Scaling of entanglement entropy in the (branching) multiscale entanglement renormalization ansatz

We investigate the scaling of entanglement entropy in both the multiscale entanglement renormalization ansatz (MERA) and in its generalization, the branching MERA. We provide analytical upper bounds for this scaling, which take the general form of a boundary law with various types of multiplicative corrections, including power-law corrections all the way to a bulk law. For several cases of interest, we also provide numerical results that indicate that these upper bounds are saturated to leading order. In particular, we establish that, by a suitable choice of holographic tree, the branching MERA can reproduce the logarithmic multiplicative correction of the boundary law observed in Fermi liquids and spin-Bose metals in D >= 2 dimensions.