Pumping Chirality in Three Dimensions

• URL: http://arxiv.org/abs/2309.15903v1
• Date: Wed, 27 Sep 2023 18:00:01 GMT
• Title: Pumping Chirality in Three Dimensions
• Authors: Lukasz Fidkowski and Matthew B. Hastings
• Abstract summary: We give a simple form for the non-trivial 3-fermion quantum cellular automaton (QCA) We relate this framing dependent phase to a pump of $8$ copies of a $p+ip$ state through the system. We discuss the pump of fewer copies of a $p+ip$ state, and describe its action on topologically degenerate ground states.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Using bosonization, which maps fermions coupled to a ${\mathbb{Z}}_2$ 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 $SU(N_ b)$, where $N_b$ 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.

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