New non-Euclidean neural quantum states from hyperbolic Lorentz recurrent architectures

In this work, we construct new non-Euclidean neural quantum states (NQS) based on hyperbolic Lorentz recurrent architectures (RNN/GRU). These constructions, together with the Poincare RNN NQS also newly constructed here, extend the class of previously introduced non-Eucllidean NQS which consists only of Poincare hyperbolic GRU. Using the Heisenberg J1J2 and J1J2J3 models consisting of 100 spins in the Variational Monte Carlo (VMC) setting, we show that the four hyperbolic RNN/GRU NQS variants are always able to furnish better representations of the ground state wavefunctions of the quantum systems than their respective Euclidean counterparts with the same architecture. In our experiments, among the four hyperbolic NQS, Lorentz RNN stands out in particular because despite having almost three times fewer parameters, it is capable of surpassing the more complex Poincare GRU and Lorentz GRU to emerge as the best overall hyperbolic NQS ansatz on many instances involving different J2 and (J2,J3) couplings. Given the findings from this work showing that the four newly constructed hyperbolic RNN/GRU NQS ansatze are able to outperform the well-established Euclidean RNN/GRU NQS in Heisenberg spin models, we establish the utility and efficiency of the hyperbolic Lorentz RNN/GRU NQS as well as the Poincare RNN/GRU NQS for future variational studies of quantum many-body systems, especially those exhibiting a hierarchical structure in the form of the different degrees of nearest-neighbor interactions.