Raman Scattering from Higgs Mode Oscillations in the Two-Dimensional Antiferromagnet ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$

We present and analyze Raman spectra of the Mott insulator ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$, whose quasi-two-dimensional antiferromagnetic order has been described as a condensate of low-lying spin-orbit excitons with angular momentum $J_{\text{eff}}=1$. In the $A_g$ polarization geometry, the amplitude (Higgs) mode of the spin-orbit condensate is directly probed in the scalar channel, thus avoiding infrared-singular magnon contributions. In the $B_{1g}$ geometry, we observe a single-magnon peak as well as two-magnon and two-Higgs excitations. Model calculations using exact diagonalization quantitatively agree with the observations. Together with recent neutron scattering data, our study provides strong evidence for excitonic magnetism in ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ and points out new perspectives for research on the Higgs mode in two dimensions.

Figure 1

(a) Raman spectra in $B_{1g}$ scattering geometry with two magnetic features $B$, $B^{\prime }$ appearing below $T_N$. The background (dashed line) is subtracted in further analysis. (b) Detailed view on the feature $B$. (c) Polarization vectors of incoming and outgoing photons with respect to the Ru lattice. In the Raman process, the exchange is modulated with opposite signs on $x$ and $y$ bonds leading to the Raman operator $\mathcal{R}\propto {\mathcal{H}}_x-{\mathcal{H}}_y$.

Figure 2

(a) Raman spectra in $A_g$ scattering geometry with two magnetic features $A$, $A^{\prime }$ appearing below $T_N$. (b) Detailed view on the feature $A$. (c) The polarization vectors in the $A_g$ setup. In this case, the exchange is modulated equally on $x$ and $y$ bonds and the Raman operator $\mathcal{R}\propto {\mathcal{H}}_x+{\mathcal{H}}_y$.

Figure 3

Fits of $T=10\mathrm{K}$ Raman spectra in $B_{1g}$ (a) and $A_g$ (b) channels using a model of phonons interacting with a magnetic continuum. The model response (blue) is compared to the experimental points (black). The obtained magnetic signal $S(\omega )$ is indicated by shading. The $A_g$ phonons marked by blue triangles are most strongly affected by the spin-phonon interaction which changes their line shape dramatically compared to the noninteracting case (red dashed line in the inset). The associated spectral-weight transfer is moderate only.

Figure 4

(a) Coordinate frames for the Ru lattice. The ordered moments point along the $b$ axis. (b) Multiplet structure of ${\mathrm{Ru}}^{4+}$ ions with a ground-state singlet $s$ with $J_{\text{eff}}=0$ and higher-lying magnetic states derived from $J_{\text{eff}}=1$ triplet. A tetragonal crystal field removes the degeneracy of the triplet states $T$ by lifting up $T_z$ [6]. An orthorhombic distortion further splits $T_{x/y}$ into $T_{a/b}=(T_x\mp T_y)/\sqrt{2}$ levels forming together with $s$ the basis for the low-energy model (shaded). (c) Modulation of the condensate energy (Mexican-hat potential) in the Raman process leading to an excitation of the amplitude mode ($A_g$ channel). Note that $\delta n_T\equiv \delta (S_x^2+S_y^2)$, i.e., the $A_g$ coupling is rotationally invariant. In contrast, the $B_{1g}$ coupling leads to the shape deformations, leaving the condensate density intact. A single magnon is excited instead of the amplitude mode. (d),(e) Raman spectra obtained by exact diagonalization on clusters with $N=16$, 18, and 20 sites [25] using $E_T=31\mathrm{meV}$, $J=7.5\mathrm{meV}$, $A=2.3\mathrm{meV}$, $\alpha =0.15$, and ${\mathrm{\Delta }}^{\prime }=4\mathrm{meV}$. The ED data for $B_{1g}$ (a) and $A_g$ (b) channels are presented in identical scales and overlayed by the magnetic $S(\omega )$ from Figs. 3, 3 (dashed line).