Entanglement entropy and superselection sectors. Part I. Global symmetries

Abstract

Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: The present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of a sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise. *

Logarithmic coefficient of the entanglement entropy of a Maxwell field

“…We address this question in the next section. A similar failure of the universal terms to be protected in the IR has been shown to happen in models with global superselection sectors [15].…”

“…See [12,27] for previous discussions where this contribution of charged particles to the Maxwell field entropy was recognized. As discussed in [15], there is a universal logarithmic correction to the mutual information for these orbifolds that shows up, even for very massive fields, once m e 1 . This is given by…”

“…This is independent of and shows there are important correlations along the surface that persist for any as long as the theory does not have charges. For the case of dominance of the area law, the flux operators are still effectively Gaussian variables since the fluctuations of the flux are produced by a large number of random independent charge fluctuations near the surface and we can apply the central limit theorem (see [15]). Since the areas of the nearby patches just add and Φ 1 Φ 2 is still given by ∼ L 12 / we get…”

“…A related problem is how a correction that depends on the details of the UV, such as the one associated with the presence of massive charges, could affect the IR result universally. This again is not restricted to the case of the Maxwell field but also happens for other models with superselection sectors [15]. The answer is that the main effect of charges is to destroy non-local correlations in some specific operators of the IR model.…”

Gravity edges modes and Hayward term

“…In this sense, this situation in gravity may look slightly analogous to topological ordered systems, where gapless degrees of freedom appear on boundaries, and therefore the classical contribution to the gravitational entropy is often called "gravity edge modes". Refer to [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] for recent discussions.…”