Optical and Hall conductivity of the two dimensional Hubbard model: effective theory description, sign-problem-free Monte Carlo simulation and applications to the cuprate superconductors

TL;DR

This paper develops an effective theory for the optical and Hall conductivities of the 2D Hubbard model, enabling sign-problem-free Monte Carlo simulations and applying them to understand cuprate superconductors.

Contribution

It introduces a sign-problem-free Monte Carlo approach based on an effective theory for local moments, applied to study optical and Hall responses in cuprates.

Findings

Optical and Hall conductivities show a two-component structure with Drude and mid-infrared features.

The sign of the Hall conductivity's Drude component can be positive or negative depending on Fermi surface details.

The framework effectively captures the impact of thermal local moment fluctuations on transport properties.

Abstract

Exact formulas for the optical conductivity and the Hall conductivity of the two dimensional Hubbard model are derived in terms of an effective theory description of the local moment fluctuation in the system. In this framework, the quantum Monte Carlo simulation of the electromagnetic response of such a strongly correlated electron system becomes sign-problem-free in many physically relevant cases. In particular, it is sign-problem-free when we assume the widely used Millis-Monien-Pines form for the phenomenological susceptibility in the effective action of the fluctuating local moment, even though these local moments are now subjected to Landau damping as a result of their coupling to the itinerant quasiparticle on the fermi surface. This is true more generally when a $\varphi^{4}$ term is included in the effective action and is thus not restricted to the Gaussian limit. Here we demonstrate the power of this framework by studying the effect of thermal fluctuation of the local moment on the optical conductivity $\sigma^{xx}(\omega)$ and the Hall conductivity $\sigma^{xy}(\omega)$ of the cuprate superconductors. Both $\sigma^{xx}(\omega)$ and $\sigma^{xy}(\omega)$ calculated are found to exhibit a two-component structure, with a Drude component at low energy and a mid-infrared component at higher energy. Depending on the relative importance of the hole pocket and the electron pocket on the reconstructed fermi surface and the coupling strength to the local moment, the Drude component in $\mathrm{Im}\sigma^{xy}(\omega)$ can be either positive or negative.(full-length abstract can be found in the main text.)