Resolution of Topology and Geometry from Momentum-Resolved Spectroscopies

Abstract

Extracting the complete quantum geometric and topological character of Bloch wavefunctions from experiments remains a challenge in condensed matter physics. Here, we resolve this by introducing the "wavefunction form factor" (WFF) matrix, a quantity directly constructible from intensities in momentum- and energy-resolved spectroscopies like ARPES and INS. We demonstrate that band topology is encoded in "spectral nodes" -- momentum-space points where the WFF determinant vanishes, providing a direct readout of topological invariants via a topological selection rule. Furthermore, when the number of independent probes exceeds the number of the target bands, our framework yields an effective band projector. This enables the determination of Wilson loop spectra and the extraction of an effective quantum geometric tensor, providing a model-independent measurement of the non-Abelian Berry curvature and quantum metric as resolved by the experimental probes.

Figures (3)

• Figure 1: Geometry of the nodal manifold enforced by the selection rule. (a) For point charges (e.g. Weyl points) of opposite signs, spectral nodes coalesce into a one-dimensional nodal arc (red) that connects the two monopole charges; along this nodal arc the ARPES/INS intensity vanishes. (b) For nodal lines (green loops), the zeros from the $\pi$ Berry phase assemble into a two-dimensional nodal membrane (red) bounded by the line and the zeros from the $Z_2$ monopole charge form a nodal tube (blue) connecting nodal lines with opposite charges.
• Figure 2: Topological selection rule and nodal arc for Weyl points. The wavefunction form factor (WFF) is calculated on different momentum space manifolds. The density plot shows the $F(\mathbf{k})$, while the vector field indicates the phase winding of $\braket{u_{n\mathbf{k}}|W_{a\mathbf{q}}}$. (a) On a sphere enclosing a charge-1 Weyl point, a single spectral node (red circle) appears, around which the phase winds by $2 \pi$. (b) On a sphere enclosing a charge-4 quadrupole Weyl point, four order-1 spectral nodes (red circles) appear. The phase winds by $2 \pi$ around each node (winding number $d=1$), satisfying the relation $\sum_i d_i =4 = C.$ (c) On a 2D slice of Brillouin zone ($\mathbf{k} = (k_x,0,k_z)$) that intersects a pair of Weyl points located at $(0,0, \pi/2 / 3\pi/2)$ with opposite charges ($C = \pm 1$), a line of nodes---the spectral nodal arc (red line)---is visible. The arc connects Weyl points with opposite charges.
• Figure 3: Spectroscopic signature of the $Z_2$ charge in $\text{Cu}_3\text{TeO}_6$ The calculated magnitude of the WFF determinant, $\det[F(\mathbf{k})]$, reveals the spectroscopic signatures of the $Z_2$ charge. (a) On a sphere enclosing the Dirac point $D_2 = (\pi,\pi,\pi)$, a spectral node is visible (red circle), confirming the selection rule for a non-trivial $Z_2$ charge. (b) The $\det[F(\mathbf{k})]$ calculated on the 2D momentum space slice defined by $\mathbf{k}=(k_1,k_1,k_2)$. A nodal arc (red line) connects the Dirac point $D_2$ with its partner $D_3$ (labeled by magenta circles). In the simulation, we choose the reciprocal lattice vector $\mathbf{G} = 2\pi\times(2,2,2)$, which lies in the [111] rotation axis which is compatible with the system's rotation symmetry. And we see the nodal arc entirely lies in the rotation axis.