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arXiv:1801.05390 · Physical review. B./Physical review. B

Gauge fixing, canonical forms, and optimal truncations in tensor networks with closed loops

Glen Evenbly

Published Aug 31, 2018• 73 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.98.085155

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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.
Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable – in contrast to the well-known undecidability for equality of PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.
Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable - in contrast to the well-known undecidability for PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.
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).
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.
Abstract Formulating non-Abelian gauge theories as a tensor network is known to be challenging due to the internal degrees of freedom that result in the degeneracy in the singular value spectrum. In two dimensions, it is straightforward to “trace out” these degrees of freedom with the use of character expansion, giving a reduced tensor network where the degeneracy associated with the internal symmetry is eliminated. In this work, we show that such an index loop also exists in higher dimensions in the form of a closed tensor network that we call the “armillary sphere”. This allows us to completely eliminate the matrix indices and reduce the overall size of the tensors in the same way as is possible in two dimensions. This formulation allows us to include significantly more representations with the same tensor size, thus making it possible to reach a greater level of numerical accuracy in the tensor renormalization group computations.
We present an overview of the key ideas and skills necessary to begin implementing tensor network methods numerically, which is intended to facilitate the practical application of tensor network methods for researchers that are already versed with their theoretical foundations. These skills include an introduction to the contraction of tensor networks, to optimal tensor decompositions, and to the manipulation of gauge degrees of freedom in tensor networks. The topics presented are of key importance to many common tensor network algorithms such as DMRG, TEBD, TRG, PEPS, and MERA.