Here we analyze the divergences of the irreducible vertex function in dynamical mean field theory, which may indicate either a non-physical 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 \textit{does not} diverge in the charge channel, wherein the irreducible vertex \textit{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 Hubbard and Anderson lattices are investigated. In general, our results support that the divergences of the irreducible vertex function do not indicate a non-physical 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 symmetric 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, which suppresses charge fluctuations.
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…
In the two-impurity Anderson model, the inter-impurity spin exchange interaction favors a spin singlet state between two impurities leading to the breakdown of the Kondo effect. We show that a local uniform magnetic field can delocalize the quasiparticles to restore the Kondo resonance. This transition is found to be continuous, accompanied by not only the divergence of the staggered (antiferromagnetic) susceptibility, but also the divergence of the uniform spin susceptibility. This may imply that the magnetic field induced quantum phase transitions in Kondo systems are in favor of the local c…
We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle irreducible vertex functions, given a cut-off in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to non-conserving approximations introduced in an earlier publication ctuation exchange approximation as well as to nonconserving approximations introducen [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single…
The temperature dependence of electric dipolar susceptibility χ_P is discussed on the basis of the Anderson-Holstein model with the use of a numerical renormalization group (NRG) technique. Note that P is related with phonon Green's function D. In order to obtain correct temperature dependence of P at low temperatures, we propose a method to evaluate P through the Dyson equation from charge susceptibility χ_c calculated by the NRG, in contrast to the direct NRG calculation of D. We find that the irreducible charge susceptibility estimated from χ_c agree with the perturbation calculation, sugge…
We investigate the effect of magnetic fields on a Kondo insulator by using the periodic Anderson model. The analysis by dynamical mean field theory combined with quantum Monte Carlo simulations reveals that the magnetic field drives the Kondo insulator to a transverse antiferromagnetic insulator at low temperatures. We calculate the staggered spin susceptibility and find its divergence signaling the antiferromagnetic instability. Further investigation of the spin correlation functions and the magnetization process clarifies how the magnetic field suppresses the Kondo singlet formation and indu…
We use the dynamical mean-field approximation to study singularities in the self-energy and a two-particle irreducible vertex induced by the metal-insulator transition of the disordered Falicov-Kimball model. We set general conditions for the existence of a critical metal-insulator transition caused by a divergence of the imaginary part of the self-energy. We calculate explicitly the critical behavior of the self-energy for the symmetric and asymmetric disorder distributions. We demonstrate that the metal-insulator transition is preceded by a pole in a two-particle irreducible vertex. We show …
Two-particle generalized susceptibilities and their irreducible vertex functions play a prominent role in the quantum many-body theory for correlated electron systems. They act as basic building blocks in the parquet formalism which provides a flexible scheme for the calculation of spectral and response functions. The irreducible vertices themselves have recently attracted increased attention as unexpected divergences in these functions have been identified. Remarkably, such singularities appear already for one of the simplest strongly interacting systems: the atomic limit of the half-filled H…
Density of states, dynamic (optical) conductivity and phase diagram of paramagnetic two-dimensional Anderson-Hubbard model with strong correlations and disorder are analyzed within the generalized dynamical mean-field theory (DMFT+Sigma approximation). Strong correlations are accounted by DMFT, while disorder is taken into account via the appropriate generalization of the self-consistent theory of localization. We consider the two-dimensional system with the rectangular "bare" density of states (DOS). The DMFT effective single impurity problem is solved by numerical renormalization group (NRG)…
The strong electron correlations caused by Coulomb interaction of impurity electrons are taken into account. The infinite series of diagrams containing irreducible Green's functions are summed. For symmetric Anderson model we establish the antisymmetry property of the impurity Green's function, formulate the exact Dyson type equation for it, find the approximate correlation function $Z_σ(iω$) and solve the integral equation which determines the full propagator of the impurity electrons. Analytical continuation of the obtained Matsubara Green's function determines the retarded one and gives the…
