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

Abstract

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 ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$, 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 ($S$) 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. 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 $\ensuremath{\nu}$ at different single-particle energies and system parameters of the NH models. We conclude that the critical exponents of the NH class ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the NH class A are $\ensuremath{\nu}=1.19\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\nu}=1.00\ifmmode\pm\else\textpm\fi{}0.04$, 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.

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