A numerical study of the energy gap of the quantum dimer-pentamer model

We present a study of the energy gap in the quantum dimer-pentamer model (QDPM) on the square lattice. This model is a generalization of the square lattice quantum dimer model (QDM), with a configuration space comprising fully-packed hard-core dimer coverings of the lattice, as well as configurations containing pentamers, where four dimers touch a vertex. Thus in the QDPM, the fully-packed, hard-core constraint of the QDM is relaxed such that the local dimer number at each vertex is fixed modulo 3; correspondingly, the local $U(1)$ gauge symmetry of the QDM Hilbert space is reduced to a local $Z_3$ gauge symmetry in the QDPM. Previous work has demonstrated the disordered quantum liquid nature of the ground state of the QDPM at the Rokhsar-Kivelson point. Here we present a study of the energy gap above the ground state at the RK point, as computed via Monte Carlo from imaginary time correlations. To investigate the possibility of $Z_3$ topological order in this system, we study both the dimer density correlations as well as a $Z_3$ generalization of $Z_2$ vision correlations. Such vision correlations have previously been shown to display the nature of the low lying excitations in $Z_2$ topologically ordered QDMs.