Universality Classes of the Anderson Transitions Driven by Non-Hermitian Disorder

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

Non-Hermitian Floquet Topological Matter-A Review

The past few years have witnessed a surge of interest in non-Hermitian Floquet topological matter due to its exotic properties resulting from the interplay between driving fields and non-Hermiticity. The present review sums up our studies on non-Hermitian Floquet topological matter in one and two spatial dimensions. We first give a bird's-eye view of the literature for clarifying the physical significance of non-Hermitian Floquet systems. We then introduce, in a pedagogical manner, a number of useful tools tailored for the study of non-Hermitian Floquet systems and their topological properties. With the aid of these tools, we present typical examples of non-Hermitian Floquet topological insulators, superconductors, and quasicrystals, with a focus on their topological invariants, bulk-edge correspondences, non-Hermitian skin effects, dynamical properties, and localization transitions. We conclude this review by summarizing our main findings and presenting our vision of future directions.

Universal scaling of higher-order spacing ratios in Gaussian random matrices

Higher-order spacing ratios are investigated analytically using a Wigner-like surmise for Gaussian ensembles of random matrices. For a kth order spacing ratio (r^{(k)},k>1) the matrix of dimension 2k+1 is considered. A universal scaling relation for this ratio, known from earlier numerical studies, is proved in the asymptotic limits of r^{(k)}→0 and r^{(k)}→∞.

Bulk-boundary correspondence in disordered non-Hermitian systems

The bulk-boundary correspondence (BBC) refers to the consistency between eigenvalues calculated under open and periodic boundary conditions. This consistency can be destroyed in systems with non-Hermitian skin effect (NHSE). In spite of the great success of the generalized Brillouin zone (GBZ) theory in clean non-Hermitian systems, the applicability of GBZ theory is questionable when the translational symmetry is broken. Thus, it is of great value to rebuild the BBC for disordered samples, which extends the application of GBZ theory in non-Hermitian systems. Here, we propose a scheme to reconstruct BBC, which can be regarded as the solution of an optimization problem. By solving the optimization problem analytically, we reconstruct the BBC and obtain the modified GBZ theory in several prototypical disordered non-Hermitian models. The modified GBZ theory provides a precise description of the fantastic NHSE, which predicts the asynchronous-disorder-reversed NHSE's directions.

Bichromatic state-dependent disordered potential for Anderson localization of ultracold atoms

ABSTRACT

The ability to load ultracold atoms at a well-defined energy in a disordered potential is a crucial tool to study quantum transport, and in particular Anderson localization. In this paper, we present a new method for achieving that goal by rf transfer of atoms in an atomic Bose-Einstein condensate from a disorder-insensitive state to a disorder-sensitive state. It is based on a bichromatic laser speckle pattern, produced by two lasers whose frequencies are chosen so that their light-shifts cancel each other in the first state and add up in the second state. Moreover, the spontaneous scattering rate in the disorder-sensitive state is low enough to allow for long observation times of quantum transport in that state. We theoretically and experimentally study the characteristics of the resulting potential.

Anderson localization and multifractal spectrum at the transition point in a two-dimensional non-Hermitian AII†system

The Anderson localization transition in a two-dimensional AII^†system is studied by eigenvalue statistics and then confirmed by the multifractal analysis of the wave functions at the transition point. The system is modeled by a two-dimensional lattice structure with real-quaternion off-diagonal elements and complex on-site energies, whose real and imaginary parts are two independent random variables. Via finite-size scaling analysis of eigenvalue spacing ratios, we find the non-Hermiticity reduces the critical disorder and give an estimate of the critical exponentν= 1.89, showing the system belongs to a new universal class other than the AII class and probably shares the same exponent with two-dimensional Hermitian DIII systems although they have different symmetries. The Anderson localization transition is further confirmed by checking the linearity in the parametric representation of the singularity strength and by checking the universality of the forms of the singularity spectra of different system sizes. The generalized dimensions are obtained asD1=1.80andD2=1.62.