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We have studied the phase diagram of the one dimensional $S=1/2$ $XXZ$ modelwith ferromagnetic nearest-neighbor and antiferromagnetic next-nearest neighborinteractions. We have applied the quantum renormalization group (QRG) approachto get the stable fixed points and the running of coupling constants. Thesecond order QRG has been implemented to get the self similar Hamiltonian. Thismodel shows a rich phase diagram which consists of different phases whichpossess the quantum spin-fluid and dimer phases in addition to the classicalN\'{e}el and ferromagnetic ones. The border between different phases has beenshown as a projection onto two different planes in the phase space.

physical review. b, condensed matter and materials physics

Phase Diagram of the One Dimensional $S=1/2$ $XXZ$ model with  Ferromagnetic nearest-neighbor and Antiferromagnetic next-nearest neighbor  interactions