Finite-size scaling on the torus with periodic projected entangled-pair states

TL;DR

This paper introduces a scalable and efficient tensor network contraction algorithm for 2D systems with periodic boundary conditions, enabling precise finite-size scaling and critical property estimation.

Contribution

The authors develop a novel linear-scaling renormalization step for tensor network contraction under periodic boundaries, improving accuracy and computational efficiency.

Findings

Comparable accuracy to state-of-the-art methods

Access to more data points at lower cost

Accurate critical property estimates when combined with scaling techniques

Abstract

An efficient algorithm is constructed for contracting two-dimensional tensor networks under periodic boundary conditions. The central ingredient is a novel renormalization step that scales linearly with system size, i.e. from $L \to L+1$. The numerical accuracy is comparable to state-of-the-art tensor network methods, while giving access to much more data points, and at a lower computational cost. Combining this contraction routine with the use of automatic differentiation, we arrive at an efficient algorithm for optimizing fully translation invariant projected entangled-pair states on the torus. Our benchmarks show that this method yields finite-size energy results that are comparable to those from quantum Monte Carlo simulations. When combined with field-theoretical scaling techniques, our approach enables accurate estimates of critical properties for two-dimensional quantum lattice systems.