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We study possible quantum ground states of the Heisenberg antiferromagnet onthe star lattice, which may be realized in the recently discovered polymericIron 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 carriesspin-5/2 and the system eventually orders magnetically at low temperatures, themagnetic ordering temperature is much lower than the estimated Curie-Weisstemperature, revealing the frustrated nature of the spin interactions.Anticipating that a lower spin analog of this material may be synthesized infuture, we investigate the effect of quantum fluctuations on the star-latticeantiferromagnet using a large-$N$ Sp($N$) mean field theory and a projectivesymmetry group analysis for possible bosonic quantum spin liquid phases. It isfound that there exist only two distinct gapped $Z_2$ spin liquid phases withbosonic spinons for non-vanishing nearest-neighbor valence-bond-amplitudes. Inparticular, the spin liquid phase which has a lower energy in thenearest-neighbor exchange model can be stabilized for relatively higher spinmagnitudes. Hence it is perhaps a better candidate for the realization ofquantum spin liquid state. We also determine the magnetic ordering patternsresulting from the condensation of the bosonic spinons in the two differentspin liquid phases. We expect these magnetic ordering patterns would directlybe relevant for the low temperature ordered phase of the Iron Acetate. Thephase diagram containing all of these phases and various dimerized states areobtained for the nearest-neighbor exchange model and its implications arediscussed.

Classification of quantum phases for the star-lattice antiferromagnet via a projective symmetry group analysis