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Register a new userExact Solution and Correlations of a Quantum Dimer Model on the Checkerboard Lattice
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We present analytic results for a special dimer model on the {em non-bipartite} and {em non-planar} checkerboard lattice that does not allow for parallel dimers surrounding diagonal links. We {em exactly} calculate the number of closed packed dimer coverings on finite checkerboard lattices under periodic boundary conditions, and determine all dimer-dimer correlations. The latter are found to vanish beyond a certain distance. We find that this solvable model, despite being non-planar, is in close kinship with well-known paradigm-setting planar counterparts that allow exact mappings to $mathbb{Z}_2$ lattice gauge theory.
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We introduce a quantum dimer model on the hexagonal lattice that, in addition to the standard three-dimer kinetic and potential terms, includes a competing potential part counting dimer-free hexagons. The zero-temperature phase diagram is studied by means of quantum Monte Carlo simulations, supplemented by variational arguments. It reveals some new crystalline phases and a cascade of transitions with rapidly changing flux (tilt in the height language). We analyze perturbatively the vicinity of the Rokhsar-Kivelson point, showing that this model has the microscopic ingredients needed for the devils staircase scenario [E. Fradkin et al., Phys. Rev. B 69, 224415 (2004)], and is therefore expected to produce fractal variations of the ground-state flux.
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We solve the monomer-dimer problem on a non-bipartite lattice, the simple quartic lattice with cylindrical boundary conditions, with a single monomer residing on the boundary. Due to the non-bipartite nature of the lattice, the well-known method of a Temperley bijection of solving single-monomer problems cannot be used. In this paper we derive the solution by mapping the problem onto one on close-packed dimers on a related lattice. Finite-size analysis of the solution is carried out. We find from asymptotic expansions of the free energy that the central charge in the logarithmic conformal field theory assumes the value $c=-2$.
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We construct the Hamiltonian description of the Chern-Simons theory with Z_n gauge group on a triangular lattice. We show that the Z_2 model can be mapped onto free Majorana fermions and compute the excitation spectrum. In the bulk the spectrum turns out to be gapless but acquires a gap if a magnetic term is added to the Hamiltonian. On a lattice edge one gets additional non-gauge invariant (matter) gapless degrees of freedom whose number grows linearly with the edge length. Therefore, a small hole in the lattice plays the role of a charged particle characterized by a non-trivial projective representation of the gauge group, while a long edge provides a decoherence mechanism for the fluxes. We discuss briefly the implications for the implementations of protected qubits.
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