Symmetries and Many-Body Excitations with Neural-Network Quantum States*

Artificial neural networks have been recently introduced as a variational ansatz for representing many-body wave functions. While previous efforts have been focused on obtaining ground states, in this work we extend the method to the study of excited states, which is an important task for many condensed matter applications. First, we give a prescription that allows us to target the lowest energy state within a symmetry sector of the Hamiltonian. Second, we give a simple algorithm to compute the low-lying states without symmetries. We demonstrate this approach on the one-dimensional spin 1/2 Heisenberg model and the one-dimensional Bose-Hubbard model and found good agreement where exact results are available. We applied our approach using both the restricted Boltzmann machine (RBM) and the feedforward neural network (FFNN). Interestingly, we obtained more accurate results using a deeper FFNN as compared with a shallower RBM with comparable number of variations parameters.