Particle theory

Abstract:

We find a strong-to-weak coupling cross-over in D 2 + 1 SU(N) lattice gauge theories that appears to become a third-order phase transition at N = ∞, in a similar way to the Gross-Witten transition in the D = 1 + 1 SU(N → ∞) lattice gauge theory. There is, in addition, a peak in the specific heat at approximately the same coupling that increases with N, which is connected to ZN monopoles (instantons), reminiscent of the first order bulk transition that occurs in D = 3 + 1 for N ≥ 5. Our calculations are not precise enough to determine whether this peak is due to a second-order phase transition at N = ∞ or to a third-order phase transition with different critical behaviour to that of the Gross-Witten transition. We investigate whether the trace of the Wilson loop has a non-analyticity in the coupling at some critical area, but find no evidence for this. However we do find that, just as one can prove occurs in D = 1 + 1, the eigenvalue density of a Wilson loop forms a gap at N = ∞ at a critical value of its trace. We show that this gap formation is in fact a corollary of a remarkable similarity between the eigenvalue spectra of Wilson loops in D = 1 + 1 and D = 2+1 (and indeed D = 3 + 1): for the same value of the trace, the eigenvalue spectra are nearly identical. This holds for finite as well as infinite N; irrespective of the Wilson loop size in lattice units; and for Polyakov as well as Wilson loops.