A Novel Inequality in Quantum Geometry

Over the last half century, topology has maintained its status as an invaluable tool in understanding condensed matter phases. Recently, more and more works have confirmed that the geometric features of wavefunctions can directly impact topological quantities. Here we unveil a new inequality between band geometry and topology, demonstrating that this feature is a general property of quantum mechanics. We then test our result against existing works, including the quantum speed limit in adiabatic evolution, the geometric contribution to electron-phonon coupling, the superfluid weight, and the minimal spread of localized Wannier functions, displaying the physical significance of our inequality.