Quantum metric induced hole dispersion and emergent particle-hole symmetry in topological flat bands*

The emergent hole dispersion in flat bands is an invaluable platform to study the interplay of quantum geometry and electron-electron interaction with a relatively simple setting. In this work, we find that the hole dispersion in ideal bands has a linear relationship with the trace of the quantum geometry tensor at every k-point for a wide range of interactions to a good approximation. Next, we give a microscopic analysis on the hole dispersion and show that the linear relationships for short-range and long-range interactions in k-space have different origins. Moreover, we show how to exploit this observation to engineer particle-hole symmetry in a Chern band with fluctuating quantum geometry. Our results will be useful for further studying the physics in particle-hole symmetric flat bands both in theory and in experiment.