Charge pumps, pivot Hamiltonians, and symmetry-protected topological phases

Generalized charge pumps are topological obstructions to trivializing loops in the space of symmetric gapped Hamiltonians. We show that given mild conditions on such pumps, the associated loop has high-symmetry points that must be in distinct symmetry-protected topological (SPT) phases. To further elucidate the connection between pumps and SPTs, we focus on closed paths, "pivot loops", defined by two Hamiltonians, where the first is unitarily evolved by the second "pivot" Hamiltonian. While such pivot loops have been studied as entanglers for SPTs, here we explore their connection to pumps. We construct families of pivot loops that pump charge for various symmetry groups, often leading to SPT phases—including dipole SPTs. Intriguingly, we find examples where nontrivial pumps do not lead to genuine SPTs but still entangle representation-SPTs (RSPTs). We use the anomaly associated with the nontrivial pump to explain the "unnecessary" criticality between these RSPTs. We also find that particularly nice pivot families form circles in Hamiltonian space, which we show are equivalent to the Hamiltonians satisfying the Dolan-Grady relation—known from the study of integrable models. This additional structure allows us to derive more powerful constraints on the phase diagram. Natural examples of such circular loops arise from pivoting with the Onsager-integrable chiral clock models, containing the aforementioned RSPT example. In fact, we show that these Onsager pivots underlie general group cohomology-based pumps in one spatial dimension. Finally, we recast the above in the language of equivariant families of Hamiltonians and relate the invariants of the pump to the candidate SPTs. We also highlight how certain SPTs arise in cases where the equivariant family is labeled by spaces that are not manifolds.