be $C_{5}=0$.

This result implies that the long-range mode cannot propagate within the infinite-strip when a uniform and constant magnetic field is perpendicularly applied at the sample’s boundaries.

We want to point out the following fact. The same property of the finite temperature polarization operator component $\Pi _{00}$ that is producing the appearance of a long-range mode in the infinite bulk, is also responsible, when it is combined with the boundary conditions, for the non-propagation of this mode in the bounded sample. It is known that the nonvanishing of ${\it \Pi }_{{\it 0}}$ at $T\neq 0$ (or equivalently, the presence of a pole $\sim 1/k^{2}$ in $\Pi _{00}$ at $T\neq 0$) guarantees the existence of a long-range mode in the infinite bulk [@8]. On the other hand, however, once ${\it \Pi }_{{\it 0}}$ is different from zero, we can use the constraint (3.24) to eliminate $C_{6}+eC_{7}$ in favor of $C_{5%
\text{ }}$ in the free-energy density of the infinite strip. Then, as we have just proved, the only stable solution of this boundary-value problem, which is in agreement with the boundary conditions, is $C_{5}=0$. Consequently, no long-range mode propagates in the bounded sample.

In the zero temperature limit $\left( \beta 