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 $σ^{xx}(ω)$ and the Hall conductivity $σ^{xy}(ω)$ of the cuprate superconductors. Both $σ^{xx}(ω)$ and $σ^{xy}(ω)$ 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 $\mathrm{Im}σ^{xy}(ω)$ can be either positive or negative.(full-length abstract can be found in the main text.)

Figures (7)

• Figure 1: The optical conductivity of the system when we assume a Gaussian action of the form Eq.74 for the thermally fluctuating local moment. Here we treat $\tilde{U}$ and $\xi$ as independent parameters, although in reality both of them may have involved temperature and doping dependence. The energy is measured in unit of $t$ and we have set $k_{B}T=0.1t$ for the electron. The $\delta$-function peak is broadened into a Lorentzian peak of width $0.03t$ in the calculation. The calculation is done on a finite cluster of square lattice with $L\times L=400$ sites and periodic boundary condition in both the $x$ and the $y$-direction.
• Figure 2: The optical conductivity of the system when we assume a non-Gaussian action of the form Eq.75 for the thermally fluctuating local moment. Such an action describes the thermal fluctuation in the renormalized classical regime of a quantum Heisenberg antiferromagnetic model defined on the square lattice. It is thought to be relevant to the electron-doped cuprate superconductors. The energy is measured in unit of $t$ and we have set $k_{B}T=0.1t$ for the electron. The $\delta$-function peak is broadened into a Lorentzian peak of width $0.03t$ in the calculation. The calculation is done on a finite cluster of square lattice with $L\times L=400$ sites and periodic boundary condition in both the $x$ and the $y$-direction.
• Figure 3: The imaginary part of the infrared Hall conductivity calculated with a Gaussian action of the form Eq.74 for the thermally fluctuating local moment. Here we have treated $\tilde{U}$ and $\xi$ as independent parameters, although both of them have involved temperature and doping dependence. The energy is measured in unit of $t$ and and we have set $k_{B}T=0.1t$ for the electron. The $\delta$-function peak is broadened into a Lorentzian peak of width $0.03t$ in the calculation. The calculation is done on a finite cluster of square lattice with $L\times L=400$ sites and periodic boundary condition in both the $x$ and the $y$-direction. The inset represents a zoom in view of the Hall conductivity spectrum. We note that the tiny negative overshoot near $\omega=0$ can be attributed to finite size effect in the anti-symmetrization procedure.
• Figure 4: The imaginary part of the infrared Hall conductivity of the system when we assume a non-Gaussian action with the form of the NLSM for the thermally fluctuating local moment. Such an action describes the thermal fluctuation in the renormalized classical regime of a quantum Heisenberg antiferromagnetic model defined on the square lattice. It is thought to be relevant to the electron-doped cuprate superconductors. The $\delta$-function peak is broadened into a Lorentzian peak of width $0.03t$. The calculation is done on a finite cluster of square lattice with $L\times L=400$ sites and periodic boundary condition in both the $x$ and the $y$-direction. The inset represents a zoom in view of the Hall conductivity spectrum. We note that the tiny negative overshoot near $\omega=0$ can be attributed to finite size effect in the anti-symmetrization procedure.
• Figure 5: The real part of the infrared Hall conductivity of the system when we assume a non-Gaussian action with the form of the NLSM and $\xi=3$ for the thermally fluctuating local moment. The calculation is done on a finite cluster of square lattice with $L\times L=400$ sites and periodic boundary condition in both the $x$ and the $y$-direction..