Probing signatures of fractionalization in the candidate quantum spin liquid ${\mathrm{Cu}}_2{\mathrm{IrO}}_3$ via anomalous Raman scattering

Long-range entanglement in quantum spin liquids (QSLs) leads to novel low-energy excitations with fractionalized quantum numbers and (in two dimensions) statistics. Experimental detection and manipulation of these excitations present a challenge particularly in view of diverse candidate magnets. A promising probe of fractionalization is their coupling to phonons. Here, we present Raman scattering results for the $S=1/2$ honeycomb iridate ${\mathrm{Cu}}_2{\mathrm{IrO}}_3$, a candidate Kitaev QSL with fractionalized Majorana fermions and Ising flux excitations. We observe anomalous low-temperature frequency shift and linewidth broadening of the Raman intensities in addition to a broad magnetic continuum, both of which, as we derive, are naturally attributed to the phonon decaying into itinerant Majoranas. The dynamic Raman susceptibility marks a crossover from the QSL to a thermal paramagnet at $\sim 120$ K. The phonon anomalies below this temperature demonstrate a strong phonon-Majorana coupling. These results provide evidence of spin fractionalization in ${\mathrm{Cu}}_2{\mathrm{IrO}}_3$.

Figure 1

Black and red curves represent the unpolarized and cross-polarized Raman spectra of ${\mathrm{Cu}}_2{\mathrm{IrO}}_3$ at three different temperatures in the spectral ranges of 70–800 ${\mathrm{cm}}^{-1}$ and 10–800 ${\mathrm{cm}}^{-1}$, respectively. The shaded region represents the low-energy magnetic continuum.

Figure 2

(a) Phonon fits of the M1-M2 doublet with Lorentzian and Breit-Wigner-Fano (BWF) line shapes, respectively. (b) Temperature evolution of the asymmetry parameter $1/\left|q\right|$ of the M2 mode. Temperature dependence of (c) phonon frequency and (d) FWHM. Blue curves are anharmonic fits to phonon frequencies and FWHMs. Shaded regions demonstrate the boundary between the normal and the Kitaev paramagnetic states.

Figure 3

(a) Symbols denote the magnetic contribution to integrated $I_{\mathrm{mid}}$ in the frequency range 120–260 ${\mathrm{cm}}^{-1}$ after subtracting the bosonic background [shown in (b)]. The blue solid curve represents fitting by the two-fermion scattering form $A+B{(1-f)}^2$, with $f=1/(1+e^{\hslash \omega /k_{B}T})$ being the Fermi distribution function. (c) Symbols denote the magnetic continuum integrated in the low-frequency region of 10–45 ${\mathrm{cm}}^{-1}$.

Figure 4

Magnetic Raman susceptibility (Mag. Raman Susc.) as a function of temperature deduced from the Kramers-Kronig relation. The shaded regions mark the boundary between the conventional and Kitaev paramagnetic states.

Figure 5

(a) and (b) [(c) and (d)] denote the interaction between the phonon and matter fermion [66] coming from the first-order (second-order) contributions to the spin-phonon coupling.

Figure 6

Imaginary part of the phonon Green's function $D$ with the frequency scaled with respect to the Kitaev coupling $J_K$. Different curves represent different temperatures. The spin-phonon coupling constants are taken as $\lambda =0.13$ and $\chi =0.2$.

Figure 7

(a) Powder x-ray diffraction pattern for ${\mathrm{Cu}}_2{\mathrm{IrO}}_3$ and its Rietveld refinement. (b) Temperature dependence of the dc magnetic susceptibility between 0.3 and 300 K. The red solid curve is the Curie-Weiss (CW) fit over the range 120–300 K. (c) Inverse susceptibility fitted to CW form (solid red line) over the range 120–300 K. The shaded regions demarcate the boundary between a conventional and Kitaev paramagnet where ${\chi }_{\mathrm{dc}}$ starts deviating from the CW behavior. (d) Temperature variation of ac susceptibility down to 2 K in addition to the dc susceptibility down to 0.3 K. (e) Low-temperature ${\chi }_{\mathrm{dc}}$ data fit to a sub-Curie law.

Figure 8

Temperature dependence of integrated background intensity (normalized) for different energy ranges.

Figure 9

(a) and (b) Fermionic fits to integrated $I_{\mathrm{mid}}$ with two different bosonic backgrounds (shown in the insets).

Figure 10

Phonon fits (after subtracting the low-frequency background) to the Raman spectra of ${\mathrm{Cu}}_2{\mathrm{IrO}}_3$ at selected temperatures. Experimental data are indicated by black circles. Blue and red curves are individual phonon modes and the cumulative fits, respectively. The magnified M1-M2 doublets are shown in the insets.

Figure 11

Temperature variation of the integrated intensity of the M3 mode, similar to that of dc magnetic susceptibility, both plotted in log-log scale.

Figure 12

The two-point unit cell has been considered along the $z$ bonds. ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ denote the two lattice vectors of the honeycomb lattice.

Figure 13

Frequency shift is calculated from the average energy of the spin system. At zero temperature, the average energy saturates to approximately $-0.43J_K$.

Figure 14

The thick and thin curves denote the dressed and bare phonon Green's function, respectively. In the last diagram, we truncate the series up to the one-loop correction.

Figure 15

The solid curves represent the propagator for “$a$” fermions.

Figure 16

Variation of phonon linewidth with (a) temperature and (b) frequency scaled with respect to the Kitaev coupling $J_K$.

Figure 17

Temperature dependence of (a) frequency shift and (b) FWHM for the M1 mode. Red squares are the experimental data after subtracting the anharmonic contribution. The smooth curves represent the theoretical curves at different values of the coupling constant. In (b), we plot only till $J_K=40$ K considering the finite bandwidth ($\sim 3J_K$) effect as shown in Fig. 16.