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arXiv:1712.04171 · Physical review. B./Physical review. B

Divergences of the irreducible vertex functions in correlated metallic systems: Insights from the Anderson impurity model

Patrick Chalupa-Gantner, Patrik Gunacker, Thomas Schäfer, Karsten Held, A. Toschi

Published Jun 22, 2018• 80 citations
arXivHTMLPDF

Abstract

In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.

DOI
10.1103/physrevb.97.245136

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In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.
We thoroughly analyze the divergences of the irreducible vertex functions occurring in the charge channel of the half-filled Hubbard model in close proximity to the Mott metal-insulator transition (MIT). In particular, by systematically performing dynamical mean-field theory (DMFT) calculations on the two-particle level, we determine the location and the number of the vertex divergences across the whole coexistence region adjacent to the first-order metal-to-insulator transition. We find that the lines in the parameter space, along which the vertex divergences occur, display a qualitatively different shape in the coexisting metallic and insulating phase, which is also associated to an abrupt jump of the number of divergences across the MIT. Physically, the systematically larger number of divergences on the insulating side of the transition reflects the sudden suppression of local charge fluctuation at the MIT. Further, a systematic analysis of the results demonstrates that the number of divergence lines increases as a function of the inverse temperature $\ensuremath{\beta}={({k}_{\mathrm{B}}T)}^{\ensuremath{-}1}$ by approaching the Mott transition in the zero-temperature limit. This makes it possible to identify the zero-temperature MIT as an accumulation point of an infinite number of vertex divergence lines, unveiling the highly nonperturbative nature of the underlying transition.
Here we analyze the divergences of the irreducible vertex function in dynamical mean field theory, which may indicate either a nonphysical breakdown of the perturbation theory or a response to some physical phenomenon. To investigate this question, we construct a quasiparticle vertex from the diverging irreducible vertex functions. This vertex describes the scattering between quasiparticles and quasiholes in a Fermi liquid. We show that the quasparticle vertex does not diverge in the charge channel, wherein the irreducible vertex does diverge; and we show that the quasiparticle vertex does diverge in the spin channel, wherein the irreducible vertex does not diverge. This divergence occurs at the Mott transition, wherein the Fermi-liquid theory breaks down. Both the half filled Hubbard and Anderson lattices are investigated. In general, our results support that the divergences of the irreducible vertex function do not indicate a nonphysical failure of the perturbation theory. Instead, the divergences are the mathematical consequence of inverting a matrix (the local charge susceptibility) which accumulates increasingly negative diagonal elements as the Hubbard interaction suppresses charge fluctuations. Indeed, we find that the first divergences of the irreducible vertex in both Hubbard and Anderson lattices occurs near the maximum magnitude of the (negative) vertex-connected part of the charge susceptibility.
An essential role for the description of correlated quantum many-particle systems is played by impurity problems. These consist of a small number of localized orbitals, where the electrons are subject to a Coulomb-interaction, coupled to noninteracting bath degrees of freedom. Impurity models are used for the description of magnetic impurities in metals, of quantum points in nanostructures, and in the context of the dynamical mean field theory (DMFT). For their solution a large number of numerical methods exists. We mention here quantum Monte Carlo methods, the numerical renormalization group, or the exact diagonalization method. In this thesis we introduce a new approach to impurity problems, which is based on the functional renormalization group (fRG). In contrast to the methods mentioned above this approach is approximative, but it can be applied with comparatively lower numerical effort. The aim of this thesis is to examine the advantages and limits of this method. An important role in this connection is played by the single impurity Anderson model (SIAM), which is already understood very well with other methods and is seen as a prototype for models with strong local correlations. The spin fluctuations of this model are governed by the so-called Kondo energy scale and a central question of this thesis is whether this scale can be reproduced with our approach. We begin with a short introduction to the Green's function formalism for correlated quantum many particle systems. Then the fRG flow equations for the one-particle irreducible vertex functions are deduced, and several approximation schemes are discussed. In chapter 4 we introduce the fRG scheme used in this thesis. At first the SIAM is mapped to a semi-infinite chain, in which the interacting orbital is given by the first site of this chain. The system is then subdivided into two parts. The first part, which is called 'core', consists of the interacting orbital and the first L bath sites. The remaining bath sites form the second part. At the beginning both parts are decoupled from each other such that the core can be solved exactly. Starting with this exact solution the coupling between the core and the remaining bath sites is switched on slowly in a renormalization group flow. This way it is possible to calculate the one-particle and two-particle correlation functions on the correlated site. We call this flow scheme 