Fluctuation of Chern Numbers in Parametric Random Matrix Models

Many Weyl semimetal materials contain a large number of Weyl points in their Brillouin zone, which prompts a description in terms of parameterized random matrices. Following early work of Wilkinson and coauthors, we consider such a model with three periodic parameters {\vec X} and investigate the eigenspectrum and the statistics of degeneracy points, to which can be associated an integer “charge”. The total charge in a “filled torus” with radial $X_3$ direction has the interpretation of a Chern number, and we investigate the statistics of this as a function of the thickness, confirming a saturation behavior beyond an initial diffusive regime. We propose a model to explain the correlations among the degeneracy points.