Mosaic spin models with topological order

We study a class of two-dimensional spin models with the Kitaev-type couplings in mosaic structure lattices to implement topological orders. We show that they are exactly solvable by reducing them to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8 close packing is investigated in detail to display the quantum phases with Abelian and non-Abelian anyons. Its topological properties characterized by Chern numbers are revealed through the edge modes of its spectrum.

Figure 1

(Color online) (a) Basic block for mosaic spin models, which consists of three branches with $x$- (black solid link), $y$- (blue dotted link), and $z$- (red double link) type couplings. (b) 6-6-6 mosaic model, i.e., Kitaev’s honeycomb model. (c) 3-12-12 mosaic model. (d) 4-6-12 mosaic model. (e) 4-8-8 mosaic model, $e$ vortices lie on squares while $m$ vortices lie on octagons. (f) and (g) The possible nonconstant terms of the effective Hamiltonian are obtained by flipping four spin pairs around an octagon (f) and a quatrefoil (g). (h) Kitaev’s toric code model is the effective model of the 4-8-8 mosaic model when $\mid J_z\mid ⪢\mid J_x\mid ,\mid J_y\mid$.

Figure 2

(Color online) (a)–(c) 4-8-8 mosaic spin models in (a) vortex-free (VF) sector, (b) vortex-half occupied (VHO) sector, and (c) vortex-full occupied (VFO) sector. (d)–(f) The corresponding phase graphs of the above lattices with gapless phase $B$ and gapped phases $A$: (d) VF, (e) VHO, and (f) VFO.

Figure 3

(Color online) (a) Thin dashed arrows describe the effective second nearest-neighbor interactions between fermions and the corresponding gauge induced by a magnetic field. (b) Profile graph of an energy spectrum with $J_x=J_y=1$, $J_z=\sqrt{2}$ along the $q_1=\pi$ axis in the absence (solid lines) and presence (dashed lines) of a magnetic field. (c) Energy spectrum of the above system with finite size along the ${\mathbf{n}}_2$ direction in the magnetic field. Two chiral edge modes crossing at $E=0$ correspond to Chern number $\pm 1$.