Pumping chirality in three dimensions

Using bosonization, which maps fermions coupled to a Z₂ gauge field to a qubit system, we give a simple form for the three-fermion nontrivial quantum cellular automaton (QCA) as realizing a phase depending on the framing of flux loops. We relate this framing-dependent phase to a pump of eight copies of a p+ip state through the system. We give a resolution of an apparent paradox, namely that the pump is a shallow depth circuit (albeit with tails), while the QCA is nontrivial. We discuss also the pump of fewer copies of a p+ip state and describe its action on topologically degenerate ground states. One consequence of our results is that a pump of n p+ip states generated by a free Fermi evolution is a free fermion unitary characterized by a nontrivial winding number n as a map from the third homotopy group of the Brilliouin Zone 3-torus to that of SU (N₁), where N₁ is the number of bands. Using our simplified form of the QCA, we give higher-dimensional generalizations that we conjecture are also nontrivial QCAs, and we discuss the relation to Chern-Simons theory.