Entanglement properties of the antiferromagnetic-singlet transition in the Hubbard model on bilayer square lattices

We calculate the bipartite Rényi entanglement entropy of an L× L× 2 bilayer Hubbard model using a determinantal quantum Monte Carlo method recently proposed by Grover [Phys. Rev. Lett. 111, 130402 (2013)]. Two types of bipartition are studied: (i) one that divides the lattice into two L× L planes and (ii) one that divides the lattice into two equal-size (L× L/2 × 2) bilayers. We compare our calculations with those for the tight-binding model studied by the correlation matrix method. As expected, the entropy for bipartition (i) scales as L2, while the latter scales with L with possible logarithmic corrections. The onset of the antiferromagnet to singlet transition shows up by a saturation of the former to a maximal value and the latter to a small value in the singlet phase. We comment on the large uncertainties in the numerical results with increasing U , which would have to be overcome before the critical behavior and logarithmic corrections could be quantified.