Approaching off-diagonal long-range order for 1+1 -dimensional relativistic anyons

We construct and study relativistic anyons in 1+1 dimensions generalizing well-known models of Dirac fermions. First, a model free is constructed then extended two ways: (i) by adding density-density interactions, as the Luttinger model, (ii) coupling to U(1)-gauge field, Schwinger model. Second, physical properties these extensions are studied. By investigating off-diagonal long-range order (ODLRO) at zero temperature, we show that anyonic statistics allows one get arbitrarily close ODLRO but this possibility destroyed gauge coupling. The latter due nonzero effective mass generated invariance, which also implies presence screening, independently statistics.