Many-Body Quantum Geometric Effects on Trapped Ultracold Bosons

Quantum geometric effects in uncorrelated systems are characterized by the Berry curvature and quantum metric. Beyond those, we propose three gauge-independent tensors describing quantum geometric effects on local interaction between correlated particles. We derive an effective hydrodynamic theory for ultracold bosons in optical lattices. Ground states and collective modes of superfluids in isotropic harmonic traps are solved for highly symmetric lattices. In a dynamic process, the amplitude and phase shift of an excited breathing mode are determined by the geometric properties of Bloch wavefunctions. We also give a tight-binding model of a bipartite square lattice with nontrivial quantum geometric effects. Our discovery advances the connections between the modern band theory and quantum many-body physics.