Quantum Matter Seminar
Professor Fidkowski's research focus is on identifying and classifying exotic phases of matter. In particular, our group focuses on phases which cannot be understood in terms of traditional symmetry breaking Ginzburg-Landau theory, such as fractional quantum Hall phases and topological insulators (TIs). Recently we have been classifying strongly interacting versions of TIs, termed ‘symmetry protected topological’ phases, using tools such as topological quantum field theory and exactly solved models. We are also interested in other settings for realizing topological order, such as at non-zero energy density or in driven (Floquet) systems, aided by many-body localization.
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).