The thesis is devoted to investigation of the properties of frustrated quantum Heisenberg antiferromagnets, which under special conditions have a dispersionless (flat) band in the one-particle energy spectrum. The model is investigated in strong magnetic fields and at low temperatures.
An s=1/2 antiferromagnetic Heisenberg model on several bilayer lattices (square, honeycomb and triangular) with magnon states from the flat band with lowest energy in the presence of a strong magnetic field is considered. Due to the localized nature of the flat-band magnon states, these systems are mapped on the classical lattice gases of hard-core objects. This mapping simplified the study of quantum models, reducing them to well-known classical ones, which allows us to draw the conclusions about the properties of the initial models. Also, the standard strong-coupling perturbation theory is applied for constructing effective Hamiltonians. These effective models allowed to investigate the phase transitions related to the ordering of localized magnons. These phase transitions belong to the different classes of universality. When completely antiferromagnetic model (square and honeycomb geometry) was considered in full frustration regime, the phase transitions that belong to the universality class of the two-dimensional Ising model were found. In the case of triangular geometry, for a completely antiferromagnetic model, the phase transition belonging to the universality class of the two-dimensional three-state Potts model, and in the case, when some interactions on the lattice are ferromagnetic — the discontinuous phase transition, which at a certain critical point reached the continuous phase transition from the universality class of the two-dimensional Ising model were found. For a completely antiferromagnetic model on a honeycomb bilayer lattice for a small deviations from the full frustration regime, a spin-flop transition, which occurs in a XXZ model with an easy axis of magnetization, was found.
On the basis of an effective model, a theory for a magnetic compound Ba2CoSi2O6Cl2 in an external magnetic field for the description of its low-temperature properties is developed. The compound can be described as a quantum spin model on a square bilayer lattice. The results of experiments for this compound have been reproduced and new predictions have been made, which require new experimental studies to confirm them. The applicability of the developed theory based on the Heisenberg quantum model on a honeycomb bilayer lattice to the magnetic compound Bi3Mn4O12(NO3) is also discussed.
For completeness, the ground state of the quantum Heisenberg antiferromagnet on the bilayers with different geometries (square, honeycomb) in the absence of magnetic field is investigated. A variational approach has been applied for that. The trial variational wave functions, which describe the state of the system for the certain relations between exchange interaction parameters, are proposed. By comparing the variational energies, the ground-state phase diagrams are constructed. The obtained results are compared with obtained recently by more sophisticated methods. Qualitative consistency and good quantitative agreement for some critical points are observed.
As a simple example, the Tasaki-Hubbard model on a sawtooth chain, where one can observe the effects of the presence of strong correlations and the lowest dispersionless (flat) band in the energy spectrum is considered. The properties of such a paramagnet in an infinitesimally small external magnetic field are studied and a comparison with the usual Curie paramagnet is made. It is found that in contrast to the usual paramagnet, the Tasaki-Hubbard paramagnet has a residual entropy. Also, such a paramagnet is more easily to magnetize.
In the thesis, the influence of the deviation from the full frustration regime and the sign of some exchange interactions on the system properties is studied. For one of the spin models, the system is investigated in a magnetic field with an arbitrary direction. The perturbation theory is extended to the case, when full Hamiltonian does not commute with the Zeeman term.
