Gluing together modular flows with free fermions

Abstract

We revisit the calculation of multi-interval modular Hamiltonians for free fermions using a Euclidean path integral approach. We show how the multi-interval modular flow is obtained by gluing together the single interval modular flows. Using this relation, we obtain an exact expression for the multi-interval modular Hamiltonian and entanglement entropy in agreement with existing results. An essential ingredient in our derivation is the introduction of the modular action. This determines the non-local field theory describing the free fermion reduced density matrix, and makes manifest it's non-local conformal symmetry and U (1) Kacs-Moody symmetry.

Entropy and modular Hamiltonian for a free chiral scalar in two intervals

“…The modular flow corresponding to this modular Hamiltonian and the entanglement entropy for several intervals have been computed in [22]. Recently, the modular Hamiltonian has also been computed using Euclidean path integral methods in [24]. In [23] it was shown that the modular flow satisfies the KMS condition.…”

“…In more generality, it is possible to identify a local part of the modular Hamiltonian which should have a large degree of universality [20,21], while not much is known about non local terms. An example of non local modular Hamiltonian which has been explicitly computed is for the vacuum state of the free massless fermion in d = 2 [22] (see also [23,24]). In this case H for several disjoint intervals has a local term proportional to the energy density and an additional non local part given by a quadratic expression in the fermion field.…”

Time development of conformal field theories associated with L 1 and L −1 operators

“…9. It would be also interesting to apply the present formalism to the calculation of the entanglement entropy for multiple sections [44][45][46][47][48][49]. These possibilities will be pursued in future research.…”

Entanglement branes, modular flow, and extended topological quantum field theory

“…For the case of the Ising model CFT, [35] showed how the entanglement entropy depends on these Cardy states and how they are mapped to entanglement boundary conditions of the microscopic model. In [36], the multi-interval modular Hamiltonian was obtained by incorporating the cutting and gluing operations that are manifest in the cobordism of figure (2). These examples give some hints for how to incorporate entanglement calculations in the framework of extended CFT.…”