Entanglement of a chiral fermion on the torus

Blanco, David; Garbarz, Alan; Pérez-Nadal, Guillem

doi:10.1007/jhep09(2019)076

Cited by 14 publications

(14 citation statements)

References 42 publications

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“…Each point is calculated by our numerical procedure up to k max = 200. Left: the dark blue curve is plotted by the exact analytic expression (6.4) derived in [18], and it fits our numerical points well. Right: the same numerical points are shown with the high-temperature and low-temperature expansions.…”

Section: Numerical Studies Of One Interval At Finite Temperature and mentioning

confidence: 66%

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An alternative method for extracting the von Neumann entropy from Rényi entropies

D’Hoker

Dong

2021

J. High Energ. Phys.

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An alternative method is presented for extracting the von Neumann entropy − Tr(ρ ln ρ) from Tr(ρn) for integer n in a quantum system with density matrix ρ. Instead of relying on direct analytic continuation in n, the method uses a generating function − Tr{ρ ln[(1 − zρ)/(1 − z)]} of an auxiliary complex variable z. The generating function has a Taylor series that is absolutely convergent within |z| < 1, and may be analytically continued in z to z = −∞ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in n. Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.

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Section: Numerical Studies Of One Interval At Finite Temperature and mentioning

confidence: 66%

“…In particular, we will study the case of ν = 3 that corresponds to the Neveu-Schwarz (NS-NS) sector. Note that η(τ ) is the Dedekind eta function defined as An exact analytic expression for the von Neumann entropy in this case was derived in [18] (see also [19]); it is 12…”

Section: Numerical Studies Of One Interval At Finite Temperature and mentioning

confidence: 99%

An alternative method for extracting the von Neumann entropy from Rényi entropies

D’Hoker

Dong

2021

J. High Energ. Phys.

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“…In a recent paper [27] (see also [28]), we computed the entanglement entropy (n = 1) of the massless Dirac field on the torus by a different method, other than the replica trick, and we obtained a result which is formally different from that of [25,26] (it involves an integral instead of a series) but nevertheless agrees with it. The starting point of the method is a general relation [29] (valid for free field theories in Gaussian states) between the entanglement entropy of a region V and the two-point function G, restricted to pairs of points in V and viewed as an op-erator which acts on functions by convolution.…”

Section: Introductionmentioning

confidence: 63%

“…This relation can be reexpressed in terms of the resolvent of G, so if one manages to compute the resolvent for the theory and state of interest one can obtain the entropy. The advantage of this method is that it yields directly the result for n = 1, without need of analytically continuing from other values of n. In [30] (see also [31]) we computed the resolvent for the massless Dirac field on the torus, and in [27] we used it to obtain the entanglement entropy.…”

Section: Introductionmentioning

confidence: 99%

Rényi entropies of the massless Dirac field on the torus

Blanco,

Chase,

Laurnagaray

et al. 2021

Preprint

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We compute the Rényi entropies of the massless Dirac field on the Euclidean torus (the Lorentzian cylinder at non-zero temperature) for arbitrary spatial regions. We do it by the resolvent method, i.e., we express the entropies in terms of the resolvent of a certain operator and then use the explicit form of that resolvent, which was obtained recently. Our results are different in appearance from those already existing in the literature (obtained via the replica trick), but they agree perfectly, as we show numerically for non-integer order and analytically for integer order. We also compute the Rényi mutual information, and find that, for appropriate choices of the parameters, it is non-positive and non-monotonic. This behavior is expected, but it cannot be seen with the simplest known Rényi entropies in quantum field theory because they are proportional to the entanglement entropy.

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“…It is worth emphasizing once more how explicit (3.30) is. Indeed, whereas alternative approaches [37,50] use the modular Hamiltonian itself to write the modular flow in terms of a set of infinitely many coupled differential equations, our method yields at once the full solution.…”

Section: Jhep12(2020)126mentioning

confidence: 99%

Resolving modular flow: a toolkit for free fermions

Erdmenger

Fries

Reyes

et al. 2020

J. High Energ. Phys.

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Modular flow is a symmetry of the algebra of observables associated to space-time regions. Being closely related to entanglement, it has played a key role in recent connections between information theory, QFT and gravity. However, little is known about its action beyond highly symmetric cases. The key idea of this work is to introduce a new formula for modular flows for free chiral fermions in 1 + 1 dimensions, working directly from the resolvent, a standard technique in complex analysis. We present novel results — not fixed by conformal symmetry — for disjoint regions on the plane, cylinder and torus. Depending on temperature and boundary conditions, these display different behaviour ranging from purely local to non-local in relation to the mixing of operators at spacelike separation. We find the modular two-point function, whose analytic structure is in precise agreement with the KMS condition that governs modular evolution. Our ready-to-use formulae may provide new ingredients to explore the connection between spacetime and entanglement.

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Entanglement of a chiral fermion on the torus

Cited by 14 publications

References 42 publications

An alternative method for extracting the von Neumann entropy from Rényi entropies

An alternative method for extracting the von Neumann entropy from Rényi entropies

Rényi entropies of the massless Dirac field on the torus

Resolving modular flow: a toolkit for free fermions

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