Transfer matrix study of the Anderson transition in non-Hermitian systems

In this paper, we present in-depth transfer matrix analyses of the Anderson transition in three non-Hermitian (NH) systems, NH Anderson, U(1) and Peierls models, that belong to NH class AI$^{\dagger}$ or NH class A. We first argue a general validity of the transfer matrix analysis, and clarify the symmetry properties of the Lyapunov exponents, scattering ($S$) matrix and two-terminal conductance in these NH models. The unitarity of the $S$ matrix is violated in NH systems, where the two-terminal conductance can take arbitrarily large values. Nonetheless, we show that the transposition symmetry of a Hamiltonian leads to the symmetric nature of the $S$ matrix as well as the reciprocal symmetries of the Lyapunov exponents and conductance in certain ways in these NH models. Finite size scaling data are fitted by polynomial functions, from which we determine the critical exponent $\nu$ at different single-particle energies and system parameters of the NH models. We conclude that the critical exponents of the NH class AI$^{\dagger}$ and the NH class A are $\nu=1.19 \pm 0.01$ and $\nu=1.00 \pm 0.04$, respectively. In the NH models, a distribution of the two-terminal conductance is not Gaussian. Instead, it contains small fractions of huge conductance values, which come from rare-event states with huge transmissions amplified by on-site NH disorders. Nonetheless, a geometric mean of the conductance enables the finite-size scaling analysis. We show that the critical exponents obtained from the conductance analysis are consistent with those from the localization length in these three NH models.