Efficient optimization of variational tensor-network approach to three-dimensional statistical systems

TL;DR

This paper introduces a split corner-transfer renormalization group scheme that significantly improves the efficiency of variational tensor network methods for three-dimensional statistical systems, enabling accurate and faster computations.

Contribution

It develops a novel contraction scheme for triple-layer tensor networks, reducing computational cost while maintaining high accuracy in 3D tensor network simulations.

Findings

Achieves results comparable to Monte Carlo simulations for 3D Ising model.

Provides a substantial speedup over previous tensor network methods.

Enables efficient gradient-based optimization in 3D tensor network applications.

Abstract

Variational tensor network optimization has become a powerful tool for studying classical statistical models in two dimensions. However, its application to three-dimensional systems remains limited, primarily due to the high computational cost associated with evaluating the free energy density and its gradient. This process requires contracting a triple-layer tensor network composed of a projected entangled pair operator and projected entangled pair states. In this paper, we employ a split corner-transfer renormalization group scheme tailored for the contraction of such a triple-layer network, which reduces the computational complexity while keeping high accuracy. Through numerical benchmarks on the three-dimensional classical Ising model, we demonstrate that the proposed scheme achieves numerical results comparable to the most recent Monte Carlo simulations, providing a substantial speedup over previous variational tensor network approaches. This makes this method well-suited for efficient gradient-based optimization in three-dimensional tensor network simulations.