Raman spectroscopy of anyons in generic Kitaev spin liquids

TL;DR

The paper develops a polarization-resolved Raman framework to detect mobile Ising anyons (visons) in a generic Kitaev spin liquid perturbed by symmetry-allowed interactions. By constructing a two-anyon effective model with braiding statistics, it derives the continuum Raman response and reveals power-law onsets near the two-anyon threshold set by the topological spin θ_σ = π/8, with subleading corrections from short-range interactions. It further shows how multiple band minima (valleys) and bound-state formation modify the onset exponents and produce sharp polarization-dependent peaks, including potential suppression in antiferromagnetic Kitaev settings. The results provide concrete, polarization-sensitive fingerprints for vison dynamics and statistics, offering actionable signatures for Raman experiments in candidate Kitaev materials and a template for other anyonic quasiparticles in quantum liquids.

Abstract

Optical probes have emerged as versatile tools for detecting exotic fractionalized phases in quantum materials. We calculate the low-energy Raman response arising from mobile, interacting Ising anyons (or visons) in the chiral Kitaev spin liquid perturbed by symmetry allowed interactions - a phase relevant to \rucl. under a magnetic field. At zero temperature, the two-anyon continuum response shows a leading power-law scaling of the intensity near the onset of the signal: $I(ω) \sim (ω-E^0_{2σ})^{\frac{1}{8}}$ for linear and parallel-circular polarization channels, where $E^0_{2σ}$ is the two-particle gap. Strong corrections due to short-range interactions arise at order $\frac{1}{4}$. For cross-circularly polarized channels, the scaling is given by $I(ω) \sim (ω-E^0_{2σ})^{|l\pm 1/8|}$, where the value of $l=0,1,2$ is determined by the number of minima in the single anyon dispersion. The exponents are directly related to the topological spin of Ising anyons $θ_σ=\fracπ{8}$, describing their exchange statistics. Our theory generalizes to spectral probes of anyonic quasiparticles with multiple band minima in other quantum liquids. Interaction between anyons may also induce bound-states, resulting in sharp peaks that show strong polarization dependence.

Figures (8)

• Figure 1: Excitations in a generic Kitaev spin liquid in the chiral phase. In the pure Kitaev model, elementary excitations are gapless Majorana fermions (dashed blue lines) and $\mathbb{Z}_2$ fluxes (visons) that are gapped and immobile (dashed orange lines). In a generic Kitaev liquid in a weak magnetic field, a topological gap $\Delta_m$ appears in the Majorana spectrum while the visons transform into Ising anyons that become dynamical degrees of freedom (solid lines).
• Figure 2: Energy of a static (bosonic) Ising anyon pair as a function of their separation $r$, plotted for different values of $\kappa$. The dashed lines are obtained by numerically fitting the data to the functional form in Eq. \ref{['eq:interaction']} with fitting parameters $\xi$ and $\lambda$ shown in the figure legends. The values are obtained for a system of linear size $L=34$ with periodic boundary conditions. Note that the anyons have a nearest-neighbor repulsive interaction but an attractive interaction at next-nearest neighbor distance.
• Figure 3: Raman scattering geometry. (a) Definitions of the vectors $d_{\langle ij \rangle_\alpha}$ and $d^\perp_{\langle ij \rangle_\alpha}$. (b) Incoming light is incident perpendicular to the honeycomb plane and the light that is reflected back is detected. Shown here is a setup where the incident and reflected light waves are right-circularly polarized. This processes transfers a finite angular momentum, $\Delta S_{ph}=2\hbar$ from the photon to the system.
• Figure 4: Illustration of vison creation processes by each (local) Raman operators given by Eq. \ref{['eq:raman_operators']}, acting on a bond at position ${\boldsymbol s}$. $\mathcal{R}^h$ creates two visons at ${\boldsymbol s}\pm {\boldsymbol \delta}^{(1)}_z$ and $\mathcal{R}^\Gamma$ creates visons at ${\boldsymbol s}\pm {\boldsymbol \delta}^{(2)}_z$. $\mathcal{R}^H$ creates four adjacent anyons which requires a higher energy cost.
• Figure 5: Effective lattice hopping model describing the relative motion of two Ising anyons in a generic Kitaev liquid. For $|{\boldsymbol r}|>\sqrt{3}$ (outside the blue hexagon), transporting one of two Ising anyons (created out of vacuum) anti-clockwise around the other results in a phase of $-\frac{\pi}{4}$. This is implemented by Peierls phases of $e^{-i\pi/24}$ multiplying the hopping amplitudes (orange dashed lines). We exclude hoppings to the origin, modeling a hard-core constraint for the anyons. This neglects a higher-order virtual process where the visons annihilate which is beyond the precision of our linear-order perturbation theory. When the anyons are close to each other, interaction effects lead to non-universal braiding phases dependent on $\kappa$. We calculate these phases and corresponding hopping rates microscopically within each hexagonal (or triangular) loops (drawn as colored polygons in (a)) up to a separation of $|{\boldsymbol r}|=\sqrt{3}$ (blue hexagon). We choose the gauge such that hopping on the yellow links carry a phase $\pi/24$ (along radially outward direction) in order to implement the mirror symmetry of the original honeycomb lattice. In (b), we plot the values of these phases along each closed loop shown in (a) (in colors) calculated explicitly on a honeycomb lattice model of linear size $L=34$. In (c), we plot the absolute values of the hopping matrix elements (normalized by the strength of the perturbation $h$ and $\Gamma$) corresponding to the sides of the colored polygons in (a), using the same colors.