Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1905.05210 · Physical review. D/Physical review. D.

Modular Hamiltonian of a chiral fermion on the torus

David Blanco, Guillem Pérez-Nadal

Published Jul 9, 2019• 29 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.100.025003

Related papers

Entanglement of a chiral fermion on the torus

David Blanco, Alan Garbarz, Guillem Pérez-Nadal

Abstract In this paper we present the detailed calculation of a new modular Hamiltonian, namely that of a chiral fermion on a circle at non-zero temperature. We provide explicit results for an arbitrary collection of intervals, which we discuss at length by checking against known results in different limits and by computing the associated modular flow. We also compute the entanglement entropy, and we obtain a simple expression for it which appears to be more manageable than those already existing in the literature.

arXiv:1906.07057•Journal of High Energy Physics•Sep 1, 2019•13 citations

Entanglement Spectrum of Chiral Fermions on the Torus

Pascal Fries, Ignacio A. Reyes

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.

arXiv:1905.05768•Physical Review Letters•Nov 19, 2019•52 citations

Entanglement and relative entropy of a chiral fermion on the torus

Pascal Fries, Ignacio A. Reyes

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.

arXiv:1906.02207•Physical review. D/Physical review. D.•Nov 19, 2019•23 citations

Modular conjugation for the chiral fermion in multicomponent regions on the torus

Nicolás Abate, Mateo Koifman

We continue the study of the Tomita-Takesaki modular conjugation for a massless Dirac field in a generic multicomponent region in $1+1$ spacetime dimensions. In this paper we focus on the computations for a thermal state on a circle, namely on the euclidean torus. By analytic continuation from the modular flow we arrive at an explicit expression for the modular conjugation in this scenario and derive its relevant limits. In contrast to the case of the vacuum on the line, this new result has a nonlocal behavior even for connected regions. It also presents a novel contribution coming from the purification one has to introduce in order to deal with a mixed state; a term that maps the algebra of operators of the region to a copy of the global one, the so-called ``second world'' algebra.

Physical review. D/Physical review. D.•Oct 5, 2023•2 citations

Modular conjugation for the chiral fermion in multicomponent regions on the torus

Nicolás Abate, Mateo Koifman

We continue the study of the Tomita-Takesaki modular conjugation for a massless Dirac field in a generic multicomponent region in $1+1$ spacetime dimensions. In this paper we focus on the computations for a thermal state on a circle, namely on the euclidean torus. By analytic continuation from the modular flow we arrive at an explicit expression for the modular conjugation in this scenario and derive its relevant limits. In contrast to the case of the vacuum on the line, this new result has a non-local behaviour even for connected regions. It also presents a novel contribution coming from the purification one has to introduce in order to deal with a mixed state: a term that maps the algebra of operators of the region to a copy of the global one, the so called 'second world' algebra.

arXiv:2307.11819•Jul 21, 2023•0 citations

Modular Hamiltonian and modular flow of massless fermions on a cylinder

Daniela Cadamuro, Markus B. Fröb, Guillem Pérez-Nadal

We determine explicitly the modular flow and the modular Hamiltonian for massless free fermions in diamonds on a cylinder in 1+1 dimensions. We consider both periodic and antiperiodic boundary conditions, the ground state in the antiperiodic case and the most general family of quasi-free zero-energy ground states in the periodic case, which depend on four parameters and are generally mixed. While for the antiperiodic ground state and one periodic ground state (the maximally mixed zero-temperature state) the modular data is known, our results for the generic ground state in the periodic case are completely new. We find that generically both the modular flow and the modular Hamiltonian are non-local, and we show that in the parametric limit where the state becomes pure the modular data becomes local. Moreover, even in the local case the modular flow generically mixes the two chiralities. This kind of behavior has not been observed previously.

arXiv:2406.19360•Jun 27, 2024•0 citations
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.
Abstract In this paper we present the detailed calculation of a new modular Hamiltonian, namely that of a chiral fermion on a circle at non-zero temperature. We provide explicit results for an arbitrary collection of intervals, which we discuss at length by checking against known results in different limits and by computing the associated modular flow. We also compute the entanglement entropy, and we obtain a simple expression for it which appears to be more manageable than those already existing in the literature.
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 continue the study of the Tomita-Takesaki modular conjugation for a massless Dirac field in a generic multicomponent region in 1+11+11+1 spacetime dimensions. In this paper we focus on the computations for a thermal state on a circle, namely on the euclidean torus. By analytic continuation from the modular flow we arrive at an explicit expression for the modular conjugation in this scenario and derive its relevant limits. In contrast to the case of the vacuum on the line, this new result has a nonlocal behavior even for connected regions. It also presents a novel contribution coming from the purification one has to introduce in order to deal with a mixed state; a term that maps the algebra of operators of the region to a copy of the global one, the so-called ``second world'' algebra.
We continue the study of the Tomita-Takesaki modular conjugation for a massless Dirac field in a generic multicomponent region in 1+11+11+1 spacetime dimensions. In this paper we focus on the computations for a thermal state on a circle, namely on the euclidean torus. By analytic continuation from the modular flow we arrive at an explicit expression for the modular conjugation in this scenario and derive its relevant limits. In contrast to the case of the vacuum on the line, this new result has a non-local behaviour even for connected regions. It also presents a novel contribution coming from the purification one has to introduce in order to deal with a mixed state: a term that maps the algebra of operators of the region to a copy of the global one, the so called 'second world' algebra.
We determine explicitly the modular flow and the modular Hamiltonian for massless free fermions in diamonds on a cylinder in 1+1 dimensions. We consider both periodic and antiperiodic boundary conditions, the ground state in the antiperiodic case and the most general family of quasi-free zero-energy ground states in the periodic case, which depend on four parameters and are generally mixed. While for the antiperiodic ground state and one periodic ground state (the maximally mixed zero-temperature state) the modular data is known, our results for the generic ground state in the periodic case are completely new. We find that generically both the modular flow and the modular Hamiltonian are non-local, and we show that in the parametric limit where the state becomes pure the modular data becomes local. Moreover, even in the local case the modular flow generically mixes the two chiralities. This kind of behavior has not been observed previously.