C-P-T anomaly matching in bosonic quantum field theory and spin chains
We consider the $O(3)$ nonlinear sigma model with the $θ$-term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $θ$, and enjoys the periodicity in $θ\rightarrow θ+2π$. At $θ=0,π$ it also has a charge-conjugation $C$-symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable $\mathbb RP^2$ manifold, after which the $2π$ periodicity of $θ$ and the $C$ symmetry at $θ=π$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation $C$, parity $P$, and time-reversal $T$ symmetries when $θ=π$. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as $C$, $P$ and $T$ are all good symmetries of the theory. We make several consistency checks with various semi-classical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity and time reversal (but not involving the $SO(3)$-spin symmetry), requiring that the ground state is never trivially gapped, even if $SO(3)$ spin symmetry is explicitly and completely broken. We also consider generalizations to $\mathbb{C}P^{N-1}$ models and show that the $C$-$P$-$T$ anomaly exists for even $N$.