Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups

Abstract

We study the entanglement for a state on linked torus boundaries in 3d ChernSimons theory with a generic gauge group and present the asymptotic bounds of Rényi entropy at two different limits: (i) large Chern-Simons coupling k, and (ii) large rank r of the gauge group. These results show that the Rényi entropies cannot diverge faster than ln k and ln r, respectively. We focus on torus links T (2, 2n) with topological linking number n. The Rényi entropy for these links shows a periodic structure in n and vanishes whenever n = 0 (mod p), where the integer p is a function of coupling k and rank r. We highlight that the refined Chern-Simons link invariants can remove such a periodic structure in n.

Entanglement on multiple S2 boundaries in Chern-Simons theory

“…The four boundary state in figure 18 is the |Ψ 2 of eq. (3.1) for n = 4 with appropriate representations and can be written as: 15 Since the braiding operator is acting even number of times, the phase factor {R1, R2, s, u} in the eigenvalue of b2 given in eq. (2.12) is raised to even power and hence becomes 1.…”

“…See Appendix of[15] for a brief review of integrable representations. A representation of SU(N ) is integrable if it satisfies l1 ≤ k, where l1 is the number of boxes in the first row of its Young tableau.…”

Multi-boundary entanglement in Chern-Simons theory with finite gauge groups

“…Another novel approach to study entanglement is to consider these theories on manifolds whose boundary itself consists of disconnected or disjoint components as shown in figure 1(b). This is usually referred as multi-boundary entanglement and has been studied in [4][5][6][7][8] in the context of Chern-Simons theories whose gauge group is a compact semi-simple Lie group (SU(N ) for example). In this work, we will study the salient features of multi-boundary entanglement structures in Chern-Simons theory with finite discrete gauge groups.…”

“…. + a r ≤ k where a i are the Dynkin labels of the highest weight of representation R. The integrable representations for all affine classical and exceptional Lie algebras are known (see the appendix of [6] for an explicit counting). Thus in this case, a basis of H T 2 will be simply |R labeled by the integrable representation R ofĝ k .…”

3d one-form mixed anomaly and entanglement entropy

“…This happens as follows. The modular S matrix can be identified as the wave function of a state on two linked tori [27][28][29]. The Hilbert space is a tensor product of two torus Hilbert spaces.…”