Universal broadening of zero modes: A general framework and identification

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 ν × ν, where ν 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.

Universal distributions from non-Hermitian perturbation of zero modes

“…This at first came as a surprise as random matrix universality is usually only established in the limit of infinite matrix dimension. This new form of universality was shown [20] to have its origin in a kind of matrix-valued central limit theorem that applies to the perturbation matrix for the zero modes. The new finite size Gaussian random matrix universality for the would-be zero modes is reached in the limit where the size of the remaining system is taken to infinity.…”

“…where l = R, I is the real and imaginary direction respectively. The parameters γ II in [20]. It follows that the broadening of the zero modes is given by the one-point correlation function of the elliptic Ginibre ensemble [35]…”

“…This corresponds to focusing on the part of the perturbation which corresponds to first order degenerate perturbation theory. The approach here follows closely the one in [20]. We may deal with both forms (I.2) and (I.3) in a unified fashion.…”

“…Although the form (III.1) resembles the second model (I.3) instead of the first one (I.2), both can be dealt here in the same way. One needs to keep in mind that the result in [20] gives a bound on α S op , where · op is the operator norm (the largest singular value). As we shall see, the corresponding bound here will depend on…”

“…The new finite size Gaussian random matrix universality for the would-be zero modes is reached in the limit where the size of the remaining system is taken to infinity. In [20], Hermitian perturbations of the original ideal Hermitian system were considered, and it was possible to shown that universal distributions of perturbed zero modes exist for all universality classes.…”