Entanglement and relative entropy of a chiral fermion on the torus

Fries, Pascal; Reyes, Ignacio A.

doi:10.1103/physrevd.100.105015

Cited by 18 publications

(14 citation statements)

References 26 publications

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“…where H(x) is the Hamiltonian density of the theory, and the speed of light has been set to unit. For a 2D CFT, the BW theorem can be extended to other geometries [25][26][27][28][29][30][31][32][33][34]. For a finite interval A = [0, ] on an infinite line in the ground state, the modular Hamiltonian is [26,29]…”

Section: The XX Spin-chainmentioning

confidence: 99%

“…In order to overcome this challenge, it was proposed in Refs. [23,24] to use the Bisognano-Wichmann (BW) theorem in quantum field theory [13,14] and its extension in conformal field theory (CFT) [25][26][27][28][29][30][31][32][33][34] to write approximate modular Hamiltonians for lattice models. From the BW modular Hamiltonian one can construct a RDM, which has been dubbed BW RDM.…”

Section: Introductionmentioning

confidence: 99%

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Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

et al. 2020

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We carry out a comprehensive comparison between the exact modular Hamiltonian and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional critical quantum spin chains. As a warm-up, we first illustrate how the trace distance provides a more informative mean of comparison between reduced density matrices when compared to any other Schatten nn-distance, normalized or not. In particular, as noticed in earlier works, it provides a way to bound other correlation functions in a precise manner, i.e., providing both lower and upper bounds. Additionally, we show that two close reduced density matrices, i.e. with zero trace distance for large sizes, can have very different modular Hamiltonians. This means that, in terms of describing how two states are close to each other, it is more informative to compare their reduced density matrices rather than the corresponding modular Hamiltonians. After setting this framework, we consider the ground states for infinite and periodic XX spin chain and critical Ising chain. We provide robust numerical evidence that the trace distance between the lattice BW reduced density matrix and the exact one goes to zero as \ell^{-2}ℓ−2 for large length of the interval \ellℓ. This provides strong constraints on the difference between the corresponding entanglement entropies and correlation functions. Our results indicate that discretized BW reduced density matrices reproduce exact entanglement entropies and correlation functions of local operators in the limit of large subsystem sizes. Finally, we show that the BW reduced density matrices fall short of reproducing the exact behavior of the logarithmic emptiness formation probability in the ground state of the XX spin chain.

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Section: The XX Spin-chainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

et al. 2020

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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: 64%

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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“…This is difficult 10 to deal with numerically for k max 1. Our approach is to use the identity 19) to rewrite (5.18) in the following way [9]:…”

Section: Jhep01(2021)042mentioning

confidence: 99%

“…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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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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Entanglement and relative entropy of a chiral fermion on the torus

Cited by 18 publications

References 26 publications

Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains

Rényi entropies of the massless Dirac field on the torus

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

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