Parameterized Families of Toric Code Phase: $em$-duality family and higher-order anyon pumping

Within the toric-code phase, we study parameterized families of topologically ordered states. We construct $1$- and $2$-parameter families of local Hamiltonians and confirm their non-triviality via topological pumping. For the $1$-parameter family, we show that the $em$-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the $2$-parameter case, we construct a ``pump of a pump’’ that transports an $S^1$-family of a system in one lower spatial dimension. Using similar methods, we also present a $1$-parameter family with a higher-order anyon pump that produces corner-localized anyon modes. These constructions provide explicit lattice realizations and concrete diagnostics of family-level topology. We use recently developed boundary algebra methods to study the non-triviality of these families.