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arXiv:1509.00782

Note on Renyi vertex contributions and twist operator weights for free scalar fields

J. S. Dowker

Published Sep 2, 2015• 3 citations
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Abstract

I give an analytical proof of the conjecture of Bueno, Myers and Witczak-Krempa regarding the relation between the universal corner contributions and twist operator conformal weights for Renyi entropy in the case of free scalar fields.
DOI
10.48550/arxiv.1509.00782

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Abstract I compute the conformal weights of the twist operators of free scalar fields for charged Rényi entropy in both odd and even dimensions. Explicit expressions can be found, in odd dimensions as a function of the chemical potential in the absence of a conical singularity and thence by images for all integer coverings. This method, developed some time ago, is equivalent, in results, to the replica technique. A review is given. The same method applies for even dimensions but a general form is more immediately available. For no chemical potential, the closed form in the covering order is written in an alternative way related to old trigonometric sums. Some derivatives are obtained. An analytical proof is given of a conjecture made by Bueno, Myers and Witczak-Krempa regarding the relation between the conformal weights and a corner coefficient (a universal quantity) in the Rényi entropy.
I compute the conformal weights of the twist operators of free scalar fields for charged Rényi entropy in both odd and even dimensions. Explicit expressions can be found, in odd dimensions as a function of the chemical potential in the absence of a conical singularity and thence by images for all integer coverings. This method, developed some time ago, is equivalent, in results, to the replica technique. A review is given. The same method applies for even dimensions but a general form is more immediately available. For no chemical potential, the the closed form in the covering order is written in an alternative way related to old trigonometric sums. Some derivatives are obtained. An analytical proof is given of a conjecture made by Bueno, Myers and Witczac--Krempa regarding the relation between the conformal weights and a corner coefficient (a universal quantity) in the Rényi entropy.
The conformal weights of spherical twist operators at non--zero (Euclidean) chemical potential are computed for free Dirac fields in arbitrary dimensions. An image technique, equivalent to replicas, is again used to obtain the nnn--fold cover quantities. The proof of a conjecture made by Bueno, Myers and Witczac--Krempa regarding the relation between the conformal weights and a corner coefficient in the Rényi entropy, given before for scalar fields, is extended to the fermion case. The variation of the weights with chemical potential indicates phase changes.
We propose a field theoretic framework for calculating the dependence of Rényi entropies on the shape of the entangling surface in a conformal field theory. Our approach rests on regarding the corresponding twist operator as a conformal defect and in particular, we define the displacement operator which implements small local deformations of the entangling surface. We identify a simple constraint between the coefficient defining the two-point function of the displacement operator and the conformal weight of the twist operator, which consolidates a number of distinct conjectures on the shape dependence of the Rényi entropy. As an example, using this approach, we examine a conjecture regarding the universal coefficient associated with a conical singularity in the entangling surface for CFTs in any number of spacetime dimensions. We also provide a general formula for the second order variation of the Rényi entropy arising from small deformations of a spherical entangling surface, extending Mezei’s results for the entanglement entropy.
In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Rényi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient σ for the EE and σn for the Rényi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for σ and σn for all n=2,3,… . The results for σ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that σ/CT=π2/24 in all 3d CFTs, where CT is the central charge. For the Rényi entropies, the ratios σn/CT do not indicate similar universality. However, in the limit n→∞, the asymptotic values satisfy a simple relationship and equal 1/(4π2) times the asymptotic values of the free energy of free scalars/fermions on the n-covered 3-sphere.