Time-reversal equivariant neural network potential and Hamiltonian for magnetic materials

This work presents Time-reversal Equivariant Neural Network (TENN) framework. With TENN, the time-reversal symmetry is considered in the equivariant neural network (ENN), which generalizes the ENN to consider physical quantities related to time-reversal symmetry such as spin and velocity of atoms. TENN-e3, as the time-reversal-extension of E(3) equivariant neural network, is developed to keep the Time-reversal E(3) equivariant with consideration of whether to include the spin-orbit effect for both collinear and non-collinear magnetic moments situations for magnetic material. TENN-e3 can construct spin neural network potential and the Hamiltonian of magnetic material from ab-initio calculations. Time-reversal-E(3)-equivariant convolutions for interactions of spinor and geometric tensors are employed in TENN-e3. Compared to the popular ENN, TENN-e3 can describe the complex spin-lattice coupling with high accuracy and keep time-reversal symmetry which is not preserved in the existing E(3)-equivariant model. Also, the Hamiltonian of magnetic material with time-reversal symmetry can be built with TENN-e3. TENN paves a new way to spin-lattice dynamics simulations over long-time scales and electronic structure calculations of large-scale magnetic materials.