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 2D are still not as successful as in 1D. 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 a new approach for approximating the expectation value of a local observable in ground states of local Hamiltonians that are represented as 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 non-trivial constraints that can be utilized via a semi-definite program to calculate rigorous lower- and upper-bounds on the expectation value. We test our approach in 1D 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.

Highlights

  • Variational tensor-network methods [1] provide a promising way for understanding the low-temperature physics of many-body condensed matter systems

  • The most natural generalization is projected entangled pairs state (PEPS) tensor network [6,7], which was introduced by Verstraete and Cirac in 2004 [6], but was used earlier under different names such as “vertex matrix product ansatz” in [8], “tensors product form ansatz” (TPFA) in [9], and “tensor product state” (TPS) in [10]

  • We introduce an approach for approximating the expectation values of local observables in a 2D PEPS tensor network

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Summary

INTRODUCTION

Variational tensor-network methods [1] provide a promising way for understanding the low-temperature physics of many-body condensed matter systems. Is polynomial in the bond dimension and the particle number, it still scales badly, which limits their practical use to small systems/resolutions (state of the art days is around 15 × 15 sites with D = 6 [25]) This has led some researchers to develop the popular simulation framework, known as the “simple update” method, in which one completely abandons the contraction of the 2D network during the variational procedure [26], essentially approximating the environment of a local tensor by a product state.

STATEMENT OF THE PROBLEM
BASIC ALGORITHM
COMMUTATOR GAUGE OPTIMIZATION
Primal problem
The dual problem
Commutator gauge optimization as a SDP
Working with constant bond dimensions
NUMERICAL TESTS
Findings
CONCLUSIONS
Full Text
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