Quantum dimer model withZ2liquid ground state: Interpolation between cylinder and disk topologies and toy model for a topological quantum bit

We consider a quantum dimer model on the kagome lattice, which was introduced recently [Phys. Rev. Lett. 89, 137202 (2002)]. It realizes a ${\mathbb{Z}}_2$ liquid phase and its spectrum was obtained exactly. It displays a topological degeneracy when the lattice has a nontrivial geometry (cylinder, torus, etc). We discuss and solve two extensions of the model where perturbations along lines are introduced: first a potential-energy term repelling (or attracting) the dimers along a line and second, a perturbation allowing to create, move, or destroy monomers. For each of these perturbations we show that there exists a critical value above which, in the thermodynamic limit, the degeneracy of the ground state is lifted from 2 (on a cylinder) to 1. In both cases the exact value of the gap between the first two levels is obtained by a mapping to an Ising chain in a transverse field. This model provides an example of a solvable Hamiltonian for a topological quantum bit where the two perturbations act as diagonal and transverse operators in the two-dimensional subspace. We discuss how crossing the transitions may be used in the manipulation of the quantum bit to simultaneously optimize the frequency of operation and the losses due to decoherence.