Non-Hermitian Disordered Systems

Abstract

Non-Hermitian disordered systems have emerged as a central arena in modern physics, with ramifications spanning condensed matter, quantum, statistical, and high energy contexts. The same principles also underlie phenomena beyond physics, such as network science, complex systems, and biophysics, where dissipation, nonreciprocity, and stochasticity are ubiquitous. Here, we review the physics and mathematics of non-Hermitian disordered systems, with particular emphasis on non-Hermitian random matrix theory. We begin by presenting the 38-fold symmetry classification of non-Hermitian systems, contrasting it with the 10-fold way for Hermitian systems. After introducing the classic Ginibre ensembles of non-Hermitian random matrices, we survey various diagnostics for complex-spectral statistics and distinct universality classes realized by symmetry. As a key application to physics, we discuss how non-Hermitian random matrix theory characterizes chaos and integrability in open quantum systems. We then turn to the criticality due to the interplay of disorder and non-Hermiticity, including Anderson transitions in the Hatano-Nelson model and its higher-dimensional extensions. We also discuss the effective field theory description of non-Hermitian disordered systems in terms of nonlinear sigma models.

Figures (5)

• Figure 1: Ginibre ensemble of non-Hermitian random matrices. (a) Complex spectrum of a $10^3 \times 10^3$ non-Hermitian random matrix drawn from the Ginibre unitary ensemble, with the spectral radius normalized to unity. (b) A complex eigenvalue $E$, its nearest-neighbor eigenvalue $E^{\text{NN}}$, and next-to-nearest-neighbor eigenvalue $E^{\text{NNN}}$. (c) Probability density of level spacings for the Ginibre ensemble (blue solid curve; Eq. \ref{['eq: Ginibre level-spacing distribution']}) and Poisson ensemble (red dashed curve; Eq. \ref{['eq: Poisson']}). The level spacing is rescaled such that its mean is unity.
• Figure 2: Diagnostics of complex-spectral statistics. (a) Probability density of complex level-spacing ratios $z$ for their modulus $\left| z \right|$ and argument $\arg z$. The numerical data are averaged over $10^4$ realizations of $10^3 \times 10^3$ matrices in the Ginibre unitary ensemble (Gaussian ensemble in class A). Red dashed curve: Poisson ensemble [$P_{\rm P}^{\rm abs} \left( \left| z \right| \right) = 2 \left| z \right|$, $P_{\rm P}^{\rm arg} \left( \arg z\right) = 1/2\pi$]. (b) Dissipative spectral form factor $K \left( \tau, \tau^{*} \right)$ in Eq. \ref{['eq: DSFF']} as a function of the modulus $\left| \tau \right|$ of complex time $\tau$ ($N=10^3$). Blue solid curve: Ginibre ensemble (class A) [$K_{\rm Gin} \left( \tau, \tau^{*} \right) \sim 1 + N \left( 2 J_1 \left( \left| \tau\right| \right)/\left| \tau \right| \right)^2 - e^{-\left| \tau\right|^2/4N}$ for $N \to \infty$]. Red dashed curve: Poisson ensemble [$K_{\rm P} \left( \tau, \tau^{*} \right) = 1 + \left( N-1 \right) e^{- \left| \tau \right|^2}$].
• Figure 3: Symmetry and non-Hermitian random matrix theory. (a) Spectral bulk, line (i.e., real or imaginary axis), and origin in the complex plane, where different symmetries are relevant. (b) Probability density of level spacings for non-Hermitian random matrices in class AI$^{\dag}$ and AII$^{\dag}$. The numerical data are averaged over $10^4$ realizations of $10^3 \times 10^3$ matrices in the Gaussian ensemble. Blue solid curve: Ginibre ensemble (class A; Eq. \ref{['eq: Ginibre level-spacing distribution']}). Red dashed curve: Poisson ensemble (Eq. \ref{['eq: Poisson']}).
• Figure 4: Complex-spectral statistics of the (a, b) damped and (c, d) dephased Ising models [$L=7$, $J=1.0$, $h_x = -1.05$, $h_z = 0.2$, $\varepsilon = 0.2$; (a, b) $\gamma = 1.75$ and (c, d) $\gamma = 0.75$]. (a, c) Complex spectrum for a single realization. (b, d) Probability density of complex level-spacing ratios $z$ for their modulus $\left| z \right|$ and argument $\arg z$. The numerical data are taken from complex eigenvalues away from the real axis and spectral edges, and averaged over $20$ disorder realizations. Blue and green dashed curves: non-Hermitian random matrices in the Gaussian ensembles for classes A and AI$^{\dag}$.
• Figure 5: Hatano-Nelson model. (a) Schematic of the nonreciprocal disordered chain. (b) Complex spectra for single realizations of disordered potentials with strength $\Delta = 0, 1.5, 3$ ($J=1.0$, $\gamma = 0.2$). (c) Renormalization-group flow. Two-parameter scaling in terms of the normal coupling parameter $1/g$ and the topological term $W$. In addition to the fixed points with $1/g = 0$, $W\in \mathbb{Z}$ (black dots), the fixed points with $1/g = 0$, $W \in \mathbb{Z} + 1/2$ (red dots) appear.