Random matrix ensembles for PT-symmetric systems
File(s)
Accepted version
Author(s)
Graefe, E-M
Mudute-Ndumbe, S
Taylor, M
Type
Journal Article
Abstract
Recently much effort has been made towards the introduction of non-Hermitian
random matrix models respecting $PT$-symmetry. Here we show that there is a
one-to-one correspondence between complex $PT$-symmetric matrices and
split-complex and split-quaternionic versions of Hermitian matrices. We
introduce two new random matrix ensembles of (a) Gaussian split-complex
Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary
sizes. They are related to the split signature versions of the complex and the
quaternionic numbers, respectively. We conjecture that these ensembles
represent universality classes for $PT$-symmetric matrices. For the case of
$2\times2$ matrices we derive analytic expressions for the joint probability
distributions of the eigenvalues, the one-level densities and the level
spacings in the case of real eigenvalues.
random matrix models respecting $PT$-symmetry. Here we show that there is a
one-to-one correspondence between complex $PT$-symmetric matrices and
split-complex and split-quaternionic versions of Hermitian matrices. We
introduce two new random matrix ensembles of (a) Gaussian split-complex
Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary
sizes. They are related to the split signature versions of the complex and the
quaternionic numbers, respectively. We conjecture that these ensembles
represent universality classes for $PT$-symmetric matrices. For the case of
$2\times2$ matrices we derive analytic expressions for the joint probability
distributions of the eigenvalues, the one-level densities and the level
spacings in the case of real eigenvalues.
Date Issued
2015-09-25
Date Acceptance
2015-07-27
Citation
Journal of Physics A: Mathematical and Theoretical, 2015, 48 (38)
ISSN
1751-8113
Publisher
IOP Publishing
Journal / Book Title
Journal of Physics A: Mathematical and Theoretical
Volume
48
Issue
38
Sponsor
The Royal Society
Identifier
http://arxiv.org/abs/1505.07810v2
Grant Number
UF130339
Subjects
math-ph
math-ph
math.MP
quant-ph
Publication Status
Published
Article Number
38FT02
Date Publish Online
2015-08-25