Electric polarization and nonlinear optical effects in noncentrosymmetric magnets

We study electric polarization and nonlinear optical effects in spin systems with broken inversion symmetry. We apply strong coupling expansion to the underlying electronic Hamiltonians, and systematically derive expressions for electric polarization in spin systems that are represented in terms of spin operators. The magnon representation of the obtained electric polarization operator allows us to compute linear and nonlinear optical responses by the standard diagrammatic method. We apply our formalism to Heisenberg model with alternating coupling constants and $J_1$-$J_2$ model with inversion symmetry breaking. We demonstrate that these inversion broken spin systems support dc current flow upon magnon excitations which arises from the shift current mechanism.