Preserving Topology while Breaking Chirality: From Chiral Orthogonal to Anti-symmetric Hermitian Ensemble

TL;DR

This paper introduces a parameter-dependent ensemble of real random matrices that transitions between chiral Gaussian orthogonal and antisymmetric Hermitian classes, preserving zero-modes and exhibiting topological features similar to topological insulators.

Contribution

It constructs a new ensemble interpolating between two symmetry classes, preserving topological zero-modes and providing explicit eigenvalue correlation functions using skew-orthogonal polynomials.

Findings

The ensemble exhibits a Pfaffian point process characteristic of topological transitions.

Explicit kernel and correlation functions are derived using skew-orthogonal polynomials.

Numerical simulations confirm analytical spectral density and eigenvalue distribution results.

Abstract

We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B$|$DI) and the ensemble of antisymmetric Hermitian random matrices (Cartan class B$|$D). It enjoys the special feature that, depending on the matrix dimension $N$, it has exactly $\nu=0$ $(1)$ zero-mode for $N$ even (odd), throughout the symmetry transition. This "topological protection" is reminiscent of properties of topological insulators. We show that our ensemble represents a Pfaffian point process which is typical for such transition ensembles. On a technical level, our results follow from the applicability of the Harish-Chandra integral over the orthogonal group. The matrix valued kernel determining all eigenvalue correlation functions is explicitly constructed in terms of skew-orthogonal polynomials, depending on the topological index $\nu=0,1$. These polynomials interpolate between Laguerre and even (odd) Hermite polynomials for $\nu=0$ $(1)$, in terms of which the two limiting symmetry classes can be solved. Numerical simulations illustrate our analytical results for the spectral density and an expansion for the distribution of the smallest eigenvalue at finite $N$.