A key characteristic of quantum spin liquids(QSL) is the presence of fractional excitations related to their entanglement properties, yet experimental verification of their statistics is missing. For example, in the potential Kitaev spin liquid, RuCl3 experiments uncovered signs of fractional particles, though not their Fermi statistics. Here we employ Raman scattering to reveal the true nature of the magnetic excitations, using improved experimental methods and analysis to remove the influence of laser heating and thermal fluctuations. Via the energy loss and gain spectra, we extract the energy and temperature dependence of the Raman susceptibility to prove RuCl3's magnetic response is given by pair creation of fermionic excitations. Furthermore, by comparing with quantum Monte Carlo (QMC) results for the exact Kitaev limit, we are able to discern the energy and temperature range where additional interaction terms are important. Our results open new directions in QSL research by providing a new way to investigate fractional excitations and the importance of terms causing spinon confinement.
We develop a theoretical description of the Raman spectroscopy in the spin-phonon-coupled Kitaev system and show that it can provide observable signatures of fractionalized excitations characteristic of the underlying spin-liquid phase. In particular, we obtain the explicit form of the phonon modes and construct the coupling Hamiltonian based on the
D3d symmetry. We then systematically compute the Raman intensity and show that the spin-phonon coupling renormalizes phonon propagators and generates the salient Fano lineshape. We find that the temperature evolution of the Fano lineshape displays two crossovers, and the low-temperature crossover shows pronounced magnetic-field dependence. We thus identify the observable effect of the Majorana fermions and the
Z2 gauge fluxes encoded in the Fano lineshape. Our results are consistent with the phonon Raman scattering experiments in the candidate material $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$.
We show how Raman spectroscopy can serve as a valuable tool for diagnosing quantum spin liquids (QSL). We find that the Raman response of the gapless QSL of the Kitaev-Heisenberg model exhibits signatures of spin fractionalization into Majorana fermions, which give rise to a broad signal reflecting their density of states, and Z(2) gauge fluxes, which also contribute a sharp feature. We discuss the current experimental situation and explore more generally the effect of breaking the integrability on response functions of Kitaev spin liquids.
Motivated by the ongoing effort to search for high-resolution signatures of quantum spin liquids, we investigate the temperature dependence of the indirect resonant inelastic x-ray scattering (RIXS) response for the Kitaev honeycomb model. We find that, as a result of spin fractionalization, the RIXS response changes qualitatively at two well-separated temperature scales,
TL and
TH, which correspond to the characteristic energies of the two kinds of fractionalized excitations,
Z2 gauge fluxes and Majorana fermions, respectively. While thermally excited
Z2 gauge fluxes at temperature
TL lead to a general broadening and softening of the response, the thermal proliferation of Majorana fermions at temperature ${T}_{H}\ensuremath{\sim}10\phantom{\rule{0.16em}{0ex}}{T}_{L}$ results in a significant shift of the spectral weight, both in terms of energy and momentum. Due to its exclusively indirect nature, the RIXS process we consider gives rise to a universal magnetic response and, from an experimental perspective, it directly corresponds to the
K-edge of
Ru3+ in the Kitaev candidate material $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$.
A prominent feature of quantum spin liquids is fractionalization of the spin degree of freedom. Fractionalized excitations have their own dynamics in different energy scales, and hence, affect finite-temperature (
T) properties in a peculiar manner even in the paramagnetic state harboring the quantum spin liquid state. We here present a comprehensive theoretical study of the spin dynamics in a wide
T range for the Kitaev model on a honeycomb lattice, whose ground state is a quantum spin liquid. In this model, the fractionalization occurs to break up quantum spins into itinerant matter fermions and localized
Z2 fluxes, which results in two crossovers at very different
T scales. Extending the previous study for the isotropic coupling case [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016)], we calculate the dynamical spin structure factor $S(\mathbf{q},\ensuremath{\omega})$, the NMR relaxation rate
1/T1, and the magnetic susceptibility $\ensuremath{\chi}$ while changing the anisotropy in the exchange coupling constants, by using the dynamical mean-field theory based on a Majorana fermion representation. We describe the details of the methodology including the continuous-time quantum Monte Carlo method for computing dynamical spin correlations and the maximum entropy method for analytic continuation. We confirm that the combined method provides accurate results in a wide
T range including the region where the spins are fractionalized. We find that also in the anisotropic cases the system exhibits peculiar behaviors below the high-
T crossover whose temperature is comparable to the average of the exchange constants: $S(\mathbf{q},\ensuremath{\omega})$ shows an inelastic response at the energy scale of the averaged exchange constant,
1/T1 continues to grow even though the equal-time spin correlations are saturated and almost
T independent, and $\ensuremath{\chi}$ deviates from the Curie-Weiss behavior. In particular, when the exchange interaction in one direction is stronger than the other two, the dynamical quantities exhibit qualitatively different
T dependences from the isotropic case at low
T, reflecting the opposite parity between the flux-free ground state and the flux-excited state, and a larger energy cost for flipping a spin in the strong interaction direction. On the other hand, when the exchange anisotropy is in the opposite way, the results are qualitatively similar to those in the isotropic case. All these behaviors manifest the spin fractionalization in the paramagnetic region. Among them, the dichotomy between the static and dynamical spin correlations is unusual behavior hardly seen in conventional magnets. We discuss the relation between the dichotomy and the spatial configuration of the
Z2 gauge fields. Our results could stimulate further experimental and theoretical analyses of candidate materials for the Kitaev quantum spin liquids.
Motivated by recent synthesis of the hyperhoneycomb material $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, we study the dynamical structure factor (DSF) of the corresponding 3D Kitaev quantum spin-liquid (QSL), whose fractionalized degrees of freedom are Majorana fermions and emergent flux loops. The properties of this 3D model are known to differ in important ways from those of its 2D counterpart---it has a finite-temperature phase transition, as well as distinct features in the Raman response. We show, however, that the qualitative behavior of the DSF is broadly dimension-independent. Characteristics of the 3D DSF include a response gap even in the gapless QSL phase and an energy dependence deriving from the Majorana fermion density of states. Since the majority of the response is from states containing a single Majorana excitation, our results suggest inelastic neutron scattering as the spectroscopy of choice to illuminate the physics of Majorana fermions in Kitaev QSLs.