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An exact solution (incomplete) of the ground-state problem for an Ising modelin an external field on a 3D honeycomb zigzag-ladder lattice with two types ofsites is found. It is shown that the geometrical frustration due to thepresence of triangle elements leads to the emergence of a variety of magneticphases. The majority of these are partially disordered (highly degenerate). Thetheoretical results are used to explain the sequence of experimentally observedphase transitions in the honeycomb zigzag-ladder magnets and to predict theappearance of new phases.

An Ising model on a 3D honeycomb zigzag-ladder lattice: a solution to  the ground-state problem and application to the SrRE$_2$O$_4$ and  BaRE$_2$O$_4$ magnets