A Group-Equivariant Autoencoder for Identifying Spontaneously Broken Symmetries in the Ising Model

We introduce the group-equivariant autoencoder (GE-autoencoder) -- a novel deep neural network method that locates phase boundaries in the Ising universality class by determining which symmetries of the Hamiltonian are broken at each temperature. The encoder network of the GE-autoencoder models the order parameter observable associated with the phase transition. The parameters of the GE-autoencoder are constrained such that the encoder is invariant to the subgroup of symmetries that never break; this results in a dramatic reduction in the number of free parameters such that the GE-autoencoder size is independent of the system size. The loss function of the GE-autoencoder includes regularization terms that enforce equivariance to the remaining quotient group of symmetries. We test the GE-autoencoder method on the 2D classical ferromagnetic and antiferromagnetic Ising models, finding that the GE-autoencoder (1) accurately determines which symmetries are broken at each temperature, and (2) estimates the critical temperature with greater accuracy and time-efficiency than a symmetry-agnostic autoencoder, once finite-size scaling analysis is taken into account.