  We study the generating functional for the massless Schwinger model in the
torus, at nonzero chemical potential and temperature. The lack of hermiticity
of the Dirac operator yields a non-trivial phase in the effective action, which
is a topological contribution induced by the chemical potential. In the sector
with no zero modes, we evaluate exactly the generating functional, the
partition function, the boson propagator and the thermally averaged Polyakov
loop. The system bosonizes at finite T and \mu, with the same mass as in
vacuum. From the solution obtained for the Schwinger model we derive also
exactly the generating functional and the partition function for the massless
Thirring model at nonzero T and \mu.
