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arXiv:2411.19312 · Journal of High Energy Physics

Smooth Perturbations to Rényi Entropy

Andrew Buchanan

Published Jul 23, 2025• 2 citations
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Abstract

A bstract A method is presented for computing the Rényi entropy of a perturbed massless vacuum on the ball via a comparison with lattice field theory. If the perturbed state is Gaussian with smoothly varying correlation functions and the perturbation parameter has units of energy, I show the coefficients for Rényi entropy are analytically computable for all values of the Rényi parameter α in odd dimensions and for integer α in even dimensions. I apply this procedure to compute coefficients for the large distant expansion for the Rényi mutual information of distant balls and the low temperature expansion for the entropy of a thermal field.

DOI
10.1007/jhep07(2025)233

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A bstract A method is presented for computing the Rényi entropy of a perturbed massless vacuum on the ball via a comparison with lattice field theory. If the perturbed state is Gaussian with smoothly varying correlation functions and the perturbation parameter has units of energy, I show the coefficients for Rényi entropy are analytically computable for all values of the Rényi parameter α in odd dimensions and for integer α in even dimensions. I apply this procedure to compute coefficients for the large distant expansion for the Rényi mutual information of distant balls and the low temperature expansion for the entropy of a thermal field.
In this paper, we study the R\'enyi entropy of 2D massless free fermions perturbed by the TT‾T\overline{T}TT term and the JT‾J\overline{T}JT term. In particular, we compute the correlators in the first-order perturbation. Three cases---the vacuum state of an infinite size system with TT‾T\overline{T}TT perturbation, the excited states of a finite size system with TT‾T\overline{T}TT perturbation, and the vacuum state of an infinite size system with JT‾J\overline{T}JT perturbation---are analyzed. We use the bosonization approach to study the perturbative expansions of R\'enyi entropy. In the bosonization language, the twist operator is known explicitly, with which the computation of correlators in the perturbative expansion can be simply performed. Moreover, we show that the TT‾T\overline{T}TT and JT‾J\overline{T}JT terms have a simple form and are similar to each other. For the first and third cases, we reproduce the known results for R\'enyi entropy using the bosonization method, while for the second case, we obtain new results for the excited states.
Abstract I first calculate the charged spherical Rényi entropy by a numerical method that does not require knowledge of any eigenvalue degeneracies, and applies to all odd dimensions. An image method is used to relate the full sphere values to those for an integer covering, n . It is shown to be equivalent to a ‘transformation’ property of the zeta-function. The n → ∞ limit is explicitly constructed analytically and a relation deduced between the limits of corner coefficients and the effective action (free energy) which generalises, for free fields, a result of Bueno, Myers and Witczak-Krempa and Elvang and Hadjiantonis to any dimension. Finally, the known polynomial expressions for the Rényi entropy on even spheres at zero chemical potential are re–derived in a different form and a simple formula for the conformal anomaly given purely in terms of central factorials is obtained.
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.
We show that for a d-dimensional CFT in flat space, the Rényi entropy S q across a spherical entangling surface has the following property: in an expansion around q = 1, the first correction to the entanglement entropy is proportional to C T , the coefficient of the stress tensor vacuum two-point function, with a fixed d-dependent coefficient. This is equivalent to a similar statement about the free energy of CFTs living on S 1 × $ \mathbb{H} $ d−1 with inverse temperature β = 2πq. In addition to furnishing a direct argument applicable to all CFTs, we exhibit this result using a handful of gravity and field theory computations. Knowledge of C T thus doubles as knowledge of Rényi entropies in the neighborhood of q = 1, which we use to establish new results in 3d vector models at large N.
We compute the R\'enyi entropies of the massless Dirac field on the Euclidean torus (the Lorentzian cylinder at nonzero 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 noninteger order and analytically for integer order. We also compute the R\'enyi mutual information, and find that, for appropriate choices of the parameters, it is nonpositive and nonmonotonic. This behavior is expected, but it cannot be seen with the simplest known R\'enyi entropies in quantum field theory because they are proportional to the entanglement entropy.