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arXiv:cond-mat/9904069 · Physical review. B, Condensed matter

Stability of self-consistent solutions for the Hubbard model at intermediate and strong coupling

V. Janiš

Published Oct 15, 1999• 52 citations
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Abstract

We present a general framework for investigating the stability of solutions being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we use explicit equations for one- and two-particle Green functions simultaneously. We formulate a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density or anomalous type. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices become unstable with respect to spin flips, before the Mott-Hubbard metal-insulator transition in the paramagnetic phase or the Kondo strong-coupling regime can be reached. Further on, we use the parquet approximation and turn it into a workable scheme with dynamical vertex corrections. We obtain a theory with two-particle self-consistence, the complexity of which is comparable with fluctuation-exchange-type approximations. We show that it is the simplest consistent theory free of spurious phases being able to correctly describe qualitatively, quantum critical points and the transition from weak to strong coupling in correlated electron systems.

DOI
10.1103/physrevb.60.11345

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We present a general framework for investigating the stability of solutions being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we use explicit equations for one- and two-particle Green functions simultaneously. We formulate a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density or anomalous type. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices become unstable with respect to spin flips, before the Mott-Hubbard metal-insulator transition in the paramagnetic phase or the Kondo strong-coupling regime can be reached. Further on, we use the parquet approximation and turn it into a workable scheme with dynamical vertex corrections. We obtain a theory with two-particle self-consistence, the complexity of which is comparable with fluctuation-exchange-type approximations. We show that it is the simplest consistent theory free of spurious phases being able to correctly describe qualitatively, quantum critical points and the transition from weak to strong coupling in correlated electron systems.
Single-band Hubbard model at criticality of the metal-insulator transition is studied using approximations derived from parquet theory. It is argued that only the electron-hole and interaction two-particle channels in the parquet algebra are relevant. A scheme is proposed how to reduce the parquet equations to a manageable form the complexity of which is comparable with single-channel approximations such as the renormalized RPA. The newly derived approximation, however, contains dynamical vertex corrections, remains self-consistent at the two-particle level and allows only for integrable singularities. A qualitatively new approach for studying two-particle singularities at zero temperature is obtained.
A two-particle self-consistency is rarely part of mean-field theories. It is, however, essential for avoiding spurious critical transitions and unphysical behavior. We present a general scheme for constructing analytically controllable approximations with self-consistent equations for the two-particle vertices based on the parquet equations. We explain in detail how to reduce the full set of parquet equations so as not to miss quantum criticality in strong coupling. We further introduce a decoupling of convolutions of the dynamical variables in the Bethe-Salpeter equations to make them analytically solvable. We connect the self-energy with the two-particle vertices to satisfy the Ward identity and the Schwinger-Dyson equation and discuss the role of the one-particle self-consistency in making the approximations reliable in the whole spectrum of the input parameters. Finally, we demonstrate the general construction on the simplest static approximation that we apply to the Kondo behavior of the single-impurity Anderson model.
In this thesis, we study quantum impurity models by means of the diagrammatic perturbation theory. First, parquet equations are simplified in the strong coupling regime and low temperatures and an analytically controllable approximation results. We apply it to both the single impurity Anderson model and the Hubbard model in the dynamical mean-field approximation and we study properties of the solution. We recover the correct three-peak density of states including the Kondo resonance and prove that it has the correct exponential scaling with increasing interaction strength. Our approximate scheme is further extended to multi-orbital models. We explicitly solve the two-orbital single impurity Anderson model and the Hubbard model and calculate effects of the Hund exchange interaction and the crystal field.
We disclose a serious deficiency of the Baym-Kadanoff construction of thermodynamically consistent conserving approximations. There are two vertices in this scheme: dynamical and conserving. The divergence of each indicates a phase instability. We show that each leads to incomplete and qualitatively different behavior at different critical points. The diagrammatically controlled dynamical vertex from the Schwinger-Dyson equation does not obey the Ward identity and cannot be continued beyond its singularity. The standardly used dynamical vertex alone cannot, hence, conclusively decide about the stability of the high-temperature phase. On the other hand, the divergence in the conserving vertex, obeying the conservation laws, does not invoke critical behavior of the spectral function and the specific heat. Moreover, the critical behavior of the conserving vertex may become spurious in low-dimensional systems. Consequently, the description of the critical behavior of correlated electrons becomes consistent and reliable only if the fluctuations of the order parameter in the conserving vertex lead to a divergence coinciding with that of the dynamical one. Published by the American Physical Society 2024
We analyse the behaviour of correlated electrons described by Hubbard-like models at intermediate coupling. We argue that with increasing interaction a pole in a generic two-particle Green function is approached. The pole signals condensation of electron-hole pairs and a metal-insulator transition at half-filling. The two-particle singularity calls for a sophisticated renormalization of the interaction strength. A self-consistent diagrammatic technique with renormalized two-particle Green functions is developed. The theory is based on a linked-cluster expansion for the thermodynamic potential with electron-electron interaction as the propagator. The simplest theory with full vertex renormalization, summing self-consistently multiple scatterings from two electron-hole channels, is proposed. We obtain an approximation with a generating functional in closed form enabling us to handle appropriately singularities in two-particle Green functions. The approximation is shown to be asymptotically exact in an external magnetic field close to the fully polarized ferromagnetic state at half-filling and zero temperature.
We extend an approximation that we developed earlier for the single-impurity Anderson model to a full-size impurity solver for models of interacting electrons with multiple orbitals. The approximation is based on parquet equations simplified by separating small and large energy fluctuations justified in the critical region of a pole in the two-particle vertex. We show that an lll-orbital model with the most general interaction is described within this approximation by $4{l}^{2}\ifmmode\times\else\texttimes\fi{}4{l}^{2}$ matrices and is Fermi liquid in the metallic phase. We explicitly calculate properties of a paramagnetic solution of a two-orbital Hubbard model with a Hund exchange and orbital splitting within the dynamical mean-field approximation. We trace the genesis of a metal-insulator transition induced by a crystal field and vanishing of the Kondo quasiparticle peak in strongly correlated orbitally asymmetric systems.