Equivariant neural network for lattice models in condensed matter systems

Machine learning (ML) is emerging as a new paradigm for scientific research and engineering. ML models have been developed to provide efficient structure-property mapping and force-field predictions in quantum molecular dynamics and quantum spin dynamics, to new a few. A crucial requirement for a faithful ML model is to preserve symmetries of the original physical systems. For example, ML models in condensed-matter lattice systems need to be invariant with respect to both discrete lattice point symmetry as well as internal symmetries associated with on-site degrees of freedom such as spins or lattice distortions. A proper descriptor, which provides a symmetry-invariant representation of a local environment, is often combined with a fully connected neural network to incorporate the symmetry requirements into ML models. An alternative approach, which has attracted enormous interest in recent years, is equivariant neural networks (ENNs). Such neural nets are characterized by the properties that neurons at each layer follow specific transformation rules of the governing symmetry groups. Here we develop a scalable ENN force-field model that incorporates the D4 point group of the square-lattice Holstein model and applied it to study large-scale coarsening dynamics of charge density waves.