Microscopic essence of the second-order nonlinear conductivity in $\mathcal{PT}$-symmetric collinear antiferromagnetic metals with magnetic toroidal moment

A magnetic toroidal moment is a fundamental electronic degree of freedom in the absence of both spatial inversion and time-reversal symmetries and gives rise to novel multiferroic and transport properties. We elucidate essential model parameters of the nonlinear transport in the space-time ($\mathcal{PT}$) symmetric collinear antiferromagnetic metals accompanying a magnetic toroidal moment. By analyzing the longitudinal and transverse components of the second-order nonlinear conductivity on a two-dimensionally stacked zigzag chain based on the nonlinear Kubo formula, we show that an effective coupling between the magnetic toroidal moment and the antisymmetric spin-orbit interaction is a source of inducing the nonlinear conductivity. Moreover, we find that the nonreciprocal longitudinal current and nonlinear Hall coefficient are largely enhanced just below the transition temperature of the antiferromagnetic ordering. We also discuss the relevance to the linear magnetoelectric effect. Our result serves as a guide for extracting essential model parameters for various multiferroic and conductive phenomena in noncentrosymmetric antiferromagnetic metals.