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PT-Symmetric $SU(2)$-like Random Matrix Ensembles: Invariant Distributions and Spectral Fluctuations

Stalin Abraham, A. Bhagwat, Sudhir Ranjan Jain

TL;DR

The paper develops a symmetry-based random-matrix framework for a PT-symmetric, non-Hermitian $2\times 2$ ensemble of normal matrices, deriving a power-law joint distribution for matrix elements under similarity invariance. By imposing a Gaussian weight for normalization, it shows that the distribution's exponents become discrete even integers, yielding an explicit form for the element distribution and enabling calculation of the nearest-neighbor level-spacing distribution. The small-spacing behavior is algebraic, with a tunable exponent $\nu=2(l+1)$ that attains the Gaussian Unitary Ensemble value when $l=0$ and becomes stronger for larger $l$, linking to quantum chaos indicators. The analysis extends to a bounded $\mathbb{R}^4$ domain, providing closed-form spacing distributions and clarifying normalization constraints, and suggests broader implications for PT-symmetric systems and solvable-model connections. Overall, the work demonstrates that PT-symmetric random matrices admit a non-Gaussian, symmetry-derived statistical structure with controllable level repulsion, offering a path to generalized ensembles and applications beyond traditional Gaussian RMT.

Abstract

We consider an ensemble of $2\times 2$ normal matrices with complex entries representing operators in the quantum mechanics of 2 - level parity-time reversal (PT) symmetric systems. The randomness of the ensemble is endowed by obtaining probability distributions based on symmetry and statistical independence. The probability densities turn out to be power law with exponents that depend on the boundedness of the domain. For small spacings, $σ$, the probability density varies as $σ^ν$, $ν\geq 2$. The degree of level repulsion is a parameter of great interest as it makes a connection to quantum chaos; the lower bound of $ν$ for our ensemble coincides with the Gaussian Unitary Ensemble. We believe that the systematic development presented here paves the way for further generalizations in the field of random matrix theory for PT-symmetric quantum systems.

PT-Symmetric $SU(2)$-like Random Matrix Ensembles: Invariant Distributions and Spectral Fluctuations

TL;DR

The paper develops a symmetry-based random-matrix framework for a PT-symmetric, non-Hermitian ensemble of normal matrices, deriving a power-law joint distribution for matrix elements under similarity invariance. By imposing a Gaussian weight for normalization, it shows that the distribution's exponents become discrete even integers, yielding an explicit form for the element distribution and enabling calculation of the nearest-neighbor level-spacing distribution. The small-spacing behavior is algebraic, with a tunable exponent that attains the Gaussian Unitary Ensemble value when and becomes stronger for larger , linking to quantum chaos indicators. The analysis extends to a bounded domain, providing closed-form spacing distributions and clarifying normalization constraints, and suggests broader implications for PT-symmetric systems and solvable-model connections. Overall, the work demonstrates that PT-symmetric random matrices admit a non-Gaussian, symmetry-derived statistical structure with controllable level repulsion, offering a path to generalized ensembles and applications beyond traditional Gaussian RMT.

Abstract

We consider an ensemble of normal matrices with complex entries representing operators in the quantum mechanics of 2 - level parity-time reversal (PT) symmetric systems. The randomness of the ensemble is endowed by obtaining probability distributions based on symmetry and statistical independence. The probability densities turn out to be power law with exponents that depend on the boundedness of the domain. For small spacings, , the probability density varies as , . The degree of level repulsion is a parameter of great interest as it makes a connection to quantum chaos; the lower bound of for our ensemble coincides with the Gaussian Unitary Ensemble. We believe that the systematic development presented here paves the way for further generalizations in the field of random matrix theory for PT-symmetric quantum systems.
Paper Structure (7 sections, 2 theorems, 51 equations)

This paper contains 7 sections, 2 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Random matrix ensemble
  3. Construction of Probability Distribution Function
  4. Determination of Normalisation Constant
  5. Level Repulsion
  6. Probability Distribution Function and Level Repulsion in a bounded Region of $\mathbb{R}^4$
  7. Concluding remarks

Key Result

Theorem 1

If $V$ is a region defined by $x_i \ge 0$, $i = 1, 2, ..., n$ and $\sum x_i \le 1$, then for $\Re \alpha_i > 0$,

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1