Deconfinement Transition for Quarks on a Line

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We examine the statistical mechanics of a 1-dimensional gas of both adjoint and fundamental representation quarks which interact with each other through 1+1-dimensional U(N) gauge fields. Using large-N expansion we show that, when the density of fundamental quarks is small, there is a first order phase transition at a critical temperature and adjoint quark density which can be interpreted as deconfinement. When the fundamental quark density is comparable to the adjoint quark density, the phase transition becomes a third order one. We formulate a way to distinguish the phases by considering the expectation values of high winding number Polyakov loop operators.