Majorana-assisted nonlocal spin correlation in quasi-one-dimensional Kitaev spin liquids

Abstract

We propose Majorana-assisted nonlocal spin correlation as a manifestation of Majorana nonlocality in quasi-one-dimensional (1D) Kitaev spin liquids. Focusing on the flux-free sector of the Kitaev honeycomb model in a quasi-1D geometry, we uncover its topological nature and show that it hosts Majorana zero modes localized at both ends, which are stabilized by finite-size-induced topology. We further show that the nonlocal Majorana fermion parity operator, $P_{\text{MF}}=iγ_{\text{L}}γ_{\text{R}}$, is mapped to a nonlocal spin-string operator, producing an end-to-end spin correlation proportional to the product of $P_{\text{MF}}$ and total fermion parity operators when local perturbations remove redundant ground-state degeneracies while preserving the Majorana and total fermion parities in the flux-free sector. Numerical calculations confirm a finite nonlocal spin correlation generated by these Majorana zero modes without any local magnetization. Our results establish a concrete signature of intrinsic Majorana nonlocality in quantum spin liquids.

Figures (9)

• Figure 1: Sketch of Majorana-assisted nonlocal correlation. (a) nonlocal electric charge transfer in a one-dimensional (1D) topological superconductor and (b) nonlocal spin correlation in a quasi-1D Kitaev honeycomb model associated with Majorana zero modes, which are represented by $\gamma_{\text{L}}$ and $\gamma_{\text{R}}$.
• Figure 2: (a) Two-dimensional honeycomb lattice structure of the Kitaev honeycomb model. (b) Schematic illustration of $\mathbb{Z}_2$ flux of hexagonal plaquette $W_p$.
• Figure 3: Phase diagram with $J_x+J_y+J_z=\text{constant}$ in (a) two-dimensional Kitaev honeycomb model Kitaev2 and (b) quasi-1D Kitaev honeycomb model with the open boundary condition in the $\bm{a}_1$ direction Tadokoro011160.
• Figure 4: Figures show the probability density of Majorana zero modes (left) and the corresponding energy spectrum (right) for the systems with the full open boundary condition, where the parameters are set to $J_{y}=1.7$, $J_{x}=1.29$, $J_{z}=0.01$ for (a) and (b) and $J_{x}=1.7$, $J_{y}=1.29$, $J_{z}=0.01$ for (c) and (d). The lattice sizes are $N_{\bm{a}_2} =10$, $N_{\bm{a}_1} =10$ for (a) and (c) and $N_{\bm{a}_2} =10$, $N_{\bm{a}_1} =5$ for (b) and (d). In the probability-density plots, the red and blue filled circles denote the $A$ and $B$ sublattices, respectively, and the circle size is proportional to $\rho_{\ell}$ defined in Eq. (\ref{['eq:edge_density']}). In the energy spectra, the purple and green dots correspond to excited states in the bulk and boundary zero modes, respectively. There are $N_{\rm MF}=20$ green dots in (a) and (c), and 10 green dots in (b) and (d).
• Figure 5: Schematic illustration of the honeycomb lattice under the full open boundary condition satisfying $N_{\bm{a}_2} > N_{\bm{a}_1}=3$. In the topological regime, Majorana zero modes (red) appear at both ends along the $\bm{a}_1$ direction. (a) When the edge is constructed such that the unit cells are preserved, six Majorana zero modes emerge. (b) When the edge termination is modified by removing a single atom at two corners, the number of Majorana zero modes is reduced by two, leaving four Majorana zero modes.