A bstract We study the entanglement structure of Abelian topological order described by p -form BF theory in arbitrary dimensions. We do so directly in the low-energy topological quantum field theory by considering the algebra of topological surface operators. We define two appropriate notions of subregion operator algebras which are related by a form of electric-magnetic duality. To each subregion algebra we assign an entanglement entropy which we coin essential topological entanglement . This is a refinement to the traditional topological entanglement entropy. It is intrinsic to the theory, inherently finite, positive, and sensitive to more intricate topological features of the state and the entangling region. This paper is the first in a series of papers investigating entanglement and topological order in higher dimensions.
A bstract We study the entanglement structure of Abelian topological order described by p -form BF theory in arbitrary dimensions. We do so directly in the low-energy topological quantum field theory by considering the algebra of topological surface operators. We define two appropriate notions of subregion operator algebras which are related by a form of electric-magnetic duality. To each subregion algebra we assign an entanglement entropy which we coin essential topological entanglement . This is a refinement to the traditional topological entanglement entropy. It is intrinsic to the theory, inherently finite, positive, and sensitive to more intricate topological features of the state and the entangling region. This paper is the first in a series of papers investigating entanglement and topological order in higher dimensions.
We study the entanglement structure of Abelian topological order described by p-form BF theory in arbitrary dimensions. We do so directly in the low-energy topological quantum field theory by considering the algebra of topological surface operators. We define two appropriate notions of subregion operator algebras which are related by a form of electric-magnetic duality. To each subregion algebra we assign an entanglement entropy which we coin essential topological entanglement. This is a refinement to the traditional topological entanglement entropy. It is intrinsic to the theory, inherently finite, positive, and sensitive to more intricate topological features of the state and the entangling region. This paper is the first in a series of papers investigating entanglement and topological order in higher dimensions.
A bstract We consider the entanglement entropy arising from edge-modes in Abelian p -form topological field theories in d dimensions on arbitrary spatial topology and across arbitrary entangling surfaces. We find a series of descending area laws plus universal corrections proportional to the Betti numbers of the entangling surface, which can be taken as a higher- dimensional version of the “topological entanglement entropy”. Our calculation comes in two flavors: firstly, through an induced edge-mode theory appearing on the regulated entangling surface in a replica path integral and secondly through a more rigorous definition of the entanglement entropy through an extended Hilbert space. Along the way we establish several key results that are of their own merit. We explain how the edge-mode theory is a novel combination of ( p − 1)-form and ( d – p − 2)-form Maxwell theories linked by a chirality condition, in what we coin a chiral mixed Maxwell theory . We explicitly evaluate the thermal partition function of this theory. Additionally we show that the extended Hilbert space is completely organized into representations of an infinite-dimensional, centrally extended current algebra which naturally generalizes 2d Kac-Moody algebras to arbitrary dimension and topology. We construct the Verma modules and the representation characters of this algebra. Lastly, we connect the two approaches, showing that the thermal partition function of the chiral mixed Maxwell theory is precisely an extended representation character of our current algebra, establishing an exact correspondence of the edge-mode theory and the entanglement spectrum.
The standard, gapped entanglement boundary condition in Chern Simons theory breaks the topological invariance of the theory by introducing a complex structure on the entangling surface. This produces an infinite dimensional subregion Hilbert space, a non-trivial modular Hamiltonion, and a UV-divergent entanglement entropy that is a universal feature of local quantum field theories. In this work, we appeal to the combinatorial quantization of Chern Simons theory to define a purely topological notion of a subregion. The subregion operator algebras are spaces of functions on a quantum group. We develop a diagrammatic calculus for the associated q-deformed entanglement entropy, which arise from the entanglement of anyonic edge modes. The q-deformation regulates the divergences of the QFT, producing a finite entanglement entropy associated to a q-tracial state. We explain how these ideas provide an operator algebraic framework for the entanglement entropy computations in topological string theory \cite{Donnelly:2020teo,Jiang:2020cqo, wongtopstring}, where a large- N limit of the q-deformed subregion algebra plays a key role in the stringy description of spacetime.
