Efficient neural-network based variational Monte Carlo scheme for direct optimization of excited energy states in frustrated quantum systems

We examine applicability of the valence bond basis correlator product state ansatz, equivalent to the restricted Boltzmann machine quantum artificial neural-network ansatz, and variational Monte Carlo method for direct optimization of excited energy states to study properties of strongly correlated and frustrated quantum systems. The energy eigenstates are found by stochastic minimization of the variational function for the energy eigenstates, which allows direct optimization of particular energy state without knowledge of the lower energy states. This approach combined with numerous tensor network or artificial neural-network ansatz wave functions then allows further insight into quantum phases and phase transitions in various strongly correlated models by considering properties of these systems beyond the ground-state properties. Also, the method is in general applicable to any dimension and has no sign instability. An example that we consider is the square lattice J1-J2 antiferromagnetic Heisenberg model. The model is one of the most studied models in frustrated quantum magnetism since it is closely related to the disappearance of the antiferromagnetic order in the high-Tc superconducting materials and there is still no agreement about the properties of the system in the highly frustrated regime near J2/J1 = 0.5. For the J1-J2 model, we write the variational ansatz in terms of the two site correlators and in the valence bond basis and calculate the lowest energy eigenstates in the highly frustrated regime near J2/J1 = 0.5 where the system has a paramagnetic phase. We find that our results are in good agreement with previously obtained results, which confirms applicability of the method to study frustrated spin systems