Form Factors of Branch-Point Twist Fields in Quantum Integrable Models and Entanglement Entropy

Abstract

In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the "replica trick" whereby the entropy is obtained as the derivative with respect to n of the trace of the n th power of the reduced density matrix of the sub-system, evaluated at n = 1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the n th power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.

Entanglement entropy in (3 + 1)-d free U(1) gauge theory

“…The replica trick has been discussed in many places, for example [21,22]. The calculations in section 6 where we compute the extractable piece are closely related to those by Donnelly and Wall [15,16], Huang [17] and Zuo [18].…”

“…one needs to continue the function Z(n) for non-integer values, and then take the derivative. Subtleties may arise in this continuation, see [22] for a discussion, but we ignore them here. Note that the above description is general and immediately applies to the U(1) theory as well.…”

Entanglement, replicas, and Thetas

“…These spin structures are the special "diagonal" higher-genus spin structures that obey the replica symmetry, namely those of the type α diag = (α, α, · · · , α), and β diag = (β, β, · · · , β), where (α, β) is the spin structure on the original torus. 4 The higher-genus answer is expressed in terms of Siegel Θ-functions with the special characteristics α diag β diag . The twist-operator approach involves computing two-point functions on the original torus, and the answer is expressed in terms of Jacobi θ-functions with spin structure (α, β).…”

“…3 The partition function vanishes for odd spin structures, so only the 2 n−1 (2 n + 1) even spin structures contribute to this sum. 4 We will often need to distinguish the original torus from the replica surface which is an n-fold copy of the original torus glued pairwise along a cut.…”

Higher spin entanglement and W N $$ {\mathcal{W}}_{\mathrm{N}} $$ conformal blocks

“…If we denote q i = diag(q i ) k , then the two point function of light operators is only non-vanishing if 31) and if this condition is satisfied the correlation function equals…”