Pumping Chirality in Three Dimensions

Matthew Hastings
Microsoft Research

Using bosonization, which maps fermions coupled to a Z_2 gauge field to a qubit system, we give a simple form for the non-trivial 3-fermion quantum cellular automaton (QCA) as a unitary operator realizing a phase depending on the framing of flux loops, building off work by Shirley et al. We relate this framing dependent phase to a pump of 8 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 non-trivial winding number n as a map from the third homotopy group of the Brilliouin Zone 3-torus to that of SU(Nb), where Nb 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. Joint work with L. Fidkowski.

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