A bstract The topological Rényi and entanglement entropies depend on the bipartition of the manifold and the choice of the ground states. However, these entanglement quantities remain invariant under a coordinate transformation when the bipartition also undergoes the identical transformation. In topological quantum field theories (TQFTs), these coordinate transformations reduce to representations of the mapping class group on the manifold of the Hilbert space. We employ this invariant property of the Rényi and entanglement entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on a torus with various bipartitions. By utilizing the replica trick and the surgery method to compute the topological Rényi and entanglement entropies, the invariant property results in Verlinde-like formulas. Furthermore, for the bipartition with interfaces as two non-intersecting torus knots, an SL(2 , ℤ) transformation can untwist the torus knots, leading to a simple bipartition with an effective ground state. This invariant property allows us to demonstrate that the topological entanglement entropy has a lower bound −2 ln D , where D is the total quantum dimensions of the system.
A bstract The topological Rényi and entanglement entropies depend on the bipartition of the manifold and the choice of the ground states. However, these entanglement quantities remain invariant under a coordinate transformation when the bipartition also undergoes the identical transformation. In topological quantum field theories (TQFTs), these coordinate transformations reduce to representations of the mapping class group on the manifold of the Hilbert space. We employ this invariant property of the Rényi and entanglement entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on a torus with various bipartitions. By utilizing the replica trick and the surgery method to compute the topological Rényi and entanglement entropies, the invariant property results in Verlinde-like formulas. Furthermore, for the bipartition with interfaces as two non-intersecting torus knots, an SL(2 , ℤ) transformation can untwist the torus knots, leading to a simple bipartition with an effective ground state. This invariant property allows us to demonstrate that the topological entanglement entropy has a lower bound −2 ln D , where D is the total quantum dimensions of the system.
Abstract We introduce a pseudo entropy extension of topological entanglement entropy called topological pseudo entropy. Various examples of the topological pseudo entropies are examined in three-dimensional Chern-Simons gauge theory with Wilson loop insertions. Partition functions with knotted Wilson loops are directly related to topological pseudo (Rényi) entropies. We also show that the pseudo entropy in a certain setup is equivalent to the interface entropy in two-dimensional conformal field theories (CFTs), and leverage the equivalence to calculate the pseudo entropies in particular examples. Furthermore, we define a pseudo entropy extension of the left-right entanglement entropy in two-dimensional boundary CFTs and derive a universal formula for a pair of arbitrary boundary states. As a byproduct, we find that the topological interface entropy for rational CFTs has a contribution identical to the topological entanglement entropy on a torus.
We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus g, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the R symbols, monodromy, and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent noncontractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.
Abstract The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S3 complements of a two-component link which is a connected sum of a knot K K and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z( MKm M K m ), which is the partition function of MKm M K m . Here MKm M K m is a closed 3-manifold associated with the knot K K m, where K K m is a connected sum of m-copies of K K (i.e., K K # K K . . . # K K ) which mimics the well-known replica method. We analayse the partition functions Z( MKm M K m ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z( MKm M K m ) can grow at most polynomially in k. On the contrary, we conjecture that Z( MKm M K m ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z( MKm M K m ) is the hyperbolic volume of the knot complement S3\ K K m. We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.
Abstract If we partition a Hilbert space we can define the entanglement entropy (both von Neumann and Renyi) between the two pieces. We consider breaking up a system into two or more distinct spatial regions and ask about the entanglement between these two regions. Studying the toric code ground state as an initial example, we show that there is an entanglement that scales as the length of the cut between the two regions, but there is also a sub-leading term, known as the topological entanglement entropy, which depends only on the topological properties of the system. While the leading term is nonuniversal, the topological term is robust and can be generalized to any other TQFT. We give three different derivations of the topological entanglement entropy.
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.
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.