The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological quantity. To this end, we set up a random matrix model for a chiral unitary system with a parametric dependence. We analytically calculate the discrete probability distribution of the winding numbers, as well as the parametric correlations functions of the winding number density. Moreover, we address aspects of universality for the two-point function of the winding number density by identifying a proper unfolding procedure. We conjecture the unfolded two-point function to be universal.

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Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding numbers for a specific parametric model of the chiral Gaussian unitary ensemble to other chiral random matrix ensembles. In particular, we address the two chiral symmetry classes, unitary (AIII) and symplectic (CII), and we analytically compute ensemble averages for ratios of determinants with parametric dependence. To this end, we employ a technique that exhibits reminiscent supersymmetric structures, while we never carry out any map to superspace.

We study here the short-time behavior of the spectral form factor for chain-like many-body systems of arbitrary length N , an important tool to distinguish chaotic and integrable dynamics. We found in the past that while the long-time behavior of the spectral form factor is universal and follows predictions by Random Matrix Theory, it becomes highly system dependent for short times, even for large N in the disorder-free case. Now, our aim is to study to which extend this observation persists if a nonzero disorder is considered.