We here investigate the entanglement structure of the ground state of a
(3+1)-dimensional U(1) quantum spin liquid, which is described by the
deconfined phase of a compact U(1) gauge theory. A gapless photon is the only
low-energy excitation, with matter existing as deconfined but gapped
excitations of the system. It is found that, for a given bipartition of the
system, the elements of the entanglement spectrum can be grouped according to
the electric flux between the two regions, leading to a useful interpretation
of the entanglement spectrum in terms of electric charges living on the
boundary. The entanglement spectrum is also given additional structure due to
the presence of the gapless photon. Making use of the Bisognano-Wichmann
theorem and a local thermal approximation, these two contributions to the
entanglement (particle and photon) are recast in terms of boundary and bulk
contributions, respectively. Both pieces of the entanglement structure give
rise to universal subleading terms (relative to the area law) in the
entanglement entropy, which are logarithmic in the system size (log L), as
opposed to the subleading constant term in gapped topologically ordered
systems. The photon subleading logarithm arises from the low-energy conformal
field theory and is essentially local in character. The particle subleading
logarithm arises due to the constraint of closed electric loops in the
wavefunction and is shown to be the natural generalization of topological
entanglement entropy to the U(1) spin liquid. This contribution to the
entanglement entropy can be isolated by means of the Grover-Turner-Vishwanath
construction (which generalizes the Kitaev-Preskill scheme to three
dimensions).Comment: 15+6 page
