Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices

Abstract

We investigate the level density σ(x) and the level-spacing distribution p(s) of random matrices M = AF ≠ M{†}, where F is a (diagonal) inner product and A is a random, real, symmetric or complex, Hermitian matrix with independent entries drawn from a probability distribution q(x) with zero mean and finite higher moments. Although not Hermitian, the matrix M is self-adjoint with respect to F and thus has purely real eigenvalues. We find that the level density σ{F}(x) is independent of the underlying distribution q(x) and solely characterized by F, and therefore generalizes the Wigner semicircle distribution σ{W}(x). We find that the level-spacing distributions p(s) are independent of q(x), and are dependent upon both the inner product F and whether A is real or complex, and therefore generalize the Wigner surmise for level spacing. Our results suggest F-dependent generalizations of the well-known Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble classes.

Optical waveguide arrays: quantum effects and PT symmetry breaking

“…Thus, symmetry properties of the disorder-induced spectrum are reflected in the disorder-averaged intensity correlation function, and not the on-site or off-diagonal nature of disorder [67]. These results also suggest that although intensity distribution, or intensity correlation function is insensitive to the disorder distribution function, higher order intensity correlations may encode signatures of different disorder distributions that have zero mean and identical variance [16,66].…”

“…The off-diagonal disorder randomly modulates the tunneling C i → C i + v i where v i is a zero-mean random variable with variance v dt . We use uniformly distributed random variables to ensure that the modulated tunneling rates remain strictly positive, although the results are independent of the type of distribution used as long as any such distribution has zero mean and identical variance [16,66]. The resultant intensity distribution is averaged over multiple M ∼ 10 4 realizations to ensure that the final results are independent of the number of disorder realizations and the probability distribution of the site or tunneling disorder.…”

Pseudosymmetric random matrices: Semi-Poisson and sub-Wigner statistics

“…The fact that real symmetric and Hermitian random matrices make two separate classes of ensembles GOE and GUE having distinct statistical properties, motivates us to separate pesudo-symmetric real matrices from the more general pseudo-Hermitian matrices [16][17][18][19]. In the present work, we invoke a lesser class of matrices which are real pseudo-symmetric under a metric η such that ηHη −1 = H t , these are essentially non-symmetric ones.…”

“…The model describes transitions from real eigenvalues to a situation in which, apart from a residual number, the eigenvalues are complex conjugate. More recently it has been shown that such ensemble of pseudo-Hermitian Gaussian matrices [17] gives rise in a certain limit to an ensemble of anti-Hermitian matrices whose eigenvalues have properties directly related to those of the chiral ensemble of random matrices [18].…”

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

“…The spectral statistics of random Hermitian Hamiltonians usually exhibits universal behavior, depending only on symmetries of the system [68]. An interesting question then naturally arises whether the spectral statistics of disordered non-Hermitian Hamiltonians also exhibits universal behavior [69][70][71][72][73][74][75][76][77][78][79][80][81][82][83].…”