  We determine the phase diagram of the Abelian-Higgs model in one spatial
dimension and time (1+1D) on a lattice. We identify a line of first order phase
transitions separating the Higgs region from the confined one. This line
terminates in a quantum critical point above which the two regions are
connected by a smooth crossover. We analyze the critical point and find
compelling evidences for its description as the product of two non-interacting
systems, a massless free fermion and a massless free boson. However, we find
also some surprizing results that cannot be explained by our simple picture,
suggesting this newly discovered critical point to be an unusual one.
