Abstract
Quantum Heisenberg chain and square lattices are important paradigms of a low-dimensional magnetism. Their ground states are determined by the strength of quantum fluctuations. Correspondingly, the ground state of a rectangular lattice interpolates between the spin liquid and the ordered collinear Néel state with the partially reduced order parameter. The diversity of additional exchange interactions offers variety of quantum models derived from the aforementioned paradigms. Besides the spatial anisotropy of the exchange coupling, controlling the lattice dimensionality and ground-state properties, the spin anisotropy (intrinsic or induced by the magnetic field) represents another important effect disturbing a rotational symmetry of the spin system. The S = 1/2 easy-axis and easy-plane XXZ models on the square lattice even for extremely weak spin anisotropies undergo Heisenberg-Ising and Heisenberg-XY crossovers, respectively, acting as precursors to the onset of the finite-temperature phase transitions within the two-dimensional Ising universality class (for the easy axis anisotropy) and a topological Berezinskii–Kosterlitz–Thouless phase transition (for the easy-plane anisotropy). Experimental realizations of the S = 1/2 two-dimensional XXZ models in bulk quantum magnets appeared only recently. Partial solutions of the problems associated with their experimental identifications are discussed and some possibilities of future investigations in quantum magnets on the square and rectangular lattice are outlined.
Highlights
The history of low-dimensional magnetism started in 1925 by the theoretical work of Ising who found an exact solution of the hypothetic system of spins arranged into a chain and oriented in one direction [1]
That at present, there are no theoretical studies of the S = 1/2 XXZ model with the extremely weak easy-plane anisotropy in the magnetic field applied within the easy plane which would provide reliable information about the field-induced spin crossover between the Ising and the XY regime
The effects of the spatial anisotropy of the exchange coupling as well as the intrinsic or magnetic-field induced spin anisotropy were discussed in the two-dimensional magnetic models and their experimental counterparts realized by the low-dimensional Cu(II) based metal-organic magnets
Summary
The history of low-dimensional magnetism started in 1925 by the theoretical work of Ising who found an exact solution of the hypothetic system of spins arranged into a chain and oriented in one direction [1]. Besides the aforementioned practical applications and solutions of fundamental problems, current low-dimensional magnetism is characterized by the search for the analogies in different fields of physics (quantum tunneling [34], Bose-Einstein condensation [35], quantum phase transitions [36,37], thermal Hall effect [38], etc.) This brief introduction to the history and current state of the low-dimensional magnetism tried to point out that this field covers extremely wide area of research comprising directions which seem to run independently until they mix together forming new qualities. We will restrict to the ground-state and finite-temperature properties of the S = 1/2 models interpolating between the quantum Heisenberg antiferromagnetic chain and the square lattice Both models represent important paradigms of the low-dimensional magnetism. The problems associated with their experimental identifications are discussed
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