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arXiv:1906.02207 · Physical review. D/Physical review. D.

Entanglement and relative entropy of a chiral fermion on the torus

Pascal Fries, Ignacio A. Reyes

Published Nov 19, 2019• 23 citations
arXivHTMLPDF

Abstract

We derive the entanglement entropy of chiral fermions on the circle at arbitrary temperature. The spin-sector contribution depends only on the total length of the entangling region, regardless of the configuration of the intervals. Thus, three-partite information provides a global indicator for the spin boundary conditions. Together with the modular Hamiltonian, our results provide a systematic way of obtaining relative entropy on the torus.

DOI
10.1103/physrevd.100.105015

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We determine the modular Hamiltonian of chiral fermions on the torus, for an arbitrary set of disjoint intervals at generic temperature. We find that, in addition to a local Unruh-like term, each point is nonlocally coupled to an infinite but discrete set of other points, even for a single interval. These accumulate near the boundaries of the intervals, where the coupling becomes increasingly redshifted. Remarkably, in the presence of a zero mode, this set of points "condenses" within the interval at low temperatures, yielding continuous nonlocality.
We consider a chiral fermion at nonzero temperature on a circle (i.e., on a torus in the Euclidean formalism) and compute the modular Hamiltonian corresponding to a subregion of the circle. We do this by a very simple procedure based on the method of images, which is presumably generalizable to other situations. Our result is nonlocal even for a single interval and even for Neveu-Schwarz boundary conditions. To the best of our knowledge, there are no previous examples of a modular Hamiltonian with this behavior.
A bstract We compute the entanglement entropy of an interval for a chiral scalar on a circle at an arbitrary temperature. We use the resolvent method, which involves expressing the entropy in terms of the resolvent of a certain operator, and we compute that resolvent by solving a problem that entails finding an analytic function on the complex torus with certain jump conditions at the interval. The resolvent is relevant by itself, since it can be used to compute any function of the reduced density matrix. We illustrate that by also computing all the Rényi entropies for the model.
In past work we introduced a method which allows for exact computations of entanglement Hamiltonians. The method relies on computing the resolvent for the projected (on the entangling region) Green's function using a solution to the Riemann-Hilbert problem combined with finite-rank perturbation theory. Here we analyze in detail several examples involving excited states of chiral fermions (Dirac and Majorana) on a spatial circle. We compute the exact entanglement Hamiltonians and an exact formula for the change in entanglement entropy due to the introduction of a particle above the Dirac sea. For Dirac fermions, we give the first-order temperature correction to the entanglement entropy in the case of a multiple-interval entangling region.
Abstract In this work, we perturbatively calculate the modular Hamiltonian to obtain the entanglement entropy in a free fermion theory on a torus with three typical deformations, e.g., TT‾ T\overline{T} TT T T ¯ deformation, local bilinear operator deformation, and mass deformation. For TT‾ T\overline{T} TT T T ¯ deformation, we find that the leading order correction of entanglement entropy is proportional to the expectation value of the undeformed modular Hamiltonian. As a check, in the high/low-temperature limit, the entanglement entropy coincides with that obtained by the replica trick in the literature. Following the same perturbative strategy, we obtain the entanglement entropy of the free fermion vacuum state up to second-order by inserting a local bilinear operator deformation in a moving mirror setting. In the uniformly accelerated mirror, the first-order and second-order correction of entanglement entropy vanishes in the late time limit. For mass deformation, we derive the entanglement entropy up to first-order deformation and comment on the second-order correction.