Asymmetric and three-dimensional extensions of the variational corner transfer matrix renormalization group (vCTMRG)

Liu, Fu, Yu, Yu, and Xie (2022) reformulated the corner transfer matrix renormalization group (CTMRG) contraction of infinite two-dimensional tensor networks as a variational bilevel optimization (vCTMRG), validated on symmetric classical models (dimer, Ising, q-state clock). Their construction, however, assumes a local tensor with reflection and rotation symmetry, so a single edge tensor and corner matrix suffice. The open problem is to extend this variational formulation to tensor networks lacking those symmetries — where left and right leading eigenvectors differ and four distinct edge and corner tensors with additional optimization conditions are required, the setting of multi-sublattice classical and quantum lattice models — and to three-dimensional lattices, where a face tensor enters and the bilevel problem becomes a trilevel optimization.

Background

The partition function of a classical lattice model (or the imaginary-time path integral of a quantum lattice model) can be written as the contraction of an infinite tensor network: each lattice site carries a local tensor $T$, and physical observables follow from this contraction, which is #P-hard in general and must be approximated. The corner transfer matrix renormalization group (CTMRG), introduced by Nishino and Okunishi (1996) and adapted to infinite lattices by Orús and Vidal (2009), approximates the network's infinite environment surrounding $T$ by a finite boundary description consisting of corner tensors $C$ (four matrices covering the quadrants) and edge tensors $E$ (rank-3 tensors covering the four sides), truncated at environment bond dimension $\chi$; CTMRG iteratively grows and re-truncates these boundary tensors. Conventional CTMRG either alternates power iterations in the two lattice directions or solves coupled fixed-point equations via canonical forms (Fishman, Vanderstraeten, Zauner-Stauber, Haegeman, and Verstraete, 2018); neither directly maximizes a quantity tied to the partition function. Liu, Fu, Yu, Yu, and Xie (2022) instead reformulated CTMRG as a variational bilevel optimization: using Baxter's variational expression for the partition function as a ratio of transfer-matrix expectation values, they solve $\max_{E} f(E, C^{*}(E))/g(C^{*}(E);E)$ subject to $C^{*}(E)=\arg\max_{C} g(C;E)$, whose solution $(E, C)$ is the fixed-point environment pursued by conventional CTMRG. They demonstrated high precision on the square-lattice dimer residual entropy, Ising, and q-state clock critical behavior. Their formulation and all benchmarks assume $T$ has reflection symmetry (mirror symmetry of the tensor's legs under exchanging the two directions of a bond) and rotation symmetry (invariance under 90-degree lattice rotations), so one edge tensor and one corner matrix, constrained symmetric, determine the whole environment. Without these symmetries — multi-sublattice classical models, chiral or stochastic models, and quantum lattice models treated by Trotter decomposition of an imaginary-time path integral, where the building-block tensor is generically asymmetric — the left and right leading eigenvectors of the transfer matrices differ, and four distinct edge tensors and four distinct corner tensors must be optimized with corresponding additional optimization conditions in the target function. In three dimensions the CTM construction needs, besides corners and edges, a face tensor on each of the six boundary faces, turning the bilevel problem into a trilevel (nested three-level) optimization; the authors note that conventional CTMRG iteration does not carry over easily to 3D, where instead coarse-graining methods such as the higher-order tensor renormalization group (HOTRG) of Xie, Chen, Qin, Zhu, Yang, and Xiang (2012) and dedicated 3D contraction schemes (Yang, Fu, Xie, and Xiang, 2023) are used. For solving the bilevel problem, the authors mention that general optimizers such as quasi-Newton methods apply when derivatives are available, but in practice propose a derivative-free alternating iteration (in the spirit of Baxter) and provide no convergence guarantee for the nested optimization — a situation analogous to the one-dimensional case of uniform matrix product states (the translation-invariant limit of one-dimensional tensor networks), which was settled by the VUMPS (variational uniform matrix product states) algorithm of Zauner-Stauber, Vanderstraeten, Fishman, Verstraete, and Haegeman (2018).

Problem Statement

Extend the variational corner transfer matrix renormalization group method of Liu et al. (2022) beyond symmetric two-dimensional classical models, as the source explicitly anticipates. Concretely: (1) For infinite two-dimensional tensor networks whose local tensor lacks reflection and rotation symmetry — so that the left and right leading eigenvectors of the transfer matrices differ — formulate the variational (bilevel) optimization problem with four distinct edge tensors and four distinct corner tensors and the corresponding additional optimization conditions in the variational target, and show that its solution reproduces the asymmetric fixed-point environment that conventional CTMRG iterations target; implement and validate it on at least one model class where these symmetries are broken, such as multi-sublattice classical models or quantum lattice models treated through their Trotterized imaginary-time tensor networks, with observables (free energy or energy, order parameters, critical points) agreeing with exact values or best available benchmarks to the accuracy vCTMRG achieves for symmetric models. (2) Formulate the three-dimensional extension, in which a face tensor is optimized alongside corner and edge tensors and the bilevel problem becomes a trilevel optimization; implement it and validate it on a three-dimensional benchmark model, with observables (free energy or energy, order parameters, critical points) agreeing with exact values or best available benchmarks. The unifying objective is a partition-function-guided variational environment optimization applicable where the original symmetric, two-dimensional vCTMRG does not apply.

References

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