A finite entanglement entropy and the c-theorem

Casini, Horacio; Huerta, Marina

doi:10.1016/j.physletb.2004.08.072

Cited by 401 publications

(640 citation statements)

References 49 publications

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“…As proven by Casini and Huerta [27] -see also [28] -a renormalized version of this quantity [29] is monotonously decreasing under the entire RG flow connecting two fixed points, and it coincides with F at fixed points. This "F-theorem" is one of the most celebrated applications of EE to QFT, and generalizes the earlier EE-based proof of the two-dimensional "ctheorem" [30,31]. Extensions of these monotonicity theorems to CFTs defined on R 1,d≥3 relying on the EE of smooth surfaces -typically spheres -have been also proposed, see e.g., [32][33][34][35].…”

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confidence: 64%

Holographic torus entanglement and its renormalization group flow

Bueno

Witczak-Krempa

2017

Phys. Rev. D

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We study the universal contributions to the entanglement entropy (EE) of 2+1d and 3+1d holographic conformal field theories (CFTs) on topologically non-trivial manifolds, focusing on tori. The holographic bulk corresponds to AdS-soliton geometries. We characterize the properties of these regulator-independent EE terms as a function of both the size of the cylindrical entangling region, and the shape of the torus. In 2+1d, in the simple limit where the torus becomes a thin 1d ring, the EE reduces to a shape-independent constant 2γ. This is twice the EE obtained by bipartitioning an infinite cylinder into equal halves. We study the RG flow of γ by defining a renormalized EE that 1) is applicable to general QFTs, 2) resolves the failure of the area law subtraction, and 3) is inspired by the F-theorem. We find that the renormalized γ decreases monotonically at small coupling when the holographic CFT is deformed by a relevant operator for all allowed scaling dimensions. We also discuss the question of non-uniqueness of such renormalized EEs both in 2+1d and 3+1d.

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confidence: 64%

Holographic torus entanglement and its renormalization group flow

Bueno

Witczak-Krempa

2017

Phys. Rev. D

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“…Similar ideas of eliminating the UV-divergence have been suggested by Casini and Huerta [16] and have also been exploited in the context of topological entropy [17] in higher dimensions. Henceforth, lengths of (sub)systems are measured in units of the lattice spacing.…”

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confidence: 71%

“…This can be used as a fingerprint for distinguishing different CFTs, as originally suggested in Ref. 16.…”

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confidence: 98%

Mutual Information and Boson Radius in ac=1Critical System in One Dimension

2009

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We study the generic scaling properties of the mutual information between two disjoint intervals, in a class of one-dimensional quantum critical systems described by the c = 1 bosonic field theory. A numerical analysis of a spin-chain model reveals that the mutual information is scale-invariant and depends directly on the boson radius. We interpret the results in terms of correlation functions of branch-point twist fields. The present study provides a new way to determine the boson radius, and furthermore demonstrates the power of the mutual information to extract more refined information of conformal field theory than the central charge. Given a microscopic model, an important and often nontrivial issue is how to obtain the effective field theory controlling its long-distance behavior. The notion of quantum entanglement, or more specifically, the entanglement entropy, has been extensively applied as a new way to address this basic matter. From a quantum ground state |Ψ , one constructs the reduced density matrix ρ A := TrĀ |Ψ Ψ| on a subsystem A by tracing out the exteriorĀ. The entanglement entropy is defined as S A := −Tr ρ A log ρ A . In 1D quantum critical systems, the entanglement entropy for an interval A = [x 1 , x 2 ] embedded in a chain exhibits a universal scaling [6,7,8,9,10,11]:where c is the central charge of the CFT and s 1 is a nonuniversal constant related to the ultra-violet (UV) cutoff. This scaling allows to determine the universal number c as a representative of the ground state structure, without having to worry about the precise correspondence between the microscopic model and the field theory.As it is well known, the central charge is not the only important number specifying a CFT. In the bosonic field theory with c = 1, the boson compactification radius R (or equivalently, the TLL parameter K = 1/(4πR 2 )) is a dimensionless parameter which changes continuously in a phase and controls the power-law behavior of various physical quantities. It is natural to ask how to identify the boson radius as a generic structure of the ground state. In this Letter, we demonstrate that the entanglement entropy can achieve this task if we consider two disjoint intervals, A = [x 1 , x 2 ] and B = [x 3 , x 4 ]. We analyze the scaling of the mutual information defined asThis measures the amount of information shared by two subsystems [12,13]. A numerical analysis of a spin-chain model reveals a robust relation between I A:B and R, irrespective of microscopic details. We compare the result with the general prediction of Calabrese and Cardy (CC) [9], and find a relevant correction to their result. Roughly speaking, the mutual information (2) may be regarded as a region-region correlator. It is known that I A:B is non-negative, and becomes zero iff ρ A∪B = ρ A ⊗ ρ B , i.e., in a situation of no correlation [14]. A motivation to consider I A:B comes from that microscopic details at short-range scales, which are often obstacles when analyzing point-point correlators, can be smoothed out over regions. As we enlar...

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“…[20]), and it appears that new ideas are needed in that case also. In holographic theories in both even and odd dimensions, there is a unified understanding of c-theorems defined using entanglement entropy [21][22][23][24][25][26]. These are only partially understood from the purely field theory perspective, and further work along these lines may shed light on the relation between scale and conformal invariance as well.…”

Section: Discussionmentioning

confidence: 99%

Scale invariance, conformality, and generalized free fields

Dymarsky

Farnsworth

Komargodski

et al. 2016

J. High Energ. Phys.

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This paper addresses the question of whether there are 4D Lorentz invariant unitary quantum field theories with scale invariance but not conformal invariance. An important loophole in the arguments of Luty-Polchinski-Rattazzi and DymarskyKomargodski-Schwimmer-Theisen is that trace of the energy-momentum tensor T could be a generalized free field. In this paper we rule out this possibility. The key ingredient is the observation that a unitary theory with scale but not conformal invariance necessarily has a non-vanishing anomaly for global scale transformations. We show that this anomaly cannot be reproduced if T is a generalized free field unless the theory also contains a dimension-2 scalar operator. In the special case where such an operator is present it can be used to redefine ("improve") the energy-momentum tensor, and we show that there is at least one energy-momentum tensor that is not a generalized free field. In addition, we emphasize that, in general, large momentum limits of correlation functions cannot be understood from the leading terms of the coordinate space OPE. This invalidates a recent argument by Farnsworth-Luty-Prilepina (FLP). Despite the invalidity of the general argument of FLP, some of the techniques turn out to be useful in the present context.

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A finite entanglement entropy and the c-theorem

Cited by 401 publications

References 49 publications

Holographic torus entanglement and its renormalization group flow

Holographic torus entanglement and its renormalization group flow

Mutual Information and Boson Radius in ac=1Critical System in One Dimension

Scale invariance, conformality, and generalized free fields

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