  We consider the space of $n \times n$ non-Hermitian Hamiltonians ($n=2$, $3$,
. . .) that are equivalent to a single $n\times n$ Jordan block. We focus on
adiabatic transport around a closed path (i.e. a loop) within this space, in
the limit as the time-scale $T=1/\varepsilon$ taken to traverse the loop tends
to infinity. We show that, for a certain class of loops and a choice of initial
state, the state returns to itself and acquires a complex phase that is
$\varepsilon^{-1}$ times an expansion in powers of $\varepsilon^{1/n}$. The
exponential of the term of $n$th order (which is equivalent to the "geometric"
or Berry phase modulo $2\pi$), is thus independent of $\varepsilon$ as
$\varepsilon\to0$; it depends only on the homotopy class of the loop and is an
integer power of $e^{2\pi i/n}$. One of the conditions under which these
results hold is that the state being transported is, for all points on the
loop, that of slowest decay.