The Anderson impurity model with a density of states $ρ(\varepsilon) \propto |\varepsilon|^r$ containing a power-law pseudogap centered on the Fermi energy ($\varepsilon = 0$) features for $0<r<1$ a Kondo-destruction quantum critical point (QCP) separating Kondo-screened and local-moment phases. The observation of mixed valency in quantum critical $β$-YbAlB$_4$ has prompted study of this model away from particle-hole symmetry. The critical spin response associated with all Kondo destruction QCPs has been shown to be accompanied, for $r=0.6$ and noninteger occupation of the impurity site, by a …
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irre…
Recent experiments measuring the emission of exciton recombination in a self-organized single quantum dot (QD) have revealed that novel effects occur when the wetting layer surrounding the QD becomes filled with electrons, because the resulting Fermi sea can hybridize with the local electron levels on the dot. Motivated by these experiments, we study an extended Anderson model, which describes a local conduction band level coupled to a Fermi sea, but also includes a local valence band level. We are interested, in particular, on how many-body correlations resulting from the presence of the Ferm…
We report the existence of a localization-delocalization transition in the classical plasma modes of a one dimensional Wigner Crystal with a white noise potential environment at T=0. Finite size scaling analysis reveals a divergence of the localization length at a critical eigenfrequency. Further scaling analysis indicates power law behavior of the critical frequency in terms of the relative interaction strength of the charges. A heuristic argument for this scaling behavior is consistent with the numerical results. Additionally, we explore a particular realization of random-bond disorder in a …
The paper deals with the nonequilibrium two-lead Anderson model, considered as an adequate description for transport through a d-c biased quantum dot. Using a self-consistent equation-of-motion method generalized out of equilibrium, we calculate a fourth-order decoherence rate $γ^{(4)}$ induced by a bias voltage $V$. This decoherence rate provides a cut-off to the infrared divergences of the self-energy showing up in the Kondo regime. At low temperature, the Kondo peak in the density of states is split into two peaks pinned at the chemical potential of the two leads. The height of these peaks …
We investigate the divergences appearing in the two-particle irreducible vertex functions of many-fermion systems with attractive on-site interactions. By means of dynamical mean-field theory calculations, we determine the location of singularity lines in the phase diagram of the attractive Hubbard model at half-filling, where the local Bethe-Salpeter equations are non invertible. We find that divergences appear both in the magnetic and in the density scattering channels. The former affect a sector of suppressed fluctuations and comply with the mapping of the physical susceptibilities of the r…
The concept of dynamical symmetries is used for formulation of the renormalization group approach to the Kondo effect in the Anderson model with repulsive and attractive interaction $U$. It is shown that the generic local symmetry of the Anderson Hamiltonian is determined by the SU(4) Lie group. The Anderson Hamiltonian is rewritten in terms of the Gell-Mann matrices of the 4-th rank, which form the set of group generators and the basis for construction of irreducible vector operators describing the excitation spectra in the charge and spin sectors. The multistage Kondo sceening is described i…
The nonequilibrium time evolution of a quantum dot is studied by means of dynamic equations for time-dependent Greens functions derived from a two-particle-irreducible (2PI) effective action for the Anderson impurity model. Coupling the dot between two leads at different voltages, the dynamics of the current through the dot is investigated. We show that the 2PI approach is capable to describe the dynamical build-up of the Kondo effect, which shows up as a sharp resonance in the spectral function, with a width exponentially suppressed in the electron self coupling on the dot. An external voltag…
The O(3) symmetric Anderson model is an example of a system which has a stable low energy marginal Fermi liquid fixed point for a certain choice of parameters. It is also exactly equivalent, in the large U limit, to a localized model which describes the spin degrees of freedom of the linear dispersion two channel Kondo model. We first use an argument based on conformal field theory to establish this precise equivalence with the two channel model. We then use the numerical renormalization group (NRG) approach to calculate both one-electron and two-electron response functions for a range of valu…
We analyze the highly non-perturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory calculations at the two-particle level. By extending the results of Schäfer, 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. These divergence lines accumulate around the critical Mott endpoint in accorda…
