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

TL;DR

This paper analyzes a quantum dimer model on the kagome lattice that exhibits a Z_2 liquid phase with topological degeneracy, exploring how perturbations affect its ground state and potential use as a topological quantum bit.

Contribution

It introduces and solves two perturbations in the kagome lattice quantum dimer model, demonstrating their effects on topological degeneracy and providing a solvable model for topological quantum computation.

Findings

Perturbations lift ground-state degeneracy above a critical value.

Exact gap values obtained via mapping to an Ising chain.

Model serves as a toy for topological quantum bits.

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.