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

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.)