Two-dimensional topological order of kinetically constrained quantum particles

We investigate how imposing kinetic restrictions on quantum particles that would otherwise hop freely on a two-dimensional lattice can lead to topologically ordered states. The kinetically constrained models introduced here are derived as a generalization of strongly interacting particle systems in which hoppings are given by flux-lattice Hamiltonians and may be relevant to optically driven cold-atom systems. After introducing a broad class of models, we focus on particular realizations and show numerically that they exhibit topological order, as witnessed by topological ground-state degeneracies and the quantization of corresponding invariants. These results demonstrate that the correlations responsible for fractional quantum Hall states in lattices can arise in models involving terms other than density-density interactions.

Figure 1

Illustration of vacancy-assisted hopping models on (a) the triangular lattice, with nearest and third-nearest neighbor hoppings as defined in Refs. [17, 18], and (b) the kagome lattice, with nearest and second-nearest neighbor hoppings as defined in Ref. [39]. Solid and dashed circles denote particles and vacancies, respectively. The hoppings (arrows) are only allowed if a vacancy is located at the position shown. Note that some of the hoppings are imaginary; see Eqs. (5) and (6).

Figure 2

Flow of eigenvalues under insertion of magnetic flux corresponding to a phase ${\phi }_2^{}$ through one of the handles of the toroidal system for (a),(b) the triangular and (c),(d) the kagome lattice. (a), (c) correspond to QHH models (i.e., $\widehat{F}={\widehat{F}}^{\mathrm{HC}}$) and (b), (d) to VAH models (i.e., $\widehat{F}={\widehat{F}}^{\mathrm{KC}}$).

Figure 3

(a) Berry curvature of the triangular-lattice model on a 42-site cluster with $N=7$ as a function of the magnetic fluxes ${\phi }_1$ and ${\phi }_2$ threading the two handles of the toroidal system; (b) error in the quantization of ${\sigma }_H^{}$ as a function of the number of subintervals in the partition of the Brillouin zone of fluxes. The unidirectionality of the stripes along ${\phi }_1$ is merely an artifact of the choice of the alignment of the two-site unit cell of the model [see Fig. 1].

Figure 4

Gap to excited states at zero flux as a function of inverse number of lattice sites $N_s^{}$ for QHH (solid triangles) and VAH (empty triangles) on (a) the triangular-lattice model and (b) the kagome-lattice model. The value of the gap is the energy difference between the highest in energy quasidegenerate FCI level and the lowest level that does not belong to the FCI state manifold.