Using Modern Equivariant Machine Learning Architectures as Effective Hamiltonians for Monte Carlo Simulations of Quantum Spin Systems

Abstract

Monte Carlo methods play a fundamental role in the numerical simulation of physical systems, particularly in statistical mechanics, quantum mechanics, and condensed matter physics. This study explores various Monte Carlo techniques and their application to different Hamiltonians, with a focus on the Ising model, the Ising-like plaquette model, and the Double Exchange model. A key objective is to compare different update strategies, including local Metropolis updates, global Wolff cluster updates, and Self Learning Monte Carlo (SLMC) updates. After demonstrating the effectiveness of SLMC using a proof-of-concept study of the Ising-like plaquette model, it is shown that SLMC in combination with network-based effective Hamiltonians can improve the simulation efficiency of complex quantum spin systems significantly. Future work might focus on extending the presented formalism to more complex systems such as the SU(3) symmetry group of QCD.