Abstract
We study the generalisation of relative entropy, the Rényi divergence Dα(ρ∥ρβ) in 2d CFTs between an excited state density matrix ρ, created by deforming the Hamiltonian, and the thermal density matrix ρβ. Using the path integral representation of this quantity as a Euclidean quench, we obtain the leading contribution to the Rényi divergence for deformations by scalar primaries and by conserved holomorphic currents in conformal perturbation theory. Furthermore, we calculate the leading contribution to the Rényi divergence when the conserved current perturbations have inhomogeneous spatial profiles which are versions of the sine-square deformation (SSD). The dependence on the Rényi parameter (α) of the leading contribution have a universal form for these inhomogeneous deformations and it is identical to that seen in the Rényi divergence of the simple harmonic oscillator perturbed by a linear potential. Our study of these Rényi divergences shows that the family of second laws of thermodynamics, which are equivalent to the monotonicity of Rényi divergences, do indeed provide stronger constraints for allowed transitions compared to the traditional second law.
Highlights
We study the generalisation of relative entropy, the Renyi divergence Dα(ρ||ρβ) in 2d CFTs between an excited state density matrix ρ, created by deforming the Hamiltonian, and the thermal density matrix ρβ
Our study of these Renyi divergences shows that the family of second laws of thermodynamics, which are equivalent to the monotonicity of Renyi divergences, do provide stronger constraints for allowed transitions compared to the traditional second law
In this paper we evaluated the Renyi divergences between the thermal density matrix and excited states created by deforming the Hamiltonian by scalar primaries, conserved currents and sine-square deformation (SSD) deformations in 2d CFTs
Summary
We review the evaluation of Renyi divergences using its path integral representation as an Euclidean quench put forward in [7]. Ρ is an excited state obtained by deforming the Hamiltonian corresponding to the thermal density matrix ρβ. The entire path integral can be thought of as performing the path integral on the thermal cylinder with the Hamiltonian HCFT deformed by the operator O coupled to a time dependent source given by μ(τ ) = (θ(τ ) − θ(τ − αβ))μ. The factors in the denominator in (2.1) can be evaluated using the path integral (2.2) and (2.4) It is clear from this formulation for evaluating Renyi divergences that we are restricted to the specific class of excited states which are obtained by deforming the original Hamiltonian HCFT. One simple check we perform is that we verify these properties are true for all the cases for which we have evaluated the Renyi divergences
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