Breaking Free with AI: The Deconfinement Transition

Employing supervised machine learning techniques, we investigate the deconfinement phase transition within $4$-dimensional $SU(2)$ Yang-Mills (YM) theory, compactified on a small circle and endowed with center-stabilizing potential. This exploration encompasses scenarios both without and with matter in either the fundamental or adjoint representations. Central to our study is a profound duality relationship, intricately mapping the YM theory onto an XY-spin model with $\mathbb Z_p$-preserving perturbations. The parameter $p$ embodies the essence of the matter representation, with values of $p=1$ and $p=4$ for fundamental and adjoint representations, respectively, while $p=2$ corresponds to pure YM theory. The logistic regression method struggles to produce satisfactory results, particularly in predicting the transition temperature. Contrarily, convolutional neural networks (CNNs) exhibit remarkable prowess, effectively foreseeing critical temperatures in cases where $p=2$ and $p=4$. Furthermore, by harnessing CNNs, we compute critical exponents at the transition, aligning favorably with computations grounded in conventional order parameters. Taking our investigation a step further, we use CNNs to lend meaning to phases within YM theory with fundamental matter. Notably, this theory lacks conventional order parameters. Interestingly, CNNs manage to predict a transition temperature in this context. However, the fragility of this prediction under variations in the boundaries of the training window undermines its utility as a robust order parameter. This outcome underscores the constraints inherent in employing supervised machine learning techniques as innovative substitutes for traditional order parameters.