Rényi entropies in the $n\to 0$ limit and entanglement temperatures

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 Rényi entropies $S_n$ 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 $n\to 0$ 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.

Figure 1

The ET propagates ballistically from the $x^0=0$ surface where it is determined by the Euclidean eikonal equations. For example, the black dots correspond to directions in which the ET at $x$ diverges. This is the universal Rindler-like behavior of ET at the boundary of a system.

Figure 2

Penrose diagram of the Vaidya metric. The entanglement temperature at a point $x$ in the asymptotic region is computed by first propagation back to the $v=2r_s$ surface. There the ET is evaluated using the Killing vector corresponding to the modular Hamiltonian of vacuum in flat space corresponding to the light-blue region. The red dot marks $x_{\mathrm{\Sigma }}$, where ${\widehat{p}}_{\mathrm{\Sigma }}$ is found by shooting a past null geodesic from $x$ in direction $\widehat{p}$.

Figure 3

The strip for general $d$ consists of two parallel walls located at $x=\pm l/2$ that extend all over the perpendicular coordinates represented collectively by $y$ in the figure. The time coordinate points outwards from the figure. The $\theta$ angle is defined for any point as the angle between $\widehat{p}$ and $x$. The $\epsilon$ regulators at $x=\pm (l/2-\epsilon )$ are also shown.