^{1}

^{,*}

^{1}

^{,†}

^{1}

^{,‡}

^{1,2}

^{,§}

^{3}.

We consider the second Rényi entropy

Entanglement entropy has become a heavily studied field of research in recent years. For example, it is widely used in quantum information theory, where many other entanglement measurements, as, e.g., mutual entropy, exist

More generally, in the context of quantum field theory, entanglement entropy can be considered as a counter of the effective number of degrees of freedom (d.o.f.), which is related to the central charge in two-dimensional conformal field theories (CFTs)

In strongly coupled quantum field theories, the entanglement entropy turns out to be notoriously hard to compute from first principles either numerically or analytically unless the models are sufficiently symmetric or low-dimensional, see, e.g.,

This nontrivial prediction was taken as a motivation to analyze the spatial dependence of the entanglement entropy in pure

In this work, we continue this line of research and present a comparative high-statistics study of the Rényi entanglement entropy in lattice gauge theories with

We work with pure

Investigations of the impact of finite volume effects, continuum extrapolation, and replica number dependence for

We are able to clearly establish the scaling of Rényi entanglement entropy with the number of gluon states

This paper is structured as follows: we start by reminding the reader of basic definitions and properties of entanglement entropy, also from a holographic perspective, in Sec.

Intuitively, entanglement entropy measures the quantum correlations of quantum fields in two complementary spatial regions

The left figure shows the bipartition of a fixed time slice

On the lattice, it seems most economical to use

These considerations make it apparent that computing higher Rényi entropies in order to improve the extrapolation to

It should be noted that the alternative formula

The entanglement and Rényi entropies typically contain nonuniversal UV divergent terms, which have to be carefully removed in order to access the universal low-energy properties of the theory. From a theoretical viewpoint, extraction of universal UV-finite terms from the entanglement entropy has been most clearly worked out for finite and scalable entangled regions, such as the interior of a sphere

However, for a realistic lattice gauge theory simulation on a square lattice it is practically impossible to implement a smooth entangling surface such as a sphere. Smooth surfaces could only be approximated as collections of polygons, which would lead to a plethora of additional contributions due to surface edges

Assuming that the lattice size

The coefficient

To extract

For slab-shaped entangled regions such an interpretation has only been confirmed by explicit calculations in free field theories

A qualitative analysis of the entanglement entropy for a slab-shaped region (see, e.g., Eq. (4.5) in

Dimensional arguments suggest that at small slab width

Let us note that a wealth of results exist for the scaling behavior of Rényi and entanglement entropies in lower, in particular two-dimensional, gapless field theories. The general lesson to be drawn from these studies is to be careful with the naive

In holographic models entanglement entropy can be calculated using the Ryu-Takayanagi formula

In the next section we discuss how to calculate the entropic

In this section we will describe how we can measure Rényi entropy on the lattice. This has already been done for

In this work, we always refer to the density matrix of the ground state, i.e.,

Powers of the reduced density matrix can be computed by gluing copies of the above lattices together, i.e.,

Visualization of the deformed geometry as it was already shown in

Using

So far, we have reduced the problem of calculating the entropic

Such differences can be calculated using the relation between the free energy and the partition function and the fundamental theorem of calculus

With this procedure, we can calculate the entropic

Generate gauge configurations with interpolating action

Measure

Integrate over

Calculate the entropic

We have shown in the previous section how to calculate Rényi entropies on the lattice. In this section we will first show the details of our lattice calculation and afterwards we are going to present results for the entropic

To implement the method described in the last section, we use a standard Wilson gauge action given by

To generate random

The parameters of the gauge field configurations which we have used in our analysis are summarized in Table

As mentioned in the last section [cf., Eq.

The entropic

Entropic

For sufficiently small values of

At larger cut lengths

On the other hand, for

The entropic

In order to check whether the transition between small-cut and large-cut regimes becomes steeper for larger

From these results we can calculate the maximal slope

While the statistics is not large enough to reach a definite conclusion, we observe a trend toward a steeper slope at

In this paper, we have calculated the second Rényi entropy

We have confirmed that at short distances the UV-finite part of the entanglement entropy scales proportional to the number of gluon states,

Let us note that since for finite

An interesting observation is that the data for

For future work, it would be of obvious interest to increase the statistics and perform a continuum limit. Going to larger

On the other hand, improving our understanding of the behavior of the entropic

Also the holographic calculations of the entanglement entropy can be brought closer to the lattice setup by considering corrections due to finite ’t Hooft coupling. Instead of entanglement entropy, one can also calculate the Rényi entropy within holographic models

N. B. was supported by an International Junior Research Group grant of the Elite Network of Bavaria. P. B. was supported by the Heisenberg Fellowship from the German Research Foundation, Project No. BU2626/3-1. A. R. and A. S. were supported by German Research Foundation (DFG) SFB/TRR-55 “Hadron Physics from Lattice QCD.” We thank the ECT* for its hospitality during the workshop “Quantum Gravity meets Lattice QFT” where part of this work was done. The numerical simulations were performed on ATHENE the HPC cluster of the Regensburg University Compute Centre. We thank Stefano Piemonte and Mohamed Anber for valuable discussions regarding