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Bulletin of the American Physical Society

Monte Carlo simulations of two-dimensional Hubbard models with string bond tensor-network states

Jeong-Pil Song, Daehyun Wee, Rudolf Torsten Clay

Published Mar 3, 2015• 0 citations
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Tensor network states (TNS) methods combined with the Monte Carlo (MC) technique have been proven a powerful algorithm for simulating quantum many-body systems. However, because the ground state energy is a highly non-linear function of the tensors, it is easy to get stuck in local minima when optimizing the TNS of the simulated physical systems. To overcome this difficulty, we introduce a replica-exchange molecular dynamics optimization algorithm to obtain the TNS ground state, based on the MC sampling technique, by mapping the energy function of the TNS to that of a classical mechanical system. The method is expected to effectively avoid local minima. We make benchmark tests on a 1D Hubbard model based on matrix product states (MPS) and a Heisenberg J1-J2 model on square lattice based on string bond states (SBS). The results show that the optimization method is robust and efficient compared to the existing results.
We describe an application of variational Monte Carlo to two-dimensional fermionic systems within the recently developed tensor-network string-bond state ansatz. We use a combination of variational Monte Carlo and stochastic optimization to optimize the matrix-product state matrices representing the ground state. We present results for a two-dimensional spinless fermion model including nearest-neighbor Coulomb interactions and determine using finite-size scaling the phase boundary between charge-ordered insulating and metallic phases. This approach can treat frustrated systems and be easily extended to fermion models with spin.
Simulating frustrated quantum magnets is among the most challenging tasks in computational physics. We apply string-bond states, a recently introduced Ansatz which combines tensor networks with Monte Carlo based methods to the simulation of frustrated quantum systems in both two and three dimensions. We compare our results with existing results for unfrustrated and two-dimensional systems with open boundary conditions and demonstrate that the method applies equally well to the simulation of frustrated systems with periodic boundaries in both two and three dimensions.