FLEX+DMFT approach to the $d$-wave superconducting phase diagram of the two-dimensional Hubbard model

TL;DR

This paper introduces FLEX+DMFT, a combined theoretical approach that captures both local and non-local correlations, revealing a dome-shaped $T_c$ dependence in the 2D Hubbard model's $d$-wave superconductivity, which was not seen with FLEX alone.

Contribution

The paper formulates FLEX+DMFT systematically and demonstrates its ability to reproduce the $T_c$ dome structure in the 2D Hubbard model, highlighting the role of local vertex corrections.

Findings

FLEX+DMFT produces a $T_c$ dome as a function of filling.

The dome originates from local vertex corrections in the self-energy.

Spectral functions show a double-peak structure indicative of Hubbard bands.

Abstract

The dynamical mean-field theory (DMFT) combined with the fluctuation exchange (FLEX) method, namely FLEX+DMFT, is an approach for correlated electron systems to incorporate both local and non-local long-range correlations in a self-consistent manner. We formulate FLEX+DMFT in a systematic way starting from a Luttinger-Ward functional, and apply it to study the $d$-wave superconductivity in the two-dimensional repulsive Hubbard model. The critical temperature ($T_c$) curve obtained in the FLEX+DMFT exhibits a dome structure as a function of the filling, which has not been clearly observed in the FLEX approach alone. We trace back the origin of the dome to the local vertex correction from DMFT that renders a filling dependence in the FLEX self-energy. We compare the results with those of GW+DMFT, where the $T_c$-dome structure is qualitatively reproduced due to the same vertex correction effect, but a crucial difference from FLEX+DMFT is that $T_c$ is always estimated below the N\'{e}el temperature in GW+DMFT. The single-particle spectral function obtained with FLEX+DMFT exhibits a double-peak structure as a precursor of the Hubbard bands at temperature above $T_c$.