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.
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.