Aspects of Quasi-Phasestructure of the Schwinger Model on a Cylinder with Broken Chiral Symmetry

We consider the $N_f$-flavour Schwinger Model on a thermal cylinder of circumference $\beta=1/T$ and of finite spatial length $L$. On the boundaries $x^1=0$ and $x^1=L$ the fields are subject to an element of a one-dimensional class of bag-inspired boundary conditions which depend on a real parameter $\theta$ and break the axial flavour symmetry. For the cases $N_f=1$ and $N_f=2$ all integrals can be performed analytically. While general theorems do not allow for a nonzero critical temperature, the model is found to exhibit a quasi-phase-structure: For finite $L$ the condensate -- seen as a function of $\log(T)$ -- stays almost constant up to a certain temperature (which depends on $L$), where it shows a sharp crossover to a value which is exponentially close to zero. In the limit $L\to\infty$ the known behaviour for the one-flavour Schwinger model is reproduced. In case of two flavours direct pictorial evidence is given that the theory undergoes a phase-transition at $T_c=0$.