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arXiv:2505.01042 · Physical review. B./Physical review. B

Raman spectroscopy of anyons in generic Kitaev spin liquids

Aprem P. Joy, Achim Rosch

Published Nov 6, 2025• 2 citations
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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 in the chiral Kitaev spin liquid perturbed by symmetry allowed interactions---a phase relevant to $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$ 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: $I(\ensuremath{\omega})\ensuremath{\sim}{(\ensuremath{\omega}\ensuremath{-}{E}_{2\ensuremath{\sigma}}^{0})}^{\frac{1}{8}}$ for linear and parallel-circular polarization channels, where ${E}_{2\ensuremath{\sigma}}^{0}$ is the two-particle gap. Strong corrections due to short-range interactions arise at order $1/4$. For cross-circularly polarized channels, the scaling is given by $I(\ensuremath{\omega})\ensuremath{\sim}{(\ensuremath{\omega}\ensuremath{-}{E}_{2\ensuremath{\sigma}}^{0})}^{|l\ifmmode\pm\else\textpm\fi{}1/8|}$, 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 ${\ensuremath{\theta}}_{\ensuremath{\sigma}}=\ensuremath{\pi}/8$, 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.

DOI
10.1103/b175-chx4

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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 in the chiral Kitaev spin liquid perturbed by symmetry allowed interactions---a phase relevant to \ensuremathα\ensuremath−RuCl3\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}\ensuremathα\ensuremath−RuCl3​ 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(\ensuremathω)\ensuremath∼(\ensuremathω\ensuremath−E2\ensuremathσ0)18I(\ensuremath{\omega})\ensuremath{\sim}{(\ensuremath{\omega}\ensuremath{-}{E}_{2\ensuremath{\sigma}}^{0})}^{\frac{1}{8}}I(\ensuremathω)\ensuremath∼(\ensuremathω\ensuremath−E2\ensuremathσ0​)81​ for linear and parallel-circular polarization channels, where E2\ensuremathσ0{E}_{2\ensuremath{\sigma}}^{0}E2\ensuremathσ0​ is the two-particle gap. Strong corrections due to short-range interactions arise at order 1/41/41/4. For cross-circularly polarized channels, the scaling is given by I(\ensuremathω)\ensuremath∼(\ensuremathω\ensuremath−E2\ensuremathσ0)∣l\ifmmode±\else\textpm\fi1/8∣I(\ensuremath{\omega})\ensuremath{\sim}{(\ensuremath{\omega}\ensuremath{-}{E}_{2\ensuremath{\sigma}}^{0})}^{|l\ifmmode\pm\else\textpm\fi{}1/8|}I(\ensuremathω)\ensuremath∼(\ensuremathω\ensuremath−E2\ensuremathσ0​)∣l\ifmmode±\else\textpm\fi1/8∣, where the value of l=0,1,2l=0,1,2l=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 \ensuremathθ\ensuremathσ=\ensuremathπ/8{\ensuremath{\theta}}_{\ensuremath{\sigma}}=\ensuremath{\pi}/8\ensuremathθ\ensuremathσ​=\ensuremathπ/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.
We calculate the Raman response for the Kitaev spin model on the $\mathcal{H}\ensuremath{-}0, \mathcal{H}\ensuremath{-}1$, and $\mathcal{H}\ensuremath{-}\ensuremath{\infty}$ harmonic honeycomb lattices. We identify several quantitative features in the Raman spectrum that are characteristic of the spin liquid phase. Unlike the dynamical structure factor, which probes both the Majorana spinons and flux excitations that emerge from spin fractionalization, the Raman spectrum in the Kitaev models directly probes a density of states of pairs of fractional, dispersing Majorana spinons. As a consequence, the Raman spectrum in all these models is gapless for sufficiently isotropic couplings, with a low-energy power law that results from the Fermi lines (or points) of the dispersing Majorana spinons. We show that the polarization dependence of the Raman spectrum contains crucial information about the symmetry of the ground state. We also discuss to what extent the features of the Raman response that we find reflect generic properties of the spin liquid phase, and comment on their possible relevance to $\ensuremath{\alpha}$-, $\ensuremath{\beta}$-, and $\ensuremath{\gamma}\ensuremath{-}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ compounds.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped Z2{Z}_{2}Z2​ fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (Kx=Ky=Kz{K}_{x}={K}_{y}={K}_{z}Kx​=Ky​=Kz​). We explore the phase diagram along two axes, an external magnetic field hhh, applied out of plane of the honeycomb, and anisotropic interactions Kz{K}_{z}Kz​ larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and h=0h=0h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped Z2{Z}_{2}Z2​ toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (eee) and magnetic (mmm) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower Kz{K}_{z}Kz​ in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
