Exact Results for Spin Dynamics and Fractionalization in the Kitaev Model

We present certain exact analytical results for dynamical spin correlation functions in the Kitaev Model. It is the first result of its kind in nontrivial quantum spin models. The result is also novel: in spite of the presence of gapless propagating Majorana fermion excitations, dynamical two spin correlation functions are identically zero beyond nearest neighbor separation. This shows existence of a gapless but short range spin liquid. An unusual, all energy scale fractionalization of a spin-flip quanta, into two infinitely massive $\pi$ fluxes and a dynamical Majorana fermion, is shown to occur. As the Kitaev Model exemplifies topological quantum computation, our result presents new insights into qubit dynamics and generation of topological excitations.

Figure 1

Elementary hexagon and “bond fermion” construction. A spin is replaced with 4 Majorana fermions ($c$, $c^x$, $c^y$, $c^z$). Bond fermion ${\chi }_{⟨23⟩}$ and spin operator are defined. $A$ and $B$ denote the sublattice index.

Figure 2

Contour plot of nonzero $⟨{\sigma }_i^z{\sigma }_j^z⟩$ in the parameter space. The distances from any point in the outer triangle to its three sides are in the ratio $J_x:J_y:J_z$. Numerical number attached with the contour is the value of $⟨{\sigma }_i^z{\sigma }_j^z⟩$. The middle triangle is the gapless phase.

Figure 3

Time evolution and fractionalization of a spin flip at $t=0$ (${\sigma }_i^z|{\mathcal{M}}_{\mathcal{G}};\mathcal{G}⟩$) at site $i$, into a $\pi$-flux pair and a propagating Majorana fermion.