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

TL;DR

The paper tackles the computational bottleneck of contracting triple-layer tensor networks in 3D classical statistical models for gradient-based variational optimization. It introduces a triple-layer split-CTMRG environment that compresses the network to a single-layer form with layer-specific environment tensors and employs a six-projector renormalization scheme, achieving an asymptotic cost of $\mathcal{O}(\chi^{3}D^{3}) \sim \mathcal{O}(D^{9})$ under $\chi \sim D^{2}$. Benchmarking on the 3D Ising model shows that the method reproduces $M(T)$, $U(T)$, and $C_v(T)$ with high fidelity and yields a critical temperature $T_c = 4.51288(13)$ and critical exponent $\beta = 0.3274(14)$ in agreement with MC and HOTRG results, while offering substantial speedups for $D \ge 5$. The approach enables scalable gradient-based optimization for larger bond dimensions and can be extended to multi-site PEPOs, fixed-point solvers, and 2D quantum or multilayer tensor-network problems, marking a significant advance in efficient 3D tensor-network simulations.

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.

Figures (16)

• Figure 1: Tensor network representation of the partition function for the 3D classical model. (a) The tensor network (TN) representation, consisting of a $\delta$ tensor on each site and a $W$ matrix on each link. (b) The TN representation consisting of a uniform tensor $O$. (c) The tensor $O$ is built by grouping the $\delta$ tensor with the connecting square root of the $W$ matrix. The square root of the $W$ tensor is obtained via singular value decomposition (SVD). The bond dimension of tensor $O$ is labeled as $d$.
• Figure 2: iPEPS ansatz and the terms for calculating the free energy and gradient. (a) The fixed point equation for the transfer matrix. The iPEPS state $\ket{\psi(A)}$ and the plane-to-plane transfer matrix $\hat{T}$ are made up of the uniform tensors $A$ and $O$, respectively. (b),(c) Triple-layer network representations involved in the calculation of free energy and its gradient.
• Figure 3: Reduced CTMRG scheme for contracting the triple-layer network. (a) The environment tensor configuration for $C$ and $T$. (b) The absorption and truncation procedure during a left move.
• Figure 4: The split-CTM environment tensor setting. The bonds of the $O$ in PEPO are denoted by dashed lines, while the bonds of the PEPS and the environment are denoted by solid lines. (a) The split-CTM environment tensor configuration for the triple-layer network. (b) The numerator involved in calculating the free energy. (c) The term involved in calculating the gradient.
• Figure 5: Left move in the split-CTMRG scheme for the triple-layer network. (a) Insertion. (b) Absorption. (c) Truncation.