Coupled Majorana modes in a dual vortex of the Kitaev honeycomb model

TL;DR

This work analyzes Majorana-bound-state physics in the vortex-full Kitaev honeycomb model with a TRS-breaking term $\\kappa$, focusing on a dual vortex that traps two Majorana modes. It develops analytic continuum perturbation theory in the small-$J$ and small-$\\kappa$ limits, derives explicit MZM wavefunctions, and decomposes the energy splitting into contributions from string and bulk terms, with careful attention to oscillatory effects that couple valleys. The authors validate the continuum results with lattice-based numerical perturbation theory and full diagonalization, finding that in the $J\\ll\\kappa$ regime the splitting is primarily set by the string term and scales as $\\epsilon \\\sim 0.566\\,\\kappa$, while in the $\\kappa\\ll J$ regime the splitting scales as $\\epsilon \\\sim 0.393\\,J$ with the string contribution dominating. The analysis clarifies how dual-vortex bound states relate to topological superconductors with even Chern numbers and provides a foundation for exploring interactions among dual vortices and lattice geometries.

Abstract

The Kitaev model is exactly solvable in terms of Majorana fermions hopping on a honeycomb lattice and coupled to a static $\mathbb{Z}_2$ gauge field, giving the possibility of $π$-vortices in hexagonal plaquettes. In the vortex-full sector and in the presence of a time-reversal-breaking three-spin term of strength $κ$, the energy spectrum is gapped and the ground state possesses an even Chern number. An isolated vortex-free plaquette acts as a ``dual vortex'' and binds a fermionic mode at finite energy $ε$ in the bulk gap. This mode is equivalent to two coupled Majorana zero modes located on the same dual vortex. In a continuum approximation, we analytically compute the Majorana wavefunctions and their coupling $ε$ in the two limits of small or large $κ$. The analytical approach is confirmed by numerical perturbation theory directly on the lattice. The latter is in excellent agreement with the full numerics on a finite-size system. We contrast our results with states bound to an isolated vortex in a topological superconductor with even Chern number.

Figures (9)

• Figure 1: Half-splitting $\epsilon$ (full line) as a function of $\kappa$ (both in units of $J$). The two limiting behaviors at small $\kappa$ ($\epsilon\simeq 0.566\, \kappa$) and small $J$ ($\epsilon\simeq 0.393\, J$) are indicated as dashed lines, see also Fig. 6 in Fuchs20. The inset shows $\epsilon/\kappa$ versus $J/\kappa$.
• Figure 2: Vortex-free sector. (a) The Kitaev gauge (i.e. $+iJ$ hopping from black to white sites) is assumed everywhere on the nearest-neighbor links of the honeycomb lattice. (b) When $J=0$, the two triangular sublattices are decoupled. We show the unit cell for the triangular sublattice of black sites. The unit cell vectors are $\mathbf{n}_3$ and $\mathbf{n}_2$ ($\mathbf{n}_1$ is the remaining nearest-neighbor vectors on the triangular lattice). The signs indicate whether the flux is $\pm \pi/2$ in a triangle.
• Figure 3: Vortex-full sector. (a) The Kitaev gauge (i.e. $+iJ$ hopping from black to white sites) is assumed everywhere on the nearest-neighbor links of the honeycomb lattice, except on flipped links that are shown in blue (i.e. $-iJ$ hopping from black to white sites). The unit cell is shown as a pink parallelogram and contains four sites labeled $A$, $B$, $C$ and $D$. (b) At $J=0$, the two triangular sublattices ($AB$ and $CD$) are decoupled. We show the unit cell for the triangular sublattice of black sites ($AB$). The flipped links are indicated by blue tails. The unit cell vectors are $\mathbf{A}_1$ and $\mathbf{A}_2$, and $\mathbf{n}_1$, $\mathbf{n}_2$, $\mathbf{n}_3$ are the nearest-neighbor vectors on the triangular lattice. The "+" signs indicate that the flux in every triangle is $+\pi/2$.
• Figure 4: (Top) Energy bands $\pm E(\mathbf{k})$ (in units of $\kappa$), see Eq. \ref{['eq:disptriang']}. (Bottom) Isoenergy lines of $E(\mathbf{k})$. Also indicated are the Brillouin zone, the reciprocal lattice vectors $\mathbf{A}_1^*$ and $\mathbf{A}_2^*$, the high symmetry points $\Gamma$, $X$, $Y$ and $M$ and the position of the two Dirac points $\mathbf{k}_\pm = \pm \mathbf{A}_2^*/4$.
• Figure 5: Dual vortex. (a) Vortex-full sector with a single vortex-free plaquette at position $0$ (shown as a gray-shaded hexagon). Flipped $J$-links along BD for the vortex-full gauge are indicated in blue. $J$-links that need to be further flipped in order to create a single vortex-free plaquette are indicated by a semi-infinite string of red crosses. The unit cell is shown in dashed green. (b) When $J=0$, the triangular sublattices are decoupled. The black triangular sublattice hosts a single "$-$" triangle in a sea of "$+$" triangles. This is the "dual vortex" (shown as a gray-shaded triangle).