Entanglement Entropy of U(1) Quantum Spin Liquids

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).