On the definition of entanglement entropy in lattice gauge theories

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.

arXiv comment29 pages, 1 figure, a comment and more references added, some typos corrected

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• Canonical Transformations and Loop Formulation of SU(N) Lattice Gauge Theories
• 't Hooft suppression and holographic entropy
• Entanglement Entropy and Duality
• Rényi Entropy for a $2d$ CFT with a gauge field: $\widehat{\mathrm{SU}}(\mathbf{N}{)}_1$ WZW theory on a branched torus
• Dynamical Edge Modes in Yang-Mills Theory
• Strong subadditivity and holography
• Superselection-Resolved Entanglement in Lattice Gauge Theories: A Tensor Network Approach