] n = ( A [σ n] KX ρ[x] n A [σ ) n] KX s n s s n s n n σ n s n n s ˆP ˆρ [X] n (3.5) 36

47 3.. NRG IN MATRIX PRODUCT STATE (MPS) FORMULATION with an arbitrary density matrix ˆρ [X] n s ns n X ρ [X] s ns n s n n s and X = K, D. s n s n A [n] KX s n [K] n [X] n s n s n A [n] KX s n Figure 3.7: Graphical representation of a backward update as given in Eq. (3.5). Figure adapted from [4]. So reduced density matrix is calculated as ˆρ [N;N ] N+N n=n+ ˆP n ˆρ,N D ρ [N;N] ss s K N ss K N s (3.53) with N ceil[log(d)/ log(d)] and N N max as number of sites until Schmidt rank has grown to full dimension D of kept space. In principle, N is also number of NRG iterations at beginning of iterative RG procedure where no states are truncated. For furr investigations reduced density matrix is diagonalized: ˆρ [N;N ] r N;N = ρ [N;N ] r r N;N. (3.54) The eigenvalues ρ [N;N ] r indicate importance of a specific linear combination r N;N of NRG eigenstates s K N for low-energy physics of later iterations. The energies of eigenbasis r N;N are given by expectation values E [N;N ] r r N;N ĤN r N;N. (3.55) It can be shown that resulting data (E r, ρ r ) are clearly correlated and small weights ρ r correspond to high energies E r, as intuitively expected (see [4] for a detailed discussion). The largest discarded weight, which describes weight missing due to discarded states at iteration N, can n be well approximated by smallest weights of reduced density matrix ˆρ [N;N ] in kept space. Therefore an a priori measure for accuracy of truncation criteria at iteration N is defined through average weights ρ [N;N ] r, that are obtained for highest energies E [N;N ] r in kept space [4]: ε Dχ N r [N;N E ] r ( χ)max(e [N;N ] with truncation at iteration N r ) without truncation at iteration N ρ [N;N ] (3.56) with typically χ.5. The overall discarded weight for whole chain is given by largest discarded weight of all iterations: ε D χ max N (εdχ N ). (3.57) 37

48 CHAPTER 3. NUMERICAL RENORMALIZATION GROUP (NRG) In DMFT calculations we will have to deal with numerically expensive models as well as with models where energy scale separation is not guaranteed due to non-constant hybridization functions or modified discretization schemes. Therefore discarded weight is especially helpful to judge accuracy of NRG data within DMFT. If it is small (< ), NRG run is reliably converged. To conclude, NRG program, developed by Andreas Weichselbaum, is implemented in a highly systematic way. The MPS formulation is based on complete basis sets and correlation functions are calculated with full density matrix, enhancing accuracy of NRG results. Furrmore, a quantitative convergence measure, discarded weight, is on hand to estimate quality of a NRG run. This is especially advantageous for DMFT applications where previous checks, like repeating a single run with different parameters, is quite borsome. Besides, our NRG program is first (and up to now only program) to use arbitrary non-abelian symmetries and thus can tackle physical problems that would not have been feasible before. 38

49 Chapter 4 Combining DMFT and NRG In this chapter we want to discuss peculiarities that emerge when NRG is used as an impurity solver within DMFT. The quantity of interest that has to be generally calculated with impurity solver is one-particle self-energy. Instead of using Dyson equation as in or approaches, it has proven to be considerably more accurate within NRG to write self-energy as ratio of a two-particle and a one-particle Green s function as introduced by R. Bulla in 998 [6]. This self-energy trick was already mentioned in derivation of self-consistency procedure. In following section we want to give a detailed explanation. Anor important issue is specific form of discretization grid for NRG run. For a constant hybridization function conventional logarithmic discretization grid leads to exponentially decaying hopping matrix elements along Wilson chain. The DMFT self-consistency procedure, however, yields a frequency-dependent input hybridization function Γ(ω) at each DMFT iteration, for which standard discretization scheme leads to Wilson chain couplings that do not necessarily decay exponentially over a certain range of sites, calling basic NRG assumption of energy separation into question. Therefore we want to debate some variations of conventional logarithmic discretization scheme in Sec Calculation of self-energy At a first glance a natural choice to calculate self-energy of effective quantm impurity model is to use Dyson equation Σ(ω) = G imp (ω) G imp (ω). While noninteracting retarded Green s function G imp (ω) = ω ε d (ω) is known exactly, interacting retarded Green s function of impurity model G imp (ω) can be calculated with impurity solver NRG and will refore display numerical errors, depending on discretization grid and broadening of raw data. Yet, taking difference of an exact and an error-prone quantity can be detrimental to accuracy of results, especially when outcome is in order of numerical error. This can happen for a Fermi liquid, where self-energy approaches zero for T close to Fermi level, since Im Σ(ω ) T. So numerical calculation is in this case less reliable for most relevant frequency range. Within DMFT it is possible that se numerical errors even increase due to iterative procedure. Therefore we will use a different approach to calculate self-energy with NRG impurity 39

50 CHAPTER 4. COMBINING DMFT AND NRG solver. In 998 R. Bulla deduced an alternative expression for self-energy of SIAM from an equation of motion ansatz [6]. It can be written as ratio of two correlation functions, with on-site correlation function and two-particle retarded Green s function Σ(ω) = F (ω) G(ω), (4.) G(ω) = ˆd ˆd ω (4.) F (ω) = [ ˆd, ĤU] ˆd ω. (4.3) Note that spin index is omitted and G(ω) G imp (ω). Ĥ U is interaction part of local impurity Hamiltonian Ĥimp. For SIAM we have ĤU = U ˆd ˆd ˆd ˆd. In Appendix C we show a detailed derivation of self-energy expression for a general multi-band Anderson model. Using FDM-NRG approach we can compute imaginary parts of retarded Green s functions G(ω) and F (ω) and get real parts of corresponding smooned spectral functions via a Kramers-Kronig transformation Re G(ω) = π P Im G(x) dx x ω. (4.4) The self-energy is n expressed as ratio of two quantities, that have been obtained numerically on equal footing. This results in a reduction of systematic to only relative errors, that enter DMFT self consistency loop. spectral function A(ω)*D discrete data for A(ω) A(ω) A(ω) after self energy trick ω/d Figure 4.: Spectral function for particle-hole symmetric one-band Hubbard model with U =, µ = U/, T =, N z = 4 and Λ =. In grey raw data is plotted as Araw(ω) 8 ω. The blue curve represents smooned spectral function A(ω) with a broadening of σ =.. The red curve is calculated with self-energy-trick as A(ω) = π Im G(ω), where G(ω) = F(ω) ω ε d (ω) Σ(ω) and Σ(ω) = G(ω). It can be clearly seen that artificial oscillations around Fermi level are removed and high-energy features come into sharper relief. 4

51 4.. THE LOGARITHMIC DISCRETIZATION GRID WITHIN DMFT Artificial oscillations, which arise due to logarithmic discretization grid, as well as broadening artefacts, like suppression of high-energy peaks in local density of states, widely cancel out using Bulla s self-energy term. This can be viewed in Fig. 4.. With R. Bulla s expression for self-energy, we can now revise self-consistency procedure introduced in Chapter (see Fig..5). The first step of a general DMFT iteration is to use frequency-dependent hybridization function obtained from preceding iteration as input for NRG impurity solver. With impurity Hamiltonian Ĥimp known, hybridization function completely determines quantum impurity model, that can n be solved with FDM-NRG approach, yielding spectral functions for calculation of self-energy. The self-energy is output function of NRG impurity solver and can be employed in calculation of new on-site lattice Green s function and afterwards in derivation of input hybridization function for next iteration. 4. The logarithmic discretization grid within DMFT Until convergence, DMFT procedure leads, at each DMFT iteration, to a new shape of frequency-dependent input hybridization function for NRG impurity solver. So energy support of hybridization function changes for every NRG run and is not known a priori. This brings us to problem how to define discretization grid for DMFT applications. As will be shown in following, efficiency of NRG calculations is strongly affected by specific choice of discretization scheme. In general, it is difficult to find a suitable grid before NRG calculation is performed. However, R. Zitko introduced an adaptive discretization method that takes actual shape of hybridization function into account [7]. 4.. Variation of conventional logarithmic discretization grid While for a standard NRG run, bandwidth can be generally chosen as [, ] only fixed quantity during DMFT loop is non-interacting density of states. Therefore its half bandwidth of D (with D set to ) is used as energy reference in following. Note that we assume semi-elliptic Be lattice density of states, where hybridization function features same shape as spectral function (up to a global constant factor). Then however, conventional discretization grid has to be expanded over a larger range of energies, as support of hybridization function will exceed non-interacting bandwidth during DMFT loop due to strong electron-electron interactions. A possible choice is ɛ n = ±Λ n, (4.5) starting with negative n n. Since we don t know energy support of hybridization function a priori, we have to choose n small enough, to ensure that whole hybridization function is sampled by grid (e.g n = 5, 4, 3,... with n = 5). This implies presence of bath states with small couplings γ n in so called star Hamiltonian of Eq. (3.7) at high representative energies ξ n. Furr, small couplings γ n arise for intervals within charge gap of an isolator or in metallic phase near MIT (see Sec. 5). While for a flat band, (physical) hopping matrix elements t N fall off exponentially along Wilson chain, different (even numerically detrimental) behavior may arise for a frequencydependent hybridization function (with partially small weight). 4

52 CHAPTER 4. COMBINING DMFT AND NRG In following, we want to study effect of small couplings in star Hamiltonian on hopping matrix elements of Wilson chain. To this end, we have to understand in more detail Lanczos tridiagonalization algorithm, that transforms star Hamiltonian Eq. (3.7) into Wilson chain Hamiltonian Eq. (3.). Following [33, pp ], we consider bath Hamiltonian of Eq. (3.5) in form Ĥ bath = nσ ξ n nσ nσ (4.6) and normalized state f σ = ˆf σ as starting vector for recursion method. ˆfσ is defined in Eq. (3.9) and denotes vacuum state. To simplify notation, we skip spin index in following. In a first step we construct a new state, f = ( P )Ĥbath f = Ĥbath f ε f, (4.7) with ε = f Ĥbath f, that is orthogonal to f, by applying operator Ĥbath to f and substracting its projection on initial state P = f f. Then we normalize it to f = t f, (4.8) with t = f f. In a second step, we calculate state f by applying a similar procedure (including orthogonalization with respect to preceding states f and f ): f = ( P )( P )Ĥbath f = Ĥbath f ε f t f, (4.9) where ε = f Ĥbath f and P = f f. The normalized state reads f = t f, (4.) with t = f f. A general step from N to N + n yields a three term relation, f N+ = t N ( P N )( P N )...( P )Ĥbath f N = t N (Ĥbath f N ε N f N t N f N (4.a) ), (4.b) with P N = f N f N, t N = f N+ f N+ and ε N = f N Ĥbath f N, since P N Ĥ bath f N for N < N. By solving above relation for Ĥbath, we obtain a one-dimensional chain representation for bath Hamiltonian as in Eq. (3.): Ĥ bath = N [ε N f N f N + t N ( f N f N+ + h.c.)]. (4.) The corresponding unitary transformation from bath Hamiltonian Eq. (4.6) to Eq. (4.) can be written in terms of states f N as U = ( f, f,...). We now define a simplified setting to study role of small couplings in star Hamiltonian within tridiagonalization procedure. We assume a constant hybridization Γ = on energy 4

