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arXiv:2508.00338 · Physical review. E

Truncating loopy tensor networks by zero-mode gauge fixing

I. V. Sokolov, Yintai Zhang, Jacek Dziarmaga

Published Nov 7, 2025• 0 citations
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Abstract

Loopy tensor networks have internal correlations that often make their compression inefficient. We show that even local bond optimization can make better use of the insight it has locally into relevant loop correlations. By cutting the bond, we define a set of states whose linear dependence can be used to truncate the bond dimension. The linear dependence is eliminated with zero modes of the states' metric tensor. The method is illustrated by a series of examples for the infinite pair-entangled projected state (iPEPS) and for the periodic matrix product state (pMPS) that occurs in the tensor renormalization group (TRG) step. In all examples, it provides better initial truncation errors than standard initialization.

DOI
10.1103/4lgp-ld2s

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Loopy tensor networks have internal correlations that often make their compression inefficient. We show that even local bond optimization can make better use of the insight it has locally into relevant loop correlations. By cutting the bond, we define a set of states whose linear dependence can be used to truncate the bond dimension. The linear dependence is eliminated with zero modes of the states' metric tensor. The method is illustrated by a series of examples for the infinite pair-entangled projected state (iPEPS) and for the periodic matrix product state (pMPS) that occurs in the tensor renormalization group (TRG) step. In all examples, it provides better initial truncation errors than standard initialization.
Truncating loopy tensor networks by zero-mode gauge fixing: the Z2Z_2Z2​ lattice gauge theory at finite temperature
Loopy tensor networks exhibit internal correlations that often render their compression inefficient. We show that even local bond optimization can more effectively exploit locally available information about relevant loop correlations. By cutting a bond, we define a set of states whose linear dependence can be identified through a zero mode of the states' metric tensor and used to truncate the bond dimension. In the absence of an exact zero mode, a linear combination of a small number of the lowest modes can instead be optimized to provide the optimal approximation to a zero mode. The truncation does not require prior gauge fixing. The method is applied to the two-dimensional finite-temperature Z2Z_2Z2​ lattice gauge theory, whose thermal-state purification is represented by an infinite projected entangled-pair state (iPEPS).
We propose a method to construct a tensor network representation of partition functions without singular value decompositions nor series expansions. The approach is demonstrated for one- and two-dimensional Ising models and we study the dependence of the tensor renormalization group (TRG) on the form of the initial tensors and their symmetries. We further introduce variants of several tensor renormalization algorithms. Our benchmarks reveal a significant dependence of various TRG algorithms on the choice of initial tensors and their symmetries. However, we show that the boundary TRG technique can eliminate the initial tensor dependence for all TRG methods. The numerical results of TRG calculations can thus be made significantly more robust with only a few changes in the code. Furthermore, we study a three-dimensional Z 2 gauge theory without gauge fixing and confirm the applicability of the initial tensor construction. Our method can straightforwardly be applied to systems with longer range and multisite interactions, such as the next-nearest neighbor Ising model. Published by the American Physical Society 2024
We describe an approach to fix the gauge degrees of freedom in tensor networks, including those with closed loops, which allows a canonical form for arbitrary tensor networks to be realized. Additionally, a measure for the internal correlations present in a tensor network is proposed, which quantifies the extent of resonances around closed loops in the network. Finally we describe an algorithm for the optimal truncation of an internal index from a tensor network, based upon proper removal of the redundant internal correlations. These results, which offer a unified theoretical framework for the manipulation of tensor networks with closed loops, can be applied to improve existing tensor network methods for the study of many-body systems and may also constitute key algorithmic components of sophisticated new tensor methods.
In usual (nonstochastic) tensor network calculations, the truncated singular value decomposition is often used for approximating a tensor, and it causes systematic errors. By introducing stochastic noise in the approximation, however, one can avoid such systematic errors at the expense of statistical errors, which can be straightforwardly controlled. Therefore in principle, exact results can be obtained even at finite bond dimension up to the statistical errors. A previous study of the unbiased method implemented in tensor renormalization group algorithm, however, showed that the statistical errors for physical quantity are not negligible, and furthermore the computational cost is linearly proportional to a system volume. In this paper, we introduce a new way of stochastic noise such that the statistical error is suppressed, and moreover, in order to reduce the computational cost we propose common noise method whose cost is proportional to the logarithm of volume. We find that the method provides better accuracy for the free energy compared with the truncated singular value decomposition when applying to tensor renormalization group for Ising model on square lattice. Although the common noise method introduces systematic error originated from a correlation of noises, we show that the error can be described by a simple functional form in terms of the number of noises, thus the error can be straightforwardly controlled in an actual analysis. We also apply the method to the graph independent local truncation algorithm and show that the accuracy is further improved.
We apply the projective truncation technique to the tensor renormalization group (TRG) algorithm in order to reduce the computational cost from $O({\ensuremath{\chi}}^{6})$ to $O({\ensuremath{\chi}}^{5})$, where $\ensuremath{\chi}$ is the bond dimension, and propose three kinds of algorithms for demonstration. On the other hand, the technique causes a systematic error due to the incompleteness of a projector composed of isometries and in addition requires iteration steps to determine the isometries. Nevertheless, we find that the accuracy of the free energy for the Ising model on a square lattice is recovered to the level of TRG with a few iteration steps even at the critical temperature for $\ensuremath{\chi}=32$, 48, and 64.
We introduce a class of tensor-network states in which physical legs are shared among local tensors, termed leg-tied tensor ansätze (LETTA). Physical leg ties encode long-range correlations directly, while a virtual matrix product state (MPS) backbone retains short-range multipartite entanglement. The linear virtual backbone allows us to develop a deterministic density matrix renormalization group-like variational optimization algorithm using exact contractions over the active tie-boundary sets and local minimization. We demonstrate the advantages of LETTA for the two-dimensional frustrated J1J_1J1​--J2J_2J2​ Heisenberg model and the three-dimensional transverse-field Ising model. Our results show that LETTA is substantially more accurate than same-bond-dimension MPS calculations and can typically reach the accuracy of much larger MPS calculations using one order of magnitude fewer variational parameters. LETTA thus opens the door for explicitly correlated tensor-network states that can encode long-range correlation beyond virtual bonds.