Median-point approximation and its application for the study of fermionic systems

TL;DR

This paper introduces a novel median-point approximation method for fermionic systems, revealing that effective interactions remain finite at phase transitions and highlighting the impact of flat bands on superconductivity in materials like twisted bilayer graphene.

Contribution

It extends the saddle-point approximation with dual bosonic fields to study fermionic systems, deriving a Bethe-Salpeter equation that accounts for effective interactions beyond the RPA.

Findings

Effective interaction remains finite at phase transitions.

Flat bands reduce kinetic energy cost of gap formation.

Increased anisotropy of interactions favors momentum-dependent order parameters.

Abstract

We consider a system of fermions with local interactions on a lattice (Hubbard model) and apply a novel extension of the Laplace's method (saddle-point approximation) for evaluating the corresponding partition function. There, we introduce dual free bosonic fields, with a propagator corresponding to an effective (renormalized) interaction with Maki-Thompson and Aslamazov-Larkin type corrections and beyond, and demonstrate that the superconducting pairing originates as an instability of the effective interaction. We derive the corresponding Bethe-Salpeter equation (instability criterion) and show that the interaction enters the equation only in its effective form to all orders, including the exchange part of the self-energy. An important implication of this result is that the effective interaction always remains finite, even at phase-transition points, directly contradicting the often used assumption of linear relationship between the interaction and susceptibility, established within the random-phase approximation. By analyzing the Bethe-Salpeter equation in the context of unconventional superconductivity, we find that the presence of a flat band close the Fermi level, found in materials such as twisted bilayer graphene, has a twofold favorable impact persisting beyond the weak-coupling approximation: a reduced kinetic energy cost of the gap formation and an increased anisotropy of the effective interaction, favoring a momentum dependent order parameter.