J. Vala, G. Kells, A.T. Bolukbasi, N. Moran
Department of Mathematical Physics
National University of Ireland, Maynooth, Ireland
We investigate low energy spectral properties of quantum lattice models which
are believed to form topologically ordered states known also as topological
phases. Our particular focus is on the Kitaev honeycomb spin-1/2 lattice model
[1]. In the absence of external magnetic field, the model is exactly solvable on a
plane and its phase diagram exhibits a gapless phase and an abelian topological
phase whose effective description is given by the Z2 x Z2 topological field
theory. As known from the perturbation theory, a weak magnetic field has no
dramatic effect on the abelian topological phase but turns the gapless phase into
a non-abelian topological phase whose effective description is given by the
SU(2)2 Chern-Simons theory. The quasiparticle excitations of this phase are
nonabelian anyons satisfying the Ising fusion rules.
We particularly study the Kitaev honeycomb lattice on torus [2]. We describe
symmetries of this model and review the perturbative mapping of its abelian
topological phase onto the Z2 x Z2 square lattice model known as the toric
code. Within the same framework, we provide the classification of finite size
effects on the model low-energy spectral properties [3]. In this context, special
attention is given to the thin-torus limit. We then investigate properties of the
model’s vortex excitations. We complete this part with discussion of topological
degeneracy of the model on torus [2].
We then proceed to numerical investigation of the non-abelian topological phase
in the perturbative limit of weak magnetic field [4] and beyond. The weak field is
modeled by an effective three body interaction term which does not commute
with the bare Hamiltonian but commutes with the vortex operators. In this
regime, we observe that the magnetic field is able to induce level crossing of
states belonging to the same vortex sector. We also investigate the model in
strong field regime modeled by the full Zeeman term which allows for dynamics
of vortices.
We conclude with discussion of the topological phase transitions in the model
and brief review of other lattice models whose low energy spectra provide
realization of topological field theories.
[1] A. Kitaev, Ann. Phys. 321, 2 (2006).
[2] G. Kells et al., Topological Degeneracy and Vortex Manipulation in the Kitaev
Honeycomb Model, submitted (2008) http://arXiv.org/abs/0804.2753 (2008).
[3] G. Kells et al., Finite-size Effects in Kitaev Honeycomb Lattice Model, in
preparation.
[4] V. Lahtinen et al, Spectrum of the Non-Abelian Phase in Kitaev’s Honeycomb
Lattice Model, Ann. Phys. (2008), in press; http://arxiv.org/abs/0712.1164
(2007).