Quantum dimer model with Z_2 liquid ground-state: interpolation between cylinder and disk topologies and toy model for a topological quantum-bit

Abstract

We consider a quantum dimer model (QDM) on the kagome lattice which was introduced recently [Phys. Rev. Lett. 89, 137202 (2002)]. It realizes a Z_2 liquid phase and its spectrum was obtained exactly. It displays a topological degeneracy when the lattice has a non-trivial geometry (cylinder, torus, etc). We discuss and solve two extensions of the model where perturbations along lines are introduced: first the introduction of a potential energy term repelling (or attracting) the dimers along a line is added, 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 transverse field. This model provides an example of solvable Hamiltonian for a topological quantum bit where the two perturbations act as a diagonal and a transverse operator in the two-dimensional subspace. We discuss how crossing the transitions may be used in the manipulation of the quantum bit to optimize simultaneously the frequency of operation and the losses due to decoherence.

11 pages, 7 (.eps) figures. Improved discussion of the destruction of the topological degeneracy and other minor corrections. Version to appear in Phys. Rev. B

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• Ordering in a frustrated pyrochlore antiferromagnet proximate to a spin liquid
• Systematic Derivation of Order Parameters through Reduced Density Matrices
• Zero-temperature Kosterlitz-Thouless transition in a two-dimensional quantum system
• Spin-1/2 Heisenberg antiferromanget on kagome: a $Z_2$ spin liquid with fermionic spinons
• Topological qubits from valence bond solids
• Emergent topological excitations in a two-dimensional quantum spin system
• Loop condensation in the triangular lattice quantum dimer model
• N=1 SQCD and the Transverse Field Ising Model