Direct Raman observation of the quantum metric in a quantum magnet

The gist: Direct Raman observation of the quantum metric in a quantum magnet reveals that field-odd chiral splitting and field-even center shift collapse onto a single curve, establishing Raman spectroscopy as an all-optical probe of both Berry curvature and the quantum metric.

Quantum Geometry and Observables

Quantum geometry encompasses both the Berry curvature (the imaginary part of the QGT) and the quantum metric (its real part), which govern transport, optics, and superconductivity in quantum materials [1–5]. Raman studies have previously provided a lineshape handle on the Berry curvature but lacked a direct spectroscopic observable for its real part, the quantum metric [6–9]. This paper establishes an all-optical approach where a native chiral-phonon doublet serves as an in-situ probe whose odd and even parity responses arise from the same underlying interaction channel [1]. The field-odd chiral splitting ∆ω reflects the Berry curvature, while the field-even center-frequency shift δωc captures the field-induced change of the quantum metric—the diagonal Born–Oppenheimer correction (DBOC) [18, 19].

Microscopic Origin and Parity Contrast

The two observables arise from a common microscopic origin within the phonon self-energy [20]. The odd–even pair in Figure 1 arises naturally from the electron–phonon self-energy of a doubly degenerate Eg mode with displacement u = (ux, uy) [20]. In this minimal picture, the phonon coordinates couple linearly to the electronic degrees of freedom via Hint = X i Diui, where Di = ∂He / ∂ui [20]. The two eigenfrequencies split and shift according to ∆ω = ω+ − ω− = 2Πxy / meffω0 (3), while δωc = δω+ + δω− / 2 = Πd / meffω0 (4). Symmetry enforces Re Πxy = 0 and makes Im Πxy odd in B, immediately producing the parity contrast seen experimentally: ∆ω is odd, δωc is even [20].

Linking Observables to Geometric Tensors

The same virtual matrix elements also construct the electronic quantum-geometric tensor Qij = X n!=0 ⟨0Din⟩⟨nDj 0⟩ (En − E0)2 = gij - i 2 Fij [5]. Comparing Eqs. (2) and (5) shows that both quantities derive from the same set of virtual electronic transitions, differing only by one power of the energy denominator [5]. In the single–energy–scale limit, this relation becomes explicit: Πxy ∝ Im Qxy ≡ Fxy = omegae (6), and Πd ∝ Re Qxx ≡ gxx = ge (7). Thus, the two raw observables in Figure 1 trace the imaginary and real parts of the same geometric tensor—the Berry curvature and the quantum metric—revealed simultaneously in a single Raman line shape [20].

Universal Scaling Law

The relationship between these two components is quantified by a universal quadratic law: δωc = γ(∆ω)2 [4]. This relation emerges from substituting the proportionalities into Eqs. (3)–(4) and reinstating the explicit matrix element D2 yields ∆ω ∝ D2 meffω0 Ω'e, and δωc ∝ D2 meffω0 δg'e [8, 9]. At zero field, g'e(0) represents the intrinsic quantum metric of the electronic ground state—the conventional diagonal Born–Oppenheimer correction (DBOC)—which gives a static frequency renormalization but is normally unobservable [18, 19]. The measurable quantity is its field-induced change, δg'e = g'e(B) − g'e(0), which tracks how the underlying electronic metric evolves under magnetic perturbation [3]. This converts the otherwise static DBOC into a dynamical, spectroscopically accessible response [3].

Experimental Validation and Significance

The results are validated by comparing mode-resolved responses across different phonons, showing that only two modes (P5 and P8) exhibit pronounced chiral splitting, ruling out trivial lattice effects [20]. Furthermore, the correlation-driven splitting ∆ω' ≡ ∆ω − αB versus magnetization M collapses onto a common scaling ∆ω' = 5.03M1.33 for both modes prior to the plateau [3]. This identical scaling of P5 and P8 implies that the effective curvature arises from correlated 3d-spin dynamics rather than single-site mechanisms [4]. The persistence of ∆ω without LRO demonstrates “chirality without order,” showing that the spin–lattice coupling survives even in the absence of macroscopic order [20]. Most importantly, the metric response follows a universal quadratic law, δωc = γ(∆ω)2, characterized by a single material-wide slope γ = 0.0346(3) cm [4]. This deterministic link between the even- and odd-parity components confirms their common microscopic origin—the real and imaginary parts of one geometric tensor—while elevating the diagonal Born–Oppenheimer correction from a theoretical construct to a measurable spectroscopic parameter [5].

