Anomaly and global inconsistency matching: $θ$-angles, $SU(3)/U(1)^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations

TL;DR

This work analyzes anomaly and global inconsistency constraints in the $SU(3)/[U(1)\times U(1)]$ nonlinear sigma model and its linearized forms, revealing how 't Hooft anomalies at special $\theta$-angles forbid trivially gapped ground states and constrain the phase diagram. By gauging the PSU(3) flavor symmetry and examining mixed anomalies with $\mathbb{Z}_3$ permutation and charge-conjugation symmetries, the authors connect LSM-type bounds to continuum field theory, and extend the framework to $SU(N)/U(1)^{N-1}$, establishing analogous anomalies and global inconsistencies. They then relate these results to $SU(N)$ WZW models, deriving anomaly-based RG-flow constraints and showing a consistent mapping between the UV sigma models and IR CFTs. The analysis is augmented by a linear sigma-model construction, a free-photon limit, and a circle-compactification scheme that preserves the anomalies, with brief discussion of 2+1D extensions and domain-wall physics, highlighting a unified approach to spin-chain physics and its field-theoretic descendants.

Abstract

We discuss the $SU(3)/[U(1)\times U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\times U(1)$ gauge theories and therefore allow for two topological $θ$-angles. These models provide a field theoretic description of the $SU(3)$ chains. We show that, for particular values of $θ$-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 $θ$-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 $\mathbb Z_3\times\mathbb Z_3\subset SU(3)/\mathbb Z_3$. 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 $θ$-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 $SU(3)$ Wess-Zumino-Witten model. We generalize the discussion to the $SU(N)/U(1)^{N-1}$ nonlinear sigma models as well as the 't Hooft anomaly of the $SU(N)$ 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.

Figures (3)

• Figure 1: (color online) The plot of the phase diagram of $SU(3)/[U(1)\times U(1)]$ 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 $PSU(3)\times \mathbb{Z}_3$ 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 counter-terms are needed to restore the corresponding $\mathsf C$-symmetries when the $SU(3)/\mathbb Z_3$ 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 non-trivial ground state, or that they are separated by a phase transition.
• Figure 2: (color online) Possible scenarios consistent with global inconsistency. The red blobs are $\mathbb{Z}_3$ symmetric points with $PSU(3)\times \mathbb{Z}_3$ '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 $PSU(3)$ 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. \ref{['sec:anomaly']}. 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.