The ground state phase diagram of a general isotropic spin-3/2 system with nearest-neighbor exchange is shown to contain unconventionally ordered spin nematic and antinematic states, as well as usual ferro- and antiferromagnetic phases. The two nematic phases have spontaneously broken rotational symmetry characterized by the long-range order of the nematic director u, as well as the broken time-reversal symmetry described by the pseudospin vector σ. Nematic phase differs from antinematic one by the type of ordering in σ vectors (uniform versus staggered). The ferromagnet-nematic and antiferromagnet-antinematic phase boundaries exhibit enhanced Sp(4) symmetry and correspond to the recently studied effective theory for spin-3/2 cold gases. We discuss optical properties and topological defects in the nematic phases.

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We analyze theoretically the novel pathway of ultrafast spin dynamics for ferromagnets with high enough single-ion anisotropy. This longitudinal spin dynamics includes the coupled oscillations of the modulus of the magnetization together with the quadrupolar spin variables, which are expressed through quantum expectation values of operators bilinear on the spin components. Even for a simple single-element ferromagnet, such dynamics can lead to a magnetization reversal under the action of an ultrashort laser pulse.

The phase II -P-$ diagram of an easy-plane strongly anisotropic antiferromagnet is studied. It is shown that in such a system a realization of the phase with the nonmagnetic order parameter, the so-called quadruple phase, is possible. The realization of this phase is caused by several purely quantum effects, moreover the mechanism of its appearance differs in principle from that of the antiferromagnet or ferromagnet. The possibility of the disappearance of the angular phase (the so-called quadruple ferromagnetic phase) is also shown.

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