Majorana Signatures in Planar Tunneling through a Kitaev Spin Liquid

Abstract

We propose a planar tunneling setup to probe vacancy-bound Majorana modes in the chiral Kitaev spin liquid. In this geometry, the inelastic tunneling conductance can be expressed directly in terms of real-space spin correlations, establishing a link between measurable spectra and the underlying fractionalized excitations. We show that spin vacancies host localized Majorana states that generate sharp near-zero-bias features, well separated from the continuum of bulk spin excitations. Compared to local STM measurements, the planar configuration naturally enhances the signal by coherently summing over multiple vacancies, reducing spatial resolution requirements. Our results demonstrate a realistic and scalable route to detect Majorana excitations in Kitaev materials.

Figures (5)

• Figure 1: (a) Illustration of spin fractionalization into Majorana fermions. (b) Kitaev honeycomb model with vacancies. Black and red bonds denote link variables $u_{ij}=+1$ and $u_{ij}=-1$, respectively. Gray-shaded plaquettes indicate $W_v=-1$, marking a $\pi$-flux bound to each vacancy. Three dangling Majorana modes $\tilde{b}^{\gamma}_j$ associated with each vacancy are also shown. (c) Schematic of the planar tunneling geometry: a substrate/Kitaev-QSL/substrate stack under bias $V$. The IETS signal $d^{2}I/dV^{2}$ directly probes the dynamical spin correlations $\sum_{jk} S^{\alpha\alpha}_{jk}(\omega)$.
• Figure 2: Different types of localized Majorana modes induced by a vacancy. The peripheral ($p$) mode has finite weight on the sites of the enlarged vacancy plaquette. The flux-induced ($f$) mode, present in the non-Abelian Kitaev spin liquid, is localized near the vacancy-bound $\mathbb{Z}_2$ flux. The $\tilde{b}$ modes originate from the fractionalization of dangling spin components adjacent to the vacancy. (b) Schematic energy spectrum of the quadratic Majorana problem, illustrating the hierarchy of localized modes involving itinerant $c$ fermions and dangling $\tilde{b}$ modes.
• Figure 3: (a) Tree diagram summarizing the decomposition of the full dynamical spin spectral weight into different components contributing to the planar tunneling response. (b) Illustration of bulk spin–spin correlations, separated into contributions from bonds "away" from vacancies (not sharing any bond with a vacancy; the closest such case is shown) and bonds "near" a vacancy (in the immediate vicinity, sharing a bond with the vacancy). The near-vacancy component reflects locally modified flux gaps and produces weak in-gap spectral weight, while the away-vacancy contribution reproduces the bulk two-flux continuum. In both cases, we include on-site and nearest-neighbor contributions to $S_{ij}^{\alpha\alpha}(\omega)$. (c) Schematic decomposition of spin correlations involving dangling spin components adjacent to vacancies into intra-vacancy and inter-vacancy components. Inter-vacancy contributions are further resolved into same-sublattice (AA/BB) and opposite-sublattice (AB/BA) channels, highlighting their distinct interference behavior in the planar tunneling response.
• Figure 4: (a)–(c) Disorder-averaged spin–spin correlation functions for different vacancy densities: (a) total correlation function $\bar{S}_{\rm total}(\omega)$. The inset shows a magnification of the near-zero-bias peak; its intensity is approximately $1\%$ of the main peak near the bulk gap $\Delta_{\rm bulk}\approx0.2J$ (dashed line). (b) Bulk contribution $\bar{S}_{\rm bulk}(\omega)$, with the corresponding flux excitation energies indicated. (c) Dangling contribution $\bar{S}_{\rm dangling}(\omega)$. The two dominant low-energy peaks arise from distinct combinations of vacancy-induced low-energy modes noteVacancies. (d)–(f) Comparison of different contributions at a fixed vacancy density $\eta=1\%$: (d) near-vacancy bulk, away-vacancy bulk, and dangling contributions. Sketches illustrate near-vacancy (top) and away-vacancy (bottom) bulk processes; the inset shows a zoom of the near-zero-bias region. (e) intra-vacancy and inter-vacancy dangling contributions. (f) Same-sublattice (AA/BB) and opposite-sublattice (AB+BA) inter-vacancy dangling contributions. The shaded region in all panels highlights the near-zero-bias window of interest. Parameters are $\kappa=0.02J$, $h=0.05J$, and Lorentzian broadening $\delta=0.01$. Each plot is individually normalized.
• Figure 5: (a) Sublattice-resolved integrated weight $\mathcal{W}_{ij}(r)$ for same-sublattice (AA/BB) versus distance $r = |\mathbf{r}_i - \mathbf{r}_j|$ (in unit cell units). The zigzag direction (filled squares) shows a period-three modulation atop an exponential envelope. In this case the fitting parameters in Eq. (\ref{['eq:fitAA_app']}) are $s = 1.0$, $A = 0.02$, $\xi = 4.92$ and $C = 0.56$. For the armchair case (filled circles) the fitting parameters are $s=1.0$, $A = 0.37$, $\xi = 4.26$ and $C = 0.11$. Zigzag (b) and Armchair (c) directions of vacancy pairs at $\bm{r}_i,\bm{r}_j$ are presented on the right panel.