Abstract:

Two-dimensional tensor networks such as projected entangled pairs states (PEPS) are generally hard to contract. This is arguably the main reason why variational tensor network methods in two dimensions are still not as successful as in one dimension. However, this is not necessarily the case if the tensor network represents a gapped ground state of a local Hamiltonian; such states are subject to many constraints and contain much more structure. In this paper, we introduce an approach for approximating the expectation value of a local observable in ground states of local Hamiltonians that are represented by PEPS tensor networks. Instead of contracting the full tensor network, we try to estimate the expectation value using only a local patch of the tensor network around the observable. Surprisingly, we demonstrate that this is often easier to do when the system is frustrated. In such case, the spanning vectors of the local patch are subject to nontrivial constraints that can be utilized via a semidefinite program to calculate rigorous lower and upper bounds on the expectation value. We test our approach in one-dimensional systems, where we show how the expectation value can be calculated up to at least 3 or 4 digits of precision, even when the patch radius is smaller than the correlation length.