We study the universal scaling behavior of the entanglement entropy of critical theories in $2+1$ dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision.
We study the Z2 topologically ordered surface state of three-dimensional bosonic SPT phases with the discrete symmetries G1 x G2. It has been argued that the topologically ordered surface state cannot be realized on a purely two-dimensional lattice model. We carefully examine the statement and show that the surface state should break G2 if the symmetry G1 is gauged. This manifests the conflict of the symmetry G1 and G2 on the surface of the three-dimensional SPT phase. Given that there is no such phenomena in the purely two-dimensional model, it signals that the symmetries are encoded anomalously on the surface of the three-dimensional SPT phases and that the surface state can never be realized on the purely two-dimensional models.
