On the definition of entanglement entropy in lattice gauge theories

TL;DR

The work tackles the challenge of defining entanglement entropy in lattice gauge theories where local gauge invariance forbids straightforward region-factorization. It proposes an extended Hilbert space approach and derives a replica-trick framework to compute the entropy, demonstrating that standard properties like strong subadditivity hold in Z_N theories. By introducing a flux representation basis, the authors decompose the reduced density matrix into boundary-flux sectors and obtain a clear entropy decomposition into a classical Shannon term and a quantum term, with exact results for topological states: S_topo(V) = (|∂V| − n_∂) log N. The results provide a robust, gauge-invariant method to study entanglement in gauge theories, including potential applications to confinement, topological order, and non-Abelian generalizations, while also highlighting caveats related to gauge fixing at boundaries.

Abstract

We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the $Z_N$ gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the $Z_N$ gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.

Figures (1)

• Figure 1: An example of the connection and charge distributions.