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

Abstract

Machine learning methods are being actively considered as a new tool of describing many body physics. However, so far, their capabilities has been only demonstrated in previously studied models, such as e.g. Ising model. Here, we consider a simple problem, demonstrating that neural networks can be successfully used to give new insights in statistical physics. Specifically, we consider 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 one. 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 after it, we test them on dimer configurations from unknown topological classes. We demonstrate that 3D lattice dimer model on a bipartite lattice can be described by an integer topological invariant ('Hopf number'), whereas lattice dimer model on a non-bipartite 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 be later verified by other (e.g. analytical) techniques.