# Universal Broadening of Zero Modes: A General Framework and Identification

## TL;DR

This paper presents a universal framework showing that small perturbations cause zero modes in Hermitian operators to broaden into Gaussian ensembles, aiding their identification in experiments across various symmetry classes.

## Contribution

It introduces a general theory unifying previous results and clarifies conditions under which zero modes broaden into Gaussian ensembles, applicable to all ten symmetry classes.

## Key findings

- Zero modes broaden into Gaussian ensembles under small perturbations.
- The broadening scale differs from bulk eigenvalues, serving as an experimental indicator.
- Results apply universally across all ten Altland-Zirnbauer symmetry classes.

## Abstract

We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian random matrix ensemble of size $\nu\times\nu$, where $\nu$ is the number of zero modes. This observation unifies and extends a number of results within chiral random matrix theory and effective field theory and clarifies under which conditions they apply. The scaling of the former zero modes with the volume differs from the eigenvalues in the bulk, which we propose as an indicator to identify them in experiments. These results hold for all ten symmetric spaces in the Altland-Zirnbauer classification and build on two facts. Firstly, the broadened zero modes decouple from the bulk eigenvalues and secondly, the mixing from eigenstates of the perturbation form a Central Limit Theorem argument for matrices.