Tensor network Monte Carlo simulations for the two-dimensional random-bond Ising model

Abstract: Disordered lattice spin systems are crucial in both theoretical and applied physics. However, understanding their properties poses significant challenges for Monte Carlo simulations. In this work, we investigate the two-dimensional random-bond Ising model using the recently proposed Tensor Network Monte Carlo (TNMC) method. This method generates biased samples from conditional probabilities computed via tensor network contractions and corrects the bias using the Metropolis scheme. Consequently, the proposals provided by tensor networks function as block updates for Monte Carlo simulations. Through extensive numerical experiments, we demonstrate that TNMC simulations can be performed on lattices as large as $1024\times 1024$ spins with moderate computational resources, a substantial increase from the previous maximum size of $64\times 64$ in MCMC. Notably, we observe an almost complete absence of critical slowing down, enabling the efficient collection of unbiased samples and averaging over a large number of random realizations of bond disorders. We successfully pinpoint the multi-critical point along the Nishimori line with significant precision and accurately determined the bulk and surface critical exponents. Our findings suggest that TNMC is a highly efficient algorithm for exploring disordered and frustrated systems in two dimensions.