Spin structures and entanglement of two disjoint intervals in conformal field theories

Abstract

Abstract. We reconsider the moments of the reduced density matrix of two disjoint intervals and of its partial transpose with respect to one interval for critical free fermionic lattice models. It is known that these matrices are sums of either two or four Gaussian matrices and hence their moments can be reconstructed as computable sums of products of Gaussian operators. We find that, in the scaling limit, each term in these sums is in one-to-one correspondence with the partition function of the corresponding conformal field theory on the underlying Riemann surface with a given spin structure. The analytical findings have been checked against numerical results for the Ising chain and for the XX spin chain at the critical point. arXiv:1511.08328v1 [cond-mat.stat-mech] 26 Nov 2015Spin structures and entanglement of two disjoint intervals in CFT 2

Entanglement, replicas, and Thetas

“…This means that the twist-operator computation, at least in the form currently known, is inadequate to compute the (finite size, finite temperature) Rényi entropy of the free Majorana (Ising)/free Dirac CFT. Subsequent to [12], issues regarding summing over fermion spin structures when computing Rényi entropies have been discussed in different contexts in [14][15][16][17][18].…”

“…It may also be useful for the study of other interesting entanglement measures such as entanglement negativity [14][15][16][36][37][38] for such systems.…”

“…In this case, the matrix B of equation (A.7) is replaced by 2B. 15 Using this matrix and repeating the above manipulations, one easily finds that the Θ-function is invariant. Thus for even n the analogue of equation (A.11) is:…”

On the mutual information in conformal field theory

“…This leads to a lot of study on the two-interval Rényi entropy in the large c CFTs, which sheds new light on the AdS 3 /CFT 2 correspondence [28][29][30][31][32][33][34][35][36]. Some relevant discussions on entanglement entropy of 2D quantum field theory can be found in [37][38][39][40][41][42][43][44][45][46][47].…”

Topological entanglement negativity in Chern-Simons theories

“…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].…”