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arXiv:2510.18350 · Physical review. D/Physical review. D.

Entanglement spectrum resolved by ( d − 2 )-form symmetries

Haruki Yagi, Zongping Gong

Published Apr 21, 2026• 0 citations
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Abstract

A rigorous analysis is presented for the entanglement spectrum of quantum many-body states possessing a higher-form group-representation symmetry generated by topological Wilson loops, which is generally non-invertible. A general framework based on elementary algebraic topology and category theory is developed to determine the block structure of reduced density matrices for arbitrary bipartite manifolds on which the states are defined. Within this framework, we scrutinize the impact of topology on the entanglement structure for low-dimensional manifolds, including especially the torus, the Klein bottle, and lens spaces. By further incorporating gauge invariance, we refine our framework to determine the entanglement structure for topological gauge theories in arbitrary dimensions. In particular, in two dimensions, it is shown for the Kitaev quantum double model that not only the topological entanglement entropy can be reproduced, but also the Li-Haldane conjecture concerning the full entanglement spectrum holds exactly.

DOI
10.1103/gsml-klfg

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A rigorous analysis is presented for the entanglement spectrum of quantum many-body states possessing a higher-form group-representation symmetry generated by topological Wilson loops, which is generally non-invertible. A general framework based on elementary algebraic topology and category theory is developed to determine the block structure of reduced density matrices for arbitrary bipartite manifolds on which the states are defined. Within this framework, we scrutinize the impact of topology on the entanglement structure for low-dimensional manifolds, including especially the torus, the Klein bottle, and lens spaces. By further incorporating gauge invariance, we refine our framework to determine the entanglement structure for topological gauge theories in arbitrary dimensions. In particular, in two dimensions, it is shown for the Kitaev quantum double model that not only the topological entanglement entropy can be reproduced, but also the Li-Haldane conjecture concerning the full entanglement spectrum holds exactly.
Symmetry is a fundamental organising principle in theoretical physics, underpinning the conservation of physical quantities and the characterisation of phases of matter. It is closely related to topology, leading to generalisations such as higher-form and non-invertible symmetries, with applications in electromagnetism, the standard model of particle physics, condensed matter, and quantum computation. This thesis harnesses generalised symmetries to derive universal results in gapped and gapless phases of matter, focusing particularly on topologically ordered systems and conformal field theories (CFTs). Topological order is characterised by long-range entangled gapped phases, namely ground states that cannot be disentangled by finite-depth quantum circuits. At low energies it is described via topological quantum field theories (TQFTs). One part of this thesis focusses on extracting patterns of long-range entanglement from the low-energy TQFT for a class of topological orders described by abelian BF theory. By considering algebras of topological operators restricted to subregions, a new measure of entanglement, termed essential topological entanglement, is introduced to refine topological entanglement entropy. This is complemented by a detailed analysis of the edge-mode spectrum of abelian BF theory and its contribution to entanglement entropy across different topologies. A key mathematical result, enabling the above analysis, is the generalisation of Kac–Moody current algebras and their representation theory to higher dimensions. On the gapless front, the thesis considers CFTs with continuous higher-form symmetries, invertible or non-invertible. It is shown that they also possess infinite-dimensional current algebras akin to those described above. Their representation theory allows for a complete characterisation of the quantum states of these theories across arbitrary spatial topologies. In four dimensions, CFTs with continuous one-form symmetries can be described in terms of free photons. Combining this with the previous analysis leads to a one-to-one correspondence between states on particular topologies and line operators, i.e. wires carrying distributions of electric and magnetic charge.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
This paper explores the rigorous formulation of entanglement spectra within the framework of C*-algebras, proposing their utility as robust topological invariants for classifying quantum phases of matter. Entanglement spectra, derived from the eigenvalues of reduced density matrices, have emerged as powerful diagnostic tools for characterizing topological order, often revealing hidden patterns beyond conventional symmetry-breaking paradigms. By embedding quantum systems and their subsystem decompositions into the abstract mathematical structure of C*-algebras, we provide a generalized and mathematically rigorous approach to define and analyze these spectra. This C*-algebraic perspective allows for a unified treatment of finite and infinite quantum systems, including those arising in quantum field theory, and naturally incorporates the algebraic notion of locality and quantum states. We demonstrate how specific features of the entanglement spectrum, such as degeneracy patterns or gaps, can be viewed as invariants under C*-algebraic deformations that preserve the underlying topological order. This framework suggests a deep connection between the algebraic properties of observables, the structure of entanglement, and the classification of topological phases, potentially linking to established mathematical tools like K-theory. The work establishes a theoretical foundation for leveraging entanglement spectra as fundamental topological indicators within a highly abstract yet physically relevant algebraic setting.
We give an exact form of the density matrix of the spin-1/2 Kitaev spin liquid represented in terms of spin operators and study the entanglement structures of the Kitaev honeycomb model within the spin framework. We show that the density matrix is naturally organized by equivalence classes of string operators associated with the underlying gauge structure of the model. With the explicit form of the density matrix, plus the exact Gauss law of the emergent gauge theory and the exact 1-form Wilson symmetry in the Kitaev model, we demonstrate the existence of the underlying symmetry-resolved block-diagonal structure of the reduced density matrix, which gives rise to the extensive degeneracy in the entanglement spectrum. The block-diagonal structure is then proven to be responsible for the separability of the entanglement entropy into the gauge and matter parts. Furthermore, we extend the formalism to subsystems with an odd number of lattice sites, revealing a relation between the entanglement spectrum and the fermion parity that is seldom mentioned in the literature.
Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.
Abstract The topological entanglement entropy is used to measure long-range quantum correlations in the ground space of topological phases. Here we obtain closed form expressions for the topological entropy of (2+1)- and (3+1)-dimensional loop gas models, both in the bulk and at their boundaries, in terms of the data of their input fusion categories and algebra objects. Central to the formulation of our results are generalized S{\mathcal {S}}S S -matrices. We conjecture a general property of these S{\mathcal {S}}S S -matrices, with proofs provided in many special cases. This includes constructive proofs for categories up to rank 5.