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Toss a handful of phosphorus into silicon, and at a certain impurity concentration the material suddenly starts conducting electricity. This is a metal-insulator phase transition, and it happens abruptly. But what exactly changes in the electron wave functions when an insulator becomes a metal? The answer can be extracted from how the material absorbs light.

The authors took the optical conductivity, how strongly an alternating electric field shakes charges, and computed its first negative moment. This quantity, like a sensitive dynamometer, measures quantum geometry: how 'spread out' the local electron wave functions are. The result is a length ℓ — the characteristic size of the region where an electron still remains itself. Away from the transition, when donor phosphorus atoms sit alone, ℓ is almost exactly the Bohr radius. That's about 2 nanometers, like a lone hydrogen atom.

As the impurity concentration approaches the critical value, the length ℓ grows. But instead of increasing smoothly, like the correlation length (the typical distance over which fluctuations decay), it makes a leap: right at the transition point it goes to infinity. This means electron states suddenly delocalize across the entire sample. Such a jump is a consequence of the fact that in the three-dimensional case, the main contribution to the sum comes from ultraviolet (that is, very fast) processes, not slow critical fluctuations. Quantum geometry turns out to be 'nearsighted' to the slow dances of charges near the transition.

This result not only provides a new experimental probe for quantum geometry in disordered systems but also sheds light on the century-old Herzfeld criterion: when the polarizability of atoms (the ability of the electron cloud to deform) reaches a critical value, the material must become a metal. The quantum-geometric correction to the Clausius-Mossotti equation, which describes the dielectric response, makes it more accurate, as if an old recipe finally got its missing spice. It turns out that the 'swelling' of donor polarizability, driven by quantum geometry, leads to the divergence of the dielectric constant, and with it, the birth of a metal.

Original abstract

as written by the authors arXiv:2606.21648

Quantum geometry has emerged as a unifying concept across condensed matter physics, underlying phenomena from nonlinear topological response to flat-band superconductivity. While usually formulated within band theory, quantum geometry remains meaningful in disordered interacting systems~\cite{resta1999electron}. Here we show that the first negative moment of the optical conductivity -- proportional to the zero temperature quantum Fisher information as a bound on the multipartite entanglement -- provides an experimental probe of quantum geometry across the three-dimensional metal-insulator quantum phase transition in phosphorus-doped silicon. We extract a quantum geometric length $$\ell$$ that characterizes the local wavefunctions. Far from the transition, this length is almost coincident with the Bohr radius of the hydrogenic phosphorus donors, reflecting their atomic-scale quantum geometry. Approaching the transition, $$\ell$$ is enhanced, but does not diverge continuously like a correlation length; it jumps discontinuously to infinity at the critical point. This reflects the UV domination of the sum rule in three dimensions that renders it insensitive to the critical fluctuations driving the diverging dielectric constant and correlation length. Its enhancement demonstrates a ``puffing" of the donor polarizability volume of quantum geometric origin, which yields a quantum geometric corrected Clausius-Mossotti description in closer agreement with the diverging dielectric response and provides a quantum mechanical foundation for the century-old Herzfeld metallization criterion.