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arXiv:2511.10080 · Letters in Mathematical Physics

The zipper condition for 4-tensors in two-dimensional topological order and the higher relative commutants of a subfactor arising from a commuting square

Yasuyuki Kawahigashi

Published Mar 18, 2026• 0 citations
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Abstract

Researchers in condensed matter physics recently study two-dimensional topological order in terms of tensor networks involving certain 3- and 4-tensors. Their 3-tensors satisfying the “zipper condition” play an important role there, and such 3-tensors can be made into certain 2-tensors by combining two wires into one. We identify their 4-tensors with bi-unitary connections in Jones’ subfactor theory in operator algebras with precise normalization constants. Then we prove that their 2-tensors satisfying the zipper condition are the same as flat fields of strings in subfactor theory which correspond to elements in the higher relative commutants of the subfactor arising from the bi-unitary connection. This is what we expect, since the zipper condition is a kind of pentagon relations, but we clarify what conditions are exactly needed for this—we do not need the flatness or the finite depth condition for the bi-unitary connection. We actually generalize their 4-tensors so that the four index sets of the 4-tensors can be all different and work on a “half-version” of the zipper condition.

DOI
10.1007/s11005-026-02069-5

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Researchers in condensed matter physics recently study two-dimensional topological order in terms of tensor networks involving certain 3- and 4-tensors. Their 3-tensors satisfying the “zipper condition” play an important role there, and such 3-tensors can be made into certain 2-tensors by combining two wires into one. We identify their 4-tensors with bi-unitary connections in Jones’ subfactor theory in operator algebras with precise normalization constants. Then we prove that their 2-tensors satisfying the zipper condition are the same as flat fields of strings in subfactor theory which correspond to elements in the higher relative commutants of the subfactor arising from the bi-unitary connection. This is what we expect, since the zipper condition is a kind of pentagon relations, but we clarify what conditions are exactly needed for this—we do not need the flatness or the finite depth condition for the bi-unitary connection. We actually generalize their 4-tensors so that the four index sets of the 4-tensors can be all different and work on a “half-version” of the zipper condition.
Abstract A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck–Mariën–Williamson–Şahinoğlu-Haegeman–Verstraete on two-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of lengthkis naturally identified with thekth higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between two-dimensional topological order and subfactor theory.
Understanding many-body quantum systems is one of the most challenging problems in contemporary condensed-matter physics. Tensor network representation of quantum states and operators are taking central stage in this pursuit and beyond. They prove to be a powerful numerical and conceptual tool, and indeed a new language altogether. This thesis investigates various aspects of these representations by focusing on two specific problems: the first half of the thesis is devoted to examining how 'stable' a tensor network representation is for two-dimensional quantum states with topological order, and the second half explores the representability of various unitary loop operators with tensor networks. In the numerical usage of the tensor networks, the tensor is varied as to find the representation of the ground states of the given Hamiltonian. In chapter two and three of this thesis we show that such a numerical program for topological phases can be 'ill-posed'. We show that tensor network can be an unstable representation for a topological phase: even an infinitesimal variation in the representation results in the loss of topological order, completely or partially. We diagnose this problem by identifying the exact causes of this instability, and find that it is only tensor variations in certain directions that result in instability, because they result in the condensation of bosonic quasi-particles of the phase. Such unstable variations are characterized by two properties: (1) they can replace a tensor in the tensor network without making the network collapse, and (2) their presence in the network represents the presence of a non-trivial topological charge. We prove that the general tensor representation of all string-net models suffer with such instabilities. We propose an exact mathematical operator to project out all such unstable variations and show its efficacy for a few models by direct calculations. Such an operator can be useful in numerical programs involving such tensor representations. We also point out that such variations play a crucial role in simulating topological phase transitions and their presence can be vital in an accurate simulation. In chapter four and five of this thesis we focus on the representability of unitary loop operators by tensor networks. Such operators not only provide an important tool in the study of dynamical process in one-dimensional systems, but also in understanding and classification of symmetry protected topological phases in two dimensions. To characterize all such operators, we find a necessary and sufficient condition for any loop tensor network operator of a given length to represent a unitary operator. In particular, it is shown that all unitary operators that map local operators to local operators (locality-preserving) can always be represented by a tensor network. Locality-preserving unitary loop operators are classified by a rational index called the GNVW index defined in Ref. [1] which measures how much information 'flows' along the loop. We define Rank-Ratio index for tensor network operators and show that it is completely equivalent to the GNVW index. Therefore, GNVW index of a unitary operator can be easily extracted from its tensor network representation. We find that, other than representing locality-preserving unitary maps, tensor networks can also represent unitary operators that map local operators to global (non-local) operators. These tensor network operators are found to have a long-ranged order similar to tensors that represent topological tensor network states in two dimensions.
Tensor Networks are a tool from quantum information theory to describe many-body quantum systems. It has been shown that states arising from Tensor Networks in 1D (Matrix Product States - MPS) accurately describe ground states of local, gapped Hamiltonians. The same project for higher dimensional systems is currently the subject of great efforts. However, quantum many-body systems in dimension 2 or greater can exhibit much richer behaviour than their one-dimensional counterparts. One of the hallmarks of this is the scarcely understood notion of topological order. It has been shown recently, that Projected Entangled-Pair States (PEPS - Tensor Networks in 2D) can describe chiral topological states, that lie in the ground state manifold of a certain class of gapless Hamiltonians. In this work, we try to perturb the Hamiltonian into a gapped chiral phase. We will then study, whether this gapped, chiral phase can be described in terms of PEPS or how the PEPS-construction has to be modified to accommodate for this kind of behaviour.
This thesis is devoted to the study of quantum many-body systems. This investigation is performed in the framework of Matrix Product States (MPS) and their generalization to higher dimensions, Projected Entangled Pair States (PEPS). In the first part of the work, we discuss the mathematical properties of such tensor network states in depth. In the first chapters we deal with one-dimensional systems, for which we use the connections between MPS and completely positive maps to prove several new results, such as the quantum version of the Wielandt's inequality or the construction of locally invariant states under a symmetry group. Chapters 5-7 are dedicated to higher-dimensional systems, for which we provide the conditions to construct invariant PEPS. The second part of the thesis is dedicated to applications in condensed matter. In chapter 8 we provide several methods of constructing quasi-solvable Hamiltonians with two-body interactions, while in chapter 9 we show that these tensor network states can be used as a laboratory for theoretical condensed matter in, for instance, the characterization of the string-order, the generalization of the Lieb-Schultz-Mattis theorem, the demonstration of new theorems relating entanglement to magnetisation or to long-range interactions, etc.
We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with Z 2 topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-1/2 lattice to a class of exactly contractible states on spin- S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how local perturbations induce finite-range correlations in this system. This class of tensor networks is readily translated onto any lattice, and we differentiate between the physical consequences of bipartite and non-bipartite lattices on the properties of the corresponding quantum states. We explicitly show this on the hexagonal, square, kagome and triangular lattices.