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 , 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 () matrix and two-terminal conductance in these NH models. The unitarity of the 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 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 . At , 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 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 at different single-particle energies and system parameters of the NH models. We conclude that the critical exponents of the NH class and the NH class A are and , 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.
5 More- Received 5 March 2021
- Revised 1 August 2021
- Accepted 9 September 2021
DOI:https://doi.org/10.1103/PhysRevB.104.104203
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