The magnetization process and adiabatic demagnetization of antiferromagnetic spin-1/2 XXZ Heisenberg clusters in the shape of regular polyhedra (tetrahedron, octahedron, cube, icosahedron and dodecahedron) are examined using the exact diagonalization method. It is demonstrated that a quantum ( xy ) part of the XXZ exchange interaction is a primary cause for the presence of additional intermediate magnetization plateaux and steps, which are totally absent in the limiting Ising case. The only exception to this rule is the spin-1/2 XXZ Heisenberg tetrahedron, which shows just a quantitative shift of the level-crossing fields related to two magnetization steps. It is shown that spin-1/2 XXZ Heisenberg regular polyhedra exhibit an enhanced magnetocaloric effect in the proximity of magnetization steps and jumps, which are accompanied with a rapid drop (rise) of temperature just above (below) the level-crossing field when the magnetic field is removed adiabatically.

The spin-$\frac{1}{2}$ Heisenberg octahedral chain with regularly alternating monomeric and square-plaquette sites is investigated using various analytical and numerical methods: variational technique, localized-magnon approach, exact diagonalization (ED), and density-matrix renormalization group (DMRG) methods. The model belongs to the class of flatband systems and it has a rich ground-state phase diagram including phases with spontaneously broken translational symmetry. Moreover, it exhibits an anomalous low-temperature thermodynamics close to continuous or discontinuous field-driven quantum phase transitions between three quantum ferrimagnetic phases, tetramer-hexamer phase, monomer-tetramer phase, localized-magnon phase, and two different spin-liquid phases. If the intraplaquette coupling is at least twice as strong as the monomer-plaquette coupling, the variational method furnishes a rigorous proof of the monomer-tetramer ground state in a low-field region and the localized-magnon approach provides exact evidence of a single magnon trapped at each square plaquette in a high-field region. In the rest of the parameter space we have numerically studied the ground-state phase diagram and magnetization process using DMRG and ED methods. It is shown that the zero-temperature magnetization curve may involve up to four intermediate plateaus at zero, one-fifth, two-fifths, and three-fifths of the saturation magnetization, while the specific heat exhibits a striking low-temperature peak in the vicinity of discontinuous quantum phase transitions.

It is evidenced that effective lattice-gas models of hard-core monomers and dimers afford a proper description of low-temperature features of spin-$\frac{1}{2}$ Heisenberg diamond and octahedral chains. In addition to monomeric particles assigned within the localized-magnon theory to bound one- and two-magnon eigenstates, the effective monomer-dimer lattice-gas model includes dimeric particles assigned to a singlet-tetramer (singlet-hexamer) state as a cornerstone of dimer-tetramer (tetramer-hexamer) ground state of a spin-$\frac{1}{2}$ Heisenberg diamond (octahedral) chain. The feasibility of the effective description is confirmed through the exact diagonalization and finite-temperature Lanczos methods. Both quantum spin chains display rich ground-state phase diagrams including discontinuous as well as continuous field-driven phase transitions, whereby the specific heat shows in vicinity of the former phase transitions an extraordinary low-temperature peak coming from a highly degenerate manifold of low-lying excitations.