Universal distributions from non-Hermitian perturbation of zero modes.

Hermitian operators with exact zero modes subject to non-Hermitian perturbations are considered. Specific focus is on the distribution of the former zero eigenvalues of the Hermitian operators. The broadening of these zero modes is found to follow an elliptic Gaussian random matrix ensemble of fixed size, where the symmetry class of the perturbation determines the behavior of the modes. This distribution follows from a central limit theorem of matrices and is shown to be robust to deformations.

Universal broadening of zero modes: A general framework and identification.

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 10 symmetric spaces in the Altland-Zirnbauer classification and build on two facts. First, the broadened zero modes decouple from the bulk eigenvalues and, second, the mixing from eigenstates of the perturbation form a central limit theorem argument for matrices.

Completing the picture for the smallest eigenvalue of real Wishart matrices.

Rectangular real N×(N+ν) matrices W with a Gaussian distribution appear very frequently in data analysis, condensed matter physics, and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of WW^{T}. The extreme eigenvalues of WW^{T} are of particular interest. We explicitly compute the distribution and the gap probability of the smallest nonzero eigenvalue in this ensemble, both for arbitrary fixed N and ν, and in the universal large N limit with ν fixed. We uncover an integrable Pfaffian structure valid for all even values of ν≥0. This extends previous results for odd ν at infinite N and recursive results for finite N and for all ν. Our mathematical results include the computation of expectation values of half-integer powers of characteristic polynomials.