Ising gauge theory at negative temperatures and spin-glasses

We study the $Z\left(2\right)\mathrm{}$ coupled spin-gauge theory at negative gauge coupling. We find two frustrated phases and a disordered phase in both three and four dimensions. All critical lines are of first order in four dimensions. In three dimensions, the frustration ↔ frustration transition as well as one of the frustration ↔ disorder lines appear to be of second order. We find that the contour in the coupling parameter space along which the unit Wilson loop vanishes intersects a transition line only at infinite negative gauge coupling and infinite spin-spin coupling. This indicates that, contrary to a recent conjecture, this contour cannot describe a transition in spin-glasses.