Deconfinement in $ \mathcal{N}=1 $ super Yang-Mills theory on $ {{\mathbb{R}}^3}\times {{\mathbb{S}}^1} $ via dual-Coulomb gas and “affine” XY-model

Abstract

We study finite-temperature N=1 SU(2) super Yang-Mills theory, compactified on a spatial circle of size L with supersymmetric boundary conditions. In the semiclassical small-L regime, a deconfinement transition occurs at T_c <<1/L. The transition is due to a competition between non-perturbative topological "molecules"---magnetic and neutral bion-instantons---and electrically charged W-bosons and superpartners. Compared to deconfinement in non-supersymmetric QCD(adj) arXiv:1112.6389, the novelty is the relevance of the light modulus scalar field. It mediates interactions between neutral bions (and W-bosons), serves as an order parameter for the Z_2^{L} center symmetry associated with the non-thermal circle, and explicitly breaks the electric-magnetic (Kramers-Wannier) duality enjoyed by non-supersymmetric QCD(adj) near T_c. We show that deconfinement can be studied using an effective two-dimensional gas of electric and magnetic charges with (dual) Coulomb and Aharonov-Bohm interactions, or, equivalently, via an XY-spin model with a symmetry-breaking perturbation, where each system couples to the scalar field. To study the realization of the discrete R-symmetry and the Z_2^{beta} thermal and Z_2^{L} non-thermal center symmetries, we perform Monte Carlo simulations of both systems. The dual-Coulomb gas simulations are a novel way to analyze deconfinement and provide a new venue to study the phase structure of a class of two-dimensional condensed matter models that can be mapped into dual-Coulomb gases. Our results indicate a continuous deconfinement transition, with Z_2^{L} remaining unbroken at the transition. Thus, the SYM transition appears similar to the one in SU(2) QCD(adj) arXiv:1112.6389 and is also likely to be characterized by continuously varying critical exponents.