Anderson transition in three-dimensional systems with non-Hermitian disorder

Abstract

We study the Anderson transition for three-dimensional (3D) N × N × N tightly bound cubic lattices where both real and imaginary parts of onsite energies are independent random variables distributed uniformly between −W/2 and W/2. Such a non-Hermitian analog of the Anderson model is used to describe random-laser medium with local loss and amplification. We employ eigenvalue statistics to search for the Anderson transition. For 25% smallest-modulus complex eigenvalues we find the average ratio r of distances to the first and the second nearest neighbor as a function of W . For a given N the function r(W ) crosses from 0.72 to 2/3 with a growing W demonstrating a transition from delocalized to localized states. When plotted at different N all r(W ) cross at Wc = 6.0±0.1 (in units of nearest neighbor overlap integral) clearly demonstrating the 3D Anderson transition. We find that in the non-Hermitian 2D Anderson model, the transition is replaced by a crossover.Anderson localization is the central concept of solid state physics for more than 60 years 1-3 . It determines electron conductivity of doped crystalline and amorphous semiconductors and many other disordered systems and is observed in experiments 4,5 .In recent years the problem of localization attracted renewed interest as research moved to formerly unexplored area of non-Hermitian systems. Random lasers 6-9 with random dissipation and amplification regions are such prototypical non-Hermitian systems. The other parts of non-Hermitian disorder physics are related to Hatano-Nelson matrices 10 , their biological applications 11,12 or to spin chains [13][14][15] . All these works focus on onedimensional systems.A simple and elegant extension of the 2D Anderson localization problem was proposed in a recent paper by Tzortzakakis, Makris and Economou (TME) 16 . They studied 50 × 50 tight-binding square lattices with real overlap energy I ij = I, and random complex onsite energies E i whose real and imaginary parts are independent random variables distributed uniformly between −W/2 and W/2. The Hamiltonian readswhere i, j in the second term are nearest neighbors, and the hard-wall boundary is employed (no bonds extended out the boundary). Below we call this non-Hermitian Hamiltonian the TME model. By calculating the participation ratio of eigenfunctions of such non-Hermitian Hamiltonian, TME noticed that they become progressively more localized when W (in units of I) grows from 1 to 5. Simultaneously the distribution function P (s) of nearest neighbor distances s between eigenvalues in the complex plane widens, which shows that the repulsion of eigenvalues weakens due to the progressive localization of eigenfunctions. This behavior is similar to what happens in the Anderson model 17 . TME, however, did not raise a question whether there is an Anderson transition or a crossover in the limit of large system.In this paper, we focus on the question of the existence of the Anderson transition in the TME model for 3D cubic and 2D square lattices. We show that in the TME m...

Wave Propagation and Localization in a Complex Random Potential with Power-law Correlations: A Numerical Study

“…Similarly to the usual Anderson model where the real part of the on-site potential is random, in the model under consideration, the decay rate of ln T increases, hence the localization is enhanced as the disorder strength W increases. These are in good agreement with the results of previous studies [31,[37][38][39], in which the transverse localization of waves in 1D, 2D and 3D optical waveguide arrays with non-Hermitian disorder has been investigated in details. Next, we study the effects of spatially correlated disorder on the longitudinal localization of waves for which the amplitude of the wave decays exponentially along the propagation direction.…”

“…In recent years, there has been considerable attention devoted to studying transport and localization properties in physical systems which are described by non-Hermitian Hamiltonians [28][29][30][31][32][33][34][35][36][37][38][39]. These kinds of Hamiltonians are usually regarded as effective Hamiltonians for which the non-Hermitian part can serve for different purposes.…”

“…The possibility for the potential to possess an imaginary part makes these non-Hermitian Hamiltonians having many unique features such as non-Hermitian delocalization transition [28,29], a transition from ballistic to diffusive transport [30], oneway scattering and transport [33,34] and topological phase transitions [35,36]. Basically, there are two classes of problems associated with non-Hermitian Hamiltonian: one in which the non-Hermitcity arises from the off-diagonal elements such as the Hatano-Nelson model, where an imaginary vector potential is applied to the Hamiltonian [28] and the other in which the non-Hermitcity originates from the complexity of on-site potentials [31,[37][38][39]. For the latter, a one-dimensional (1D) non-Hermitian lattice model with randomness only in the imaginary part of the on-site potential has been studied, with the result that the eigenstates of such a model are localized, but the properties of localization are attributed to be qualitatively different from those of the usual Anderson model [31].…”

“…They demonstrated that the complex disordered potential acts in a dual way: its time-reversal symmetry breaking favors more delocalized states, while its disorder nature favors more localized ones. Very shortly after, Huang and Shklovskii extended the 2D square lattice model to 3D cubic one [39]. The authors showed that there exists a transition from delocalized to localized states as in the usual 3D Anderson model.…”

Localization landscape for Dirac fermions

“…The non-Hermitian Anderson Hamiltonian [26,27] H = −∇ 2 + V 1 (r) + iV 2 (r) (16) has been studied in the context of a random laser [28]: a disordered optical lattice with randomly varying absorption and amplication rates, described by a complex dielectric function V 1 + iV 2 . On a d-dimensional square lattice (lattice constant a), the discretization of −∇ 2 → a −2 d i=1 (2−2 cos k i a) produces a spectral band width of W 0 = 4d/a 2 .…”