$\mathrm{SU}(N)\ensuremath{\bigotimes}\mathrm{SU}(N)$ spin systems in two dimensions are studied for $N=2, 3, \mathrm{and} 4$ on 10\ifmmode\times\else\texttimes\fi{}10, 20\ifmmode\times\else\texttimes\fi{}20, and 30\ifmmode\times\else\texttimes\fi{}30 lattices with use of Monte Carlo methods. Variant forms of the action are considered. In addition to crossover phenomena, unexpected first-order transitions in multidimensional phase diagrams are discovered. These transitions are related to the presence of ${Z}_{N}$ vortices which experience short-range interactions. Contrasts with $\mathrm{SU}(N)$ gauge theories in four dimensions are drawn.$\mathrm{SU}(N)\ensuremath{\bigotimes}\mathrm{SU}(N)$ spin systems in two dimensions are studied for $N=2, 3, \mathrm{and} 4$ on 10\ifmmode\times\else\texttimes\fi{}10, 20\ifmmode\times\else\texttimes\fi{}20, and 30\ifmmode\times\else\texttimes\fi{}30 lattices with use of Monte Carlo methods. Variant forms of the action are considered. In addition to crossover phenomena, unexpected first-order transitions in multidimensional phase diagrams are discovered. These transitions are related to the presence of ${Z}_{N}$ vortices which experience short-range interactions. Contrasts with $\mathrm{SU}(N)$ gauge theories in four dimensions are drawn.

Physical Review LettersDec 21, 198114 citations

Vortices in the Su(n) X Su(n) Spin Systems in Two Dimensions.

Robert J. Kares

Jan 1, 19820 citations

Anomaly Matching and Symmetry-Protected Critical Phases in SU(N) Spin Systems in 1+1 Dimensions

Yuan Yao, Chang-Tse Hsieh, Masaki Oshikawa

We study (1+1)-dimensional SU(N) spin systems in the presence of global SU(N) rotation and lattice translation symmetries. Knowing the mixed anomaly of the two symmetries at low energy, we identify, by the anomaly matching argument, a topological index for the spin model-the total number of Young-tableau boxes of spins per unit cell modulo N-characterizing the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases. As an example of the latter case, we show that only a class of SU(N) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries. Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry. Our results agree with several examples known in previous studies of SU(N) models, and predict a general constraint on the structure factor which is measurable in experiments.We study (1+1)-dimensional SU(N) spin systems in the presence of global SU(N) rotation and lattice translation symmetries. Knowing the mixed anomaly of the two symmetries at low energy, we identify, by the anomaly matching argument, a topological index for the spin model-the total number of Young-tableau boxes of spins per unit cell modulo N-characterizing the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases. As an example of the latter case, we show that only a class of SU(N) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries. Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry. Our results agree with several examples known in previous studies of SU(N) models, and predict a general constraint on the structure factor which is measurable in experiments.

arXiv:1805.06885Physical Review LettersOct 31, 201957 citations

On the phase transitions in quantum spin systems

R. Micnas, K. A. Chao

Physics Letters AJul 1, 19853 citations

Generalized symmetry enriched criticality in (3+1)d

Benjamin Moy

We construct two classes of continuous phase transitions in 3+1 dimensions between gapped phases that break distinct generalized global symmetries. Our analysis focuses on SU(N)/\mathbb{Z}_N S U ( N ) / ℤ N gauge theory coupled to N_f N f flavors of Majorana fermions in the adjoint representation. For N N even and sufficiently large odd N_f N f , upon imposing time-reversal symmetry and an SO(N_f) S O ( N f ) flavor symmetry, the massless theory realizes a quantum critical point between a gapped phase in which a \mathbb{Z}_N ℤ N one-form symmetry is completely broken and a phase where it is broken to \mathbb{Z}_2 ℤ 2 , leading to \mathbb{Z}_{N/2} ℤ N / 2 topological order. We characterize the possible patterns of symmetry fractionalization in these phases and provide an explicit lattice model that exhibits the transition. The critical point has an enhanced symmetry, which includes non-invertible analogues of time-reversal symmetry. Enforcing a non-invertible time-reversal symmetry and the SO(N_f) S O ( N f ) flavor symmetry, for N N and N_f N f both odd, we demonstrate that this critical point can appear between a topologically ordered phase and a phase that spontaneously breaks the non-invertible time-reversal symmetry, furnishing an analogue of deconfined quantum criticality for generalized symmetries.We construct two classes of continuous phase transitions in 3+1 dimensions between gapped phases that break distinct generalized global symmetries. Our analysis focuses on SU(N)/\mathbb{Z}_N S U ( N ) / ℤ N gauge theory coupled to N_f N f flavors of Majorana fermions in the adjoint representation. For N N even and sufficiently large odd N_f N f , upon imposing time-reversal symmetry and an SO(N_f) S O ( N f ) flavor symmetry, the massless theory realizes a quantum critical point between a gapped phase in which a \mathbb{Z}_N ℤ N one-form symmetry is completely broken and a phase where it is broken to \mathbb{Z}_2 ℤ 2 , leading to \mathbb{Z}_{N/2} ℤ N / 2 topological order. We characterize the possible patterns of symmetry fractionalization in these phases and provide an explicit lattice model that exhibits the transition. The critical point has an enhanced symmetry, which includes non-invertible analogues of time-reversal symmetry. Enforcing a non-invertible time-reversal symmetry and the SO(N_f) S O ( N f ) flavor symmetry, for N N and N_f N f both odd, we demonstrate that this critical point can appear between a topologically ordered phase and a phase that spontaneously breaks the non-invertible time-reversal symmetry, furnishing an analogue of deconfined quantum criticality for generalized symmetries.

arXiv:2507.15925SciPost PhysicsDec 3, 20250 citations

$\mathrm{SU}(N)\ensuremath{\bigotimes}\mathrm{SU}(N)$ spin systems in two dimensions are studied for

N=2, 3, \mathrm{and} 4

on 10\ifmmode\times\else\texttimes\fi{}10, 20\ifmmode\times\else\texttimes\fi{}20, and 30\ifmmode\times\else\texttimes\fi{}30 lattices with use of Monte Carlo methods. Variant forms of the action are considered. In addition to crossover phenomena, unexpected first-order transitions in multidimensional phase diagrams are discovered. These transitions are related to the presence of

{Z}_{N}

vortices which experience short-range interactions. Contrasts with

\mathrm{SU}(N)

gauge theories in four dimensions are drawn.