The Mott transition — where electrons in a metal collectively refuse to conduct, forming an insulator — is conventionally tuned by bandwidth. Narrow bands mean stronger correlations, and when the interaction energy exceeds the kinetic energy, the electrons localize. The textbook picture is an energy competition: kinetic energy wants delocalization, Coulomb repulsion wants localization.
Ding & Claassen (2602.22548) demonstrate that the geometric properties of electronic wavefunctions — the quantum metric, measuring how wavefunctions change shape across momentum space — can independently drive Mott transitions and control the competition between ferromagnetic and antiferromagnetic order. In the Kane-Mele-Hubbard model, tuning quantum geometry while holding the bandwidth fixed triggers metal-insulator transitions and switches the magnetic ground state.
This is not quantum geometry as a secondary effect or a correction to the standard picture. It is a complementary mechanism — a separate dial that turns the same transition. Two systems with identical bandwidths and identical interaction strengths can be on different sides of the Mott transition because their wavefunctions have different geometric structure. The geometry of the quantum states matters independently of their energetics.
The general principle: when a transition is understood through one control parameter (bandwidth), finding that a second parameter (wavefunction geometry) independently controls the same transition reveals that the original parameter was a proxy for a more general condition. The bandwidth tuned the Mott transition because it changed the effective wavefunction overlap — but geometry changes overlap too, through a different route. The transition responds to the overlap, not to the specific mechanism that adjusts it. Understanding which variable is fundamental requires finding the second dial.