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Phase Transitions at Finite Temperature and Dimensional Reduction for Fermions and Bosons

In a recent Letter we discussed the fact that large-$N$ expansions and computer simulations indicate that the universality class of the finite temperature chiral symmetry restoration transition in the 3D Gross-Neveu model is mean field theory. This was seen to be a counterexample to the standard 'sigma model' scenario which predicts the 2D Ising model universality class. In this article we present more evidence, both theoretical and numerical, that this result is correct. We develop a physical picture for our results and discuss the width of the scaling region (Ginzburg criterion), $1/N$ corrections, and differences between the dynamics of BCS superconductors and Gross-Neveu models. Lattices as large as $12 \times 72^2$ are simulated for both the $N=12$ and $N=4$ cases and the numerical evidence for mean field scaling is quite compelling. We point out that the amplitude ratio for the model's susceptibility is a particulartly good observable for distinguishing between the dimensional reduction and the mean field scenerios, because this universal quantity differs by almost a factor of $20$ in the two cases. The simulations are done close to the critical point in both the symmetric and broken phases, and correlation lengths of order $10$ are measured. The critical indices $β_{mag}$ and $δ$ also pick out mean field behavior. We trace the breakdown of the standard scenario (dimensional reduction and universality) to the composite character of the mesons in the model. We point out that our results should be generic for theories with dynamical symmetry breaking, such as Quantum Chromodynamics. We also simulated the $O(2)$ model on $8 \times 16^3$ lattices to establish that our methods give the results of dimensional reduction in purely bosonic