  The $PT$ symmetric potential $V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$ has
a completely real spectrum for $\lambda\le 1$, and begins to develop complex
eigenvalues for $\lambda>1$. At the symmetry-breaking threshold $\lambda=1$
some of the eigenvectors become degenerate, giving rise to a Jordan-block
structure for each degenerate eigenvector. In general this is expected to
result in a secular growth in the amplitude of the wave. However, it has been
shown in a recent paper by Longhi, by numerical simulation and by the use of
perturbation theory, that for a broad initial wave packet this growth is
suppressed, and instead a saturation leading to a constant maximum amplitude is
observed. We revisit this problem by explicitly constructing the Bloch
wave-functions and the associated Jordan functions and using the method of
stationary states to find the dependence on the longitudinal distance $z$ for a
variety of different initial wave packets. This allows us to show in detail how
the saturation of the linear growth arises from the close connection between
the contributions of the Jordan functions and those of the neighbouring Bloch
waves.
