Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model

We provide a comprehensive microscopic understanding of the nucleation of topological quantum liquids, a general mechanism where interactions between non-Abelian anyons cause a transition to another topological phase, which we study in the context of Kitaev's honeycomb lattice model. For non-Abelian vortex excitations arranged on superlattices, we observe the nucleation of several distinct Abelian topological phases whose character is found to depend on microscopic parameters, such as the superlattice spacing or the spin-exchange couplings. By reformulating the interacting vortex superlattice in terms of an effective model of Majorana fermion zero modes, we show that the nature of the collective many-anyon state can be fully traced back to the microscopic pairwise vortex interactions. Due to Ruderman-Kittel-Kasuya-Yosida-type sign oscillations in the interactions, we find that longer-range interactions beyond the nearest neighbor can influence the collective state and, thus, need to be included for a comprehensive picture. The omnipresence of such interactions implies that corresponding results should hold for vortices forming an Abrikosov lattice in a $p$-wave superconductor, quasiholes forming a Wigner crystal in non-Abelian quantum Hall states, or topological nanowires arranged in regular arrays.

Figure 1

Microscopics of the interaction between a pair of vortices in Kitaev's honeycomb lattice model: In agreement with the Ising anyon fusion rule $\sigma \times \sigma =1+\psi$, the non-Abelian vortices can combine into two possible collective states with energies $E_1$ and $E_{\psi }$. The splitting $\epsilon \left(d\right)=E_{\psi }-E_1$ between them decreases exponentially with vortex separation and shows characteristic oscillation due to interference effects. When $\epsilon \left(d\right)>0(<0)$, the microscopics of the system energetically favor the state where the vortices combine into the trivial (fermionic) excitation. The plot is for $J_x=J_y=J_z=1$ and $K=1/20$, which corresponds to a coherence length of $\xi \approx 2.5$. The continuous curve has been obtained by suitably tuning the spin-exchange couplings to simulate the continuous transport of the vortices.[5]

Figure 2

Phase diagram of Kitaev's honeycomb model (a) in the absence of a vortex superlattice and (b) in the presence of $D=1,2,3$ superlattices. We have parametrized the phase diagram by ${\sum }_{\gamma }{J}_{\gamma }=1$ and have set $K=0.03$. The plots show the lowest energy gap ${\Delta }_v$ and Chern numbers $\nu$ along a cut ($1/3⩽J_z⩽1/2$ when $J_x=J_y$) indicated by the dashed red line in (a). In the absence of vortices, the continuous phase transition between the Abelian TC phase ($\nu =0$) and the non-Abelian Ising phase ($\nu =-1$) occurs at $J_z=1/2$. In the presence of a vortex superlattice, we again find a continuous phase transition but which occurs now always at some $J_z^c<1/2$ separating the TC phase ($J_z>J_z^c$) from the nucleated Abelian phase ($J_z<J_z^c$). The exact value of $J_z^c\left(D\right)$ can also be traced back to the vortex-vortex interactions with there being no simple direct relation between the spacing $D$ and the critical value.[19]

Figure 3

The characteristic band structure of the nucleated topological phases with ${\Psi }_v^{\pm }\left({\Psi }_f^{\pm }\right)$ denoting the low-energy (high-energy) vortex (fermion) bands. The actual plot is for the $D=1$ vortex superlattice on a cylinder (open boundary conditions in the $x$ direction). Consistent with the nucleated Chern number $\nu =-2$ phase, the spectral flow shows two edge modes per edge (red and green lines denoting different edges) emanating from the four gapped Dirac cones.

Figure 4

(a) A vortex (blue squares) superlattice of spacing $D$ (here, $D=2$) and (b) the corresponding effective model in terms of Majorana zero modes tunneling on the triangular superlattice. The effective hopping amplitudes $t_1$ (solid lines) and $t_{\sqrt{3}}$ (dashed lines) are identified with the energy splittings $\epsilon \left(D\right)$ and $\epsilon \left(D\sqrt{3}\right)$ at the corresponding vortex separations, respectively. The fluxes in the effective model are chosen such that plaquettes indicated as $T_1,T_{\sqrt{3}}$, and $T_{1,\sqrt{3}}$ are assigned fluxes ${\Phi }_{T_1}=\frac{\pi }{2},{\Phi }_{T_{\sqrt{3}}}=-\frac{\pi }{2}$, and ${\Phi }_{T_{1,\sqrt{3}}}=\frac{\pi }{2}$, respectively, that are consistent with the enclosed area.

Figure 5

Left: The phase diagram of the effective Majorana fermion zero-mode model (here, $t_1=\cos \theta$ and $t_{\sqrt{3}}=\sin \theta$) as characterized by the Chern numbers ${\nu }_M$ for the flux assignment $({\Phi }_{T_1},{\Phi }_{T_{\sqrt{3}}},{\Phi }_{1,T_{\sqrt{3}}})=(\frac{\pi }{2},-\frac{\pi }{2},\frac{\pi }{2})$. The squares and the circles denote phase transitions at $|t_1|=|t_{\sqrt{3}}|$ and $t_1=-2t_{\sqrt{3}}$, respectively. Right: The lowest gap in the energy spectrum ${\Delta }_M$ along the path shown in the left panel for $t_{\sqrt{3}}=-1$.

Figure 6

The energy gaps ${\Delta }_v\left(K\right)$ of the nucleated phases in the honeycomb model (black circles) and the gaps ${\Delta }_M[\epsilon (D,K),\epsilon (D\sqrt{3},K)]$ predicted by the effective model (red squares) for $D=1,2$, and 3 vortex superlattices. $\nu$ is the Chern number observed in the honeycomb model, and ${\widehat{\nu }}_M={\nu }_M-1$ is the one predicted by the Majorana model from the interactions. Note the quantitative similarity of the sequence of phases for the $D=3$ superlattice to that of Fig. 5 (the finite gap around $K=0.0585$ is a finite-size effect). All the data are for $J_x=J_y=J_z=1/3$, and the values of the pairwise energy splittings $\epsilon \left(d\right)$ have been obtained by restricting the relevant two vortex sectors in a finite system of $40\times 40$ plaquettes (3200 sites) on a torus.

Figure 7

Left: The unit cell for the effective Majorana model with both nearest- ($t_1$) and next-nearest- ($t_{\sqrt{3}}$) neighbor tunneling. The first couple all sites of the lattice, whereas, the latter couple only sites belonging to one of three distinct sublattices denoted by circles, squares, and diamonds. When $t_{\sqrt{3}}=0$, the unit cell consists of a black and a white site irrespective of their sublattice labels, whereas, for $t_1=0$, the unit cell consists of six sites—a black and a white site from each sublattice. Right: For every site $(i,j)$ in the unit cell, one has six independent gauge choices—three on the $t_1$ lattice and three on the respective $t_{\sqrt{3}}$ sublattice. When calculating the flux per plaquette, one has to take into account the overall $i$ factor in the tunneling. We assume a convention that the phase of the hopping between any two sites is given by $i(-i)$ when it is along (against) the shown orientations.