We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n-component link inside the threesphere S 3 . The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U (1) k ) we give a general formula for the entanglement entropy associated to an arbitrary (m|n−m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU (2) k , we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a "W-like" entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have "GHZ-like" entanglement (i.e., tracing out one torus does lead to a separable state).

Abstract:We study the entanglement entropy between (possibly distinct) topological phases across an interface using an Abelian Chern-Simons description with topological boundary conditions (TBCs) at the interface. From a microscopic point of view, these TBCs correspond to turning on particular gapping interactions between the edge modes across the interface. However, in studying entanglement in the continuum Chern-Simons description, we must confront the problem of non-factorization of the Hilbert space, which is a standard property of gauge theories. We carefully define the entanglement entropy by using an extended Hilbert space construction directly in the continuum theory. We show how a given TBC isolates a corresponding gauge invariant state in the extended Hilbert space, and hence compute the resulting entanglement entropy. We find that the sub-leading correction to the area law remains universal, but depends on the choice of topological boundary conditions. This agrees with the microscopic calculation of [1]. Additionally, we provide a replica path integral calculation for the entropy. In the case when the topological phases across the interface are taken to be identical, our construction gives a novel explanation of the equivalence between the left-right entanglement of (1+1)d Ishibashi states and the spatial entanglement of (2+1)d topological phases.

We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU (2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U (1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU (2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S 3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure -i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have Wlike entanglement -i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified SL(2, C) Chern-Simons theory, which is closely related to 3d Einstein gravity with a negative cosmological constant. In the limit of small Newton constant, we discuss how the entanglement structure is controlled by the Neumann-Zagier potential on the moduli space of hyperbolic structures on the link complement.

The exact renormalization group (ERG) for O(N ) vector models (at large N ) on flat Euclidean space can be interpreted as the bulk dynamics corresponding to a holographically dual higher spin gauge theory on AdS d+1 . This was established in the sense that at large N the generating functional of correlation functions of single trace operators is reproduced by the on-shell action of the bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. In this paper, we extend the ERG formalism to the wave functionals of arbitrary states of the O(N ) vector model at the free fixed point. We find that the ERG flow of the ground state and a specific class of excited states is implemented by the action of unitary operators which can be chosen to be local. Consequently, the ERG equations provide a continuum notion of a tensor network. We compare this tensor network with the entanglement renormalization networks, MERA, and its continuum version, cMERA, which have appeared recently in holographic contexts. In particular the ERG tensor network appears to share the general structure of cMERA but differs in important ways. We comment on possible holographic implications.

The nonclassical symmetries of a class of Burgers' systems are considered. This study was initialized by Cherniha and Serov with a restriction on the form of the nonclassical symmetry operator. In this paper we remove this restriction and solve the determining equations to show that (1) a new form of a Burgers' system exists that admits a nonclassical symmetry and (2) a Burgers' system exists that is linearizable.