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

Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual ions in a crystal are subjected essentially to nearestneighbor interactions. In contrast, the systems studied in this paper have all their degrees of freedom coupled to each other. Thus, the problem of linearized small oscillations of such systems involves two full positive-definite and non-commuting matrices, as opposed to the sparse matrices associated with disordered crystals. Consequently, the familiar methods for determining the averaged vibrational spectra of disordered crystals, introduced many years ago by Dyson and Schmidt, are inapplicable for highly connected disordered systems. In this paper we apply random matrix theory (RMT) to calculate the averaged vibrational spectra of such systems, in the limit of infinitely large system size. At the heart of our analysis lies a calculation of the average spectrum of the product of two positive definite random matrices by means of free probability theory techniques. We also show that this problem is intimately related with quasi-hermitian random matrix theory (QHRMT), which means that the 'hamiltonian' matrix is hermitian with respect to a non-trivial metric. This extends ordinary hermitian matrices, for which the metric is simply the unit matrix. The analytical results we obtain for the spectrum agree well with our numerical results. The latter also exhibit oscillations at the high-frequency band edge, which fit well the Airy kernel pattern. We also compute inverse participation ratios of the corresponding amplitude eigenvectors and demonstrate that they are all extended, in contrast with conventional disordered crystals. Finally, we compute the thermodynamic properties of the system from its spectrum of vibrations. In addition to matrix model analysis, we also study the vibrational spectra of various multi-segmented disordered pendula, as concrete realizations of highly connected mechanical systems. A universal feature of the density of vibration modes, common to both pendula and the matrix model, is that it tends to a non-zero constant at vanishing frequency.

show abstract

“…of (15), according to (16), in the large-N limit. We can then obtain the desired averaged density of eigenvalues…”

Section: B Random Matrix Modelmentioning

confidence: 99%

Dynamics of disordered mechanical systems with large connectivity, free probability theory, and quasi-Hermitian random matrices

Feinberg

Riser

2021

Annals of Physics

View full text Add to dashboard Cite

show abstract

“…Thus, the eigenvalues of φ are either real, or come in complex-conjugate pairs. See [14] for a recent discussion of (real asymmetric) PH random matrices.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-Hermitian Random Matrix Models: General Formalism

Feinberg,

Riser

2021

Preprint

View full text Add to dashboard Cite

Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite dimensional subspace. As such, they can be used, in the usual spirit of random matrix theory, to model chaotic or disordered P T -symmetric quantum systems, or their gain-loss-balanced classical analogs, in the phase of broken P T -symmetry. The eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. In this paper we introduce a family of pseudo-hermitian random matrix models, depending parametrically on their metric. We apply the diagrammatic method to obtain its averaged resolvent and density of eigenvalues as explicit functions of the metric, in the limit of large matrix size N . As a concrete example, which is essentially an ensemble of elements of the non-compact unitary Lie algebra, we choose a particularly simple set of metrics, and compute the resulting resolvent and density of eigenvalues in closed form. The spectrum consists of a finite fraction of complex eigenvalues, which occupy uniformly two two-dimensional blobs, symmetric with respect to the real axis, as well as the complimentary fraction of real eigenvalues, condensed in a finite segment, with a known non-uniform density. The numbers of complex and real eigenvalues depend on the signature of the metric, that is, the numbers of its positive and negative eigenvalues. We have also carried thorough numerical analysis of the model for these particular metrics. Our numerical results converge rapidly towards the asymptotic analytical large-N expressions.

show abstract

“…Upon truncation to finite vector spaces, pseudo-hermitian operators turn into pseudohermitian matrices. See [12] for a recent discussion of (real asymmetric) pseudo-hermitian random matrices.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-hermitian random matrix theory: a review

Feinberg,

Riser

2021

Preprint

View full text Add to dashboard Cite

We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N , planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various 'phase transitions' associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations.

show abstract

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

Cited by 7 publications

References 29 publications

Dynamics of disordered mechanical systems with large connectivity, free probability theory, and quasi-Hermitian random matrices

Dynamics of disordered mechanical systems with large connectivity, free probability theory, and quasi-Hermitian random matrices

Pseudo-Hermitian Random Matrix Models: General Formalism

Pseudo-hermitian random matrix theory: a review

Contact Info

Product

Resources

About