Gapless excitations in non-Abelian Kitaev spin liquids with line defects

We show that line defects in a non-Abelian Kitaev spin liquid harbor gapless one-dimensional Majorana modes if the interaction across the defect falls below a critical value. Treating the weak interaction at the line defect within a mean-field approximation, we determine the critical interaction strength as a function of the external magnetic field. In the gapless regime, we use the low-energy effective field theory to calculate the spin-lattice relaxation rate for a nuclear spin near the defect and find a cubic temperature dependence that agrees with experiments in the Kitaev material $\alpha {\text{-RuCl}}_3$.

Figure 1

Kitaev honeycomb model with a line defect. Black and white circles represent the sublattices. Nearest-neighbor $x$-, $y$-, and $z$-bonds are colored in red, green, and blue, respectively. (a) The coupling $\tilde{J}$ along the defect is represented by dashed lines. The vectors ${\mathbf{e}}_{x,y,z}$ indicate the bond directions in the plane. (b) In $\alpha \text{−}{\mathrm{RuCl}}_3$, the Cartesian axes with unit vectors $\widehat{\mathbf{x}}$, $\widehat{\mathbf{y}}$, and $\widehat{\mathbf{z}}$ are defined by the vertices of the ligand octahedra. The magnetic field forms an angle $\theta$ with $\widehat{\mathbf{z}}$.

Figure 2

Energy gap for the Majorana modes bound to the line defect as a function of the modified coupling $\tilde{J}$. Here we fix the magnetic field along the [111] direction with $\left|\mathbf{h}\right|=0.8J$. The dashed line indicates the critical coupling ${\tilde{J}}_c$. The inset shows the spectrum below and above ${\tilde{J}}_c$. The green lines refer to defect modes and the continuum to gapped bulk modes.

Figure 3

Critical value of the interaction at the line defect as a function of the magnetic field. The field is fixed along the [111] direction ($\theta ={tan}^{-1}\sqrt{2}$ and $\varphi =\pi /4$). Inset: Critical coupling as a function of $\theta$ for fixed $\left|\mathbf{h}\right|=0.5J$ and $\varphi =\pi /4$.