Band geometry from position-momentum duality at topological band crossings

We show that the position-momentum duality offers a transparent interpretation of the band geometry at the topological band crossings. Under this duality, the band geometry with Berry connection is dual to the free-electron motion under gauge field. This identifies the trace of quantum metric as the dual energy in momentum space. The band crossings with Berry defects thus induce the dual energy quantization in the trace of quantum metric. For the $\mathbb{Z}$ nodal-point and nodal-surface semimetals in three dimensions, the dual Landau level quantization occurs owing to the Berry charges. Meanwhile, the two-dimensional (2D) Dirac points exhibit the Berry vortices, leading to the quantized dual axial rotational energies. Such a quantization naturally generalizes to the three-dimensional (3D) nodal-loop semimetals, where the nodal loops host the Berry vortex lines. The ${\mathbb{Z}}₂$ monopoles bring about additional dual axial rotational energies, which originate from the links with additional nodal lines. Nontrivial band geometry generically induces finite spread in the Wannier functions. While the spread manifest quantized lower bounds from the Berry charges, logarithmic divergences occur from the Berry vortices. The band geometry at the band crossings may be probed experimentally by a periodic-drive measurement.