Footprints of the Kitaev spin liquid in the Fano lineshapes of the Raman active optical phonons

Abstract

We develop a theoretical description of the Raman spectroscopy in the spin-phonon coupled Kitaev system and show that it can provide intriguing 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 Hamiltonians based on $D_{3d}$ symmetry. We then systematically compute the Raman intensity and show that the spin-phonon coupling renormalizes phonon propagators and generates the salient Fano linshape. 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 $Z_2$ gauge fluxes encoded in the Fano lineshape. Our results explain several phonon Raman scattering experiments in the candidate material $α$-RuCl$_3$.

Figures (7)

• Figure 1: (a) Crystal structure of $\alpha$-RuCl$_3$. The unit cell shown in blue dashed lines is defined by $\mathbf{n_1}=(\sqrt{3},0)$ and $\mathbf{n_2}=(\frac{3}{2},\frac{\sqrt{3}}{2})$ and includes two Ru$^{3+}$ and six Cl$^-$ ions. $\mathbf{M_{x,y}}=(\pm \frac{\sqrt{3}}{2},\frac{1}{2})$ and $\mathbf{M_z}=(0,-1)$ are nearest neighbor vectors. The sites $\mathbf{r},\mathbf{r'},\mathbf{r"}$ form a generic three-spin link $\braket{\mathbf{r},\mathbf{r'},\mathbf{r"}}_{yx}$ as described in the text. (b) Visualization of the eigenmodes of $E^1_g$ and $E^2_g$ phonons in $xy$ plane, obtained by linear representation theory (see Sec. A of SM).
• Figure 2: Panel (a): $I^{xx}$ and $I^{xx}_\textmd{exp}$ are, respectively, the stratified Monte Carlo (strMC) simulated Raman intensity and the experimental intensity from Ref. Sandilands2015 at $T=0.22$ and $\kappa=0$. By fitting $I^{xx}$ to the experimental intensity $I^{xx}_\textmd{exp}$, the best-fit model parameters are obtained: $\omega_\Gamma = [7.31, 10.10]$, $\lambda_\Gamma = [0.25, 0.52]$, $\mu_\Gamma = [0.38, 1.00]$, $\nu = -0.63$. Panels (b-d): The temperature dependence of the $E_g^2$ peak curve parameters obtained from the asymmetric Lorentzian fitting: $1/|q|$, $\gamma$ and $\omega^{\textmd{r}en}$. $T_l$ and $T_h$ are two crossover temperatures. In panel (b), the computed $\gamma$ has been offset by a background line width obtained at $T=10^{1.5}$). This background line width mainly originates from the artificial broadening $\delta_\textmd{ph}$ as shown in Sec. D of SM. The red dots are experimental line width $\gamma_\textmd{exp}$, obtained from Ref. Sandilands2015. The two green vertical dashed lines in (e) indicate $T=5$ K and 150 K. The unit conversion we use here is $J\approx 23$ K.
• Figure 3: The magnetic field dependence of curve parameters $1/|q|$ and $\gamma$ of two phonon peaks $E_g^1$ and $E_g^2$ in the computed Raman spectrum are shown in (a,b) and (c,d), respectively. The purple dots denote experimental data from Ref. Wulferding2020 measured at $T=2$ K, i.e. log$T=-1.1$. The corresponding theoretical curve is also colored purple. The line width $\gamma$ has been offset by the background contribution (see caption of Fig. \ref{['fig: FullModelFano']} for the reasoning). The inset of (b) shows the density of state of Majorana fermions at various $\kappa$Feng2020. The conversion from $\kappa$ in the unit of $J$ to magnetic field $B$ in the unit of Tesla follows from Kitaev2006: $\kappa = \frac{(\mu_B B)^3}{\Delta_\textmd{flux}^2}$, $\mu_B$ is the Bohr magneton and $\Delta_\textmd{flux} = 0.27 J$ is the flux energy, and $J\approx 23$ K.
• Figure S1: Unit cell of $\alpha-$RuCl$_3$ with labeled ions.
• Figure S2: The Feynman diagrams of the phonon Raman vertices: (a) $\mu_\Gamma R_{\Gamma m}$ (b) $\mathcal{P}_{\Gamma m, L}$ (c) $\mathcal{P}_{\Gamma m, R}$.

Theorems & Definitions (1)

• Theorem 1