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arXiv:1605.01633 · Journal of Physics A Mathematical and Theoretical

Quantum revivals in free field CFT

J. S. Dowker

Published Jan 27, 2017• 2 citations
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Abstract

Abstract The recent work by Cardy (arXiv: 1603.08267 ) on quantum revivals and higher dimensional CFT is revisited and enlarged upon for free fields. The expressions for the free energy used here are those derived some time ago. The calculation is extended to spin–half fields for which the power spectrum involves the odd divisor function. An explanation of the rational revivals for odd spheres is given in terms of wrongly quantised fields and modular transformations. Comments are made on the equivalence of operator counting and eigenvalue methods, which is quickly verified.

DOI
10.1088/1751-8121/aa5360

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Abstract The recent work by Cardy (arXiv: 1603.08267 ) on quantum revivals and higher dimensional CFT is revisited and enlarged upon for free fields. The expressions for the free energy used here are those derived some time ago. The calculation is extended to spin–half fields for which the power spectrum involves the odd divisor function. An explanation of the rational revivals for odd spheres is given in terms of wrongly quantised fields and modular transformations. Comments are made on the equivalence of operator counting and eigenvalue methods, which is quickly verified.
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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.
We develop a functional-analytic framework for modular constraints in two-dimensional conformal field theory based on the reflection symmetry [Formula: see text]. This symmetry isolates a natural reflection-odd subspace of [Formula: see text] in which all deviations from modular self-duality live. For any modular-invariant partition function, the vacuum-heavy decomposition determines a uniquely defined reflection-odd modular defect [Formula: see text] whose completed Mellin transform satisfies an odd functional equation. Pairing D with compactly supported reflection-odd test functions yields a structured hierarchy of linear functionals. Under mild assumptions on the light spectrum, these pairings obey positivity conditions that lead to smeared Cardy inequalities comparing the heavy spectrum to the Cardy density through positive Laplace kernels. We also show that the vanishing of all such pairings is equivalent to averaged Cardy saturation in [Formula: see text], giving a complete reflection-odd characterization of modular self-duality. This provides a simple and flexible functional basis for modular bootstrap analyses, replacing ad hoc constructions with a unified reflection-odd hierarchy suited for both analytic arguments and numerical implementation.
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.