Weyl Loops in Random Topological Semimetals

In topological nodal line semimetals, bands cross along one-dimensional curves in the three-dimensional Brillouin zone, referred to as Weyl loops. They are usually protected by symmetry, and give rise to topological surface states guaranteed by pi-quantized Zak phases. For realistic systems, it may be difficult to locate the Weyl loops and topological surface states in the Brillouin zone. Therefore, we extend the analysis of a parametric random matrix model proposed by Walker and Wilkinson, limited to real symmetric matrices, to find the statistics of their distributions. By numerically locating the Weyl loops, we analyze the of their sizes and numbers, as well as the pattern of the surface states, which lie on regions of the two-dimensional surface Brillouin zone. We also explore statistical mechanic models that simulate our findings.