Abstract
Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.
Highlights
The concept of long-range order is crucial to characterize phase transitions taking place in physical systems as a function of temperature and described by means of an order parameter [1]
(1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins
The appearance of long-range order is the manifestation of the occurrence of a phase transition and is related to a continuous or a discrete broken symmetry in magnetic systems depending on the model used to describe the underlying physics of the phase transition (e.g., Ising model, Potts model, XY model, Heisenberg model, etc.)
Summary
The concept of long-range order is crucial to characterize phase transitions taking place in physical systems as a function of temperature and described by means of an order parameter [1]. These recent findings and observations have stimulated several theoretical studies about the role played by the biquadratic contribution on the dynamical properties of ferromagnetic films coupled via antiferromagnetic Heisenberg exchange coupling On this basis, it is interesting to extend the Hohenberg–Mermin–Wagner theorem to quantum systems modeled via a generalized quantum Heisenberg Hamiltonian including both isotropic bilinear and biquadratic exchange short-range interactions. This is an important result because a novel ground-state giving no long-range order is added to the phase diagram of a quantum ferromagnet Note that, in this respect, there is a proof given by Thorpe in the 70s who demonstrated the absence of long-range order at finite T in 1D and 2D ferromagnetic lattices modeled by a generalized quantum Heisenberg Hamiltonian containing isotropic spin-spin exchange interactions to all orders [24]. For the systems studied in [24,25,26,27], the analysis was mainly mathematical and the physical effects of the biquadratic exchange interaction were not investigated
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