Electrical Access to Ising Anyons in Kitaev Spin Liquids

We show that spin-spin correlations in a non-Abelian Kitaev spin liquid are associated with a characteristic inhomogeneous charge density distribution in the vicinity of ${\mathbb{Z}}_2$ vortices. This density profile and the corresponding local electric fields are observable, e.g., by means of surface probe techniques. Conversely, by applying bias voltages to several probe tips, one can stabilize Ising anyons (${\mathbb{Z}}_2$ vortices harboring a Majorana zero mode) at designated positions, where we predict a clear Majorana signature in energy absorption spectroscopy.

Figure 1

(a) Two-vortex state in the Kitaev model, where filled hexagons represent ${\mathbb{Z}}_2$ vortices with $W_p=-1$. Compared to the uniform g.s. with all $W_p=+1$, the gauge fields $u_{⟨jk{⟩}_{\alpha }}$ are reversed for bonds crossed by a string (solid black line) connecting both vortices. Different bond types, $\alpha =(x,y,z)$, are depicted in red, green, and blue, respectively. Spin-spin correlations, see Eq. (4), near a vortex become spatially isotropic in the thermodynamic limit and with all other vortices far away, i.e., $C_1^{\prime }=C_1$. (b) A local charge imbalance ${\rho }_j$ arises from virtual orbital-dependent hopping around triangles with one bond of each type $\alpha$, with amplitude $t$ for nearest-neighbor bonds (solid lines) and $t^{\prime }$ for next-nearest-neighbor bonds (dashed lines). For the site with index $j$, the yellow triangle gives a contribution ${\rho }_j\propto 2C_1$.

Figure 2

Charge density profile ${\rho }_j$ near a vortex (shaded hexagon) and the corresponding electrostatic potential $V_{\mathrm{vort}}(z)$ for $\kappa =0.2K$ and parameters (7). (a) Color-scale plot of ${\rho }_j/{\rho }_{\max }$, see Eq. (8). (b) $|{\rho }_j|/{\rho }_{\max }$ vs in-plane distance $r$ from the vortex center on a semilogarithmic scale. (c) $V_{\mathrm{vort}}(z)$ vs perpendicular distance $z$, with $V_{\mathrm{vort}}(z=0)\simeq -0.118\mathrm{meV}$ at the vortex center.

Figure 3

(a) Setup with four probe tips held at the same voltage $V_0$. By applying a weak ac voltage $V_{\mathrm{ac}}(t)$ to one tip, the energy absorption probability (11) can be measured. This quantity provides information about the low-energy QSL excitation spectrum. (b) Energy gap of the lowest four-vortex state $\mathrm{\Delta }{E}_{4v}$ (red) and two-fermion state $\mathrm{\Delta }{E}_{2f}$ (blue) vs tip voltage parameter ${\xi }_0=e{V}_0/[(1-3\eta )U]$. We use the parameters in Eq. (7), $\kappa =0.1\mathrm{K}$, $V_{\mathrm{ac}}=0$, $L=30$, with all tips far away from each other. The shaded region shows the two-fermion continuum without vortices.

Figure 4

Line shape $A(\omega )$ of the absorption spectrum (11) for the setup in Fig. 3. We use the parameters (7), ${\xi }_0=0.2$, $\kappa =0.2K$, $L=48$, $d=2$ (see inset), and different numbers $N_V$ of vortices below the tips: $N_V=0$ (green), $N_V=2$ (blue, including a vortex at the first tip), and $N_V=4$ (red curve). The green arrow indicates the $N_V=0$ two-fermion continuum threshold $\omega =\mathrm{\Delta }{E}_{2f}$. The red arrow marks the MZM peak at $\omega ={ϵ}_M(d)$, possible only for $N_V=4$. For the continuum part, $\delta$ functions are broadened by the maximal level spacing, $(\delta \omega {)}_{N_V=0}\approx 8.7\times {10}^{-3}\mathrm{K}$ and $(\delta \omega {)}_{N_V=2,4}\approx 5.5\times {10}^{-3}\mathrm{K}$. For the MZM peak, $\delta \omega =5\times {10}^{-4}\mathrm{K}$.