Cluster State as a Noninvertible Symmetry-Protected Topological Phase.

We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics.

Quantization of Axion-Gauge Couplings and Noninvertible Higher Symmetries.

We derive model-independent quantization conditions on the axion couplings (sometimes known as the anomaly coefficients) to the standard model gauge group [SU(3)×SU(2)×U(1)_{Y}]/Z_{q} with q=1, 2, 3, 6. Using these quantization conditions, we prove that any QCD axion model to the right of the E/N=8/3 line on the |g_{aγγ}|-m_{a} plot must necessarily face the axion domain wall problem in a postinflationary scenario. We further demonstrate the higher-group and noninvertible global symmetries in the standard model coupled to a single axion.

Noninvertible Time-Reversal Symmetry.

In gauge theory, it is commonly stated that time-reversal symmetry only exists at θ=0 or π for a 2π-periodic θ angle. In this Letter, we point out that in both the free Maxwell theory and massive QED, there is a noninvertible time-reversal symmetry at every rational θ angle, i.e.

Noninvertible Global Symmetries in the Standard Model.

We identify infinitely many noninvertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the U(1)_{A} axial symmetry because of the Adler-Bell-Jackiw anomaly, for every rational angle 2πp/N, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic U(1) gauge field.