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Collinear Néel-type ordering in partially frustrated lattices

We consider two partially frustrated S = 1/2 antiferromagnetic spin systems on the triangular and pentagonal lattices. In an elementary plaquette of the two lattices, one bond has exchange interaction strength $α$ ($α\leq 1$) whereas all other bonds have exchange interaction strength unity. We show that for $α$ less than a critical value $α_{c}$, collinear Néel-type ordering is possible in the ground state. The ground state energy and the excitation spectrum have been determined using linear spin wave theory based on the Holstein-Primakoff transformation.