53 4.. THE LOGARITHMIC DISCRETIZATION GRID WITHIN DMFT support [, ] and conventional NRG discretization procedure that was described in Sec To simulate small weights of hybridization function, we replace original couplings γ x Λ x in intervals I x at (positive and negative) representative energies ξ x ±Λ x by much smaller values γ x ξ x. The normalized and symmetric initial state f n exhibits a component u () x γ x (up to normalization) with respect to introduced small coupled level ( at positive and negative representative energy ξ x, respectively), to which we refer as x-component from now on. To obtain f we multiply f with Ĥbath, which is diagonal in matrix representation. By symmetry (with respect to positive and negative energies), Ĥbath f is already orthogonal to f. Thus x-component of f reads u () x ξ x t u () x u () x, (4.3) with normalization constant t as in Eq. (3.). The x-component of second state is essentially given by x-component of initial state f, since it can be expressed as u () x = t (ξ x u () x t u () x ) t t u () x (4.4) according to Eq. (4.b) with N =. This implies that x-component of third state, ) u (3) x = (ξ t x u () x t u () x ) ( t t t t t u () x, (4.5) is comparable in magnitude to u () x. From Eq. (4.b) we deduce a general relation for x-component of state N: u (N+) x = (ξ t x u (N) x t N u N x ). (4.6) N So we essentially obtain following even-odd behavior as long as ξ x /t N : u (N+) x t N t N t N t N t N t N u (N ) x u (N ) x N (N ) Λ u x Λ 3 for N + even u (N ) x for N + odd. (4.7) Iterating u (N) x n leads to u (N+) x N Λ 4 u () x for N + even Λ 4 u () x for N + odd. (4.8) While x-components start at u () x for even iterations, y begin at lower values u () x u () x for odd iterations. As soon as u (Neven) x and u (N odd) x become comparable at iteration N ξx, evenodd behavior stops. N ξx can be obtained from ansatz Λ N ξx 4 u () x Λ 3N ξx 4 u () x, (4.9) with Eq. (4.3), resulting in condition ξ x t Nξx, where t N N = Λ N+, or equivalently N ξx ln ξ x ln Λ. (4.) 43

54 CHAPTER 4. COMBINING DMFT AND NRG N ξx thus corresponds approximately to Wilson chain site whose energy scale is equal to ξ x. For N > N ξx (or analogously ξ x > t N ) different behavior can be observed. Now second term of Eq. (4.6), that scales like t N /t N = Λ, can be ignored compared to first part, which contains factor ξ x t N ξx Λ N Nξx. (4.) t N t N So x-component grows exponentially for N > N ξx : u (N+) x Starting from Λ N N ξx we arrive at expression u (N) x u (N ξx x ) for even and odd Wilson chain sites N, since N N N ξx = N =N ξx t N N N ξx ξ x N = Λ N N ξx u (N) x. (4.) N N N Λξ ξx x =±.46 3 N =N ξx u (N ξx x ) Λ 4 (N N ξx ) (4.3) N = 4 (N N ξ x )(N N ξx ) γ = 55 4 (N 7 N ξ x ) x. (4.4) The behavior of u (N) x is depicted in Fig. 4. (upper panel), where we γ x =have 85 6 chosen two different small values γ x in intervals I x with corresponding representative energies ξ x = ± ξ =±.46 N x (N) u x t N t (N) N u x u x (N) 4 (b) 3 3 ξ x =± N ξ x 6 ξ x (a) N ξx N ξx N ξx N ξx N ξx (c) Γ= on [,] x γ x = 5 8 γ x = 4 γ x = 7 Γ= on [,] x γ x = 5 8 γ x = 4 γ x = 55 7 γ x = 7 γ x = 85 6 Γ= on [,] x γ x = 5 8 γ x = 4 ξ x (c) γ x = (a) γ x = 7 γ x = 85 6 (b) N3 4 5 N ξ =±.46 3 x ξ x Figure 4.: The black curves in upper and lower panel show x-components u x (N) and hopping matrix elements t N for a constant hybridization function Γ = on an energy support [-,] (D = N), ξx respectively. We use conventional discretization grid with Λ = and z =. For N ξx colored curves hybridization was set to different low values in intervals with representative energies ξ x = ± The brown fit curves, denoted by (a), (b) and (c), (c) correspond to relations Eq. (4.8) ξ x and Eq. (4.3). t N N

55 4.. THE LOGARITHMIC DISCRETIZATION GRID WITHIN DMFT For yellow curve, we explicitly show that scaling laws Eq. (4.8) (fit (a) for even and fit (b) for odd in Fig. 4. (upper panel)) and Eq. (4.3) (fit (c)) apply. For a flat band, a coupling γ x ξ x essentially leads to a hopping matrix element of height ξ x at site N ξx, where (see black curve in Fig. 4. (lower panel)). Here second domain of u (N) x, which ξ x t Nξx is characterized by Eq. (4.3), does not emerge, as can be reviewed in Fig. 4. (upper panel) for black curve. For small couplings γ x ξ x hopping matrix element of height ξ x is obtained at a later site N P, which is determined by condition u (N P ) x in Eq. (4.3), leading to N P ln γ x ln Λ + const. (4.5) Thus a peak arises in Wilson chain couplings at site N P. This is depicted in Fig. 4. (lower panel), where we have plotted hopping matrix elements for different small couplings γ x in intervals I x with energies ξ x = ± The onset N ξx and height of peaks is determined by absolute value of representative energies ξ x (red bars in Fig. 4.). The additional shift of peaks is proportional to square root of chain site, representing energy scale of corresponding coupling γ x. Naively one might have expected a linear relation between N P and γ x. However superexponential behavior for u (N) x in Eq. (4.3) results in a square root relation N P γ x. In Fig. 4.3 we plot N P versus x P ln γ x /ln Λ and can explicitly confirm Eq. (4.5). 45 ξ x =± linear fit: N P =. x P +4.6 N P x P Figure 4.3: The hybridization Γ = on an energy support [-,] is set to different low values in intervals with representative energies ξ x = ± We plot peak positions N p for corresponding small values of coupling γ x (blue circles) and find a linear dependence between N p and x P ln γ x /ln Λ (red fit) with a slope of roughly. Within DMFT we might have to deal with small couplings γ x in outermost intervals. So we now set weight of hybridization Γ to small values for intervals I x with representative energies ξ x = ±.75. For this case first domain of u (N) x, which is determined by Eq. (4.8), is missing, as immediately N > N ξx applies. Thus peaks emerge already within first 45

56 CHAPTER 4. COMBINING DMFT AND NRG sites of chain, even for merest coupling γ x 6 (see Fig. 4.4). Due to little weight of hybridization function in outermost intervals ±[, /] (colored curves), bandwidth effectively reduces to energy support [ /, /]. Therewith, it is advisable to stop discretization already at ±/, which leads to hopping matrix elements (black curve), that does not feature any peak and fall off faster. (b) (a) (b) 4 Γ= on ξ ± [ /,/] x γ = 5 x γ x = 5 8 ξ x =±.75 4 Γ= on [ /,/] x γ x = 5 γ x = 5 8 ξ x =±.75 γ x = 45 4 γ x = 45 4 t N (rescaled) t N γ x = 65 γ Γ= x = 85 6 on [ /,/] x γ = 5 x γ x = 5 8 t N (rescaled) γ x = 65 γ x = γ x = 45 4 γ x = 65 γ x = N N Figure 4.4: We consider a constant hybridization function Γ = on an energy support [-,] using conventional discretization grid (where D is set to ) with Λ = and z =. For colored curves hybridization was set to different low (partially effectively zero) values in outermost intervals with representative energies ξ x = ±.75. The black curve illustrates hopping matrix elements for a reduced energy support [-/,/]. Panel (b) depicts corresponding rescaled Wilson chain couplings. The bottom line of this study is that full numerical bandwidth should not be chosen arbitrarily large. Especially outer intervals with high absolute values for representative energies ξ x and relatively small couplings γ x lead to peaks in hopping matrix elements at beginning of Wilson chain, roughly around site N where truncation should set in. However basic NRG assumption of energy scale separation is n not guaranteed in this region and truncation should be postponed until hopping matrix elements fall off again. Thus much more states have to be kept to achieve accurate results. For more complex systems, like two-band models, calculations can become so expensive that y are hardly feasible. Note however that peaks at later Wilson chain sites are less problematic, since truncation can be performed before peak arises. At sites where Wilson chain couplings increase, we n may keep all states and truncate again afterwards. To avoid small couplings in star Hamiltonian, it is advisable to introduce a slightly altered discretization grid (which will be used throughout this sis). This grid has a fixed outer interval border, ɛ, (4.6a) that is large enough to include all high-energy weight of hybridization function. The first interval border (determined by n Z ) is n chosen such that coupling strength 46

57 4.. THE LOGARITHMIC DISCRETIZATION GRID WITHIN DMFT γ in outermost interval is maximally one order smaller than square root of overall hybridization: ɛ = Λ (n+z). (4.6b) The remaining intervals follow conventional logarithmic discretization scheme ɛ n = Λ (n+z) n = n +, n +,.... (4.6c) Note that D = and n can be negative. To motivate this altered discretization grid, we plot Wilson chain couplings for spectral function of a specific DMFT calculation (black curve in Fig. 4.6) for different choices of n. The spectral function corresponds to asymmetric one-band Hubbard model for U/D = 4, µ/d =.568 and T/D =.5 (see Sec. 5.3). As DMFT calculation is performed for an infinite-dimensional Be lattice hybridization has (up to an overall constant factor) exactly same form as spectral function. (a) (b) t N /D hopping matrix elements n =4 n =3 n = n = n = n = n = n = 3 n = 4 t N (rescaled) rescaled hopping matrix elements n =4 n =3 n = n = n = n = n = n = 3 n = N 5 5 N Figure 4.5: Panel (a) shows hopping matrix elements for spectral function of Fig. 4.6 for different widths of two outermost intervals, which are determined by interval borders ±ɛ = ±Λ (n) with z = and Λ =. Panel (b) depicts corresponding rescaled Wilson chain couplings. For black curve first interval border is chosen as ɛ = ( 4) = 6, n ɛ = 8. So we get about five intervals with weak coupling γ (compared to square root of overall hybridization). In case of n = 3 this reduces to about only three intervals, while we have no outer interval with small couplings for n =. As shown before, weak coupled intervals lead to growing hopping matrix elements in Wilson chain. This is quite extreme for case of n = 4 where outermost intervals lead to an increase of Wilson chain couplings for first few sites and inner weak coupled intervals ensure that this behavior is retained until site N = 5. So truncation is not justified for about 6 NRG iterations. For n = couplings of outer intervals contain more spectral weight and hopping matrix elements show a smooth decay, as expected. Correspondingly, convergence of rescaled hopping matrix elements to (shown in Fig. 4.5 (b)) is achieved for much earlier sites. 47

