Parity of Chern numbers in the Kitaev honeycomb model and the sixteenfold way

In two dimensions, topological phases of free Majorana fermions coupled to a ${\mathbb{Z}}_2$ gauge field are known to be classified according to the Chern number $\nu \in \mathbb{Z}$. Its value mod 16 specifies the type of anyonic excitations. In this paper, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are $\nu =\pm 7$. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.

Figure 1

Pictorial representation of the Hamiltonian $H$ for the standard gauge where $u_{jk}=+1$ if $j$ is a white site and $k$ is a black site. This gauge leads to the vortex-free configuration ($w_p=+1$ for all $p$). Black links correspond to nearest-neighbor hoppings $J$, and dashed red links correspond to next-nearest-neighbor hoppings $\kappa$.

Figure 2

Triangular vortex configuration with vortex density $\rho =1/3$. Gray (white) plaquettes indicate plaquettes with (without) a vortex. In this case, $P_w=-1$ so that the Hamiltonian unit cell (thin rhombus) is twice larger that the geometric unit cell spanned by the vectors ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$. The blue cross indicates the inversion center. Black thin (green thick) links correspond to $u_{jk}=+1(u_{jk}=-1)$ if $j$ is a white site and $k$ is a black site. The gauge transformation $\mathcal{G}$ is shown as $\pm$ (red symbols) for sites within the Hamiltonian unit cell. The dashed line indicates the string of flipped links between the two gray plaquettes inside the unit cell (see Appendix pp1-s1 for a systematic construction).

Figure 3

Dual vortex configuration $\rho =2/3$ (see Fig. 2 for notations). In this case, $P_w=+1$, so that the Hamiltonian unit cell coincides with the geometric unit cell (see also Appendix pp1-s2).

Figure 4

Dual vortex configuration $\rho =3/4$ (see Fig. 2 for notations). In this case, $P_w=-1$ so that the Hamiltonian unit cell is twice larger that the geometric unit cell. The dashed line indicates the string of flipped links connecting the two white plaquettes inside the unit cell (see also Appendix pp1).

Figure 5

Schematic energy spectrum of dilute configurations. When gray (white) plaquettes are strongly dilute in a vortex-free (vortex-full) background, the bandwidth of the low-energy bands $W$ vanishes, $\mathrm{\Delta }$ is given by the gap of the vortex-free (vortex-full) sector, and $\delta$ vanishes (remains finite).

Figure 6

Energy gaps $\mathrm{\Delta }$ (black) of the vortex-full configuration (see Appendix pp3 for an analytical expression) and $\delta$ (green) of isolated white plaquettes in a vortex-full background as a function of $\kappa$. Both gaps vanish for $\kappa =0,1/2$ whereas $\mathrm{\Delta }\sim 2\sqrt{3}\kappa$ and $\delta =0.7865\left(1\right)$ in the large-$\kappa$ limit.

Figure 7

Triangular vortex configurations ($P_w=-1$) with $\rho =1/7$ (top) and $\rho =1/12$ (bottom). The dashed line indicates the string of flipped links (green) between two vortices. The inversion center lies in the middle of a link for $\rho =1/7$ and in the center of a white plaquette for $\rho =1/12$. In both cases, the gauge transform $\mathcal{G}$ ($\pm$ red symbols) is antisymmetric ($g_{-j}=-g_j$) so that ${\mathcal{P}}^2=-\mathbb{1}$.

Figure 8

Dual vortex configurations with $\rho =6/7$ (top) and $\rho =11/12$ (bottom). The dashed line indicates the string of flipped links (green) between two white plaquettes. For $\rho =6/7(P_w=+1)$, the gauge transform $\mathcal{G}$ ($\pm$ red symbols) is antisymmetric ($g_{-j}=-g_j$) so that ${\mathcal{P}}^2=-\mathbb{1}$. By contrast, for $\rho =11/12(P_w=-1)$, the gauge transform $\mathcal{G}$ is symmetric ($g_{-j}=g_j$) so that ${\mathcal{P}}^2=+\mathbb{1}$.