^{*}

^{3}.

We show that Rényi entropies of subregions can be used to distinguish when the entire system is in a microcanonical ensemble from when it is in a canonical ensemble, at least in theories holographically dual to gravity. Simple expressions are provided for these Rényi entropies in a particular thermodynamic limit with high energy density and fixed fractional size of the subregion. Holographically, the Rényi entropies are determined by the areas of cosmic branes inserted into the bulk spacetime. They differ between a microcanonical and a canonical ensemble because the two ensembles provide different boundary conditions for the gravitational theory under which cosmic branes lead to different backreacted geometries. This is in contrast to the von Neumann entropy which is more coarse-grained and does not differentiate microcanonical ensembles from canonical ensembles.

In a chaotic quantum system, the eigenstate thermalization hypothesis

This raises an important question: are there more

The main purpose of this Letter is to answer this second question in holographic theories by showing that the Rényi entropies

Our main technical tool is a simple, geometric prescription derived in Refs.

This holographic prescription for Rényi entropies works in the large

Finding conical defect solutions with given boundary conditions is a well-defined task, albeit a technically difficult one in general cases without a symmetry. Here, we will simplify this task by focusing on a particular thermodynamic limit with the entire system size

Our main results in the thermodynamic limit with high energy density are as follows:

1. In a thermal state at inverse temperature

2. In a microcanonical ensemble at fixed energy

In a QFT, a thermal state

In the high temperature limit that we are interested in, the bulk geometry

(a) Bulk geometry

Let us first study the Rényi entropies of the entire system. The holographic prescription

Now consider the Rényi entropy of a subregion

Bulk geometry

According to Eq.

Even though our results do not require the boundary QFT to have additional symmetries, it is worth considering the special case where it is a

In a microcanonical ensemble, the state of the entire system is characterized by a fixed total energy

We now show that the required bulk solution is precisely

According to Eq.

The microcanonical Rényi entropy

In the special case of a

We have shown holographically that Rényi entropies of any subsystem of finite fractional size can be used to diagnose the state of the entire system. In particular, they differ between a microcanonical and a canonical ensemble for the entire system. This is in sharp contrast to the von Neumann entropy which is more coarse-grained and cannot distinguish these two types of ensembles. From the holographic perspective, this is because Rényi entropies are determined from bulk cosmic branes which lead to different backreacted geometries due to distinct boundary conditions in these two types of ensembles, whereas the von Neumann entropy is determined from extremal surfaces that only see the same semiclassical geometry. This distinction between Rényi entropies and the von Neumann entropy is reminiscent of a thermodynamic explanation of a similar phenomenon given in Ref.

The holographic prescription

Our results have been obtained in the large

For simplicity we have worked in the thermodynamic limit with high energy density so that Rényi entropies satisfy a volume-law scaling in the sense that

Even though we have focused in the previous section on a microcanonical ensemble at fixed energy

I thank Anatoly Dymarsky, Thomas Faulkner, Tarun Grover, Donald Marolf, Henry Maxfield, and Huajia Wang for useful discussions. This work was supported in part by the U.S. Department of Energy under Grant No. DE-SC0019139 and by funds from the University of California. I am also grateful to the KITP for hospitality during part of the development of this work. The KITP was supported in part by the National Science Foundation under Grant No. PHY-1748958.