The thermodynamics of low dimensional spin-1/2 Heisenberg ferromagnets (HFM) in an external magnetic field is investigated within a second-order two-time Green function formalism in the wide temperature and field range. A crucial point of the proposed scheme is a proper account of the analytical properties for the approximate transverse commutator Green function obtained as a result of the decoupling procedure. A good quantitative description of the correlation functions, magnetization, susceptibility, and heat capacity of the HFM on a chain, square and triangular lattices is found for both infinite and finite-sized systems. The dependences of the thermodynamic functions of 2D HFM on the cluster size are studied. The obtained results agree well with the corresponding data found by Bethe ansatz, exact diagonalization, high temperature series expansions, and quantum Monte Carlo simulations.

A model of a close-packed polytype with a random stacking fault structure is used to interpret the anomalies of the thermodynamic properties of the disordered (glassy) phase in solid HCP 4He in the so-called supersolid state. The temperature dependence of the phonon pressure is calculated theoretically, and compared to experimental data. A quantitative agreement between the theory and the experiment is achieved.

A model is proposed which gives an analytical description of the dynamics of collective excitations of two-dimensional close-packed atomic crystal lattices (atomic monolayers on substrates). The model takes into account both the interaction between atoms of the layer and the interaction of the layer with the substrate. The phonon spectra are found for an ideal triangular lattice and for a triangular lattice with a uniform distortion along one of the close-packed directions in the plane of the layer. The temperature dependence of the heat capacity is constructed for crystalline structures of both the commensurate and incommensurate types (in relation to the substrate). The theoretical results obtained are used for a detailed discussion and interpretation of the published experimental data on the spectra of lattice excitations and the heat capacity of monolayers of rare gases, including He3 and He4, on various types of substrates.

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