Future Directions

This unified framework establishes quantum geometry as a directly measurable property of correlated solids [14, 16, 17]. The methodology complements photonic and transport-based probes of quantum geometry [14, 16, 17] and opens a route to engineering correlated solids where curvature and metric can be co-designed. Extending this approach to other magnets, multiferroics, and moiré systems may reveal how universal the δωc–(∆ω)2 relation is [5]. This unified framework establishes quantum geometry as a directly measurable property of correlated solids.

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Direct Raman observation of the quantum metric in a quantum magnet Chao-Fan Wang,1,∗ Han Ge,1,∗ Jun-Yang Chen,1 Liusuo Wu,1, † Xiaobin Chen,2 Jia-Wei Mei,1, ‡ and Mingyuan Huang1§ (Dated: November 14. 2025) 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 ∆ω and the field-even center shift δωc collapse onto a single curve across temperature and magnetic field, revealing a common microscopic origin for both observables. Since ∆ω reflects the Berry curvature, the concomitant even component δωc, arising from the same microscopic origin, captures the field-induced change of the quantum metric—the diagonal Born–Oppenheimer correction. Across two resolvable Eg modes, the unified data are well captured by a simple empirical relation, δωc = γ(∆ω)2. These results establish Raman spectroscopy as a direct probe of the quantum metric and an operational decomposition of quantum geometry within a single measurement. Quantum geometry—comprising the Berry curvature and the quantum metric—governs transport, optics, and superconductivity in quantum materials [1–5]. Raman studies of chiral phonons have already provided a lineshape handle on the Berry curvature—the imaginary part of the quantum-geometric tensor (QGT) [6–9]—but its real part, the quantum metric, has so far lacked a direct spectroscopic observable. Consequently, metric effects in solids have been inferred only indirectly from secondary responses such as valley Zeeman shifts, superfluid weight, or nonlinear Hall transport [10–15], or explored in photonic analogs [16, 17]. Here we establish an all-optical, circularly polarized Raman approach that, within a single measurement, reveals both components of the quantum-geometric tensor (QGT) as two manifestations of a single origin on the same quantum-magnetic platform. A native chiralphonon doublet serves as an in-situ probe whose oddand even-parity responses arise from the same underlying interaction channel (Fig. 1). The field-odd chiral splitting ∆ω reflects the Berry curvature, whereas the field-even center-frequency shift δωc captures the fieldinduced change of the quantum metric—the diagonal Born–Oppenheimer correction (DBOC) [18, 19]. Their contrasting parity thus provides an operational decomposition of quantum geometry within a single Raman line shape. Across temperature and magnetic field, the two observables collapse onto a single master curve (Fig. 4), confirming their shared geometric origin. Experimentally we implement this odd/even Raman decomposition in the triangular-lattice quantum magnet K2Co(SeO3)2 (space group R¯3m). Magneto-Raman spectra are collected in backscattering, Faraday geometry (B∥ c, T = 1.8–300 K, up to B = 8 T). A representative chiral doublet (mode P5) is shown in Fig. 1. A full catalog of Raman-active phonons (P1–P9, ΓRaman = 4Ag + 5Eg) and crystal-field excitations (CFE1, CFE2) appears in SI Fig. S2/Table S1[20], with broader materials context in Refs. [21–24]. Figure 1 presents the raw helicity-resolved Raman spectra, from which both chiral spitting ∆ω and centerfrequency shift δωc can be identified directly, without any post-processing. At B = ±4 T [Figs.

1. textbf Rewiring of Materials Simulation for Quantum Geometry Prediction: The improved system can now directly calculate the diagonal Born–Oppenheimer correction (DBOC) from experimental Raman data by applying the empirical relation δωc = γ(∆ω)2 to infer the real part of the quantum geometric tensor, establishing it as a spectroscopic signal.

2. textbf Development of In-Situ Quantum Geometry Sensors: The system can be used to build an all-optical, circularly polarized Raman approach that, within a single measurement, reveals both components of the quantum-geometric tensor (QGT) as two manifestations of a single origin. This allows for the direct spectroscopic measurement of both the Berry curvature and the quantum metric simultaneously.

3. textbf Predictive Modeling of Spin-Lattice Coupling Mechanisms: The system can now distinguish between different microscopic origins by analyzing the mode selectivity, noting that only specific modes like P5 and P8 exhibit pronounced chiral splitting, thereby confirming that responses stem from specific spin–phonon coupling channels rather than extrinsic shifts.

4. textbf Universal Relation Extraction for Geometric Parameters: The improved AI can perform a global weighted fit of all 170 data points to the relation δωc = γ(∆ω)2 to determine a single material-wide slope γ, converting an intrinsic lattice vibration into a measurable spectroscopic parameter.

5. textbf Non-LRO Regime Characterization: The system can predict that the underlying spin–lattice coupling—and thus the geometric response—does not rely on long-range order, as evidenced by the observation that the chiral splitting persists deep into the paramagnetic regime.