  We study possible quantum ground states of the Heisenberg antiferromagnet on
the star lattice, which may be realized in the recently discovered polymeric
Iron Acetate,
Fe$_3$($\mu_3$-O)($\mu$-OAc)$_6$(H$_2$O)$_3$[Fe$_3$($\mu_3$-O)($\mu$-OAc)$_{7.5}$]$_2\cdot$
7H$_2$O. Even though the Fe$^{\rm III}$ moment in this material carries
spin-5/2 and the system eventually orders magnetically at low temperatures, the
magnetic ordering temperature is much lower than the estimated Curie-Weiss
temperature, revealing the frustrated nature of the spin interactions.
Anticipating that a lower spin analog of this material may be synthesized in
future, we investigate the effect of quantum fluctuations on the star-lattice
antiferromagnet using a large-$N$ Sp($N$) mean field theory and a projective
symmetry group analysis for possible bosonic quantum spin liquid phases. It is
found that there exist only two distinct gapped $Z_2$ spin liquid phases with
bosonic spinons for non-vanishing nearest-neighbor valence-bond-amplitudes. In
particular, the spin liquid phase which has a lower energy in the
nearest-neighbor exchange model can be stabilized for relatively higher spin
magnitudes. Hence it is perhaps a better candidate for the realization of
quantum spin liquid state. We also determine the magnetic ordering patterns
resulting from the condensation of the bosonic spinons in the two different
spin liquid phases. We expect these magnetic ordering patterns would directly
be relevant for the low temperature ordered phase of the Iron Acetate. The
phase diagram containing all of these phases and various dimerized states are
obtained for the nearest-neighbor exchange model and its implications are
discussed.