58 CHAPTER 4. COMBINING DMFT AND NRG A(ω)*D spectral function n =4 U/D=4, µ/d=.568, T/D=.5 A raw / ω * n = 4 A raw / ω * n =4 n =3 n = n = n = n = n = n = 3 n = ω/d Figure 4.6: Spectral function for asymmetric one-band Hubbard model for U/D = 4, µ/d =.568 and T/D =.5, calculated with DMFT and NRG (Λ =, N max = 35 and N z = ) using different discretization grids. To find a good choice for width of outermost two intervals, determined by interval borders ±ɛ = ±Λ (n+z), different values of n are tested. While grids n = 4 to n = yield same spectral function, grids with too broad outer intervals n = to n = 4 result in less accurate functions. In inset accurate spectral function (black curve) and outcome for n = 4 (orange curve) are plotted toger with corresponding raw data to demonstrate origin of wavy feature in deviating spectral functions. A natural question that arises at this stage is, how large can outermost interval be chosen? As depicted in Fig. 4.5 an even broader interval (n > ) results in faster decaying Wilson chain couplings. However, first interval borders n lie in region of quasiparticle peak (as shown for case n = by dashed brown vertical lines in Fig. 4.6) and grid does no longer allow a proper resolution of transition between so called Hubbard bands and quasiparticle peak at Fermi level (read Chapter 5 for an explanation of characteristic form of local spectral function for Hubbard model). This situation is presented in Fig While grids for n = 4 to n = essentially yield same spectral function (even so efficiencies of calculations differ quite strongly), a too large outer interval, that cuts into quasiparticle peak, leads to deviations from expected spectral function. In principle, one could properly adapt broadening procedure to an enlarged outer interval (in order to smooth wavy features of deviating spectral functions), however at some point too much information is lost by a too coarse grid and also accuracy of quasiparticle peak is affected. This can also be checked by plotting filling factor N of spectral function (see Eq. (5.)) for different grids, specified by n (see Fig. 4.7). Clearly N starts to deviate from a stable value for about n. Altoger altered grid as suggested in Eq. (4.6) with properly chosen n constitutes a reasonable compromise between computational effort and required accuracy. Neverless it is based on heuristic arguments. The resolution of spectral function can be improved by averaging over different discretization meshes. Note that z-averaging is performed such that outermost intervals increase with z > which is different to standard scheme presented in Eq. (3.8). Yet this procedure does not recover true continuum limit. 48

59 4.. THE LOGARITHMIC DISCRETIZATION GRID WITHIN DMFT N dependence of filling factor on discretization grid.85 n =4 n =3.84 n = n = n.83 = n = n =.8 n = 3 n = /6 /8 /4 / ε n Figure 4.7: Filling factors N of spectral functions shown in Fig. 4.6, which are calculated for different discretization grids determined by interval borders ±ɛ = ±Λ n. 4.. Zitko s adaptive logarithmic discretization grid In 9 R. Zitko proposed an adaptive discretization method that takes shape of hybridization function into account and is additionally based on requirement to reproduce input hybridization function after z-averaging [7]. The logarithmic discretization scheme incorporates actual form of hybridization function in a way that it is denser in regions with more weight and shows broader intervals, when function is very low, thus avoiding intervals with weak coupling and consequently unsolicited peaks in hopping matrix elements. Moreover, couplings and representative energies of star Hamiltonian are calculated by solving differential equations, that ensure that z-averaging generates original hybridization function. In following, we want to derive se differential equations to determine adaptive grid and corresponding representative energies and couplings (focusing on positive frequencies for convenience). To this end, we define a general logarithmic discretization grid with mesh points as in [7]: with continuity constraint ɛ z =, ɛ z n > ɛ z n+, (4.7a) (4.7b) ɛ z n Λ (n+z) n, (4.7c) z [, ], (4.7d) ɛ n = ɛ n+ (4.7e) for all n. The representative energies of intervals (now denoted by E n ) n follow same asymptotic behavior E n Λ (n+z). (4.8a) 49

60 CHAPTER 4. COMBINING DMFT AND NRG Furrmore, boundary condition and continuity constraint E = E n = E n+ (4.8b) (4.8c) have to be fulfilled for a meaningful z-averaging procedure [7]. For every logarithmic discretization mesh, continuous density of states of conduction band as seen by impurity, A (ω) = π Im ˆf ˆf ω = Γ(ω) π /ξ, (4.9) with spin index omitted and normalization as in Eq. (3.) is n given by a set of δ-peaks, ξ = dε Γ(ω) π, (4.3) à (z, ω) = n w z nδ(ω E z n), (4.3) with normalized weights for all z: n= wn z = = A (ω)dω. (4.3) We now claim that spectral density A is reproduced by taking average (integral) over z: A (ω) = This can be rewritten in following form à (z, ω)dz. (4.33) A (ω) = n w z nδ(ω E z n)dz = w z n E z n/ z, (4.34) where n and z are determined implicitly by ω = E z n. As E z n is a strictly decreasing function of z, this choice is unique. We now introduce a continuous grid parameter, x = n + z, x [; + ) (4.35) and accordingly continuous functions ɛ(x), E(x) and w(x), that automatically fulfill continuity constraints given above. ɛ(x) and E(x) are monotonically decreasing functions that have to satisfy boundary conditions ɛ(x) = for x and lim ɛ(x) =, x (4.36) E() = and lim E(x) =. x (4.37) 5

61 4.. THE LOGARITHMIC DISCRETIZATION GRID WITHIN DMFT Eq. (4.34) n reads A (ω) = w(x) de(x)/dx (4.38) with x = R(ω) and inverse function R[E(x)] = x. The normalization condition still holds for continuous version n= w(n + z) = = and can be used to reformulate w(x): A (ω)dω = A (ω) de(x) dx dx = w(x)dx (4.39) ɛ(x) w(n + z) = A (ω)dω = n= n= ɛ(x+) A (ω)dω (4.4) w(x) = ɛ(x) ɛ(x+) A (ω)dω. (4.4) So, we arrive at following differential equation for calculation of representative energies E(x) ɛ(x) de(x) ɛ(x+) = A (ω)dω (4.4) dx A [E(x)] with initial condition E() =. Up to this point, logarithmic discretization grid has not been determined yet. Thereby, we are free to adapt exact location of mesh points as long as asymptotic behavior ɛ(x) Λ x holds. (The additional factor Λ is introduced for convenience.) This is equivalent to demand that de(x) = Λ x ln Λ C(x, ɛ) (4.43) dx with arbitrary, but strictly positive function C(x, ɛ) with non-zero limit λ = lim x ɛ C(x, ɛ). (4.44) For C(x, ɛ) = this yields conventional discretization scheme as given in Eq. (3.8). However, motivated by previous considerations, present goal is to define an adaptive grid that is less dense in regions of weak hybridization. To this purpose, we define C(x, ɛ) = F (x) A (ɛ) (4.45) and consider special case of F (x) F as in [7]. Then F must be chosen as F = A (ω)dω = /, such that conventional discretization grid is recovered for a flat band with A (ω) = /. Due to /A (ɛ) dependence, C(x, ɛ) diverges for A (ɛ) and corresponding energy regions will not contain any discretization mesh points, as desired. All in all, we can use Eq. (4.4) and Eq. (4.43) toger with a convenient ansatz for an adaptive logarithmic grid and corresponding representative energies, ɛ(x) = g(x)λ x, (4.46) E(x) = f(x)λ x, (4.47) 5

62 CHAPTER 4. COMBINING DMFT AND NRG and arrive at following differential equations for functions g(x) and f(x): dg(x) [ ] dx = ln Λ g(x) C(x, g(x)λ x ), (4.48) df(x) dx = ln Λ g(x)λ x g(x+)λ A x (ω)dω Λ x A [f(x)λ x, (4.49) ] with initial conditions g() = and f() = Λ. First Eq. (4.48) is solved and n used for Eq. (4.49). In [7] R. Zitko gives a short introduction to implementation of discretization equation solver, that he provides on his webpage The couplings of star Hamiltonian are n calculated as usual with Eq. (3.) as square root of integrated weights in different intervals. To visualize effect of Zitko s adaptive discretization scheme, we use his solver for same spectral function as in Fig (a) (b) spectral function spectral function.8 A(ω) adaptive grid g(x) g(x) used for adaptive grid.8 A(ω) adaptive grid g(x) g(x) used for adaptive grid A(ω)*D.6 A(ω)*D ω/d ω/d Figure 4.8: Panel (a) shows negative frequency part of spectral function (black curve) for asymmetric one-band Hubbard model with parameters U/D = 4, µ/d =.568 and T/D =.5, panel (b) positive frequency range. In order to observe effect of R. Zitko s adaptive discretization scheme, frequency axis is plotted logarithmically. The adaptive interval borders are shown by dashed green vertical lines. The blue crosses on blue curve give values of g(x), which have been used to obtain adaptive mesh points. Furr, y indicate positions of conventional discretization intervals. In Fig. 4.8 this spectral function for asymmetric one-band Hubbard model with parameters U/D = 4, µ/d =.568 and T/D =.5 (black curves) and corresponding adaptive grid for z = (dashed green lines) are plotted for positive and negative frequencies, respectively. It can be clearly seen that for smaller spectral weight intervals are enlarged. This is particularly visible for first interval in panel (a) and second interval in panel (b) of Fig. 5

