Phase diagram of the lattice G(2) Higgs Model

We study the phases and phase transition lines of the finite temperature G(2) Higgs model. Our work is based on an efficient local hybrid Monte-Carlo algorithm which allows for accurate measurements of expectation values, histograms and susceptibilities. On smaller lattices we calculate the phase diagram in terms of the inverse gauge coupling $\beta$ and the hopping parameter $\kappa$. For $\kappa\to 0$ the model reduces to G(2) gluodynamics and for $\kappa\to\infty$ to SU(3) gluodynamics. In both limits the system shows a first order confinement-deconfinement transition. We show that the first order transitions at asymptotic values of the hopping parameter are almost joined by a line of first order transitions. A careful analysis reveals that there exists a small gap in the line where the first order transitions turn into continuous transitions or a cross-over region. For $\beta\to\infty$ the gauge degrees of freedom are frozen and one finds a nonlinear O(7) sigma model which exhibits a second order transition from a massive O(7)-symmetric to a massless O(6)-symmetric phase. The corresponding second order line for large $\beta$ remains second order for intermediate $\beta$ until it comes close to the gap between the two first order lines. Besides this second order line and the first order confinement-deconfinement transitions we find a line of monopole-driven bulk transitions which do not interfer with the confinement-deconfinment transitions.