Direct probing the quantum geometric tensor for bosonic collective excitations

Abstract

The quantum geometric tensor (QGT), whose real and imaginary parts define the quantum metric and Berry curvature, encodes the intrinsic geometry of quantum states. While electronic QGT has been directly observed and linked to various phenomena like electron-phonon coupling, its bosonic analogue remains both theoretically and experimentally unexplored. We demonstrate that the dynamical structure factor directly encodes the full QGT throughout the Brillouin zone, establishing it as a sensitive probe of both quantum metric and Berry curvature. Applying this framework, we uncover clear geometric signatures in a twofold quadruple Weyl phonon in BaPtGe and the node-line magnon in Gd. Our results establish a general, direct route to measuring quantum geometry in bosonic systems, a crucial step toward elucidating its impact on condensed matter phenomena.

Figures (4)

• Figure 1: Schematic illustration of the direct measurement of the Berry curvature and quantum metric tensor via the dynamical structure factor $S(\mathbf{Q},\omega)$. (a) Experimental geometry for measuring $S(\mathbf{Q},\omega)$. (b) Measured dynamical structure factor $S(\mathbf{Q},\omega)$. (c) Pseudospin texture reconstructed from $S(\mathbf{Q},\omega)$. (d) Berry curvature and quantum metric tensor derived from the pseudospin texture.
• Figure 2: (a) Crystal structure and (b) Brillouin zone for BaPtGe. The gray square cross section perpendicular to [111], denoted as the $\alpha$ plane, is centered at (0.1, 0.1, 0.1) with side length 0.2. (c) The phonon dispresion of the TQW for BaPtGe, and the associated circularly polarized atomic motions for the two phonon branches. (d) The heat-map distribution of $\cos\langle \mathbf{d},\mathbf{V}\rangle=\sin\theta_{\mathbf{k}}\sin\phi_{\mathbf{k}}$ obtained from the $k\cdot p$ model, plotted on the $\alpha$-plane in the Brillouin zone $\mathbf{G}^{V_y}=(3, 1, 0)$. (e) Pseudospin texture of $\mathbf{d}$ on a sphere enclosing the TQW. (f) Distribution of the QMT component $g^{xx}$ on the $\alpha$-plane. (g) Distribution of the QMT trace and the Berry curvature on a sphere enclosing the TQW, both obtained from the pseudospin texture and represented in polar coordinates for the upper band.
• Figure 3: (a) DSF with $\mathbf{G}=(0,3,3)$ and (b) QMT component $g^{xx}$ for the TQW in BaPtGe along three high-symmetry lines, obtained from first-principles calculations. It is evident that the QMT is zero along the $C_2$-invariant line $\Gamma-X$.
• Figure 4: (a) Crystal structure and magnetic order of Gd. (b) Brillouin zone and the location of node-line magnon. DSF for the node-line magnon with $\mathbf{G}=(0,0,2)$ on (c) a $q$-plane with the constant-energy surface ($E_{cut}$) in momentum space and (d) along the $\Gamma-K$. (e)-(f) $g^{yy}$ obtained by the DSF in (c) and (d), respectively.