63 4.. THE LOGARITHMIC DISCRETIZATION GRID WITHIN DMFT 4.8. The corresponding function g(x) is plotted in blue. It has to be noticed that energy support, used in Zitko s derivation, has to be rescaled to energies larger than in order to be consistent with DMFT choice of D = for non-interacting lattice density of states of width D. The blue crosses indicate position of original grid on frequency axis and determine values for g(x) on vertical axis. (a) (b) t N /D hopping matrix elements n =4 n =3 n = n = n = n = n = n = 3 n = 4 adaptive t N (rescaled) rescaled hopping matrix elements n =4 n =3 n = n = n = n = n = n = 3 n = 4 adaptive N 5 5 N Figure 4.9: Hopping matrix elements calculated with R. Zitko s differential equation solver. In panel (a) Wilson chain couplings of adaptive grid are shown by thick green curve, rescaled version of hopping matrix elements is found in panel (b) (thick green curve). The dotted colored curves show couplings of Fig. 4.5 which are obtained via discretization scheme presented in previous section. As requested, corresponding hopping matrix elements fall off along Wilson chain (green thick curve in Fig. 4.9 (a)). They decrease even faster than couplings for adequate choice n = of discretization scheme, presented in previous section. Analogously we find rescaled Wilson chain couplings (green thick curve in Fig. 4.9 (b)), where peaks are shifted to much earlier sites compared to cases n = 4 to n = (dotted green, blue and black colored curves). To obtain smooned spectral function in case of an adaptive grid, width of broadening kernel should take mesh density into consideration, since less discrete data is produced in energy regions with low spectral weight. In [7] benefits of adaptive grid are studied by direct comparison of a test input function A with NRG output of same quantity. It is clearly shown that discretization artefacts can be drastically reduced (for this specific case). Neverless some open questions remain with regard to adaptive grid. A major issue is that re exists no clear preference how to incorporate mesh density in width of broadening kernel. As R. Zitko s discretization scheme has been considered towards end of master s term this problem remains to be studied in future. 53

64 CHAPTER 4. COMBINING DMFT AND NRG 54

65 Chapter 5 One-band Hubbard model For a first quality check of our DMFT program with NRG impurity solver, we want to investigate one-band Hubbard model which was already introduced in Sec.... It is most basic model, that is able to describe fundamental physics of strongly correlated materials: Ĥ = µ ˆn iσ + tĉ iσĉjσ + U ˆn i ˆn i. (5.) iσ i ij σ The kinetic energy is characterized by hopping amplitude t and interaction is given by local Coulomb repulsion U. µ is chemical potential of lattice. A general illustration of one-band Hubbard model is given in Fig Dieter Vollhardt Fig. : Schematic illustration of interacting electrons in a solid in terms of Hubbard model. Figure The 5.: ions Sketch appear of only one-band as a rigidhubbard lattice (here model. represented The strongly-correlated as a square lattice). material The electrons, is depicted as a cubic, which two-dimesional have a mass, lattice. a negative Conduction charge, and electrons a spin ( with or ), spin move up from or down one lattice can hop site between to next localized states (orbitals) with a hopping at lattice amplitude sites i t. and The j quantum with an dynamics amplitude thus t. leads When to two fluctuations electrons in meet occupation on one site, y interact of via lattice Coulomb sites as repulsion indicated U. by The time time sequence. sequence When indicates two electrons quantum meet mechanical on a lattice fluctuations site in occupation for one lattice site: eir it is unoccupied, singly occupied (with an electron of spin up (which is only possible if y have opposite spin because of Pauli exclusion principle) y or down) or doubly occupied. The interplay of hopping and repulsion can lead to interesting many-body encounter an interaction U. Alatticesitecaneirbeunoccupied,singlyoccupied( or ), or phenomena like Mott-Hubbard metal-insulator transition. Figure taken from [, Chapter ]. doubly occupied.. Hubbard model The simplest model describing interacting electrons in a solid is one-band, spin-/ Hubbard 55 model [7 9] where interaction between electrons is assumed to be so strongly screened that it is taken as purely local. The Hamiltonian consists of two terms, kinetic energy Ĥ and interaction energy ĤI (here and in following operators are denoted by a hat):

66 CHAPTER 5. ONE-BAND HUBBARD MODEL The conduction electrons with spin σ =, can hop between localized states (orbitals) at lattice sites i and j with an amplitude t. Thus, fluctuations in occupation of lattice sites arise as depicted by time sequence in Fig. 5.. In principle, a site can be unoccupied, singly occupied with electrons of spin up or down, respectively, or doubly occupied with electrons of spin up and down (due to Pauli s exclusion principle). When two electrons sit on same site, y experience a repulsion of strength U. So a competition between kinetic and local interaction part of Hamiltonian emerges. While kinetic energy favors electrons to move, leading to doubly occupied sites, Coulomb interaction penalizes double occupation by an energy cost of U. Therefore overall electron movement is highly correlated and determined by ratio U/t. But also filling factor N of lattice plays a major role. It is given by number of electrons N e divided by number of lattice sites N l : N = N e N l. (5.) Two simple cases are N =, where no itinerant electrons exist, and N =, where all sites are completely filled and hopping is not possible, as well. So, se two filling factors denote insulators, as expected from conventional band ory (for a band that is eir not or completely filled). However, this common picture does not work in general for strongly correlated materials. Let s consider integer filling N = (half-filled or particle-hole symmetric case), where we have one electron per site, on average. If ratio U/t is small, electrons move between different lattice sites and we find a metal. However, if U/t is big enough, energetic cost for doubly occupied sites can grow so large, that electrons get localized. In real materials this effect is triggered by absence of sufficient screening of Coulomb interaction in integer filling scenario [, Chapter 3]. This heuristic explanation shows how an insulating state (called Mott insulator) can arise for a half-filled band due to strong correlations. Yet, it has to be mentioned that an insulating state does not appear for any non-integer filling factor (and finite t). For those filling regimes singly, doubly or non-occupied sites always coincide with result that total energy can be lowered via hopping processes [, Chapter 3]. U = t =, µ = U A(!) A(!)! U U! Figure 5.: Sketch of local spectral function for half-filled one-band Hubbard model. In left panel Fermi gas limit U = is shown, in right panel atomic limit t = is depicted. While electrons are delocalized for left case, y are completely localized for right figure. Adapted from [, Chapter 3]. 56

67 5.. THE MOTT-HUBBARD METAL-INSULATOR TRANSITION (MIT) The final state for half-filled Hubbard model is governed by ratio U/t (or to be more precise, by U/D with half bandwidth D of non-interacting lattice density of states). For Fermi gas limit U = spectral function of system is equal to non-interacting lattice density of states A(ω) = N B k δ(ω ε k). In case of a Be lattice (with infinite coordination) we find a semi-elliptical spectral function as sketched in Fig. 5.. In atomic limit t = lattice problem separates into a sum of isolated atomic systems. It can be easily shown that only δ-peak excitations at µ and µ + U arise in this case [, Chapter 3]. But what happens for intermediate values of U/D? As indicated above, one expects a transition between delocalized and localized limits shown in Fig The Mott-Hubbard metal-insulator transition (MIT) In 998 correlation induced transition between a paramagnetic metal and a paramagnetic insulator, referred to as Mott-Hubbard metal-insulator transition (MIT), was studied by R. Bulla, A.C. Hewson and Th. Pruschke, which was one of first investigations using NRG method as impurity solver within DMFT [6]. (Magnetic order is assumed to be suppressed in this model, hence it is frustrated. For systems with dimension larger than two, an insulating antiferromagnetic phase generally hides MIT of unstable paramagnetic phase [, Chapter 3].) The Mott transition has been one of early successes of DMFT approach [3]. As a first quality test we want to reproduce MIT as performed in [6]. (a).8.6 = U=. U=. U=.85 U=.93 U=4. (b) spectral function U/D=. U/D=. U/D=.85 U/D=.93 U=4. A(ω).4 A(ω)*D ω 4 4 ω/d Figure 5.3: MIT illustrated by spectral functions for different values of U, µ = U/ and T for half-filled one-band Hubbard model. Panel (a) is taken from [6]. In panel (b) our results (obtained with Λ =, N z = 4 and σ =.3 and N z = 8, σ =. for U/D =.85) are plotted. The MIT is displayed by shape of spectral function (or local density of states) A(ω) = π Im G latt(ω), which is calculated with our DMFT+NRG program (see panel (b) 57

68 CHAPTER 5. ONE-BAND HUBBARD MODEL of Fig. 5.3, results obtained in [6] are presented in panel (a)). Here we concentrate on half-filled (particle-hole symmetric) one-band Hubbard model at T =. To demonstrate MIT, Coulomb interaction is increased from U/D = to U/D = 4. For DMFT self-consistency procedure we use semi-elliptical Be lattice density of states with half bandwidth D. The self-consistency loop starts with a flat band as input function. Its strength is chosen such that spectral function after first DMFT iteration shows a (quasiparticle) peak at Fermi level and thus metallic behavior. We call this a metallic input hybridization in contrast to insulating input, where a spectral function with gap around Fermi level is obtained after first DMFT iteration. Convergence is reached after about 8 iterations (except for U/D =.85). For small U (U/D = ) we obtain a spectral function that is similar to non-interacting case (Fermi gas limit in Fig 5. (left panel)). It clearly features a coherent quasiparticle peak around Fermi level. This is associated with a metallic phase. For intermediate U (U/D = ) a typical three-peak structure develops in spectrum with a lower and an upper Hubbard band, which are centered at ±U/, and a quasiparticle peak at ω =. While central peak corresponds to coherent quasiparticle excitations, Hubbard bands emerge from incoherent atomic-like excitations (as in Fig. 5. (right panel), but broadened by finite hopping amplitude). This three-peak structure is a characteristic feature of strongly correlated materials, that also occurs for non-integer filling (for sufficiently small temperatures). It reminds of three-peak spectrum of conventional SIAM (with Kondo peak at ω = ). Yet, it has to be emphasized that three-peak structure for strongly correlated materials arises from a lattice model, that is mapped self-consistently onto an impurity Anderson model. The Kondo peak in strong coupling regime of SIAM originates from local impurity spins, that are completely screened by bath electrons. In case of a lattice model local moments and screening are both associated with lattice electrons [, Chapter 3]. spectral function.6 U/D=.6, µ/d=.3, T/D=.6e 8 metallic input insulating input.5 A(ω)*D ω/d Figure 5.4: Spectral functions for half-filled one-band Hubbard model in coexistence region U c, < (U/D =.6) < U c, with µ = U/, T, N z = 8, σ =./.3 (blue curve / red curve) and Λ =. The blue curve was obtained with a metallic input hybridization, red curve with an insulating one. 58

