Renormalized perturbation theory for Fermi systems: Fermi surface deformation and superconductivity in the two-dimensional Hubbard model

TL;DR

This paper develops a renormalized perturbation theory for Fermi systems, applied to the 2D Hubbard model, revealing superconductivity, Fermi surface deformations, and instabilities near van Hove singularities.

Contribution

It introduces a renormalized perturbation expansion that explicitly accounts for Fermi surface shifts and superconductivity in the Hubbard model, including second-order numerical solutions.

Findings

Superconductivity with d-wave symmetry near half-filling in the repulsive Hubbard model.

Fermi surface deformations and Pomeranchuk instability near van Hove singularities.

Weakly momentum-dependent s-wave superconductivity in the attractive Hubbard model.

Abstract

Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the non-interacting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical soutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second order calculation yeilds s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.