New topological invariants in crystalline quantum states

While crystalline topological invariants have been mathematically classified in detail, it is still unclear how to fully measure these invariants in a given lattice model or ground state wave function. Here we discuss two non-perturbative approaches to do so in (2+1)-dimensional fermionic topological states with the symmetries of a square lattice and U(1) charge conservation. The first approach is to measure the response to inserting symmetry defects such as lattice disclinations and dislocations; the second is to measure expectation values of partial symmetry operators in the ground state. Our results apply in particular to topological insulators and Chern insulators, and are robust to interactions. As an application, we study the square lattice Hofstadter model and compute several new crystalline invariants, which fully characterize the model.