Probing Kitaev models on small lattices

We address the following important question: how to distinguish Kitaev models experimentally realized on small lattices from other nontopological interacting spin models. Based on symmetry arguments and exact diagonalization, we show that a particularly characteristic pattern of spin-spin correlations survives despite finite size, open boundary, and thermal effects. The pattern is robust against small residual perturbing interactions and can be utilized to distinguish the Kitaev interactions from other interactions such as antiferromagnetic Heisenberg interactions. The effect of external magnetic field is also considered and found to be not critical.

Figure 1

(Color online) Spin-spin correlation functions of the 16-site Kitaev model in a pure ground state. The coupling strengths are $J_x=0.3$, $J_y=0.4$, and $J_z=1.0$. Solid red (dashed blue) bond denotes negative (positive) correlation. Bond thickness is proportional to the magnitude of the correlation. Empty bonds denote zero correlations.

Figure 2

(Color online) Top left: honeycomb lattice of 16 sites and three types of bonds. Others: spin-spin correlations in the low-temperature thermal equilibrium state. All other components such as $⟨{\sigma }^x{\sigma }^y⟩$ vanish identically.

Figure 3

(Color online) Spin-spin correlations for the anisotropic Heisenberg model $H={\sum }_{⟨bw⟩}{\sum }_{\alpha }{J}_{\alpha }{\sigma }_b^{\alpha }{\sigma }_w^{\alpha }$ with parameters $J_x=0.3$, $J_y=0.4$, and $J_z=1.0$. This is dramatically different from the case of Fig. 2 of Kitaev model.

Figure 4

(Color online) Spin-spin correlation functions on a lattice of 16 sites. Top three panels are pure Kitaev model and the Bottom three panels are pure antiferromagnetic Heisenberg model. Middle six panels are Kitaev model with residue Heisenberg interactions with strength $J_h$. The parameters for Kitaev model are $J_x=0.3$, $J_y=0.4$, and $J_z=1.0$.

Figure 5

(Color online) Spin-spin correlations and spin moment in the low-temperature thermal equilibrium state when a uniform magnetic field $B_z=0.1$ along the $z$ direction is applied. In the right-bottom panel, red (blue) denotes negative (positive) moment. The size of dot denotes the magnitude of spin moment. $⟨{\sigma }_2^z⟩=0.78$.

Figure 6

(Color online) Magnetization ${\sum }_i⟨{\sigma }_i^z⟩/16$ in Kitaev model as a function of uniform external magnetic field $B_z$ along the $z$ direction.

Figure 7

(Color online) Spin-spin correlations in the gapless phase of Kitaev model on a lattice of 16 sites.

Figure 8

(Color online) Correlations in toric code model of Eq. (2). Top three panels are pure toric code model. The second and third rows are toric mode with uniform antiferromagnetic Heisenberg interactions. The last row is pure Heisenberg model on square lattice. The coupling strength of toric code is $J=1$.