Diagnosis of universal geometric responses of fractional quantum Hall liquids

Geometric response of topological quantum liquids is expected to reveal rich phenomenon. However, explicit demonstration or identification of the geometric response in a microscopic model is very challenging. Here, we demonstrate that Dehn-twist deformation is able to reveal both the universal modular properties and the microscopic features. We provide numerical evidences for various fractional quantum Hall (FQH) states, including fermionic and bosonic Laughlin states, Hierarchy states, Halperin states and Moore-Read states, by means of exact diagonalization calculations. We conclusively show, geometric transformation applied on torus geometry gives rise to a viscosity related Berry phase, which reflects the geometric metric of elementary FQH droplets. It also captures intrinsic modular information like topological spin and chiral central charge. These findings not only provide a unified description of Berry phase induced by geometric deformation, but also provide a systematical way to inspect the incompressibility of a gapped topological order in the projected Landau level.