Edge theory approach to topological entanglement entropy, mutual information, and entanglement negativity in Chern-Simons theories

Abstract

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 non-contractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders. CONTENTS

Topological entanglement negativity in Chern-Simons theories

“…Our method applies to a Chern-Simons gauge theory defined on an arbitrary oriented (2+1)-dimensional spacetime manifold. Our results agree with the edge theory approach in a recent work [35]. …”

“…Recently, the entanglement negativity has been extensively studied in conformal field theories [11][12][13], quantum spin chain systems [14][15][16][17][18][19][20], coupled harmonic oscillators in one and two dimensions [21][22][23][24][25][26][27], free fermion systems [28][29][30][31][32], topological ordered systems [33][34][35], and holographic entanglement [36][37][38]. Furthermore, the entanglement negativity has JHEP09(2016)012 also been studied in the non-equilibrium case [39][40][41][42] as well as the finite temperature case [39,43,44].…”

Universal Aspects of Entanglement in Quantum Field Theory and String Theory

“…Modular invariance dictates that |B satisfies the Cardy condition, which implies that |B is a particular linear combination of the Ishibashi states, with the coefficients given by elements of the modular S-matrix. Unfortunately, the state |B does not reproduce the known entanglement entropies in Chern Simons theory, and the authors of [89] showed how to obtain the correct entropies by relaxing the Cardy condition.…”

“…The appearance of physical edge modes in the entanglement spectrum of topological phases was first discussed in [65], [82], [70] and the computation of topological entanglement entropy in terms of left-right entanglement of boundary states was done in [22]. The reference [89], noted a problem with the approach of [70], where the reduced density matrix was obtained by a quantum quench in which region A and B are disconnected suddenly. The initial condition for such a quench is given by a conformally invariant boundary state |B satisfying…”

“…The series is badly behaved as → 0 due to the infinite temperature limit, but can be computed by applying a modular transform [89]. …”