The identification and manipulation of charge-neutral fractionalized quasi-particles, in particular chiral edge modes (CEM), is a long-standing quest in physics. Remarkably, the microscopically mediated interaction between light and charge-neutral excitations in Mott-Hubbard insulators can take an identical form to the Raman coupling between light and particles with electric charge. However, since CEMs are Raman-inactive due to conservation of lattice momentum, Raman probes have been deemed unsuitable for their identification. Here, using the Kitaev quantum spin liquid (KSL) as an illustrative example, we demonstrate that the long-range correlated disorder inherent to a closed edge can lead to a Raman circular dichroism (RCD) signal that avoids suppression by linear and angular momentum selection rules, and exhibits a dependence on experimentally tunable length and energy scales that are characteristic of CEM. Having calculated the low-frequency RCD response of generic KSL, we argue that the interaction of the chiral matter fermion with the $\Ztwo$ boundary charge leaves a unique fingerprint of the KSL via the anisotropic Zeeman field dependence.
Motivated by growing evidence for the significance of the third-nearest-neighbor Heisenberg (J3J_3J3​) interaction in candidate Kitaev materials, we investigate the dynamical properties of the Kitaev spin liquid (KSL) under a J3J_3J3​ perturbation, focusing on its spin dynamical structure factor (DSF) and Raman scattering. Within a self-consistent parton mean-field plus random-phase approximation framework, we find that J3J_3J3​ induces coherent, paramagnon-like collective modes that coexist with a high-energy Majorana continuum in the spin DSF. The softening of these modes with increasing ∣J3∣|J_3|∣J3​∣ signals a quantum phase transition to magnetic order. Remarkably, magnetic ordering sets in at a common critical J3J_3J3​ for both ferromagnetic (K<0K<0K<0) and antiferromagnetic (K>0K>0K>0) Kitaev models, with the resulting ordered states forming exact dual pairs under a four-sublattice duality transformation that maps (K,J3)→(−K,J3)(K,J_3) \rightarrow (-K,J_3)(K,J3​)→(−K,J3​). An external magnetic field further softens the preexisting paramagnon modes, thereby enhancing magnetic order. Perturbative Raman calculations show that while the Kitaev-like Raman vertex probes only itinerant matter Majorana fermions, the response from the J3J_3J3​-like vertex features both matter Majoranas and visons. Four-vison excitations produce a sharp peak accompanied by a two-fermion continuum, whereas two-vison excitations yield a continuum closely resembling the single-matter-fermion density of states. These results provide a unified perspective on the dynamical signatures of J3J_3J3​-perturbed KSL and are helpful for interpreting experimental spectra in candidate Kitaev materials with sizable J3J_3J3​ interactions.
We show that neutral anyonic excitations have a signature in spectroscopic measurements of materials: The low-energy onset of spectral functions near the threshold follows universal power laws with an exponent that depends only on the statistics of the anyons. This provides a route, using experimental techniques such as neutron scattering and tunneling spectroscopy, for detecting anyonic statistics in topologically ordered states such as gapped quantum spin liquids and hypothesized fractional Chern insulators. Our calculations also explain some recent theoretical results in spin systems.
Quantum spin liquids (QSLs) host a variety of fractionalized particles. In Kitaev's paradigmatic honeycomb model a spin-fractionalizes intoflux due to emergentgauge field and matter Majorana fermions. Although these excitations have well-defined dynamics in the integrable limit, their direct experimental identification is notoriously challenging: realistic materials inevitably host additional symmetry-allowed interactions that break integrability and hybridize gauge and matter sectors, while magnetic fields, which are often required to suppress competing order and stabilize a putative QSL regime, further entangle the responses of different fractionalized quasiparticles and may even drive the system into field-induced spin-liquid phases that are not adiabatically connected to the integrable limit. A prominent example is the quantum Majorana metal, in which the distinct dynamics of fractionalized Majorana fermions can become directly visible in scattering. This review highlights recent progress on these related questions: in which field-stabilized QSL regimes and nearby emergent phases, and under what conditions, can the response of a specific fractionalized quasiparticle be isolated and positively understood, thereby clarifying the existence and the experimental scope of putative spin liquids? We review the progress on these questions across Abelian, non-Abelian, and emergent quantum phases under magnetic field that are not perturbatively connected to the integrable limit. We connect these field-induced dynamical phenomena to concrete experimental observables, relevant for neutron scattering, resonant inelastic x-ray scattering, and pump-probe spectroscopy, techniques that are capable of identifying the nature of different magnetic-field-induced phases and resolving specific types of fractionalized particles, including Majoranas andfluxes.