Mott-insulator phases of spin-3/2 fermions in the presence of quadratic Zeeman coupling

We study the influence of the quadratic Zeeman effect in the Mott-insulator phases of hard-core spin-3/2 fermions. We show that contrary to spinor bosons, any quadratic Zeeman coupling preserves a $SU(2)\otimes SU(2)$ symmetry, leading for large-enough quadratic Zeeman coupling to an isotropic pseudo-spin-1/2 Heisenberg antiferromagnet. Depending on the scattering lengths, on 1D lattices the quadratic Zeeman coupling can induce either a Kosterlitz-Thouless transition between a gapped dimerized spin-3/2 phase and a gapless pseudo-spin-1/2 antiferromagnet, or a commensurate-incommensurate transition from a gapless spin-liquid into the pseudo-spin-1/2 antiferromagnet. Similar arguments allow to foresee corresponding transitions on ladder type and square lattices. We analyze various observables which should reveal in experiments these phases.