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arXiv:2309.07699 · Physical review. B./Physical review. B

Failure of the Baym-Kadanoff construction to consistently match quantum dynamics with thermodynamic critical behavior

V. Janiš, Vladislav Pokorný, Šimon Kos

Published Feb 29, 2024• 2 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physrevb.109.075171

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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
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We present a construction of a mean-field theory for thermodynamic and spectral properties of correlated electrons reliable in the strong-coupling limit. We introduce an effective interaction determined self-consistently from the reduced parquet equations. It is a static local approximation of the two-particle irreducible vertex, the kernel of a potentially singular Bethe-Salpeter equation. The effective interaction enters the Ward identity from which a thermodynamic self-energy, renormalizing the one-electron propagators, is determined. The dynamical Schwinger-Dyson equation with the thermodynamic propagators is then used to calculate the spectral properties. The thermodynamic and spectral properties of correlated electrons are in this way determined on the same footing and in a consistent manner. Such a mean-field approximation is analytically controllable and free of unphysical behavior and spurious phase transitions. We apply the construction to the asymmetric Anderson impurity and the Hubbard models in the strong-coupling regime.
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.
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