  We investigate the properties of antiferromagnetic spin-S ladders with the
help of local Berry phases defined by imposing a twist on one or a few local
bonds. In gapped systems with time reversal symmetry, these Berry phases are
quantized, hence able in principle to characterize different phases. In the
case of a fully frustrated ladder where the total spin on a rung is a conserved
quantity that changes abruptly upon increasing the rung coupling, we show that
two Berry phases are relevant to detect such phase transitions: the rung Berry
phase defined by imposing a twist on one rung coupling, and the twist Berry
phase defined by twisting the boundary conditions along the legs. In the case
of non-frustrated ladders, we have followed the fate of both Berry phases when
interpolating between standard ladders and dimerized spin chains. A careful
investigation of the spin gap and of edge states shows that a change of twist
Berry phase is associated to a quantum phase transition at which the bulk gap
closes, and at which, with appropriate boundary conditions, edge states appear
or disappear, while a change of rung Berry phase is not necessarily associated
to a quantum phase transition. The difference is particularly acute for regular
ladders, in which the twist Berry phase does not change at all upon increasing
the rung coupling from zero to infinity while the rung Berry phase changes 2S
times. By analogy with the fully frustrated ladder, these changes are
interpreted as cross-overs between domains in which the rungs are in different
states of total spin from 0 in the strong rung limit to 2S in the weak rung
limit. This interpretation is further supported by the observation that these
cross-overs turn into real phase transitions as a function of rung coupling if
one rung is strongly ferromagnetic, or equivalently if one rung is replaced by
a spin 2S impurity.