69 5.. THE MOTT-HUBBARD METAL-INSULATOR TRANSITION (MIT) When we furr increase U width of quasiparticle peak decreases and vanishes above a critical value U c, /D.93. Up to U c, height of quasiparticle peak is pinned at a fixed value ( Luttinger pinning ) [34]. For U > U c, a gap develops between two Hubbard bands, centered at ±U/, which becomes broader when U grows to larger values. So for U > U c, we find an insulating phase. If we start with a metallic input for DMFT loop width of peak around ω = vanishes exponentially slowly with DMFT iterations (close to transition point). While metal to insulator transition occurs at a critical value U c,, transition from an insulating to a metallic phase is characterized by a lower critical value U c, < U c, (U c,.5) [7]. In Fig. 5.4 we show spectral function for U/D =.6. While we find a quasiparticle peak, starting from metallic side (blue curve), we get an insulator (red curve) for same value of U, starting with an insulating input. The physical solution in coexistence (or hysteresis) region U c, < U < U c, is metallic one, as it exhibits lower ground state energy (see discussion in [7]). T T c Bad Metal Metal (Fermi Liquid) U (T) c Semi Conductor U (T) c U c U c U c Mott Insulator U Figure FIGURE 5.5: 9. Paramagnetic Paramagneticphase phases diagram of Hubbard for model frustrated within one-band DMFT, displaying Hubbard schematically model within DMFT. Figure spinodal taken lines from of []. Mott insulating and metallic mean-field solutions(dashed),first-ordertransition line (plain) and critical endpoint. The shaded crossoverlinesseparatingdifferenttransportregimes discussed in Sec.3 are also shown. The Fermi-liquid to bad metal crossover line corresponds to quasiparticle coherence scale and is a continuation of spinodal U c (T ) above T c.thecrossoverinto In Fig. 5.5 we show a schematic (paramagnetic) phase diagram for one-band (frustrated) insulating state corresponds to continuation of U c spinodal. Magnetic phases are not displayed Hubbard and depend model, on plotted degree of for frustration. general temperatures Figure from Refs. and [86] interaction and [87]. strengths U. Below critical endpoint (T c, U c ) model displays two possible solutions (metal or Mott insulator) in hysteresis region U c, (T ) < U(T ) < U c, (T ), which is confined by dashed spinodal lines U c, Indeed, (T ) (where in V O 3 as insulating well as inphase organics, vanishes) and critical U c, (T temperature ) (where corresponding Fermi liquidtoproperties disappear). endpoint of The red first-order line gives Mott transition first-orderline phase is atransition factor ofboundary 5towhere smaller than free energy of bare bolectronic solutions becomes bandwith. equal [9]. For temperatures above T c different transport regimes can occur. The blue shaded crossover line separating Fermi liquid and bad metal regions (see Sec. 5.3 for details) continues border U c, (T ) above T c. Analogously, blue shaded crossover line of insulating state is an extension of U 4.3. Physical properties of correlated c, (T ) for T > T metallic state: c []. DMFT To summarize results of Fig. 5.3, MIT is monitored by existence of a quasiparticle peak in spectral function. As can confronts be seen, experiments curves obtained in [6] (panel (a)) coincide with our DMFT+NRG results (panel (b)). Even bump at around ω/d =.8 arises for U/D =.93, origin ofthree which peaks: is not clearified evidence yet. from photoemission In Fig., we reproduce early photoemission spectra of some d transition metal oxides, from pioneering work of Fujimori and coworkers [88]. This work established experimentally, more than ten years ago, 59 existence of well-formed (lower) Hubbard bands in correlated metals, in addition to low-energy quasiparticles. This experimental study and oretical prediction of a 3-peak structure from DMFT [6] came independently around same time. However, back in 99, existence of a narrow quasiparticle peak in A(ω) resembling DMFT results was, to say least, not ob-

70 CHAPTER 5. ONE-BAND HUBBARD MODEL (a).... U=. U=. U=.85 U=.85 U=.93 U=.93 (b).5 imaginary part of self energy U/D=. U/D=.85 U/D=.93 ImΣ(ω) ImΣ(ω) Σ(ω)/D Im(Σ(ω))/D Σ(ω)/D ω ω..4.. ω/d 4 6 Figure 5.6: MIT illustrated by imaginary part of self-energy for different values of U, µ = U/ and T for half-filled one-band Hubbard model. Panel (a) is taken from [6]. In panel (b) our results (obtained with Λ =, N z = 4 and σ =.3 and N z = 8, σ =. for U/D=.85) are plotted. The inset shows region around Fermi level ω = in more detail. For metallic phase Fermi liquid behavior (Im Σ(ω = ) = ) can be observed. Also structure of self-energy is found to be qualitatively equal as can be observed in Fig 5.6 and Fig In panels (a) we show again results of [6], while panels (b) display our solutions. The self-energy exhibits peaks in its imaginary and real part at frequencies, that correspond to small weights in imaginary and real part of lattice Green s function, respectively, which is revisable for Im Σ(ω) and A(ω) in Fig. 5.6 and Fig Therefore a three-peak structure in A(ω) results in a two-peak structure in Im Σ(ω). When quasiparticle peak vanishes with U U c,, two peaks in Im Σ(ω) approach and merge to a single δ-peak for insulating phase (plotted in red in Fig. 5.6 (b), but not shown in Fig. 5.6 (a)). The zoom in region around ω = (see insets of Fig. 5.6) shows slight differences between curves of panel (a) and our results. While black curves for U/D = are in good agreement, blue curve exhibits sharper features in our case, leading to a thinner quasiparticle peak near transition point. For red curve shape near ω = is more flat in panel (b). Near ω = we had to reduce imaginary part of self-energy artificially down to zero for metallic phase. The NRG calculations exhibit a (systematic) small error of about.%, which may slightly grow during DMFT procedure. This potentially leads to a positive self-energy, that has to be reduced again to ensure causality for DMFT loop. In Sec. 5.3 we will focus on exact quantitative results near ω = and introduce a strategy to deal with this problem in a controlled way. In general, it is possible to furr reduce error of DMFT+NRG calculations by increasing numerical effort (such as performing more z-shifts or keeping more states, especially at first sites of Wilson chain). These options have not been exhausted for MIT. Neverless we can state that Im Σ(ω = ) = (within a certain error margin) for metallic phase, which is connected to Fermi-liquid behavior (see next section for an 6

71 5.. THE MOTT-HUBBARD METAL-INSULATOR TRANSITION (MIT) introduction to Fermi liquid ory). The real part of self-energy (in Fig 5.7) displays a negative slope at ω = for U < U c, (black and blue curves). In insulating phase slope diverges (red dashed curves). 3 ÍÎÏÐÏÑÒÓÔ 3456 ôõö Šˆ ˆ +*+ ˆ ˆ (-*+ ˆ ˆ ˆ ˆ (,*+ ˆ ˆ ()*+ ˆ ˆ +*++ (+*-+ (+*,+ (+*)+ (+*.+ Œ Ž 789:; Œ Ž 789:<= Œ Ž 789:>? (+*/+ (+*,+(+*-+ +*++ +*-+ Šˆ ˆ (.*+ ]ba ()*+(,*+(-*++*+ ˆ ˆ -*+ ˆ ˆ,*+ ˆ )*+.*+ ˆ /*+ *+ ]^_` ]^ ]a_` a a_` ^ ^_` hjk defgfhijk ba ca ^a a ]^a ]ca lemn pmlq rs qte uens]evelwx yjkzc_a yjkzc_{` yjkzc_ b Figure 5.7: MIT illustrated by real part of self-energy for different values of U, µ = U/ and T for half-filled one-band Hubbard model. Panel (a) is taken from [6]. In panel (b) our results (obtained with Λ =, N z = 4 and σ =.3 and N z = 8, σ =. for U/D=.85) are plotted. The Fermi liquid character of metallic phase is confirmed by fixed point regime of corresponding energy flow diagrams. In Fig. 5.8 (a) RG-flow for U/D =, µ/d =, T/D, z = and Λ = is shown. Clearly system flows to strong coupling or Fermiliquid fixed point, as also observed in [6]. For U > U c, we have an insulating phase which is associated with localized electrons and correspondingly unscreened local moments. In this sense, expression local moment fixed point seems appropriate for insulators of fully frustrated Hubbard model [6]. (Note that for models which are not fully frustrated, local moments order to an antiferromagnetic phase below Néel temperature []). However, corresponding RG-flow converges to a finite size spectrum, despite gap in spectral function (see Fig. 5.8). This suggests that fixed point regime at end of Wilson chain is not based on a physical but rar on a numerical effect: In insulating phase couplings of star Hamiltonian almost vanish for inner intervals in gap, leading to a region with pronounced peaks in hopping matrix elements (indicated by red bar in Fig. 5.9). Presumably Wilson chain effectively decouples in that region. The second part of chain may n be viewed as a basically independent subsystem that barely influences NRG results at impurity. Thus, Wilson chain could be chosen significantly shorter. To confirm this assumption we repeated calculation for U/D = 4 with a chain length of N max = 3 and obtained same result, as shown in inset of Fig

72 CHAPTER 5. ONE-BAND HUBBARD MODEL (a) (b) E s N (even spectrum) E s N (odd spectrum) ΔE N 5 U/D=, µ/d=, T/D=.6e N ground state E g = N ΔE even ΔE odd min(e D ) max(e K ) 4 N K rnrg.m based on mex routine NRGWilsonQS() E s N (even spectrum) E s N (odd spectrum) ΔE AWb, Feb 6, U/D=4, µ/d=, T/D=.6e N 5 ground state 4 E g = N ΔE even ΔE odd N min(e D ) max(e K ) 7 3 N K U=, ε d =, Γ=., B=, Λ=, N=7, z=.5, D=89 Figure 5.8: Energy flow diagrams for half-filled one-band Hubbard model for (a) metallic phase represented by U/D = and (b) insulating phase with U/D = 4. Furr µ = U/, T, Λ = and z = are used. In panel (a) energies flow to Fermi liquid fixed point. The red bar in panel (b) indicates region where Wilson chain for insulating phase seems to effectively decouple due to infinitesimal couplings in star Hamiltonian. The fixed point regime after red bar may n be uncontrolled as based on numerical noise. E denotes energy shift of ground state energy to zero. Although one-band Hubbard model does not contain any material specific details (such as several orbitals and Hund s coupling terms) and DMFT method is based on approximation of a local (or momentum-independent) self-energy, combination of both already yields a profound understanding of MIT for whole parameter range U/D (as shown in this section) and also for all values of temperature T. In MIT spectral weight is transferred from quasiparticle peak to Hubbard bands. For strongly correlated materials this transfer can be induced by temperature, pressure or doping. The MIT is found in materials with quite different crystal structures, such as vanadium oxide (V O 3 ), nickel selenium sulfide (NiS x Se x ) or (layered) organic compounds []. In many transition metal oxides (such as NiO) Mott mechanism explains ir unconventional transport properties. Although d-band is only partially filled for se materials, y are not good metals as expected by conventional band ory. Due to strong correlations even insulators arise for partially filled bands. Anor highly interesting example of transition metal oxides are superconducting cuprates. 6

