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.

Figure 1

A lattice dimer model consists of a dimer lattice lattice and dimers placed on its bonds in such a way that every site is attached to exactly one dimer. (a) Two dimer configurations are considered equivalent if they can be connected to each other by a series of local flips. (b) The simplest dimer configuration, which is not equivalent to trivially aligned dimers, is a hopfion. To perform our study we used (c) a $4\times 4\times 4$ lattice with trivially aligned dimers, (d) a hopfion placed on a lattice of the same size and surrounded by trivially aligned dimers, and (e) two hopfions stacked on the same lattice and also surrounded by trivially aligned dimers.

Figure 2

A dimer lattice is parametrized by an antisymmetric matrix $M_{ij}$, whose components are equal to $\pm i$ in such a way that a product of the matrix components around a plaquette is equal to $-1$. (a) Arrangement of signs of $i$ along a given plaquette, which are marked by arrows. (b) The same arrangement within the 3D lattice.

Figure 3

Outputs of the neural network as functions of the number of hidden units applied to trivial configurations (0), generated from one hopfion (1), two of them (2), a mirror-reflected hopfion ($-1$), and a mirror-reflected pair of hopfions ($-2$). The solid line corresponds to the neural network trained on a small data set ($\sim {10}^4$ training samples, 1500 epochs), containing configurations 0 and 1. Its accuracy for configurations with negative Hopf numbers can be improved by either increasing the training data set (dashed line refers to the case of $\sim 6\times {10}^6$ training samples, 145 epochs) or by retraining the neural network on configurations 0, 1, and $-1$, as shown by the dotted line ($\sim {10}^4$ samples, 200 epochs).

Figure 4

Outputs of the neural network trained on configurations 0 and 1 for a $4\times 4\times 4$ lattice with periodic boundary conditions ($\sim 6\times {10}^5$ samples, 500 epochs).

Figure 5

An example of a stacked triangular lattice forming a (a) trivial dimer configuration and (b) a hopfion. (c)–(h) Six possible local flips on a stacked triangular lattice that preserve $\mathrm{sgn}\mathrm{Pf}\left(M_{ij}{n}_{ij}\right)$.

Figure 6

Outputs of the neural network trained on configurations 0 and 1 for a $4\times 4\times 4$ stacked triangular lattice with open boundary conditions ($\sim 1.6\times {10}^6$ samples, 400 epochs). One can see that samples 0, 2, $-2$ are indistinguishable. In the same way samples 1 and $-1$ are also indistinguishable. Thus, we conclude that the neural network learns a $Z_2$ invariant equal to the parity of the number of hopfions.