Synergetic effect of spin-orbit coupling and Zeeman splitting on the optical conductivity in the one-dimensional Hubbard model

Abstract

We study how the synergetic effect of spin-orbit coupling (SOC) and Zeeman splitting (ZS) affects the optical conductivity in the one-dimensional Hubbard model using the Kubo formula. We focus on two phenomena: (1) the electric dipole spin resonance (EDSR) in the metallic regime and (2) the optical conductivity in the Mott-insulating phase above the optical gap. In both cases, we calculate qualitatively the effects of SOC and ZS and how they depend on the relative angle between the SOC vector and the magnetic field direction. First, we investigate the spin resonance without electron correlation (the Hubbard parameter $U=0$). Although, neither SOC nor ZS causes any resonance by itself in the optical conductivity, the EDSR becomes possible when both of them exist. The resulting contribution to the optical conductivity is analyzed analytically. The effect of $U$ on the spin resonance is also studied with a numerical method. It is found that at half filling, the resonance is first enhanced for small $U$ and then suppressed when the optical gap is large enough. In the strong coupling limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$ at half filling, we also refer to the resonance between the lower and upper Hubbard bands appearing at $\ensuremath{\omega}\ensuremath{\sim}U$, above the optical gap. A large magnetic field tends to suppress the signal while it is recovered thanks to SOC depending on the relative angle of the magnetic field.