Quantum dimer model with $Z_2$ liquid ground-state: topological degeneracy, interpolation between cylinder and disk and toy model for a topological quantum-bit
Quantum Dimer Models (QDM) were introduced by Rokhsar and Kivelson [1] in the context of resonating valence-bond (RVB) theories for the high-temperature superconductors. These models provide simplified descriptions of the singlet bonds (dimer) dynamics in some quantum disordered spin-1/2 antiferromagnets. In particular, QDM can offer simple realizations[2,3] of Z2 liquid ground-state without any broken symmetry. Those liquids have the remarkable property that the ground-state degeneracy depends on the genus (cylinder, torus, etc) of the surface where the system is defined. In the example of Ref.[3] (QDM on a kagome lattice), the full spectrum of the model can be obtained exactly (excitations are non-interacting ‘visons’) and its topological degeneracy can be seen in a simple way. We will discuss two extensions of this 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 there exist a critical value above which 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 simple mapping to an Ising chain in transverse field. This model provides an example of solvable Hamiltonian for a topological quantum bit[4,5] (two-level system) and 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 to due to de-coherence.
[1] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988).
[2] R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B 65, 024504 (2002).
[3] G. Misguich, D. Serban, V. Pasquier, Phys. Rev. Lett. 89, 137202 (2002).
[4] A. Yu. Kitaev, Annals Phys. 303, 2 (2003) [quant-ph/9707021].
[5] L. B. Ioffe, M. V. Feigel’man, A. Ioselevich, D. Ivanov, M. Troyer and G. Blatter, Nature 415, 503 (2002).
SPhT, CEA/Saclay
