Universal low-temperature behavior of two-dimensional lattice scalar chromodynamics

Abstract

We study the role that global and local non-Abelian symmetries play in two-dimensional (2D) lattice gauge theories with multicomponent scalar fields. We start from a maximally $\mathrm{O}(M)$-symmetric multicomponent scalar model. Its symmetry is partially gauged to obtain an $\mathrm{SU}({N}_{c})$ gauge theory (scalar chromodynamics) with global $\mathrm{U}({N}_{f})$ (for ${N}_{c}\ensuremath{\ge}3$) or $\mathrm{Sp}({N}_{f})$ symmetry (for ${N}_{c}=2$), where ${N}_{f}>1$ is the number of flavors. Correspondingly, the fields belong to the coset ${S}^{M}/\mathrm{SU}({N}_{c})$ where ${S}^{M}$ is the $M$-dimensional sphere and $M=2{N}_{f}{N}_{c}$. In agreement with the Mermin-Wagner theorem, the system is always disordered at finite temperature and a critical behavior only develops in the zero-temperature limit. Its universal features are investigated by numerical finite-size scaling methods. The results show that the asymptotic low-temperature behavior belongs to the universality class of the 2D ${\mathrm{CP}}^{{N}_{f}\ensuremath{-}1}$ field theory for ${N}_{c}>2$ and to that of the 2D $\mathrm{Sp}({N}_{f})$ field theory for ${N}_{c}=2$. These universality classes correspond to 2D statistical field theories associated with symmetric spaces that are invariant under $\mathrm{Sp}({N}_{f})$ transformations for ${N}_{c}=2$ and under $\mathrm{SU}({N}_{f})$ for ${N}_{c}>2$. These symmetry groups are the same invariance groups of scalar chromodynamics, apart from a U(1) flavor symmetry that is present for ${N}_{f}\ensuremath{\ge}{N}_{c}>2$, which does not play any role in determining the asymptotic behavior of the model.