Robust lattice-tailored corner transfer matrix renormalization group for non-positive tensor networks on Archimedean lattices

The corner transfer matrix renormalization group (CTMRG) approximates the contraction of infinite two-dimensional tensor networks by growing boundary matrix product states and corner matrices and truncating through projectors at environment bond dimension $\chi$. Lukin and Sotnikov constructed native, geometry-respecting CTMRG algorithms for the triangular, kagome, honeycomb, square-octagon, star, ruby, square-hexagon-dodecahedron (SHD), and dice lattices, but their constructions are confined to networks with positive bulk tensors, the setting of classical statistical mechanics, with anisotropic tensors and enlarged unit cells treated only in isolated instances, and they suffer a high-$\chi$ instability on the lattices updated through nonsymmetric factorizations, where the effective density matrix scales as a high power of the corner matrix (order of magnitude $S\propto C_6^6$) and its eigenvalues underflow machine precision. The open problem is to complete this framework into a robust general environment solver: characterize the applicability of symmetric (Cholesky, matrix-fractional-power) factorizations and provide stable biorthogonal constructions for general non-positive bulk tensors as they arise in infinite projected entangled-pair state (iPEPS) double layers; design stable high-precision projector extraction at large $\chi$; extend systematically to anisotropic tensors, enlarged unit cells, and the uncovered lattices of coordination number five; and integrate the native constructions into variational iPEPS optimization, with comparison against mapping-based contraction and boundary-MPS alternatives such as VUMPS and the computation of long-range correlation functions suggested in the source's outlook as further directions. Work in 2024 to 2026 resolved several instances, namely variational triangular CTMRG, QR-based square- and honeycomb-lattice CTMRG, and honeycomb-mapped iPEPS studies of star, maple-leaf, and ruby Heisenberg antiferromagnets, but the positivity restriction remains unresolved for the native constructions on kagome, star, ruby, and dice, and the high-$\chi$ instability on ruby, SHD, and dice.

Background

Two-dimensional (2D) tensor networks represent partition functions of classical lattice models and expectation values of infinite projected entangled-pair states (iPEPS), the variational ansatz for ground states of 2D quantum lattice models in which a local tensor with virtual bond dimension $D$ is repeated over the infinite plane; evaluating an observable requires contracting a double-layer network of a bra and a ket, whose local tensors are generally neither positive nor Hermitian. Exact contraction of such networks is computationally intractable, so one uses iterative truncating schemes. The corner transfer matrix renormalization group (CTMRG), introduced by Nishino and Okunishi for the square lattice and adapted by Orús and Vidal to the contraction of infinite projected entangled-pair states, approximates the environment of a finite patch by boundary matrix product states (bMPS, matrix-product-state tensors placed on the open legs of the patch) and corner matrices at the patch corners; local tensors are repeatedly absorbed into this boundary while the boundary bond dimension is truncated to a control parameter $\chi$ (the environment bond dimension) through projectors extracted from enlarged corner or density matrices. Non-square lattices have mostly been treated by mapping them onto a square lattice with an enlarged unit cell, which inflates intermediate bond dimensions, obscures lattice-specific quantities such as corner (entanglement) spectra, and forfeits the point-group symmetries of the lattice. Lukin and Sotnikov then constructed native CTMRG algorithms that respect the geometry of the triangular, kagome, honeycomb, square-octagon, star, ruby, square-hexagon-dodecahedron (SHD), and dice lattices — seven of the Archimedean lattices, the vertex-transitive tilings of the plane by regular polygons, together with the dice lattice, the dual of the kagome tiling — with a general recipe: define the unique bMPS on the lattice, define the corner matrices as intersections of bMPS, derive the corner updates, and use the corner tensors to build the environments that determine the truncation projectors. They benchmarked the result on classical Ising models. Three restrictions keep these constructions short of iPEPS-grade generality. First, on several of the lattices — kagome, star, ruby, and dice, in addition to the triangular lattice where the remedy is demonstrated — the update steps involve symmetric factorizations, such as Cholesky decompositions of corner matrices or matrix fractional powers like $C_6^{3/2}$, which exist only for positive semidefinite matrices; this is guaranteed by the positive Boltzmann weights of classical models but fails for iPEPS double layers. The source notes, conversely, that the honeycomb, square-octagon, and SHD algorithms do not contain symmetric factorizations and can therefore be applied to arbitrary bulk tensors without positivity restrictions. The paper demonstrates the remedy (bMPS built from distinct left and right tensors, biorthogonal truncation projectors, nonsymmetric factorizations) only for the triangular lattice, testing it on random symmetric non-positive bulk tensors. Second, exactly on the lattices updated through nonsymmetric factorizations (ruby, SHD, dice), the effective density matrix entering projector extraction scales as a high power of the corner matrix, of order $S\propto C_6^6$, so away from criticality its eigenvalues underflow below machine precision (about $10^{-16}$) once $\chi$ is large and the extracted projectors become unreliable; the authors mitigate this by ramping $\chi$ gradually and stopping when the eigenvalues of $S$ fall below a floor such as $10^{-14}$, and they suggest that more precise projector schemes developed for the square lattice by Fishman, Vanderstraeten, Zauner-Stauber, Haegeman, and Verstraete might transfer. Third, the constructions are given for isotropic tensors in minimal unit cells, apart from isolated instances (a two-site honeycomb unit cell; anisotropic triangular q-state Potts tests); the lattices of coordination number $z=5$ (five nearest neighbors per site) that the paper does not cover (maple-leaf, Shastry-Sutherland, trellis) are naturally anisotropic and have generally non-symmetric density matrices, so biorthogonalization would be needed at every step. The paper's outlook additionally lists the completely variational formulation of these algorithms, their comparison with VUMPS (the variational uniform matrix product state method of Vanderstraeten et al., an alternative environment solver that variationally optimizes the boundary MPS of the 2D network instead of growing corner matrices), and the use of projector structure to compute long-range two- and three-point correlation functions. Later work has narrowed the problem on specific fronts. QR-based CTMRG, which replaces eigen- and singular-value decompositions of enlarged corner matrices by local QR decompositions of smaller network segments and is compatible with automatic differentiation (algorithmic differentiation through the contraction code, which supplies energy gradients for variational optimization), was developed for the square lattice, with its $C_{4v}$ point-group symmetry, by Zhang, Yang, and Corboz and extended to the honeycomb lattice, with $C_{3v}$ symmetry, by Yang and Corboz, giving order-of-magnitude speedups, improved stability, and variational iPEPS optimization with long-range correlation functions on the honeycomb lattice; these authors explicitly state that the extension to further geometries such as triangular and kagome is non-trivial and open. Naumann, Eisert, and Schmoll built a triangular-lattice CTMRG tailored to PEPS on the basis of Lukin and Sotnikov's proposal, with a stabilized two-stage projector extraction and automatic differentiation for direct variational energy minimization on the native triangular lattice (and on kagome through mild coarse-graining). Frustrated quantum models on the star, maple-leaf, and ruby lattices have since been studied with iPEPS by mapping those lattices onto honeycomb-type tensor networks with multi-site unit cells, using honeycomb CTMRG rather than the native star, ruby, or dice constructions.

