Rényi entanglement entropies in quantum dimer models: from criticality to topologicalorder

Abstract

Thanks to Pfaffian techniques, we study the Rényi entanglement entropies and the entanglementspectrum of large subsystems for two-dimensional Rokhsar–Kivelson wavefunctionsconstructed from a dimer model on the triangular lattice. By including a fugacityt on some suitable bonds, one interpolates between the triangular lattice (t = 1) and thesquare lattice (t = 0). The wavefunction is known to be a massive topological liquid for t > 0 whereas it is a gapless critical state att = 0. We mainly consider two geometries for the subsystem: that of a semi-infinitecylinder 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 extensiveterm—proportional to the length of the boundary—and a universal subleading constantsn(t). Fitting these cylinder data (up to a perimeter ofL = 32 sites) providessn with a very highnumerical accuracy (10 − 9 at t = 1 and 10 − 6 at t = 0.5). In the topological liquid phase we find sn(t > 0) = − ln2,independent of the fugacity t and the Rényi parameter n. At t = 0 we recover a previously known result, for n < 1 and sn(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 entropysnKP(t0) = − ln2 in the whole massivephase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Someresults for the gapless limit are discussed.