The SU(3)/Z3 QCD(adj) deconfinement transition via the gauge theory/“affine” XY-model duality

Abstract

Earlier, two of us and M. Ünsal [1] showed that a class of 4d gauge theories, when compactified on a small spatial circle of size L and considered at temperatures β−1 near the deconfinement transition, are dual to 2d “affine” XY-spin models. We exploit this duality to study the deconfinement phase transition in $ \mathrm{SU}(3)/{{\mathbb{Z}}_3} $ gauge theories with n f > 1 massless adjoint Weyl fermions, QCD(adj) on $ {{\mathbb{R}}^2}\times \mathbb{S}_{\beta}^1\times \mathbb{S}_L^1 $ . The dual “affine” XY-model describes two “spins” — compact scalars taking values in the SU(3) root lattice. The spins couple via nearest-neighbor interactions and are subject to an “external field” perturbation preserving the topological $ \mathbb{Z}_3^t $ and a discrete $ \mathbb{Z}_3^{{{d_{\upchi}}}} $ subgroup of the anomaly-free chiral symmetry of the 4d gauge theory. The equivalent Coulomb gas representation of the theory exhibits electric-magnetic duality, which is also a high-/low-temperature duality. A renormalization group analysis suggests — but is not convincing, due to the onset of strong coupling — that the self-dual point is a fixed point, implying a continuous deconfinement transition. Here, we study the nature of the transition via Monte Carlo simulations. The $ \mathbb{Z}_3^t\times \mathbb{Z}_3^{{{d_{\upchi}}}} $ order parameter, its susceptibility, the vortex density, the energy per spin, and the specific heat are measured over a range of volumes, temperatures, and “external field” strengths (in the gauge theory, these correspond to magnetic bion fugacities). The finite-size scaling of the susceptibility and specific heat we find is characteristic of a first-order transition. Furthermore, for sufficiently large but still smaller than unity bion fugacity (as can be achieved upon an increase of the $ \mathbb{S}_L^1 $ size), at the critical temperature we find two distinct peaks of the energy probability distribution, indicative of a first-order transition, as has been seen in earlier simulations of the full 4d QCD(adj) theory. We end with discussions of the global phase diagram in the β-L plane for different numbers of flavors.