Enhancement of exciton radius near a band-gap closing through quantum geometry
Exciton engineering traditionally focuses on modifying semiclassical material properties, such as the effective mass and dielectric screening, while largely overlooking the quantum geometry of the underlying electron and hole Bloch states. This approximation is adequate for many materials but breaks down near a topological band-gap closing, where the quantum metric around the band extrema becomes strongly enhanced. In this regime, Bloch states at different momenta become less similar, reducing the projected electron--hole Coulomb matrix elements and consequently weakening exciton binding. We demonstrate this mechanism in a spin--orbit-coupled Lieb-lattice model tuned toward a topological phase transition. The suppressed Coulomb matrix elements narrow the exciton wavefunction in momentum space, leading to an enlarged exciton radius in real space. This increase in exciton size produces an experimentally accessible enhancement of the weak-field diamagnetic response. Our results show that quantum geometry can fundamentally reshape exciton properties near a topological phase transition, revealing a previously underexplored route for engineering excitonic states.