The temperature dependences of the magnetization, internal energy and specific heat in a three-sublattice ferrimagnet or ferromagnet with |J ab | = |J bc | = J ca are calculated within the framework of the linear spin-wave approximation, by employing retarded Green's functions. For both the ferromagnet and the ferrimagnet, the internal energy and the specific heat decrease with increasing J /J and/or the value of the spins. For fixed values of S a , S b , S c and J /J , the internal energy and the specific heat increase, whereas the sublattice magnetization decreases with increasing temperature θ . The three-sublattice ferrimagnet has some particular characteristics which are not shown by the systems with two sublattices. For ferrimagnets, the antiferromagnetism of the system becomes weaker with increasing J /J . The sublattice magnetization at low temperatures (also the magnetization M 0 at 0 K) of a ferrimagnet increases with increasing J /J for fixed values of S a , S b and S c . The effects of the spins S a (S c ) and S b on the magnetizations of other sublattices differ. The characteristics of the a-sublattice are the same as those of the c-sublattice, due to their similarity as well as the symmetry of the system. The behaviours of the b-sublattice are different from those of the aand c-sublattices, due to the asymmetry of the three-sublattice system. The spin-value dependences of the spin deviation m per spin (and also the energy for the zero-point quantum fluctuation) of the system are different for different sublattices. These differences are ascribed to the asymmetry of the three-sublattice systems, which leads to the new intrinsic properties of the systems.

The magnon energy band in a ferrimagnetic superlattice with three layers in a unit cell is studied by employing retarded Green's functions and the spin-wave method. Two modulated energy gaps ∆ 13 ω and ∆ 23 ω are evaluated systematically, which exist in the magnon energy band along the K x -direction perpendicular to the plane of the superlattice. It is revealed that the energy gap ∆ 13 ω has a direct relation with the symmetry among the spin quantum numbers and the interlayer exchange couplings, while the energy gap ∆ 23 ω relates to the symmetry among these spin quantum numbers only. These symmetries differ from the symmetry of crystallographic point groups. We define the magnetically structural symmetry that is dominated mainly by the magnetic parameters. The absence of the energy gap at a certain condition means that the system has a high magnetically structural symmetry. The magnetically structural symmetry of the superlattice, which is an intrinsic property, strongly affects the magnon energy band structure and thus the magnetic behaviors of the system. Furthermore, two complete bandgaps are observed to extend through the Brillouin zone (referred to as "magnonic crystal") in this superlattice system.

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By employing retarded Green functions, the spin-wave spectrum and the layer-sublattice magnetization in Heisenberg ferrimagnetic three-layer superlattices and three-layer systems are calculated within the framework of the linear spin-wave approximation. The effects of the interlayer exchange constants and the intralayer exchange constants on the magnetic properties of the two systems are compared with those for the corresponding three-sublattice bulk ferrimagnets. It is found that all the differences between the magnetic properties of these systems originate from the differences between the exchange couplings in the three dimensions of the systems. The quantum correlations, such as the competition, cancellation, and transmission of the effects of the exchange couplings, are important for the magnetic properties of the systems. The asymmetry of the systems plays an important role in the zero-point quantum fluctuations and, correspondingly, in the layer-sublattice magnetizations of the layers.

The magnon energy band in a three-layer ferromagnetic superlattice is studied by using the linear spinwave approach and Green's function technique. It is found that two modulated energy gaps exist in the magnon energy band along K x direction perpendicular to the plane of the superlattice. The spin quantum numbers and the interlayer exchange couplings all affect the two energy gaps. The disappearance of one of the energy gaps corresponds to a high symmetry among the interlayer exchange couplings, while the vanishing of another energy gap corresponds to a high symmetry of the system, which belongs to the type III Shubnikov group or the polychromatic group of magnetic group. This magnetically structural symmetry includes not only the symmetry of crystallographic point-groups or space-groups, but also the symmetry of strength and direction of spins. Comparing with the previous results for a three-layer ferrimagnetic superlattice, it is concluded that the energy gap of the ferromagnetic/ferrimagnetic superlattices is dominated by different symmetries originated from ferromagnetic/antiferromagnetic interlayer exchange couplings.