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

The Anderson transition driven by non-Hermitian (NH) disorder has been extensively studied in recent years. In this paper, we present in-depth transfer matrix analyses of the Anderson transition in three NH systems, NH Anderson, U(1), and Peierls models in three-dimensional systems. The first model belongs to NH class AI(double dagger), whereas the second and the third ones to NH class A. We first argue a general validity of the transfer matrix analysis in NH systems, and clarify the symmetry properties of the Lyapunov exponents, scattering (5) 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. Using the transfer matrix method, we construct a phase diagram of the NH Anderson model for various complex single-particle energy E. At E = 0, the phase diagram as well as critical properties become completely symmetric with respect to an exchange of real and imaginary parts of on-site NH random potentials. We show that the symmetric nature at E = 0 is a general feature for any NH bipartite-lattice models with the on-site NH random potentials. Finite size scaling data are fitted by polynomial functions, from which we determine the critical exponent v at different single-particle energies and system parameters of the NH models. We conclude that the critical exponents of the NH class AI(double dagger) and the NH class A are v = 1.19 +/- 0.01 and v = 1.00 +/- 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.

Automated summary

The paper presents a detailed transfer matrix analysis of the Anderson transition driven by non-Hermitian disorder in three NH systems. It discusses the general validity of the transfer matrix analysis in NH systems and analyzes the symmetry properties of the Lyapunov exponents, scattering matrix, and two-terminal conductance in these NH models. The study shows violations of unitarity in the S matrix in NH systems and the symmetric nature of the S matrix, Lyapunov exponents, and conductance in certain NH models.