Entanglement entropy and superselection sectors I. Global symmetries

TL;DR

The paper analyzes how global superselection sectors (DHR) modify entanglement entropy in quantum field theories, by comparing the field algebra F with the invariant observable algebra O under a global symmetry G. It develops an information-theoretic framework where the difference in mutual information I_F−I_O is captured by a relative entropy, bounded below by intertwiners and bounded above by twists; in many cases the bounds saturate to log|G|, revealing a universal topological contribution tied to the group structure. The authors extend the analysis to continuous groups, spontaneous symmetry breaking, excited states, thermal states, and the replica trick, providing concrete examples in free fermion theories and 2d CFTs, and connect these ideas to holographic entanglement entropy, bit threads, and edge modes, suggesting that holography realizes a sub-theory with a large number of superselection sectors. The work unifies several seemingly disparate results on EE across topologies, excitations, and dimensions, offering a conceptual route to understanding holography as an entropy-based manifestation of a rich SS structure. In Part II, the framework is expected to be extended to local (gauge) symmetries, BF sectors, and more intricate holographic settings.

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.

Figures (7)

• Figure 1: Mutual information between two regions separated by a strip of width $\epsilon$ (shaded region in the figure). For $\epsilon$ wide enough the typical charge anticharge fluctuations are not sensed by mutual informations of both models (left panel). When the width $\epsilon$ becomes small enough to allow for charge anticharge fluctuations to occur on each side of the wall with enough probability (right panel), the mutual information of ${\cal F}$ will take into account these correlations while the neutral model ${\cal O}$ will not.
• Figure 2: The intertwiner $V_1 V_2^\dagger$ commutes with the additive algebra of the exterior of the two balls $W_1$ and $W_2$. It cannot be formed additively with the algebras of these balls. The twist $\tau$ cannot be formed additively on the exterior of the two balls but commmutes with the algebras of the balls. The intertwiner and the twist do not commute with each other.
• Figure 3: Two complementary regions $A$ and $B$ with non trivial topology. There is an independent set of intertwiners and twists for each connected component of the common boundary between $A$ and $B$. In the figure the number of connected components of the boundary is $n_\partial=4$.
• Figure 4: Localization of the wave packet $\left|\alpha\left(x\right)\right|^{2}$ for $\lambda=0$ and different values of $\sigma=\frac{1}{5},\frac{2}{5},1$.
• Figure 5: Localization of the wave packet $\left|\alpha\left(x\right)\right|^{2}$ for $\sigma=1$ and different values of $\lambda=0,1,2,3$.