Problem Statement

Complete the lattice-tailored CTMRG framework of Lukin and Sotnikov into a robust, general environment solver for two-dimensional tensor networks on Archimedean lattices and their relatives, removing the restrictions that currently confine it to networks with positive bulk tensors, with anisotropic and enlarged-unit-cell networks treated only in isolated instances. The common objective uniting the deliverables is that the lattice-geometry-respecting CTMRG constructions should be usable, at iPEPS-grade tensor content and at high precision, across the lattice zoo rather than case by case with ad hoc fixes; the deliverables are linked by this objective, since the non-positivity and stability properties are prerequisites for trustworthy variational applications. Concretely: (a) characterize the range of applicability of the symmetric (Cholesky, matrix-fractional-power) factorization variants and construct stable biorthogonal or otherwise nonsymmetric alternatives valid for general non-positive bulk tensors, as arise in iPEPS double layers, on the covered geometries whose updates rely on symmetric factorizations, beyond the triangular-lattice demonstration (kagome, star, ruby, dice); (b) design projector-extraction schemes that remain stable at large environment bond dimension $\chi$ on the lattices whose updates involve high matrix powers (ruby, SHD, dice, where order-of-magnitude $S\propto C_6^6$ and eigenvalue underflow below machine precision corrupt the projectors), in the concrete sense that projector extraction remains reliable in the regime where the current gradual-$\chi$-ramp workaround must be halted at an $S$-eigenvalue floor such as $10^{-14}$, replacing that workaround, for example by QR-based or rescaled decompositions in the spirit of the square-lattice schemes of Fishman et al. or the QR-based CTMRG of Zhang, Yang, and Corboz; (c) extend the constructions systematically to anisotropic tensor networks, enlarged unit cells, and the naturally anisotropic uncovered lattices with coordination number $z=5$ (maple-leaf, Shastry-Sutherland, trellis), where density matrices are generally non-symmetric at every step, going beyond the isolated instances now available (a two-site honeycomb unit cell; anisotropic triangular q-state Potts tests); and (d) integrate the native constructions into variational iPEPS optimization (for example via automatic differentiation) beyond the triangular lattice; comparison against mapping-based contraction and against the boundary-MPS alternative VUMPS, and the computation of long-range two- and three-point correlation functions through the projector structure, are directions suggested in the source's outlook rather than mandatory benchmark requirements. A resolution may proceed lattice by lattice, but completion requires that the general construction principle be stated in checkable form, specifying for the geometries whose updates rely on symmetric factorizations the algebraic condition that decides admissibility of each such step and the biorthogonal or otherwise nonsymmetric update rule that replaces it where it is inadmissible, that the non-positivity property (a) be demonstrated on at least one of the lattices whose updates rely on symmetric factorizations (kagome, star, ruby, dice), beyond the triangular-lattice demonstration, and that the high-$\chi$ stability property (b) be demonstrated on at least one of the lattices with high matrix-power updates (ruby, SHD, dice). Directions (c) and (d) pursue the same objective but are not, by themselves, completion requirements.

References

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