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Can someone please let me know if there is some reference for the calculation of entanglement entropy of $U(1)$ lattice gauge theory? I have seen a few references where Z2 lattice gauge theory has been dealt with. Also these references suggest that since in a lattice gauge theory the degrees of freedom live on links hence the Hilbert space of states can't be decomposed into a direct product of states belonging entirely to a region, say A and its complement B because of links which cross the boundary of the region of interest.

My question is can I not gauge fix the link variables crossing the boundary of the region of interest so that link variables belonging solely to the complement region can be traced over unambiguously to obtain the reduced density matrix.

Also how do I write down a gauge invariant ground state wave function for a U(1) lattice gauge theory in order to define the density matrix?

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Regarding the definition of entanglement entropy in lattice gauge theory, it's true that the Hilbert space can't be expressed as a product for the reason you describe: the link variables cross the boundary between two regions. Although you can fix the link variables on the boundary, this won't capture all the entropy because the links on the boundary are subject to vacuum fluctuations and therefore should contribute to the entanglement entropy. The solution (as shown in http://arxiv.org/abs/1109.0036) involves a direct sum over all possible values of the link variables on the boundary. The entropy then breaks up into a sum of the entropy of each block in the direct sum, plus an entropy associated with the fluctuating boundary links.

The tricky part is finding the ground state wave function, and I don't know whether the U(1) theory can be solved exactly. Perhaps it is possible in 2+1 dimensions using electric-magnetic duality? It may be that the best you can do is approximate the ground state using numerical methods. See http://arxiv.org/abs/1007.4145 for an algorithm that is adapted for this purpose.

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