A. Consistent description of quantum criticality in systems with correlated electrons

A consistent and reliable description of the low-temperature behavior of strongly correlated electron systems has not yet been reached in spite of decades of intensive research in this field. Most properties of weakly and moderately coupled electrons in metals are captured in a sufficient extent by the Fermi-liquid theory. The problems arise when one tries to extend Fermi-liquid solutions to the strong-coupling regime. Fermi liquid becomes unstable due to quantum dynamical fluctuations and the electron system approaches a quantum critical point for a sufficiently strong interaction.  Non-perturbative approaches are then needed.

            Analytic approaches are generally based on a many-body perturbation, diagrammatic expansion in the interaction strength. To apply the diagrammatic expansion beyond weak electron correlations one needs to sum infinite series of specific classes of diagrams and make the approximations non-perturbative and self-consistent. Self-consistency cannot be introduced in an arbitrary manner. The canonical way how to achieve conserving and thermodynamically consistent approximations was outlined by Baym and Kadanoff. Even if we guarantee mass conservation in the theory with renormalized one-particle functions in the Baym-Kadanoff construction, we are unable to match the irreducible vertex derived from the self-energy via the functional Ward identity with the full vertex used in the Schwinger-Dyson equation. The renormalization in the Baym-Kadanoff construction does not renormalize the bare interaction. Consequently, there is no direct control of singularities in the Bethe-Salpeter equations for the two-particle functions.

            The Baym and Kadanoff method works wit one self-energy but two vertex functions, one from the Ward identity and one from the Schwinger-Dyson equation. The Ward identity is needed to keep the solutions conserving while the Schwinger-Dyson equation is the actual quantum dynamical equation. Both vertex functions display singularities in the strong-coupling regime and the Baym and Kadanoff construction becomes inconsistent in quantum criticality.

         We modified the construction of self-consistent approximations in that we start from a the two-particle irreducible vertex of the singular Bethe-Salpeter equation and take it as the generating functional. To avoid spurious singularities in the Bethe-Salpeter equations we use a two-particle self-consistency scheme based on the parquet equations. We adopted (reduced) the parquet equations in such a way that the critical behavior in Bethe-Salpeter equations is not completely suppressed. It happens for the full set of the parquet equations with the bare interaction as its input due to the overweighed two-particle self-consistency there.

            We use the irreducible vertex from the singular Bethe-Salpeter equation and use it to determine a thermodynamic self-energy. Since the full Ward identity cannot be solved exactly, we resorted to an approximate solution, linear in the symmetry-breaking field of the critical point of the Bethe-Salpeter equation. In this way the critical behavior of the two-particle response function (susceptibility) generates a symmetry-breaking in the thermodynamic self-energy. The thermodynamic self-energy constructed in this way is then used to renormalize the one-electron propagators in the Bethe-Salpeter equations. The thermodynamically renormalized propagators are used in the Schwinger-Dyson dynamical equation to determine the spectral self-energy. The latter is then used as the true physical self-energy. The spectral self., richer in the dynamical structure than the thermodynamic one, shares the same critical behavior with the thermodynamic one. In this way a qualitative consistency between the thermodynamic and dynamical, spectral properties has been achieved. An unprecedented qualitatively consistent description of quantum criticality of correlated electrons has been achieved.

            The description of correlated electrons in the critical region of singularities in Bethe-Salpeter equations has been formulated quite generally, but quantitatively only a static approximation on the irreducible vertex from the singular Bethe-Salpeter equation was performed for the symmetric single-impurity Anderson model, where the strong-coupling Kondo behavior was qualitatively correctly reproduced in thermodynamic and spectral functions [1]. This construction was extended to a mean-filed approximation, a generalization of the Hartree weak-coupling solution, to extended models [2].  Such a static approximation is restricted to zero temperature and high spatial dimensions. Presently we work on a dynamical extension of the static solution, both within the local (mean-field) approach as well with non-local corrections covering spatial fluctuations. The latter are of principal importance in low-dimensional systems.      

 

References

[1]   V. Janiš, A. Kauch, and V. Pokorný, Thermodynamically consistent description of criticality in models of correlated electrons, Phys. Rev. B 95, 045108 (2017) 1-14.

[2]   V. Janiš, V. Pokorný, and A. Kauch Mean-field approximation for thermodynamic and spectral functions of correlated electrons: Strong coupling and arbitrary band filling, Phys. Rev. B 95, 165113 (2017) 1-12.

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