  We describe a new class of systems exhibiting return point memory (RPM) that
are different from those discussed before in the context of ferromagnets. We
show numerically that one dimensional random Ising antiferromagnets have RPM,
when configurations evolve from a large field. However, RPM is violated when
started from some stable configurations at finite field unlike in the
ferromagnetic case. This implies that the standard approach to understanding
ferromagnetic RPM systems will fail for this case. We also demonstrate RPM with
a set of variables that keep track of spin flips at each site. Conventional RPM
for the spin configuration is a projection of this result, suggesting that spin
flip variables might be a more fundamental representation of the dynamics. We
also present a mapping that embeds the antiferromagnetic chain in a two
dimensional ferromagnetic model, and prove RPM for spin exchange dynamics in
the interior of the chain with this mapping.

Statistical Mechanics