C-P-T anomaly matching in bosonic quantum field theory and spin chains

TL;DR

This work identifies a mixed 't Hooft anomaly among charge conjugation $C$, parity $P$, and time reversal $T$ in the 1+1D $O(3)$ nonlinear sigma model at $\theta=\pi$, uncovered by gauging the discrete symmetries on the nonorientable manifold $\mathbb{RP}^2$. The authors demonstrate that topological charge becomes half-integer on $\mathbb{RP}^2$, leading to $C$-symmetry breaking under these backgrounds and a nontrivial anomaly that cannot be canceled by local counterterms. Anomaly matching then implies that the ground state cannot be trivially gapped when $C$, $P$, and $T$ are preserved, with checks against the solvable XYZ model and semiclassical analyses supporting the picture. The paper further extends the CPT anomaly to $\mathbb{C}P^{N-1}$ models for even $N$ and discusses implications for microscopic spin chains, showing the anomaly persists even upon breaking the full $SO(3)$ symmetry, thereby generalizing the LSM-type constraints beyond conventional symmetry groups.

Abstract

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 $N$.

Figures (5)

• Figure 1: A cartoon of a spin chain without $\bmod 2$ translational symmetry. The red/blue links indicate that the interactions are different. The dashed line indicates the axis around which parity is a good symmetry.
• Figure 2: A figure of a half-instanton configuration on a disk $\mathbb RP^2$, which is topologically a disk whose opposite points on the edges are identified up to a transformation $\bm n\rightarrow -\bm n$. The red arrows represent the unit vector $\bm n$. It is clear that the configuration depicted has a half-integer winding number.
• Figure 3: A figure illustrating the twisted boundary condition for reflection symmetries with the $\mathbb{C}P^1$ representation.
• Figure 4: A cartoon of a spin chain.
• Figure 5: The depiction of an open $2$-torus and contours.