Quantum dimer model with ${\mathbb{Z}}_2$ liquid 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.

Figure 1

A dimer covering on the kagome lattice (fat bonds). The corresponding representation with arrows living on the bonds of the hexagonal lattice (dashed lines) is displayed.

Figure 2

Kagome lattice on a cylinder with a cut $\Delta$ going from one edge of the cylinder to the other. The dual cut ${\Delta }^{*}$ passes through the centers of the triangles and winds around the cylinder.

Figure 3

Dimer covering of the kagome lattice in the vicinity of the cut $\Delta$ (dashed line). The arrows next to $\Delta$ [appearing in Eq. (2)] are shown. The signs in the hexagons Int., $h_1,\dots ,h_L$, Ext. indicate the values of the corresponding pseudospins ${\sigma }^z$ (with the assumption that the reference configuration has no dimer crossing $\Delta$).

Figure 4

Cylinder (full lines) and loops (dashed lines) along which a dimer configuration $c$ differs from the reference. The signs indicate the value of ${\sigma }^z$ in each domain.

Figure 5

Top: Kagome lattice in the vicinity of the cut ${\Delta }^{*}$ (dashed line). Bottom: Mixed dimer-monomer configuration and the arrow representation.

Figure 6

Effect of a change of boundary conditions on the ground-state energy of an ICTF with $N=10$ spins as a function of the (ferromagnetic) exchange $\lambda$. Circles: closed chain with periodic and antiperiodic boundary conditions. Thin line: ground-state energy difference between open chain with ferromagnetic $(\uparrow \cdots \uparrow )$ and antiferromagnetic $(\downarrow \cdots \uparrow )$ boundary conditions. Thick line: $N=\infty$ case (same result for the open and closed chains).

Figure 7

Contour in the complex plane used to define the integral $I_C$. The crosses on the real axis indicate the poles of $1∕\sin \left(Lz\right)$ and the fat segments on the imaginary axis indicate the branch cuts of $f\left(z\right)$. The region $B$ is sent to $\mathrm{Im}\left[z\right]=\pm \infty$.