73 5.. FERMI-LIQUID THEORY (a) (b) hopping matrix elements.5 rescaled hopping matrix elements.3 t N /D A(ω)*D ω/d t N (rescaled) U/D=4, µ/d=, T/D=.6e 8 3 N U/D=4, µ/d=, T/D=.6e 8 3 N Figure 5.9: Hopping matrix elements for insulating phase of half-filled one-band Hubbard model with U/D = 4. As for energy flow diagram µ = U/, T, Λ = and z = are used. Panel (a) shows non-rescaled Wilson chain couplings and corresponding spectral function in inset, panel (b) depicts rescaled Wilson chain couplings. The red bar indicates region where pronounced peaks arise due to infinitesimal couplings in star Hamiltonian. 5. Fermi-liquid ory In order to understand nature of quasiparticle excitations and to classify metallic and insulating phase of an MIT quantitatively, we examine structure of retarded lattice Green s function and corresponding self-energy in more detail (following derivations given in [35, Chapter 5]). The one-particle retarded Green s function in DMFT is given as G k (ω) = (5.3) ω + µ ε k Σ(ω) with chemical potential µ, dispersion relation ε k and momentum-independent selfenergy Σ(ω) taking local interactions into account. Quantum labels as for spin are neglected for now. The Fermi wave number k F is determined as ε kf µ =. The influence of interaction part of Green s function can be investigated by separating real and imaginary part of self-energy explicitly, G k (ω) = ω + µ ε k Re Σ(ω) i Im Σ(ω), (5.4) which leads to a renormalized Fermi wave number k F, defined by µ + Re Σ(ω) =. For ε kf small energies and k close to k F inverse of retarded Green s function can be expanded around k = k F and ω = : [ G k (ω) ω ω ω Re Σ(ω) ω= (k k F ) k ε k k= kf Im Σ(ω)]. (5.5) 63

74 CHAPTER 5. ONE-BAND HUBBARD MODEL The result is n brought into a form similar to non-interacting Green s function which leads to with quasiparticle weight inverse quasiparticle lifetime for small ω and effective energy G k(ω) = G k (ω) = Z = ω + µ ε k + iδ, (5.6) Z ω ε k + i τ(ω) Furrmore we introduce renormalized chemical potential (5.7) ω Re Σ(ω) ω=, (5.8) τ (ω) = Z Im Σ(ω) (5.9) ε k = Z(k k F ) k ε k k= kf. (5.) µ eff = µ Re Σ(ω = ), (5.) which cancels in above expression due to definition of renormalized Fermi wave number k F. The imaginary part of self-energy was not expanded explicitly. The quasiparticle weight is determined by slope of real part of self-energy at ω =. As shown in previous section, slope is a direct indicator of MIT. For metallic phase we get a negative slope at ω = and thus a quasiparticle weight Z (, ], for insulating phase slope diverges, leading to Z =. Hence, Z reflects weight of quasiparticle peak in spectral function. The weight of Hubbard bands is n given as Z. This interpretation can be furr motivated by evaluating k-dependent spectral function which is given as a δ-peak at energy ε k µ, A k (ω) = π Im G k(ω) (5.) A k (ω) = δ(ω (ε k µ)), (5.3) in non-interacting case. In interacting case it is a function with a Lorentzian shaped coherent quasiparticle peak of weight Z at position ε k and of width τ (ω) (for k close to k F and very small ω) and an additional incoherent part of weight Z: A k (ω) = π Z τ(ω) (ω ε k ) + ( τ(ω) ) + A k(ω). (5.4) While first term is given as π Im G k (ω), second term includes spectral weight that was neglected by performing a first order expansion around k = k F and ω =. Within local DMFT Z is equal to mass renormalization. In analogy to free electrons with dispersion relation ε k = k /m effective energy ε k in Eq. (5.) is usually written as ε k = (k k F ) k F m, (5.5) 64

75 5.3. FERMI LIQUID PHYSICS AND RESILIENT QUASIPARTICLES which leads to m = Z. (5.6) m The inverse lifetime of quasiparticles for small ω, given by Eq. (5.9), determines height and width of Lorentzian quasiparticle peak. Due to interactions, excitations exhibit finite lifetimes as electrons exchange momenta in scattering processes and thus change ir quantum state [36, Chapter 4]. Im Σ(ω) can refore be associated with transport scattering rate. For very long lifetimes (and weak excitations) it is possible to represent excitations of an interacting system as well-defined non-interacting quasiparticles with properly identified excitation energies. This concept, called Fermi liquid ory, was developed by L.D. Landau in late 95 s and has been furr refined, since. It is basic oretical background behind widely used free-electron model (that gives surprisingly good results especially for description of metals). It is based on a one-to-one correspondence between long-lived excitations (quasiparticles) of an interacting Fermi liquid and low-energy excitations of a free Fermi gas [35]. The physical picture of a Landau quasiparticle can be associated with an electron that is surrounded by a screening cloud of particle-hole excitations or analogously density fluctuations, retaining quantum labels of electrons (like spin quantum number) but changing its effective parameters (such as mass) [36, Chapter 4]. This leads to fact that quasiparticles can be labeled by same quantum numbers as original fermions (if corresponding operators still commute with full Hamiltonian). Furrmore y follow Fermi statistics. The influence of interactions is considered by a renormalization of physical parameters, such as electron mass which is replaced by an effective mass m. It can be derived that inverse lifetime of a Landau particle and thus imaginary part of self-energy near ω = is given as Im Σ(ω, T ) [ω + (πt ) ] (5.7) for a Fermi liquid [6]. T denotes temperature. So for small temperatures and near Fermi surface lifetime of excitations becomes large and interacting single particle Green s function approaches that of free particles, which explains success of Fermi liquid ory in description of interacting systems [35]. In previous section about MIT imaginary part of self-energy was found to go to zero for ω = and T = in complete metallic phase, which can thus be described by a Fermi liquid. 5.3 Fermi liquid physics and resilient quasiparticles While we studied MIT in a primarily qualitative way, we now want to examine accuracy of our calculations quantitatively. This is done for asymmetric (hole-doped with N =.8 or doping δ = %) one-band Hubbard model that was only recently investigated by X. Deng, J. Mravlje, R. Zitko, M. Ferrero, G. Kotliar and A. Georges, using DMFT (with semi-elliptical Be lattice density of states of half bandwidth D) in combination with continuous time Quantum Monte Carlo simulations (CT-QMC) and NRG [6]. They used this rar simple setting to study unconventional transport properties of metals with strong electron correlations, which are still poorly understood oretically. For conventional metals resistivity increases with temperature due to phonon scattering 65

76 tion. Our data reveal that regime occurs at Brinkm is a measure of kinetic e quantum sca (above a certain, small temperature below which electron-electron scattering is dominant).degeneracy In a quasiparticle picture this can be explained by a growing number of rmally-induced scattering DMFT, both QP weigh events reducing mean-free-path l. A resistivity maximum is n reached when l becomes are proportional to doping Z shorter than lattice spacing a, since at this point quasiparticle description of transport very low T < TFL, F breaks down. This limit, where l a (or similarly kf l ), is known as At Mott-Ioffe-Regel limit and associated with a temperature TMIR. determined TFL from sever For many strongly correlated materials, resistivity at high temperatures exceeds this limit, onset of T resistivity ( which is often called bad-metallic behavior. For very small temperatures (T < TFL ) y Im (!, )/T vs.!/t (see follow Fermi-liquid behavior. In general TFL is much lower than TMIR : in Sr RuO4Tfor example, TFL K and TMIR 8K [6]. While transport can be described by scaling Landau s quasiparticles of Z(T = )/Z(T ) vs for T < TFL, it has to be clarified how to think of transport for T > TFL. In [6] especially T well-defined, '.5soD, also proport intermediate regime TFL < T < TMIR is investigated. A main result is thatfl called much smaller prefactor than resilient quasiparticle (RQP) excitations, can be found clearly above temperature range in which Fermi liquid ory is valid. These excitations gradually disappearresistivity in crossover is way sma at to TFL bad metal regime (T > TMIR ). See Fig. 5. for an overview of different transport regimes. low-t expansion yields: (T and hence (TFL )/ MIR '. There is thus a wide tem TMIR in which transport does ior, although resistivity is sti MIR. Right above TFL, imately linearly (with a nega feature is observed at a tem Fig. c), above which high in which (T ) has linear tem a liquid positive intercept), as can Figure 5.: The different transport regimes as a function of doping δ: Fermi behavior is found for T < TFL (blue region), bad metal regime is indicated in red with red points as MIR limit, pansion [5, 6]. FIG.. intermediate (a) Temperature-dependence of resistivity for white region is RQP regime. The crossover to bad metal is gradual. Figure taken several doping levels 4, as in all figures). The The above findings raise t from [6], more details(u/d are given= re. MIR value as defined in text is reached at a temperature are charge carriers in TMIR indicated by plain arrows. Inset: resistivity at low temtfl. T.including TMIR? We depic Furrmore y focus on precise temperature-dependence of self-energy peraturesvs.determination T / D revealing T behavior (dashed line) of TFL. Contrary to previous assumptions, it was recently shown (see e.g. function A resolved spectral below TFL arrow). (b) Determination a c superconductors, which FL byhigh-t [37])(empty that Fermi liquid physics is highly relevantofin Tcuprate tures as energy distribution c scaling plot of Z(T! )of vs. / D. Here, Z(Tof) Fermi-liquid underlines )/Z(T importance an T exact understanding properties of strongly ent "k s). Also shown is m correlated (!, T )/@!. (c) The di erent regimes: FL (blue) our main goal is to confirm low-temperature results of [6]. These We focusresults on validreveal a remar for T < THere FL, bad metal (red) and intermediate RQP regime. ity of Fermi-liquid behavior for small temperatures but also reveal existence of resilient The crossover into bad metal is gradual: onset of red excitations exist throughout quasiparticles above TFL. We compare our results with plots in [6] and have additionally shading corresponds to with optical waythéoretique, above Ecole FL scale. Our a direct collaboration Michel spectroscopy Ferrero from signatures Centre de Physique discussedpolytechnique, in text, France, while red points indicate where who provided continuous time Monte MIR Carlo dataparticle for some temperatures is a pragmatic one: w is reached. The with thinmf dashed lineinindicates calculations knee inare (T ). (labeled when used our plots). All performed with semi-elliptical a well-resolved peak in v Be lattice density of states of half bandwidth D, Λ =, Nz = 8 and σ =. for last to aoflower Hubbard DMFT iterations. Our NRG results show excellent convergence with a addition discarded weight 5 order. Hubbard band (UHB). For T < TFL (e.g. T /D = a semicircular density-of-states (DOS) with a half-width are seen close to Fermi en D, and corresponding sum-rule preserving 66 expression of long-lived Landau QPs. Fo 3/ ( ) = () ( /D). In following, resistivity and T /D =. in Fig. ), will be expressed in units of MIR value defined as RQPs are visible mostly on / e ()/~D. This is consistent with critechapter 5. ONE-BAND HUBBARD MODEL

