Theory of hopping magnetoresistance induced by Zeeman splitting

We present a study of hopping conductivity for a system of sites that can be occupied by more than one electron. At a moderate on-site Coulomb repulsion, the coexistence of sites with occupation numbers 0, 1, and 2 results in an exponential dependence of the Mott conductivity upon Zeeman splitting ${\mathrm{\mu }}_{\mathit{B}}$H. We show that the conductivity behaves as lnσ=(T/${\mathit{T}}_0$ ${)}^{1/4}$F(x), where F is a universal scaling function of x=${\mathrm{\mu }}_{\mathit{B}}$H/T(${\mathit{T}}_0$/T${)}^{1/4}$. We find F(x) analytically at weak fields, x≪1, using a perturbative approach. Above some threshold ${\mathit{x}}_{\mathrm{th}}$, the function F(x) attains a constant value, which is also found analytically. The full shape of the scaling function is determined numerically, from a simulation of the corresponding ‘‘two-color’’ dimensionless percolation problem. In addition, we develop an approximate method which enables us to solve this percolation problem analytically at any magnetic field. This method gives a satisfactory extrapolation of the function F(x) between its two limiting forms.