Contributor info: Dr Yuya Tanizaki

Philippe Lecheminant, Yuya Tanizaki, Keisuke Totsuka

SciPost Phys. 18, 183 (2025) · published 10 June 2025 | Toggle abstract · pdf

A general strategy is proposed to explore the low-energy properties of two-dimensional nonlinear $\sigma$ models with $\theta$ terms. We demonstrate its application to nonlinear $\sigma$ models with the target space $SU(N)/H$, which include $\mathbb{C}P^{N-1}$, complex Grassmannian manifolds as well as the flag $SU(N)/U(1)^{N-1}$ and $SU(N)/SO(N)$ manifolds. By analyzing the symmetry and its anomaly content, we realize these nonlinear $\sigma$ models by considering specific deformations of the $SU(N)_1$ conformal field theory. For the flag-manifold $SU(N)/U(1)^{N-1}$ and $SU(N)/SO(N)$ models, those deformations are shown to correspond to the marginal current-current operator with the specific sign which leads to a massless renormalization group flow to the $SU(N)_1$ fixed point. In contrast, a massive regime with a two-fold ground-state degeneracy is found for the $\mathbb{C}P^{N-1}$ ($N >2$) and the Grassmannian nonlinear $\sigma$ models at $\theta=\pi$.

Aleksey Cherman, Theodore Jacobson, Yuya Tanizaki, Mithat Ünsal

SciPost Phys. 8, 072 (2020) · published 5 May 2020 | Toggle abstract · pdf

We show that $2$d adjoint QCD, an $SU(N)$ gauge theory with one massless adjoint Majorana fermion, has a variety of mixed 't Hooft anomalies. The anomalies are derived using a recent mod $2$ index theorem and its generalization that incorporates 't Hooft flux. Anomaly matching and dynamical considerations are used to determine the ground-state structure of the theory. The anomalies, which are present for most values of $N$, are matched by spontaneous chiral symmetry breaking. We find that massless $2$d adjoint QCD confines for $N >2$, except for test charges of $N$-ality $N/2$, which are deconfined. In other words, $\mathbb Z_N$ center symmetry is unbroken for odd $N$ and spontaneously broken to $\mathbb Z_{N/2}$ for even $N$. All of these results are confirmed by explicit calculations on small $\mathbb{R}\times S^1$. We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when $N$ is even. Furthermore, for most values of $N$, $2$d massive adjoint QCD describes a non-trivial symmetry-protected topological (SPT) phase of matter, including certain cases where the number of interacting Majorana fermions is a multiple of $8$. As a result, it fits into the classification of $(1+1)$d SPT phases of interacting Majorana fermions in an interesting way.