Abstract
The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the R\'enyi 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 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(M_K_m), which is the partition function of M_K_m. Here M_K_m is a closed 3-manifold associated with the knot K_m, where K_m is a connected sum of m-copies of K (i.e., K#K…#K) which mimics the well-known replica method. We analyse the partition functions Z(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(M_K_m) can grow at most polynomially in k. On the contrary, we conjecture that Z(M_K_m) for SO(3) group shows an exponential growth in k, where the leading term of ln Z(M_K_m) is the hyperbolic volume of the knot complement S³ K_m. We further propose that the R\'enyi 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.