Abstract
We consider the second R\'enyi entropy ${S}^{(2)}$ in pure lattice gauge theory with $SU(2)$, $SU(3)$, and $SU(4)$ gauge groups, which serves as a first approximation for the entanglement entropy and the entropic $C$-function. We compare the results for different gauge groups using scale setting via the string tension. We confirm that at small distances $l$ our approximation for the entropic $C$-function $C(l)$, calculated for the slab-shaped entangled region of width $l$, scales as ${N}_{c}^{2}\ensuremath{-}1$ in accordance with its interpretation in terms of free gluons. At larger distances $l$ $C(l)$ is found to approach zero for ${N}_{c}=3$, 4, somewhat more rapidly for ${N}_{c}=4$ than for ${N}_{c}=3$. This finding supports the conjectured discontinuity of the entropic $C$-function in the large-$N$ limit, which was found in the context of AdS/CFT correspondence and which can be interpreted as transition between colorful quarks and gluons at small distances and colorless confined states at long distances. On the other hand, for $SU(2)$ gauge group the long-distance behavior of the entropic $C$-function is inconclusive so far. There exists a small region of lattice spacings yielding results consistent with ${N}_{c}=3$, 4, while results from other lattice spacings deviate without clear systematics. We discuss several possible causes for discrepancies between our results and the behavior of entanglement entropy in holographic models.
Highlights
Entanglement entropy has become a heavily studied field of research in recent years
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) [4]
OðN2cÞ and Oð1Þ scalings of the entropy at short and large distances appears to be smooth for our data for all Nc which we consider, we find indications that for Nc 1⁄4 4 the entanglement entropy changes faster in the transition region than for Nc 1⁄4 3
Summary
Entanglement entropy has become a heavily studied field of research in recent years It is widely used in quantum information theory, where many other entanglement measurements, as, e.g., mutual entropy, exist [1], as well as in connection with the gauge/gravity duality [2]. It can be used as a universal order parameter of quantum phase transitions, as it is done, e.g., for 2 þ 1 dimensional topological field theories, where one relates entanglement entropy to the quantum dimension [3].
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