Yugo Onishi, MIT

Universal bounds in topological phases

Topology plays a fundamental role in condensed matter physics. Even when ground states share the same symmetries, they can still be topologically distinct, giving rise to distinct topological phases of matter. These phases are characterized by topological invariants, such as the Chern number. Topological invariants can manifest in physical observables, such as the quantized Hall conductance in Chern insulators.

In this work, we explore another fundamental aspect of topological phases: the existence of universal bounds on physical observables, which we term topological bounds. Specifically, we establish bounds on two physical observables in Chern insulators: the static structure factor and the energy gap. The bound on the static structure factor is determined solely by the Chern number, while the bound on the energy gap is determined by the Chern number and the electron density. Remarkably, these bounds apply universally to all insulators, including strongly correlated systems such as fractional Chern insulators. As an example, we apply these bounds to moiré transition metal dichalcogenides. We also interpret the topological bounds as geometric relations with quantum geometry.

Our results provide a new perspective on topological phases of matter, demonstrating that the topological invariants can universally manifest as topological bounds.