On a more fundamental level, the topological phases arise from the geometric properties of the quantum wave function, which provide deep information about quantum states beyond the E-k dispersion.  Prominent quantum geometrical properties include quantum metric, Berry curvature, Berry connection, etc. These quantum geometrical properties capture the coherences between orbitals, atomic positions, and spins within the unit cell. These quantum geometrical properties are expected to strongly modify the motion and dynamics of quasiparticles, giving rise to highly unconventional (sometimes very surprising) responses to external driving. However, apart from a few areas such as the intrinsic anomalous Hall in magnets, in general, how quantum geometry affects the responses of quantum materials to external fields is largely unknown. Therefore, searching for new quantum geometrical responses is an important frontier in the research of quantum materials. 

     Recent theoretical advances show that quantum geometry can manifest dramatically in the nonlinear responses to external electric or magnetic field, especially in the low frequency regimes where ℏω corresponds to important low-energy physics. Indeed, our recent mid-infrared photocurrent measurements have detected the chirality of Weyl fermions in the Weyl semimetal TaAs, and our recent nonlinear transport studies have mapped out the nontrivial Berry curvature properties of monolayer and bilayer WTe2. Many other nonlinear responses have been proposed and await exploration. Moreover, even the theory is evolving quickly with new proposals constantly emerging. 

   The nonlinear optical measurements can be generalized into high magnetic field quantum Hall and fractional quantum Hall regimes. Recent transport and STM experiments on quantum Hall states of ABA graphene and bismuth single crystals have revealed very intriguing spontaneous rotational symmetry breaking (a nematic phase). Our group will develop optical/THz SHG at low temperatures under magnetic field to explore the rich symmetry breaking in the quantum Hall states of graphene and other novel quantum materials.