Topological entanglement and hyperbolic volume

Abstract

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 S 3 complements of a two-component link which is a connected sum of a knot K $$ \mathcal{K} $$ 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 $$ {M}_{{\mathcal{K}}_m} $$ ), which is the partition function of M K m $$ {M}_{{\mathcal{K}}_m} $$ . Here M K m $$ {M}_{{\mathcal{K}}_m} $$ is a closed 3-manifold associated with the knot K $$ \mathcal{K} $$ m , where K $$ \mathcal{K} $$ m is a connected sum of m-copies of K $$ \mathcal{K} $$ (i.e., K $$ \mathcal{K} $$ # K $$ \mathcal{K} $$ . . . # K $$ \mathcal{K} $$ ) which mimics the well-known replica method. We analayse the partition functions Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ) 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 $$ {M}_{{\mathcal{K}}_m} $$ ) can grow at most polynomially in k. On the contrary, we conjecture that Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ) is the hyperbolic volume of the knot complement S 3 \ K $$ \mathcal{K} $$ 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.

Keywords

• Chern-Simons Theories
• Conformal Field Theory
• Topological Field Theories
• Wilson, ’t Hooft and Polyakov loops