Renyi entanglement entropies in quantum dimer models: from criticality to topological order

Thanks to Pfaffian techniques, we study the Renyi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar-Kivelson wavefunctions constructed from a dimer model on the triangular lattice. By including a fugacity t on some suitable bonds, one interpolates between the triangular lattice (t - 1) and the square lattice (t - 0). The wavefunction is known to be a massive Z(2) topological liquid for t > 0 whereas it is a gapless critical state at t = 0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder and the disc-like set-up proposed by Kitaev and Preskill (2006 Phys. Rev. Lett. 96 110404). In the cylinder case, the entropies contain an extensive term-proportional to the length of the boundary-and a universal subleading constant s(n)(t). Fitting these cylinder data (up to a perimeter of L = 32 sites) provides s(n) with a very high numerical accuracy (10(-9) at t = 1 and 10(-6) at t = 0.5). In the topological Z(2) liquid phase we find s(n)(t > 0) = -ln 2, independent of the fugacity t and the Renyi parameter n. At t = 0 we recover a previously known result, s(n)(t = 0) = -1/2 ln(n)/(n - 1) for n < 1 and s(n)(t = 0) = -ln(2)/(n - 1) for n > 1. In the disc-like geometry-designed to get rid of the boundary contributions-we find an entropy s(n)(KP)(t0) = -ln 2 in the whole massive phase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Some results for the gapless limit R-n(KP)(t -> 0) are discussed.