Anomaly and global inconsistency matching: $\theta$ angles, $SU\left(3\right)/U{\left(1\right)}^2$ nonlinear sigma model, $SU\left(3\right)$ chains, and generalizations

We discuss the $SU\left(3\right)/\left[U\right(1)\times U(1\left)\right]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U\left(1\right)\times U\left(1\right)$ gauge theories and therefore allow for two topological $\theta$ angles. These models provide a field theoretic description of the $SU\left(3\right)$ chains. We show that, for particular values of $\theta$ angles, a global symmetry group of such systems has a 't Hooft anomaly, which manifests itself as an inability to gauge the global symmetry group. By applying anomaly matching, the ground-state properties can be severely constrained. The anomaly matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin chain from which the field theory descends, and it forbids a trivially gapped ground state for particular $\theta$ angles. We generalize the statement of the LSM theorem and show that 't Hooft anomalies persist even under perturbations which break the spin-symmetry down to the discrete subgroup ${\mathbb{Z}}_3\times {\mathbb{Z}}_3\subset SU\left(3\right)/{\mathbb{Z}}_3$. In addition, the model can further be constrained by applying global inconsistency matching, which indicates the presence of a phase transition between different regions of $\theta$ angles. We use these constraints to give possible scenarios of the phase diagram. We also argue that at the special points of the phase diagram the anomalies are matched by the $SU\left(3\right)$ Wess-Zumino-Witten model. We generalize the discussion to the $SU\left(N\right)/U{\left(1\right)}^{N-1}$ nonlinear sigma models as well as the 't Hooft anomaly of the $SU\left(N\right)$ Wess-Zumino-Witten model, and show that they match. Finally, the $(2+1)$-dimensional extension is considered briefly, and we show that it has various 't Hooft anomalies leading to nontrivial consequences.

Figure 1

The plot of the phase diagram of $SU\left(3\right)/\left[U\right(1)\times U(1\left)\right]$ nonlinear sigma model. The ${\mathbb{Z}}_3$-symmetric points are shown with blobs, and the blue blobs show that there is no 't Hooft anomaly for $PSU\left(3\right)\times {\mathbb{Z}}_3$ while the red ones show that there is the 't Hooft anomaly. The global inconsistency lines in the $({\theta }_1,{\theta }_3)$ plane for the symmetries ${\mathsf{C}}_1,{\mathsf{C}}_2,{\mathsf{C}}_3$ are sketched as red, blue, and green lines. The solid, dashed, and dot-dashed lines indicate that different counterterms are needed to restore the corresponding $\mathsf{C}$-symmetries when the $SU\left(3\right)/{\mathbb{Z}}_3$ symmetry is gauged, indicating that there is a global inconsistency between different-type lines (e.g., between solid and dashed). The inconsistency can be saturated either by at least one of these lines having a nontrivial ground state, or that they are separated by a phase transition.

Figure 2

Possible scenarios consistent with global inconsistency. The red blobs are ${\mathbb{Z}}_3$ symmetric points with $PSU\left(3\right)\times {\mathbb{Z}}_3{}^{\prime }\mathrm{t}$ Hooft anomaly, and the origin (blue blob) is the ${\mathbb{Z}}_3$ symmetric point without anomaly. Blank regions painted with different colors (light blue, orange, green) all correspond to trivially gapped phases, but they are different as SPT phases protected by $PSU\left(3\right)$ symmetry. (Left) The global inconsistency is matched by the spontaneous breaking of $\mathsf{C}$ on thick gray lines. (Right) The global inconsistency is matched by the phase transitions lines (gray curves) separating distinct $\mathsf{C}$-symmetric trivial vacua.

Figure 3

The energy density of the ground state as a function of the two $\theta$ parameters. Notice that the same picture emerges as discussed in Sec. 3. On the 3D plot on the right, it is clear that level crossings occur at the $\mathsf{C}$-symmetric lines, which meet at ${\mathbb{Z}}_3$-cyclic permutation symmetric points which carry a 't Hooft anomaly.