The Green-function technique, termed the irreducible Green functions (IGF) method, that is a certain reformulation of the equation-of motion method for double-time temperature dependent Green functions is presented. This method was developed to overcome some ambiguities in terminating the hierarchy of the equations of motion of double-time Green functions and to give a workable technique to systematic way of decoupling. The approach provides a practical method for description of the many-body quasi-particle dynamics of correlated systems on a lattice with complex spectra. Moreover, it provides…
We show that in two dimensions (2D) a systematic expansion of the self-energy and the effective interaction of the dilute electron gas in powers of the two-body T-matrix T_0 can be generated from the exact hierarchy of functional renormalization group equations for the one-particle irreducible vertices using the chemical potential as flow parameter. Due to the interference of particle-particle and particle-hole channels at order T_0^2, in 2D the ladder approximation for the self-energy is not reliable beyond the leading order in T_0. We also discuss two-body scattering in vacuum in arbitrary d…
A self-consistent theory for two-particle fluctuations with renormalized irreducible vertices is proposed. Using the Parquet formalism, we construct the fully antisymmetric full vertex in terms of the two-particle fluctuations in the charge, the spin and the particle-particle channels on an equal footing to satisfy the Pauli principle. The fluctuations are determined self-consistently, which are reflected into the one-particle self-energy via the Schwinger-Dyson equation. We demonstrate the application of the present theory to the impurity Anderson model and the Hubbard model on a square latti…
We review two analytical approaches in Dynamical Mean-Field Theory (DMFT) based on a perturbation theory expansion over the electron hopping to and from the self consistent environment. In the first approach the effective single impurity Anderson model (SIAM) is formulated in terms of the auxiliary Fermi-fields and the projection (irreducible Green's function) technique is used for its solution. A system of the DMFT equations is obtained that includes as simple specific cases a number of known approximations (Hubbard-III, AA, MAA, ...). The second approach is based on the diagrammatic techniqu…
Taking the competition and the mutual screening of various bosonic fluctuations in correlated electron systems into account requires an unbiased approach to the many-body problem. One such approach is the self-consistent solution of the parquet equations, whose numerical treatment in lattice systems is however prohibitively expensive. In a recent article it was shown that there exists an alternative to the parquet decomposition of the four-point vertex function, which classifies the vertex diagrams according to the principle of single-boson exchange (SBE) [F. Krien, A. Valli, and M. Capone, ar…
We investigate the non-Fermi-liquid (NFL) behavior of the impurity Anderson model (IAM) with non-Kramers doublet ground state of the f$^2$ configuration under the tetragonal crystalline electric field (CEF). The low energy spectrum is explained by a combination of the NFL and the local-Fermi-liquid parts which are independent with each other. The NFL part of the spectrum has the same form to that of two-channel-Kondo model (TCKM). We have a parameter range that the IAM shows the $- \ln T$ divergence of the magnetic susceptibility together with the positive magneto resistance. We point out a po…
Non-perturbative effects of short-range spatial correlations at the two-particle level
Max-Planck-Institut für Festkörperforschung（马克斯·普朗克固体研究所）；Institute of Solid State Physics, TU Wien（维也纳工业大学固体物理研究所）；Dipartimento di Fisica, Università di Trieste（的里雅斯特大学物理系）
We present here two alternative schemes designed to correct the high-frequency truncation errors in the numerical treatment of the Bethe-Salpeter equations. The schemes are applicable to all Bethe-Salpeter calculations with a local two-particle irreducible local, which is relevant, e.g., for the dynamical mean-field theory (DMFT) and its diagrammatic extensions. In particular, within a purely diagrammatic framework, we could extend existing algorithms for treating the static case in the particle-hole sector to more general procedures applicable to all bosonic frequencies and all channels. Afte…
We construct a new functional for the single particle Green's function, which is a variant of the standard Baym Kadanoff functional. The stability of the stationary solutions to the new functional is directly related to aspects of the irreducible particle hole interaction through the Bethe Salpeter equation. A startling aspect of this functional is that it allows a simple and rigorous derivation of both the standard and extended dynamical mean field (DMFT) equations as stationary conditions. Though the DMFT equations were formerly obtained only in the limit of infinite lattice coordination, th…
Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identities among vertices originating from Goldstone's theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green's functions, three- and four-point vertices, and two-particle Green's functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared…
