Equivariant symmetry breaking*

Equivariant neural networks (ENNs) have been shown to be extremely useful in many applications involving some underlying symmetries. However, many physical systems undergo spontaneous symmetry breaking phase transitions, such as octahedral tilting in perovskites. So given a highly symmetric input, we may want a less symmetric output. Naively, an ENN always gives an output of equal or higher symmetry and is unable to perform such a task. We realize that the issue is we need a degenerate set of symmetrically related outputs, each of which is equally valid and we just want to sample one of them. But many equivariant frameworks are not designed to give such a set of objects. Here, we propose the idea of symmetry breaking sets (SBS). Rather than redesign existing networks to output symmetrically degenerate sets, we design sets of symmetry breaking objects which we feed into our network. In order to achieve full generality, we impose that our SBSs only depend on the symmetry $S$ we want to break. Further, we derive additional conditions imposed by equivariance under a group $G$. Ideally, there is a one to one correspondence between elements in our equivariant SBS and the outputs. We prove that when this ideal case happens is equivalent to finding complements of normal subgroups.