Probing topological properties of a three-dimensional lattice dimer model with neural networks

Machine learning methods are being actively considered as a new tool for describing many-body physics. However, so far, their capabilities have been demonstrated only in previously studied models, such as the Ising model. Here, we consider a simple problem demonstrating that neural networks can be successfully used to give new insights into statistical physics. Specifically, we consider a three-dimensional (3D) lattice dimer model which consists of sites forming a lattice and bonds connecting the neighboring sites in such a way that every bond can be either empty or filled with a dimer and the total number of dimers ending at one site is fixed to be 1. Dimer configurations can be viewed as equivalent if they are connected through a series of local flips, i.e., simultaneous "rotation" of a pair of parallel neighboring dimers. It turns out that the whole set of dimer configurations on a given 3D lattice can be split into distinct topological classes, such that dimer configurations belonging to different classes are not equivalent. In this paper we identify these classes by using neural networks. More specifically, we train the neural networks to distinguish dimer configurations from two known topological classes, and afterward, we test them on dimer configurations from unknown topological classes. We demonstrate that a 3D lattice dimer model on a bipartite lattice can be described by an integer topological invariant (Hopf number), whereas a lattice dimer model on a nonbipartite lattice is described by Z(2) invariant. Thus, we demonstrate that neural networks can be successfully used to identify new topological phases in condensed-matter systems, whose existence can later be verified by other (e.g., analytical) techniques.