Generation of chirality and orbital magnetization by Stone-Wales-type lattice defects in the Kitaev spin liquid

TL;DR

This work identifies Stone-Wales-type local defects in the gapless Kitaev honeycomb spin liquid as a route to locally break time-reversal symmetry and generate chirality, with Majorana fermions mediating a long-range ferromagnetic coupling between defect chiralities. A T-matrix analysis and flux energetics show Lieb-flux configurations as ground states, while real-space probes (local Chern marker and scalar spin chirality) reveal monopole- and dipole-like chirality patterns around defects. Periodic defect arrays yield a Majorana gap and a quantized Chern number $C=\pm1$ for certain flux sectors, and the long-range interactions drive a finite-temperature transition to a non-Abelian chiral spin liquid with $T_c$ scaling as $T_c\propto n_d$ and depending on the exponent $\gamma$ of the interaction; as $\gamma\to2$, $T_c$ diverges. Defect perturbations can tune $\gamma$ and thus enhance or suppress $T_c$, offering a platform to study chiral spin liquids in 2D Dirac systems with fluctuating Ising impurities and to identify spin liquids with lattice defects.

Abstract

In this work we extend our study of the effect of certain crystallographic defects on the spin-1/2 Kitaev honeycomb spin liquid (arXiv:2511.19409), focusing on its gapless phase and contrasting with the gapped phase. We identify a Stone-Wales (SW) local defect consisting of a 90$^\circ$ bond rotation that preserves Kitaev bond labels for edge-sharing octahedra and thereby enables exact solvability. These SW-type defects involve odd-sided plaquettes with $\pm π/2$ fluxes, but can be created locally. An isolated defect hosts a time-reversal pair of ground-state flux configurations with large net chirality. Certain excitations are also chiral. The chirality manifests in Majorana local Chern marker and in scalar spin chirality, producing electronic orbital magnetization. T-matrix analysis and numerics at finite defect density $n_d$ show that defect chiralities generate a topological gap of $11 n_d$ protecting a Chern number $C=\pm 1$. Emergent ferromagnetic long range Ising interactions $r^{-γ}$ with $2<γ< 3$ between defect chiralities lead to a finite temperature $T_c$ phase transition into the chiral spin liquid. The $T_c$ is proportional to $n_d$ and diverges when $γ\rightarrow 2$. We also consider additional solvable impurity potentials and find that $γ$ can be reduced to below $2.3$ and correspondingly enhance $T_c$. Our results offer applications to 2D Dirac cone systems with a finite density of fluctuating Ising magnetic impurities and to identifying spin liquids with lattice defects.

Figures (14)

• Figure 1: Stone-Wales (SW) type defects in Kitaev honeycomb lattices from edge-sharing octahedra. Top: a SW defect is created by a $\pi/2$ bond rotation on the honeycomb lattice. Bond color corresponds to Kitaev interaction axis $x,y,z$. Bottom: in-principle geometric realizability with edge sharing octahedra. Each magnetic site (black disk) is coordinated by six ligands forming the vertices of an octahedron. The color of the edge shared by two octahedra defines the Kitaev interaction axis on the corresponding bond. When two octahedra are rotated by $\pi/2$ around the (0,0,1) axis normal to the red edges, their shared edge remains red and the Kitaev bond axis is exactly preserved. The rotation distorts octahedra and produces local out-of-plane buckling, here drawn without any relaxation and depicted via two disconnected octahedra; real material-dependent atomic relaxation would elastically smooth out the distortions over a few unit cells and generate a material dependent local perturbation $\delta H_r$.
• Figure 2: Flux gap for SW defects at finite density $n_d$. We compute the total energy of a defect superlattice at density $n_d$ (number of defects per sites), for the cases where the defects carry Lieb-flux, PT-flux, or Uniform-flux. The Lieb-flux pair of states remain the ground states at all densities, as shown by the flux excitation gaps of the other two states. Flux gap is computed as energy per defect; note the log-linear scale.
• Figure 3: Creating SW defects by Majorana fermion impurity potentials. Here we consider a SW defect represented by an impurity potential that arises from a $+\pi/2$ rotation of the central bond. The resulting parameters are shown for each of the three flux cases. Parameters $t_1$, $t_1'$ and $t_2$ are taken to be positive and associated with oriented bonds as shown. Generic values of $t_1\neq 1$ spoil the solvable three-coloring, which is restored only in the $t_1=1$ limit via removal of the corresponding nearest neighbor bonds. The Uniform-flux pattern is obtained from (c) by $t_1'=1$. These additive impurity potentials generate the same fluxes as shown in Figs. \ref{['fig_BiancoResta_SW']} and \ref{['fig_uniform']}.
• Figure 4: Majorana gap and Chern number for different fluxes in array of SW defects. The defect density $n_d$ is obtained by placing one defect per unit cell of linear size $l_u=n_d^{-1/2}$. We set $t_1=t_1'=t_2=1$ corresponding to the unperturbed defects. The Majorana gap is shown in the main panel. The Lieb states (circles) always show a finite gap scaling linearly with $n_d$ (dashed line). The PT-fluxes (triangles) remain gapless for all $n_d$. The gap for the Uniform-flux defects (squares) behaves in a complicated non-monotonic manner and can become very small for some densities. The inset shows the Chern number $C$ for Lieb and Uniform-fluxes (circles and solid squares respectively). Even for small finite gaps the integrated Berry curvature converges to the quantized Chern number with increasing integration mesh ($36\times 36$, $72\times72$ in red, blue respectively). The sign of the Chern number is determined by the imaginary flux signs: here Lieb-flux defects are taken with $\mu^z=1$ (5577: $++--$ in units of $\pi/2$) and show $C=1$. Uniform-flux defects are taken with $\mu^z_U=-1$ (fluxes $++++$ in units of $\pi/2$) and show $C$ taking any of the three values $-1,0,1$.
• Figure 5: Local marker contribution of defects to chirality: Local Chern marker pattern generated for (a) Lieb-flux, and (b) PT-flux with sign of fluxes corresponding to $\mu^z=1$ and $\mu^z_\text{U}=1$, respectively. Local marker on each site is shown as disk with corresponding area. Max/min values are shown in brackets on color scale. The marker is largest near the defect core and decays away from it. For (a) the contribution is always positive in the bulk, while for (b) it is symmetrically distributed about zero. These local marker patterns can be viewed as arising from monopolar (a) and dipolar (b) distributions of chirality over the constituent disclinations.