Quantum Matter Seminar

Starting with a 3 dimensional lattice Hilbert space with an onsite action of a $U(1)$ `vector' symmetry $U(1)_V$, we construct a non-onsite action of an axial symmetry $U(1)_A$ that has a mixed anomaly with $U(1)_V$. This mixed anomaly is reflected in the non-conservation of $U(1)_A$ in the presence of topologically non-trivial $U(1)_V$ gauge field configurations. We discuss the similarities and differences from the usual case of axial and vector $U(1)$ symmetries of a Dirac fermion, in particular highlighting the role of the mixed $U(1)$ - gravitational anomaly. We then use this to construct a lattice operator version of the non-invertible discrete chiral symmetry that remains once the $U(1)_V$ is gauged, and discuss connections between this non-invertible symmetry and quantum cellular automata (QCA).