R\'enyi entropies in the $n\to0$ limit and entanglement temperatures

TL;DR

This paper explores how entanglement temperatures relate to Re9nyi entropies in quantum field theories, revealing that in the limit as n approaches zero, these entropies can be computed from solutions to eikonal equations, connecting high-temperature states to relativistic Boltzmann dynamics.

Contribution

It establishes a novel formula linking the ne0 0 limit of Re9nyi entropies to solutions of eikonal equations, extending understanding of entanglement in quantum field theories.

Findings

Re9nyi entropies at ne0 0 can be derived from eikonal equations.

High-temperature states follow a relativistic Boltzmann equation with conserved currents.

Special symmetric cases reduce to perfect fluid equations.

Abstract

Entanglement temperatures (ET) are a generalization of Unruh temperatures valid for states reduced to any region of space. They encode in a thermal fashion the high energy behavior of the state around a point. These temperatures are determined by an eikonal equation in Euclidean space. We show that the real-time continuation of these equations implies ballistic propagation. For theories with a free UV fixed point, the ET determines the state at a large modular temperature. In particular, we show that the $n \to 0$ limit of R\'enyi entropies $S_n$, can be computed from the ET. This establishes a formula for these R\'enyi entropies for any region in terms of solutions of the eikonal equations. In the $n\to 0$ limit, the relevant high-temperature state propagation is determined by a free relativistic Boltzmann equation, with an infinite tower of conserved currents. For the special case of states and regions with a conformal Killing symmetry, these equations coincide with the ones of a perfect fluid.