Wegner flow-equations for electromagnetic response of many-body Hamiltonians

We formulate a systematic approach for investigating the low-energy description of electromagnetic responses. Beginning with a generic microscopic Hamiltonian, we employ flow equations to extract a low-energy effective response theory. This approach permits the systematic, order-by-order inclusion of contributions from coupling to high-energy degrees of freedom. We explicitly compute and detail the leading-order results and then apply these findings to the study of electromagnetic responses in a variety of multi-band electron systems, including twisted bilayer/trilayer moir'e graphene.