In two-dimensional systems with quadratic band touching, weak interactions give rise to an exponentially small gap in the spectrum that cannot be captured by any finite order of perturbation theory. Researchers have shown that the singular quantum geometry of such a system turns this elusive scale into algebraically growing responses — in the quantum metric and the optical moment.
The work was carried out by Stefano Bilamur, René Meyer, Johanna Erdmenger and Domenico Di Sante. The preprint was published on 28 September 2026 on arXiv under number 2609.34607. The authors analytically examined the massive quadratic band touching and then tested the results on a microscopic model of an interacting kagome lattice, where the mass of the quantum anomalous Hall phase arises spontaneously due to loop currents.
The quantum metric forms a hot spot in momentum space, whose integral over the Brillouin zone diverges logarithmically. When the gap is dynamically generated by interactions, this integral already grows algebraically. Through the Souza–Wilkins–Martin inverse-frequency sum rule, the same effect manifests itself in the negative first longitudinal optical moment, although the ordinary optical conductivity remains on the order of e²/ℏ, and the absorption threshold is exponentially small.
Imagine a tiny crack in ice: by itself it is almost imperceptible, but the special geometry of the surface turns it into a long fracture line visible from afar. This is exactly how quantum geometry amplifies invisible nonperturbative scales.
The results point to a new mechanism that goes beyond perturbation theory and allow a fresh look at correlated two-dimensional materials. According to the preprint, this approach may explain the enhancement of responses in real systems with quadratic band touching.
Quantum geometry acts as an asymptotic amplifier of nonperturbative interaction scales.
