We study the phase diagram of one-dimensional quantum ferrimagnets by using a numerical exact diagonalization of a finite size system along with a field-theoretical non-linear σ model of the quantum ferrimagnets at zero temperature and its effective description in the presence of the external magnetic field in terms of the quantum XY-model. The low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states where in the intermediate

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... tic field (h_c1 < h < h_c2), it is an XXZ+h model with easy plane anisotropy, which possess the spiral (superfluid) states that carry the dissipationless spin-supercurrent. We derive the critical exponents, and then will study the stability of the XY spiral state against these spin-supercurrents and the hard axis fluctuations. We will show a first order phase transition from the easy plane spiral state to a saturated ferromagnetic state occurs at h=h_c2 if the spin-supercurrent reaches to its critical value.