We consider the entanglement entropy arising from edge-modes in Abelian p-form topological field theories in d dimensions on arbitrary spatial topology and across arbitrary entangling surfaces. We find a series of descending area laws plus universal corrections proportional to the Betti numbers of the entangling surface, which can be taken as a higher-dimensional version of the "topological entanglement entropy." Our calculation comes in two flavors: firstly, through an induced edge-mode theory appearing on the regulated entangling surface in a replica path integral and secondly through a more rigorous definition of the entanglement entropy through an extended Hilbert space. Along the way we establish several key results that are of their own merit. We explain how the edge-mode theory is a novel combination of (p−1)-form and (d−p−2)-form Maxwell theories linked by a chirality condition, in what we coin a "chiral mixed Maxwell theory." We explicitly evaluate the thermal partition function of this theory. Additionally we show that the extended Hilbert space is completely organized into representations of an infinite-dimensional, centrally extended current algebra which naturally generalizes 2d Kac-Moody algebras to arbitrary dimension and topology. We construct the Verma modules and the representation characters of this algebra. Lastly, we connect the two approaches, showing that the thermal partition function of the chiral mixed Maxwell theory is precisely an extended representation character of our current algebra, establishing an exact correspondence of the edge-mode theory and the entanglement spectrum.
In this thesis we will explore entanglement of various subsystems in quantum field theory, and its uses for describing underlying structure of states. We begin this journey by focusing on topological quantum field theory in chapters 1 and 2. These chapters are based heavily upon the papers [1] and [2, 3], respectively. In the introduction to chapter 1 we take the opportunity to recall the definition of bipartite entanglement entropy in quantum field theory and describe, briefly, how the entropy associated to connected spatial subregions provides a signature of topological order at the level of the wave-function. In particular, for a gapped phase with topological order, we recapitulate how the sub-leading correction to the “area-law” of the entanglement entropy contains coarse information about the topological phase while also being insensitive to the short-distance details of the underlying theory. We then present results that show that this sub-leading correction can also detect the class of interactions that glue a subregion to its complement. In principle, the subregion could be a different topological phase than its complement, and so we say that the entanglement entropy provides a signature of the interface between topological phases. In showing this, we discuss the classes of interactions between gapped topological phases and how they are represented as boundary conditions in the low-energy effective topological field theory, which in this case is an Abelian Chern-Simons theory. Using these boundary conditions, we perform path-integral calculations of the entanglement entropy using the replica trick and show that the universal contribution does indeed depend on the class of boundary conditions, and the gapped topological phases of the subsystem and its complement. To finish this chapter we reformulate this problem at the level of the Hilbert space and show how these universal contributions are related to resolving a generic ambiguity in defining subsystems of field theories with gauge invariance. We continue in the theme of topological field theories for chapter 2. Here we will shift focus from subregions of a connected spatial slice to disconnected spatial regions in an effort to explore, firstly, how long-range entanglement is encoded in topological field theory, and secondly, how that entanglement is encoded amongst multiple parties. We begin with a brief introduction of multi-party entanglement and recall the classification of three-party entanglement. Moving on to our specific setup, we describe the construction of states on multiple non-intersecting torus boundaries. Generically, these are states on link complements where the torus boundaries can be thought of as the edges of thickened closed strings entwined and knotted inside a compact manifold, which we will take to be the three-sphere. The wave-functions of these states are represented by various link invariants (such as the Gauss linking number for Abelian links, and Jones polynomial for SU(2)). We will exhibit several examples of multi-component links in both the Abelian and the SU(2) theories, and afterwards discuss generic features of the entanglement of link states in Chern-Simons theories with a compact group. In this discussion, we will conjecture a classification (albeit broad) of links via multi-partite entanglement of their corresponding states. We will end the chapter moving away from compact groups and discuss hyperbolic links in SL(2,C) Chern-Simons theory in the semi-classical limit. In particular, utilizing the generalized volume conjecture, we will show that the entanglement structure of hyperbolic links is greatly constrained in this semi-classical limit. Finally, in chapter 3 we switch gears and discuss renormalization of free bosonic field theory. The content of this chapter is drawn from the work of [4]. Motivated by work establishing duality between the singlet sector of O(N) vector models at large N and higher spin gauge theory on Anti de Sitter (AdS) space, we develop an exact renormalization group (ERG) framework for discussing the renormalization of a large class of O(N) singlet states. We will show that for the ground state and the excited states in this class the ERG is implemented by the action of unitary operators (that can be taken to be local). Consequently, the ERG framework can be regarded as a continuum tensor network. We contrast this tensor network with some well known tensor network renormalization schemes such as the muti-scale entanglement renormalization ansatz (MERA) and its proposed continuum version (cMERA). We also comment on the nature of this network with ordinary Wilsonian renormalization. One of the central features of the ERG network is how it acts on the momentum space entanglement of the field theory. In particular, we argue that for excited states, the ERG disentangles modes that lie just above and below the UV cutoff.
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~) GSD A ˜ , where GSDA~ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity 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.