Rényi entropies in the n →0 limit and entanglement temperatures

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→0 limit of Rényi entropies Sn, can be computed from the ET. This establishes a formula for these Rényi entropies for any region in terms of solutions of the eikonal equations. In the n→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.