77 5.3. FERMI LIQUID PHYSICS AND RESILIENT QUASIPARTICLES (a) x 3 (b) x Im Σ(ω)/D 3 3 iteration n+ iteration n+ iteration n+3 iteration n+4 iteration n+5 changed Im Σ(ω).. (c) x 3 ω/d U/D=4, µ/d=.568, T/D=.5 Im Σ(ω)/D 3 iteration n+ iteration n+ iteration n+3 iteration n+4 iteration n+5 changed Im Σ(ω).. imaginary part of self energy (d) ω/d. DMFT and NRG polynomial fit Im Σ(ω)/D Im Σ(ω)/D 3 changed Im Σ(ω) with method (a) changed Im Σ(ω) with method (b). ω/d.. U/D=4, µ/d=.59, T/D=e ω/d Figure 5.: Reduction of imaginary part of self-energy for Im Σ(ω) > in case of asymmetric ( N =.8) one-band Hubbard model with U/D = 4, µ/d =.568, T/D =.5. Near ω = and for low T imaginary part of self-energy can become slightly positive due to small numerical errors and has to be lowered to zero. In panel (a) and (b) black dash-dot curves show two different possibilities to reduce Im Σ(ω) > for solid black curve. The reduction is repeated in every DMFT iteration until convergence is reached for sure in iteration n + 5 (n is number of preceding iterations, depending on input hybridization). In panel (c) we plot converged results for both strategies, which turn out to be essentially equal. These observations legitimize to define maximum value Im Σ(ω =, T = ) as new zero energy point and to shift curves for all or temperatures T > accordingly. Panel (d) illustrates, how we determine Im Σ(ω =, T = ) using a polynomial fit. The temperature region for Fermi liquid physics can be determined by examining onset of T -resistivity (as performed in [6], Fig. ), but also by scaling behavior of imaginary part of self-energy as given in Eq. (5.7), Im Σ(ω, T )D T = A[(π) + (ω/t ) ] (5.8) 67

78 CHAPTER 5. ONE-BAND HUBBARD MODEL for small frequencies ω and scaling of quasiparticle weight Z(T = )/Z(T ). (5.9) For T = we should find that Im Σ(ω =, T = ) =. However, as already mentioned before, Im Σ(ω) can exceed this value in our calculations and become slightly positive. As a positive imaginary part of self-energy is detrimental during DMFT iterations, we lower Im Σ(ω) > artificially. In Fig. 5. two different possible ways of reducing Im Σ(ω) are presented. In panel (a) Im Σ(ω) is rescaled to lower values in a certain frequency region around Im Σ(ω) > after each DMFT iteration. In panel (b) we use a rugged method and just set whole frequency range Im Σ(ω) > to Im Σ(ω) =. Interestingly, final outcome is not affected by precise way of changing Im Σ(ω) >, as can be seen in panel (c). In addition, numerical error is systematic and converges within DMFT to a stable final shape (see panel (a) and (b) in Fig. 5.). So despite changing Im Σ(ω) > artificially down to zero, DMFT outcome of next iteration is exactly same as in previous iteration when convergence is reached. So we redefine maximum value Im Σ(ω =, T = ) (which is determined using a polynomial fit in panel (d) of Fig. 5.) as new zero energy point and shift curves for all or temperatures accordingly. Note that calculation for T > should be performed with same NRG parameters (Λ, E keep, N z ) as for T = run to ensure a similar systematic error. T K /D=.9 self energy scaling Im Σ(ω=)/D T FL /D=. 3 4 DMFT and NRG fit: Im Σ(ω=)=A(πT) MF 3 T/D Figure 5.: Im Σ(ω =, T ) versus T for doped one-band Hubbard model with N =.8 and U/D = 4. For T < T FL.D Fermi liquid behavior is valid (blue fit) with A 6. The green crosses are Monte Carlo data from our collaboration with Michel Ferrero. Furr Kondo temperature T K is shown, which was deduced from T = result. Now we can identify constant A in Eq. (5.8) by plotting Im Σ(ω =, T ) versus T and simultaneously assess temperature T FL, below which Fermi liquid behavior is valid, using a logarithmic scale for both axes. As long as a linear relation is obtained in Fig. 5., Eq. (5.8) holds and A can be read off from blue fit as A 6 ( value Im Σ(ω = ) =, which is not shown in log-log plot, was also used to determine A). In [6] a value close to δ is 68

79 5.3. FERMI LIQUID PHYSICS AND RESILIENT QUASIPARTICLES reported, where δ denotes doping. So our result for A seems to be appropriate. T FL is n estimated as T FL.D. However, since most of values Im Σ(ω =, T ), for which Fermi liquid behavior applies, reside within energy range of manual shift Im Σ(ω =, T = ), fit has to be taken with caution. Neverless, fact that a systematic Fermi liquid relation can still be observed, justifies our procedure. Furr, we reproduced results shown in Fig. 5. for T < T FL with different NRG settings (higher DLM OPQRSTUV H EFFF KFF JFF IFF WXYZDX[X\]^ W_`Ya[] VUTbFcFFHG VUTbFcFFG VUTbFcFE VUTbFcFH VUTbFcFG VUTbFcE def H gpquvr H h HFF F DEF DG F QUV G EF Figure 5.3: Scaling of imaginary part of self-energy Im Σ(ω, T )D/T versus ω/t for doped one-band Hubbard model with N =.8 and U/D = 4. Panel (a) is taken from [6] with A close to Z. Panel (b) shows our results, which are in good agreement with left plot. The Fermi liquid relation Eq. (5.8) with A 6 is valid for T < T FL.D In Fig. 5.3 (b) we study Fermi liquid scaling relation Eq. (5.8) for a frequency range around ω = for different temperatures. Below T FL.D Fermi liquid scaling law applies as all curves follow shape of parabola given by red dashed fit. The data is compared to Fig. 5.3 (a), which shows results obtained in [6]. Evidently we can reproduce scaling law of panel (a) to high accuracy, despite manual shift, that had to be performed to ensure causality. As can be observed in both panels of Fig. 5.3, imaginary part of self-energy starts to differ distinctly from scaling relation for T FL.D and stronger deviations are generally found for positive frequency axis, revealing particle-hole asymmetry of system. The scaling of quasiparticle weight Z(T = )/Z(T ) is shown in [6] for several dopings (see Fig. 5.4(a)). We concentrate on a doping δ =. in Fig. 5.4 (b). T FL.5δD =.D coincides with temperature scale, where Z(T ) starts to deviate from Z(T = ) =.. Mass renormalization is given by Z(T = ) = m m =. δ. In Fig. 5.5 (b) (and additionally in (a)) we plot renormalized chemical potential µ eff (T ) = µ(t ) Re(ω =, T ). Note that chemical potential is temperature dependent and has to be determined numerically to ensure a filling of N =.8 (see Sec for details). 69

80 CHAPTER 5. ONE-BAND HUBBARD MODEL "#$% '$"( )* (+#,#%*#-#"./!! Figure 5.4: Scaling plot of quasiparticle weight Z(T = )/Z(T ) with T FL.5δD for doped one-band Hubbard model (U/D = 4) for different dopings δ in panel (a) and δ =. in panel (b). Panel (a) is adapted from [6], panel (b) depicts our results. For Fermi liquid region renormalized chemical potential µ eff (T ) is essentially fixed at effective chemical potential µ eff (T = ) (dashed blue line in Fig. 5.5 (a) and (b)). For higher temperatures (in RQP regime) effective chemical potential increases rapidly. (a) (b) MF NRG temperature dependence of µ eff MF NRG µ eff /D.. T FL /D=... 3 T/D Figure 5.5: Temperature dependence of renormalized effective chemical potential µ eff (T ) = µ(t ) Re(ω =, T ) for doped one-band Hubbard model with N =.8 and U/D = 4. While panel (a) (adapted from [6]) shows behavior mainly for larger temperatures, we concentrate on lower temperatures in panel (b). For T < T FL, µ eff is fixed at effective chemical potential µ eff (T = ). 7

81 5.3. FERMI LIQUID PHYSICS AND RESILIENT QUASIPARTICLES This can be observed for results of [6] in Fig. 5.5 (a) (dark blue data points) and also for our results (red crosses in panel (a) and (b)). While our NRG results are in good agreement with QMC data of Michel Ferrero (green crosses in panel (a) and (b)) for low temperatures, re are stronger deviations from dark blue curve in panel (a) for high temperatures. Neverless, we observe same qualitative behavior. (a) A(ω)*D frequency, hole-like rate than electron-like curves of Im vs. T a crossing point. Abov creasing function of fr excitations with finite those at! =. Thes provide largest co (b) spectral function intermediate RQP regi pends weakly on temp T/D=.5.6 T/D=.5 behavior in Fig. 3 b) an T/D=. T/D=. bandwidth and at mos.4 T/D=.5 considerations on a QP. FL regime in c tions, see [6]. The strong electron ω/d teresting consequences < with states are stro Figure 5.6: Spectral functions for different temperatures for doped one-band Hubbard!model aswedetermined by hn i =.8 and U/D = 4. Panel (a) is taken from [6], panel (b) depicts our results. Note that did not plot a curve for T /D =, but instead for T /D =., as we want to focus on RQP regime in this inflates to a larger v figure. Well-resolved quasiparticle peaks between an upper and a lower Hubbard band exist clearly above volume [3]. From F Fermi liquid temperature TFL.D and gradually disappear when entering bad metal regime. deviation from lineari at low!) defines two In [6] TMIR is found to be approximately on order of doping δd. As already shown, Fermiand! for el hole-like + liquid physics does not apply in this intermediate regime TFL < T < TMIR. Neverless, well kinks in QP dispersi resolved quasiparticle peaks between an upper and a lower Hubbard band exist clearly above studies [7, 8]. We n Fermi liquid temperature TFL and gradually disappear when entering bad metal regime. only! sets scal The resilient quasiparticles are visible in Fig. 5.6 (a) and (b) for T /D <.. Above!+ TMIR (!+ Panels ' TFL ). Using qu Hubbard satellites remain (yellow curves in panel (a) or (b) and red curve in panel (a)). (a) and (b) of Fig. 5.7 show corresponding imaginary parts of self-energy.previous While studies [9, 3 panels (a) are taken from [6], panels (b) show our results. Please note that we did not plotof particle-ho portance FIG.. (a) Temperature evolution of total DOS for a curve for T /D =, but instead for T /D =., as we want to focus on RQP regime. =.. (b) Momentum- resolved spectral functions. The insulators. For T < TFLshaded we findarea very [-5k long-lived Landau quasiparticles near ω =. At finite frequencies and B T,5kB T ] indicates states with a signifia sensitive probe of for T < TMIR contribution electron-hole asymmetry leads to higher absolute values of imaginary part cant to transport. of self-energy and refore to higher scattering rates for hole-like excitations (ωseebeck < ). coefficient ( Strikingly, subleadi in low-frequency ex of Q(T ) at low-t by a to a naive FL ory 7 would only retain term anticipated in Ref. [3] tively important. The rate of RQPs, dis