'hybridization flow'. The flow equations are formulated in an effective theory on the first bath site outside the core, which turns out to be advantageous compared to other implementations of the hybridization flow. In chapter 5 this hybridization flow method is applied to the SIAM with semi-elliptic bath density of states. We discuss the differences which arise for different core-sizes (L = 0,1,2,3) and for different truncations of the fRG flow equations. In the cases L=0 and L=2 the local Fermi liquid properties of the SIAM and the associated Kondo scale can not be reproduced in both approximation schemes. For the core sizes L=1 and L=3 the ground state of the core is already a spin singlet state in the beginning of the flow and the Fermi liquid properties are reproduced. The Kondo energy scale, which determines the width of the central quasi-particle peak and the size of the local spin-susceptibility, is however not accurately resolved. This is partly due to the finite temperature used in our implementation such that for larger interaction strength the temperature is above the Kondo energy scale. The application of the hybridization flow method to quantum impurity problems in the context of the DMFT is discussed in chapter 6. In this approach lattice models with a local interaction, like the Hubbard model, are mapped to an effective impurity problem for a single lattice site embedded in a dynamical mean field representing the influence of the other electrons. This scheme is exact in infinitely many dimensions and a nonperturbative approximation method for finite-dimensional systems. In particular the Mott metal-insulator transition in the Hubbard model is successfully described by this method. One class of quantities, which is easily accessible from the hybridization flow scheme, is given by the local two-particle correlation functions. These play an essential role in non-local extensions of the DMFT. Therefore we mainly focus on these quantities. At first we apply our flow scheme to the Hubbard model on the Bethe lattice in infinite dimensions and calculate the local one-particle irreducible vertex functions in the insulating and the metallic phase in good agreement with previous calculations, which use the exact diagonalization method as impurity solver. After this we extend our flow scheme to a cluster DMFT method, which includes short-range antiferromagnetic correlations and calculate again the local and non-local vertex functions for the two-dimensional Hubbard model. We compare these to the results of the single-site DMFT.
We analyze the highly nonperturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory (DMFT) calculations at the two-particle level. By extending the results of Sch\"afer et al. [Phys. Rev. Lett. 110, 246405 (2013)] we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. By comparing our numerical data for the Hubbard model with analytical calculations for exactly solvable systems of increasing complexity [disordered binary mixture (BM), Falicov-Kimball (FK), and atomic limit (AL)], we have (i) identified two different kinds of divergence lines; (ii) classified them in terms of the frequency structure of the associated singular eigenvectors; and (iii) investigated their relation to the emergence of multiple branches in the Luttinger-Ward functional. In this way, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, single energy scale ${\ensuremath{\nu}}^{*}$ below which perturbation theory is no longer applicable, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the nonperturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.
Identifying the fingerprints of the Mott-Hubbard metal-insulator transition may be quite elusive in correlated metallic systems if the analysis is limited to the single particle level. However, our dynamical mean-field calculations demonstrate that the situation changes completely if the frequency dependence of the two-particle vertex functions is considered: The first nonperturbative precursors of the Mott physics are unambiguously identified well inside the metallic regime by the divergence of the local Bethe-Salpeter equation in the charge channel. In the low-temperature limit this occurs for interaction values where incoherent high-energy features emerge in the spectral function, while at high temperatures it is traceable up to the atomic limit.
In the last decades, multifaceted manifestations of the breakdown of the self-consistent perturbation theory have been identified for the many-electron problem. Yet, the investigations have been so far mostly limited to paramagnetic states, where symmetry breaking is not allowed. Here, we extend the analysis to the spontaneously symmetry-broken antiferromagnetic (AF) phase of the repulsive Hubbard model. To this aim, we calculated two-particle quantities using dynamical mean-field theory for the AF-ordered Hubbard model and studied the possible occurrence of divergences of the irreducible vertex functions in the charge and spin sectors. Our calculations pinpoint the divergences in the AF phase diagram, showing that while the onset of AF order mitigates the breakdown of the perturbation expansion, it does not fully prevent it. Moreover, we have been able to link the changes in the dynamical structure of the corresponding generalized susceptibilities to the physical crossover from a weak-coupling (Slater) to a strong-coupling (Heisenberg) antiferromagnet, which takes place as the interaction strength is gradually increased. Finally, we discuss possible physical consequences of the irreducible vertex divergences in triggering phase-separation instabilities within the AF phase and elaborate on the implications of our findings for two-dimensional systems, where the onset of a long-range AF order is prevented by the Mermin-Wagner theorem.