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Progress of Theoretical Physics

Localized Spin Wave State in a One-Dimensional Heisenberg Ferromagnet with a Boundary and Impurity Spins

Y. Tsuchida

Published Jan 1, 1976• 4 citations
Publisher

Abstract

Problems of the localized spin wave states in a semi-infinite one-dimensional Heisenberg ferromagnet with impurity spins are studied by the scattering matrix method. The rigorous expressions for the energy and the wave function of the localized spin wave state are obtained in the presence of a boundary.
DOI
10.1143/ptp.55.10

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The ground state for S 0 =1 and the low-lying excited states including localized states for S 0 =1/2 of a linear Heisenberg ferromagnet of spin 1/2 containing an antiferromagnetic impurity spin are calculated, where S 0 is a magnitude of the impurity spin. In these cases they belong to the subspace spanned by the two spin deviation kets. The two magnon states are solved. The method is based on the excitation from the rigorous one magnon localized state. The wave functions and the averages of the Z component of each spin are calculated. Numerical calculations for a ring consisting of twelve spins are performed. The results are in good agreement with analytical results for the ground state energy and localized levels.
Disordered one-dimensional XYZ Heisenberg ferromagnets with substitutional impurities are studied. The substitutional spins give rise to a random distribution of the exchange couplings. By introducing a new transformation for spin operators, the XYZ Heisenberg Hamiltonian can be exactly diagonalized and expressed by Bose operators. Being restricted by the spin wave approximation, the Hamiltonian obtained is shown to be similar to the Anderson Hamiltonian for electronic systems. We have shown that the magnon localization should be destroyed at the middle of the band due to the effects of commensurability of the magnon wavelength and lattice constant. The magnon density of states and the localization length are shown to have singularities at the middle of the band irrespective of the strength of disorder. The middle of the magnon band can be reached not only by changing the magnon energy but also by varying the external magnetic field which facilitates the observation of the phenomenon.
Disordered one-dimensional XYZ Heisenberg ferromagnets with substitutional impurities are studied. The substitutional spins give rise to a random distribution of the exchange couplings. By introducing a new transformation for spin operators, the XYZ Heisenberg Hamiltonian can be exactly diagonalized and expressed by Bose operators. Being restricted by the spin wave approximation, the Hamiltonian obtained is shown to be similar to the Anderson Hamiltonian for electronic systems. We have shown that the magnon localization should be destroyed at the middle of the band due to the effects of commensurability of the magnon wavelength and lattice constant. The magnon density of states and the localization length are shown to have singularities at the middle of the band irrespective of the strength of disorder. The middle of the magnon band can be reached not only by changing the magnon energy but also by varying the external magnetic field which facilitates the observation of the phenomenon.
Magnetic impurities coupled antiferromagnetically to a one-dimensional Heisenberg model are studied by numerical diagonalization of chains of finite clusters. By calculating the binding energy and the correlation function, it is shown that a local singlet develops around each impurity. This holds true for systems with a single impurity, with two impurities, and for impurities forming a lattice. The local character of the singlet is found to be little affected by the presence of other impurity spins. A small effective interaction is found between a pair of impurity spins, which oscillates depending on impurity distances. For impurity lattices, the energy spectrum shows a gap which is found to be much smaller than the binding energy per impurity if the coupling constants are small. For larger coupling constants, it increases to the same order of magnitude as the binding energy, indicating that a local singlet is broken to create excited states. Impurity lattices with ferromagnetic couplings are also studied and their connection to the Haldane problem is discussed.