With the advances of quantum simulators in implementing various quantum many-body states, it is important to find efficient ways to characterize and measure many-body states, without resorting to full quantum state tomography. Specifically, in contrast to electronic materials, where the measurements are mainly within the linear-response paradigm, quantum simulators offer unique access to the full wave function that inspires novel probing approaches. In this talk, I discuss how various quantities, such as entanglement spectrum, symmetry-protected topological invariants, and fractional many-body Chern number could be extracted. In the latter case, we show how such an invariant can be measured, using a single wave function, without the knowledge of the Hamiltonian. This should be contrasted to the conventional way, where on requires a family of many-body wave functions parameterized by twist angles in order to calculate the Berry curvature.
Science Advance 6, 3666 (2020)