Cross-correlated topological response of fragile and higher-order topological states

Twisted two-dimensional electronic systems can display various competing ordered states, possessing rich topological properties. Apart from correlated insulators and chiral superconductors, supporting net Chern numbers for occupied states, it is also possible to realize higher-order and fragile topological states. Usually, higher-order and fragile topological states do not support gapless edge modes. In this work, we probe the quantized, topological response of such phases. We identify the precise nature of spin and orbital dependent twisted boundary conditions required for quantized pumping of electric charge. Many d-wave ordered states in particle-hole and particle-particle channels are shown to possess such topological properties.