82 FIG.. (a) Temperature evolution of total DOS for =.. (b) Momentum- resolved spectral functions. The shaded area [-5kB T,5kB T ] indicates states with a significant contribution to transport. CHAPTER 5. ONE-BAND HUBBARD MODEL (a) (a) (b) (b) imaginary part of self energy Im Σ(ω)/D FIG Self-energy and particle-hole asymmetry. (a) Im (!) T/D=.5 for di erent temperatures. (b) Temperature dependence T/D=.5 of 3 Im (!c ) for!c =.5,.4,...,. (turquoise), T/D=.!c =. T/D=. (thick black) and! =.,.,...,.5 (green). Below c T/D=.5.5 dashed line Z(T )Im. T. (c) Re (!) at T /D =.5, two kinks (arrows). and.5 previous studies [9, 3] have a portance of particle-hole asymm insulators. A sensitive probe of partic Seebeck coefficient (rmopower Strikingly, subleading particl in low-frequency expansion o of Q(T ) at low-t by a factor of to a naive FL ory estimate would only retain terms! + anticipated in Ref. [3] and is sh tively important. The plateau b rate of RQPs, discussed ab for Q(T ) still increasing in an to T ' T. At a higher tempe regime Q(T ) changes sign and, metal regime, approaches sim D. T U (Fig. 4 ). The atomi successfully describes rm thus be taken as anor fingerp accuracy of approximate formul been tested also in or studies It is interesting to observe h port regimes relate to rmody Fig. 4 we display entropy S K(T ), and Curie constant T local (q-integrated) magnetic sus as well as Curie constant rea.5 ω/d Figure 5.7: Imaginary part of self-energy for different temperatures for doped one-band Hubbard model with hn i =.8 and U/D = 4. Panel (a) is taken from [6], in panel (b) our results are shown. For T < TFL transport is dominated by excitations at ω =. In RQP regime re exists a certain temperature (denoted by T.8D in [6]), above which low-energy electron-like excitations with finite ω > dominate transport, as se excitations exhibit larger lifetimes (smaller values for Im Σ(ω)) than those at ω =. This can be tracked in Fig. 5.7 (a) for T /D =. and in our results in (b) for T /D =. and T /D =.. The plateau for ω > leads to inverse lifetimes that only depend weakly on temperature for T > T [6]. To summarize, we have demonstrated accuracy of our DMFT+NRG calculations by investigating Fermi liquid physics of hole-doped one-band Hubbard model with hn i =.8 and U/D = 4. We have also found resilient quasiparticles for T > TFL and revealed same properties as in [6]. This clearly shows potential of our program. 7

83 Chapter 6 Two-band Hubbard-Kanamori model Although one-band Hubbard model can explain general phenomena of strongly correlated materials, like Mott physics described in previous Chapter, it is clear that one has to extend Hubbard Hamiltonian to account for more complex material specific aspects. Typically several orbitals (bands) need to be included with properly chosen hopping integrals, Coulomb repulsion and Hund s coupling between orbitals. 6. Realistic modeling of strongly correlated materials A recently developed approach to investigate realistic materials is LDA+DMFT framework, which was shortly introduced in Sec.... While LDA method provides ab initio manybody model Hamiltonians based on density-functional ory (DFT), DMFT calculations include strong electronic correlation effects, which are not yet incorporated within LDA. Let s consider an example to understand basic concept of (realistic) modeling of strongly correlated materials, following [, Chapter 6] and [38]. A prevalent lattice structure of transition-metal compounds is cubic perovskite with KCuF 3 as a common representative (see Fig 6. (a)). The transition metal atom (Cu) is situated at center of a F-octahedron and surrounded by cubically arranged K-atoms. In principle, nominal valence electrons of KCuF 3 are distributed among atomic orbitals of given atoms as K + (4s ), F (p 6 ) and Cu + (3d 9 4s ). However, a cubic crystal symmetry at Cu sites leads to a splitting of 5-fold degenerate 3d-orbitals into lower energy (3-fold degenerate) t g and partially filled (-fold degenerate) e g -manifold with higher energy. The former consists of d xy, d xz and d yz orbitals and latter of d x y and d 3z r orbitals, which are depicted in Fig. 6. (b). So we get (t g 6 e 3 g ) instead of (3d 9 4s ) for Cu. The experimental structure of KCuF 3 (R in Fig. 6. (a)) reveals that a perfect cubic crystal field (I δ= in Fig. 6. (a), where δ now denotes cubic crystal field distortion) is an idealization. A more realistic description has to account for Jahn-Teller distortion of F-octahedron (due to coupling between electrons and lattice), resulting in CuF bonds of different length (short (s) and long (l) bonds) in xy-plane, that additionally alternate along x- and y-direction of pseudocubic axes (shown at corner of Fig. 6. (a)) [38]. Besides, conventional tetragonal cell is compressed. These deviations from a perfect cubic crystal symmetry furr split e g doublet into lower energy orbital d 3l r and higher energy orbital d s y. The latter leads to orbital pattern of Fig. 6. (a). 73

84 ungzentrum J ulich, 545 J ulich, Germany, University of Hamburg, Jungiusstrasse 9, 355 Hamburg, Germany (Dated: August 6, 8) ite KCuF3 is considered archetype of an orbitally-ordered system. mean-field ory (DMFT) method, we investigate mechanism for material. We show that purely electronic Kugel-Khomskii superne leads to a remarkably large transition temperature of TKK6. 35 K. CHAPTER TWO-BAND erimentally believed to persist to at least 8 K. Thus Jahn-Teller bilizing orbital-order at such high temperatures. HUBBARD-KANAMORI MODEL LDA+DMFT w,7.7.+a,7..hf (a) homskii showed orbital degrees rise to orbitalexchange mechnow believed to tronic and magoxide Mott inb repulsion is a r it just enor really drives he archetype of ]. In this 3d9 ompletely filled fold degenerate e first scenario ds split parnon-degenerate ulsion, U, n favoring ocappens in some ordering is ulomb repulsion e to crystalario purely rising from and Jahn-Teller In this picture portance []. s evident from hich show [6, 7] ahedra are sta mev per forude larger than GGA+DMFT s, suggesting in ay a small role ure of this sysly indicate that imated in LDA (b) R l Rδ R y Iδ y (x,y,z) x γ=.95 δ=4.4% z K Cu F z s c/ x 6.33 (y,x,-z) γ=.95 δ=.4% γ= δ=-.4% Fig. 5: The s (first row), py, pz, px (second row), and dxy, dyz, d3z r, dxz, dx y (last row) FIG. : (Color online) Left: Crystal structure and orbital- real harmonics. order in a-type [] KCuF3. Cu is at center of F octhaedra enclosed in a K cage. The conventional cell is tetragonal with Figure 6.: (taken from [38]) illustrates definitions crystal orbital-order a-type x =structure r sin θ cos φ, yand = r sin θ sin φ, z = r cos in θ, we can express pseudo-cubic axes Using axes a, b, c, wherepanel a = b, c (a) =.95 a. The real harmonics (Fig.of 5) a as F-octahedron and surrounded are defined x =transition (a+b)/, y =metal ( a+b)/, z = c/. All l =,,at KCuF The atomand(cu) is situated center 3. as Cu sites are equivalent. For sites long (short) bond l (s) conventional cell is described by axes a, b and c, by cubically arranged K-atoms. The tetragonal! is along y (x). Vice versa for sites. Orbital $ (see table I), while axes, at Right: left corner, are + b)/, y = ( a + b)/ and s = ygiven = Y as x = =(a 4π occupied by pseudocubic one hole, is shown for shown each site. Jahn! 3 zteller = c/. is labeled to distortion (measured by Jahn-Teller distortions sites, measuredstructure by δ = (l s)/(l + s)/ with y/r py =R. y Due = i (Y + Y ) = Theatexperimental 4π!. + R is experimental structure, Rfeatures δ and Iδ long (l) and short (s) bonds δand = γ(l= c/a s)/(l s)/), F-octahedron in xy-plane. Furr, 3 z/r = y = Y = 4π two ideal structures with reduced distortions, and I is cubic. pz it is compressed, which is indicated by γ = c/a <. The structure R! at right side of panel (a) 3 px = y = (Y Y ) = 4π x/r corresponds to site in left illustration, while F-octahedron at site! is rotated in xy-plane i 5 xy/r dxy = y = (Y an Y )idealized = 4π by an angle of 9due degrees. Iδ with δ =rar (nothan distortions) denotes structure with a perfect or GGA, probably to self-interaction,! i 5 yz/r 3d-orbitals into dyz = y of = (Y 5-fold + Y ) = degenerate cubic crystal field at super-exchange Cu-atoms, leading to a splitting identifying Kugel-Khomskii as driv4π! ing mechanism for orbital-order. This is supported by ab5 (3-fold degenerate) tg and partially filled (-fold eg -manifold. d3z degenerate) = 4π 3Panel (3z r (b) )/r (taken from [, r = y = Y! initio Hartree-Fock (HF) calculations which give results Chapter 6] depicts atomic orbitals (real harmonics). The first row shows s-orbital, second row 5 akin to LDA+U [9]. Moreover, in superexchange sce- dxz = y = (Y Y ) =! 4π xz/r illustrates p-orbitals (px, py and pz ), third row figures five d-orbitals (d, d xy yz, d3z r, dxz 5 nario it remains to be explained why TOO 8 K [], dx y = y = (Y + Y ) = 4π (x y )/r y ). and d x more than twenty times 3D antiferromagnetic (AFM) critical temperature, TN 38 K [, ], a surprising fact if magnetic- and orbital-order were driven by same super-exchange mechanism. Using LDA approach one can calculate band structure for KCuF3, which is shown in Fig. 6. (a). The black curves denote empty s-like bands of Cu and K, as well as filled p-like bands of F. The tg -like bands of Cu are depicted in red. As y are completely filled, y do not hybridize with eg -like bands, which are located around Femi level, suggesting system to be metallic. However KCuF3 is actually isolating. At this point importance of combining LDA predictions with DMFT calculations becomes clear. As electron-electron interactions are only considered in sense of a static mean-field within LDA, this method fails to describe strongly correlated materials. For DMFT calculations it is essential to work with a minimal model Hamiltonian, that can still be solved numerically. Neverless, crucial material specific degrees of freedom have to be maintained. Based on idea of Wilson s renormalization group, it is possible to construct low energy models with renormalized physical parameters, by integrating out high-energy degrees of freedom. In context of LDA this procedure is known as massive downfolding and results in few-band model Hamiltonians, that can n be solved with DMFT. 74