arXiv:2201.13358

Entanglement entropy and $C_T$ for monodromy defects of fields on odd-dimensional spheres

J. S. Dowker

Published Jan 31, 2022• 0 citations

Abstract

The effect of a spherical monodromy defect on the entanglement entropy and central charge $C_T$ of a free conformal scalar field propagating on an odd-dimensional sphere is investigated. As on even spheres the central charge becomes negative for a range of the flux parameter, $δ$, with possible implications for reflection positivity. The work is mostly numerical but closed forms are found for the $Z_2$ monodromy ($δ=1/2$) yielding an explicit $d\to\infty$ limit, $C_T\to -6/π$

DOI: 10.48550/arxiv.2201.13358

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Renyi entropy for spherical monodromy of free scalar fields in even dimensions

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arXiv:2201.13358

Entanglement entropy and $C_T$ for monodromy defects of fields on odd-dimensional spheresEntanglement entropy and $C_T$ for monodromy defects of fields on odd-dimensional spheres

J. S. Dowker

Published Jan 31, 2022• 0 citations

arXiv HTML PDF

Abstract

The effect of a spherical monodromy defect on the entanglement entropy and central charge

C_T

of a free conformal scalar field propagating on an odd-dimensional sphere is investigated. As on even spheres the central charge becomes negative for a range of the flux parameter,

δ

, with possible implications for reflection positivity. The work is mostly numerical but closed forms are found for the

Z_2

monodromy (

δ=1/2

) yielding an explicit

d\to\infty

limit,

C_T\to -6/π

DOI: 10.48550/arxiv.2201.13358

Related papers

Renyi entropy for spherical monodromy of free scalar fields in even dimensions

J. S. Dowker

The effect of a co--dimension--2 spherical monodromy defect on the free--field theory of a conformal scalar is investigated. The conformal anomaly is computed (in two ways) and thence the Rényi and entanglement entropies. The $C_T$ central charge is found to be negative for a range of monodromy flux.The calculations are pursued for any even dimensional sphere. A Possible conflict with existing results is raised.The effect of a co--dimension--2 spherical monodromy defect on the free--field theory of a conformal scalar is investigated. The conformal anomaly is computed (in two ways) and thence the Rényi and entanglement entropies. The $C_T$ central charge is found to be negative for a range of monodromy flux.The calculations are pursued for any even dimensional sphere. A Possible conflict with existing results is raised.

arXiv:2112.09031Dec 16, 20211 citation

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A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term, ${\left(-1\right)}^{\frac{d-1}{2}}\ F(A)$ − 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term, $$ {\left(-1\right)}^{\frac{d-1}{2}}\ F(A) $$ − 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.

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Abstract We study co-dimension two monodromy defects in theories of conformally coupled scalars and free Dirac fermions in arbitrary d dimensions. We characterise this family of conformal defects by computing the one-point functions of the stress-tensor and conserved current for Abelian flavour symmetries as well as two-point functions of the displacement operator. In the case of d = 4, the normalisation of these correlation functions are related to defect Weyl anomaly coefficients, and thus provide crucial information about the defect conformal field theory. We provide explicit checks on the values of the defect central charges by calculating the universal part of the defect contribution to entanglement entropy, and further, we use our results to extract the universal part of the vacuum Rényi entropy. Moreover, we leverage the non-supersymmetric free field results to compute a novel defect Weyl anomaly coefficient in a d = 4 theory of free $\mathcal{N}$ N = 2 hypermultiplets. Including singular modes in the defect operator product expansion of fundamental fields, we identify notable relevant deformations in the singular defect theories and show that they trigger a renormalisation group flow towards an IR fixed point with the most regular defect OPE. We also study Gukov-Witten defects in free d = 4 Maxwell theory and show that their central charges vanish.Abstract We study co-dimension two monodromy defects in theories of conformally coupled scalars and free Dirac fermions in arbitrary d dimensions. We characterise this family of conformal defects by computing the one-point functions of the stress-tensor and conserved current for Abelian flavour symmetries as well as two-point functions of the displacement operator. In the case of d = 4, the normalisation of these correlation functions are related to defect Weyl anomaly coefficients, and thus provide crucial information about the defect conformal field theory. We provide explicit checks on the values of the defect central charges by calculating the universal part of the defect contribution to entanglement entropy, and further, we use our results to extract the universal part of the vacuum Rényi entropy. Moreover, we leverage the non-supersymmetric free field results to compute a novel defect Weyl anomaly coefficient in a d = 4 theory of free $$ \mathcal{N} $$ N = 2 hypermultiplets. Including singular modes in the defect operator product expansion of fundamental fields, we identify notable relevant deformations in the singular defect theories and show that they trigger a renormalisation group flow towards an IR fixed point with the most regular defect OPE. We also study Gukov-Witten defects in free d = 4 Maxwell theory and show that their central charges vanish.

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Renyi entropy and C_T for p-forms on even spheres

J. S. Dowker

Renyi entropy and central charge, $C_T$ , are calculated for a coexact p--form on an even sphere with particular reference to the conformally invariant case. It is shown, for example, that the entanglement entropy is minus the standard conformal anomaly with no `shift' being required. The shift necessary for a conformal p--form, when using a hyperbolic technique, is predicted, on a numerical basis, to be (minus) the entanglement entropy of a conformal (p-1)-form. The central charges agree numerically with a general formula of Buchel {\it et al}.Renyi entropy and central charge, $C_T$, are calculated for a coexact p--form on an even sphere with particular reference to the conformally invariant case. It is shown, for example, that the entanglement entropy is minus the standard conformal anomaly with no `shift' being required. The shift necessary for a conformal p--form, when using a hyperbolic technique, is predicted, on a numerical basis, to be (minus) the entanglement entropy of a conformal (p-1)-form. The central charges agree numerically with a general formula of Buchel {\it et al}.

arXiv:1706.04574Jun 14, 20173 citations

Entanglement entropy and

C_T

for monodromy defects of fields on odd-dimensional spheres

The effect of a spherical monodromy defect on the entanglement entropy and central charge

C_T

of a free conformal scalar field propagating on an odd-dimensional sphere is investigated. As on even spheres the central charge becomes negative for a range of the flux parameter,

δ

, with possible implications for reflection positivity. The work is mostly numerical but closed forms are found for the

Z_2

monodromy (

δ=1/2

) yielding an explicit

d\to\infty

limit,

C_T\to -6/π

C_T

central charge is found to be negative for a range of monodromy flux.The calculations are pursued for any even dimensional sphere. A Possible conflict with existing results is raised.

C_T

, is also calculated. Comparison with existing results is made and it is shown how these can be obtained from the values here.

A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term,

{\left(-1\right)}^{\frac{d-1}{2}}\ F(A)

− 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.

\mathcal{N}

N = 2 hypermultiplets. Including singular modes in the defect operator product expansion of fundamental fields, we identify notable relevant deformations in the singular defect theories and show that they trigger a renormalisation group flow towards an IR fixed point with the most regular defect OPE. We also study Gukov-Witten defects in free d = 4 Maxwell theory and show that their central charges vanish.

Renyi entropy and central charge,

C_T

, are calculated for a coexact p--form on an even sphere with particular reference to the conformally invariant case. It is shown, for example, that the entanglement entropy is minus the standard conformal anomaly with no `shift' being required. The shift necessary for a conformal p--form, when using a hyperbolic technique, is predicted, on a numerical basis, to be (minus) the entanglement entropy of a conformal (p-1)-form. The central charges agree numerically with a general formula of Buchel {\it et al}.