Massive multi-flavor Schwinger model at finite temperature and on compact space

The multi-flavor Schwinger model on $R^1$ at finite temperature $T$ is mathematically equivalent to the model on $S^1$ at $T=0$. The latter is reduced to a quantum mechanical system of $N-1$ degrees of freedom. Physics sensitively depends on the parameter $m/T$. Finite temperature behavior of the massive Schwinger model is quite different from that of the massless Schwinger model.