Monte Carlo simulation with tensor network states

We demonstrate that Monte Carlo sampling can be used to efficiently extract the expectation value of projected entangled pair states with a large virtual bond dimension. We use the simple update rule introduced by H. C. Jiang et al. [Phys. Rev. Lett 101, 090603 (2008)] to obtain the tensors describing the ground state wave function of the antiferromagnetic Heisenberg model and evaluate the finite size energy and staggered magnetization for square lattices with periodic boundary conditions of linear sizes up to L = 16 and virtual bond dimensions up to D = 16. The finite size magnetization errors are 0.003(2) and 0.013(2) at D = 16 for a system of size L = 8,16, respectively. Finite D extrapolation provides exact finite size magnetization for L = 8, and reduces the magnetization error to 0.005(3) for L = 16, significantly improving the previous state-of-the-art results.