Direct Raman observation of the quantum metric in a quantum magnet

Abstract

The quantum geometric tensor (QGT) unifies the Berry curvature (its imaginary part) and the quantum metric (its real part), yet Raman studies of chiral phonons have so far accessed only the former. We perform circularly polarized Raman spectroscopy on the quantum magnet K2Co(SeO3)2, where the field-odd chiral splitting and the field-even center-frequency shift collapse onto a single curve across temperature and magnetic field, revealing a common microscopic origin for both observables. Since the chiral splitting reflects the Berry curvature, the concomitant even component, arising from the same microscopic origin, captures the field-induced change of the quantum metric, corresponding to the diagonal Born-Oppenheimer correction. Across two resolvable Eg modes, the unified data are well captured by a simple empirical quadratic relation. These results establish Raman spectroscopy as a direct probe of the quantum metric and an operational decomposition of quantum geometry within a single measurement.

Figures (4)

• Figure 1: Chiral phonon mode P5 at $T=1.8K$.a,b, Helicity-resolved Raman spectra at $B=\pm4T$ (LCP/RCP in blue, RCP/LCP in red). Each cross-helicity channel predominantly excites one branch of the chiral-phonon doublet. Finite depolarization gives residual intensity in the opposite channel Porto1966. c, Field-dependent normalized intensity map showing the continuous evolution of the two helicity branches, characterized by an odd-in-$B$$\Delta\omega$ and a concurrent even-in-$B$$\delta\omega_c$.
• Figure 2: Mode-selective coupling to quantum geometry. Under identical optical and thermal conditions ($T=1.8$ K, $B=4$ T; LCP/RCP and RCP/LCP channels), two $E_g$ modes (P5 and P8) display robust chiral splitting, whereas three $E_g$ modes (P1, P3, P6) remain inert.
• Figure 3: Chiral splitting without LRO and mode-resolved scaling.a,b, P5 (sharp, on-site) retains a robust chiral splitting above $T_{\mathrm{N}}$: raw spectra (a) and a map of $\Delta\omega(B,T)$ (b) both show persistence deep into the paramagnetic regime. The dashed line represents the critical transition temperatures from the paramagnetic state to the $uud$ 1/3-magnetization plateau phaseSM_link. c,$\Delta\omega(B)$ for P5 (solid symbols) and P8 (empty symbols); the $uud$ plateau provides a linear window to extract the Zeeman slopes $\alpha$. d, Correlation-driven splitting $\Delta\omega'\equiv\Delta\omega-\alpha B$ versus magnetization $M$ collapses onto $\Delta\omega'=5.03 M^{1.33}$ for both modes prior to the plateau.
• Figure 4: Material-wide quadratic relation linking metric and curvature.a, Nonsplitting reference modes (P1, P3, P6 of $E_g$ and P9 of $A_g$) show negligible center shifts across field, isolating $\delta\omega_{\mathrm{c}}$ to the chiral interaction channel. b, Mode-dependent center shifts $\delta\omega_{\mathrm{c}}(B)$ for the chiral modes P5 (on-site) and P8 (non-local). c, Upper: Even-in-$B$$\delta\omega_{\mathrm{c}}$ plotted against $(\Delta\omega)^2$ collapses onto the quadratic law $\delta\omega_{\mathrm{c}} = \gamma(\Delta\omega)^2$ with a common slope $\gamma$, unifying the metric and curvature readouts (symbols: P5 solid, P8 empty). Lower: Residuals of the quadratic collapse. Together, these panels demonstrate that both metric and curvature responses originate from a single microscopic interaction channel.