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Journal of the Physical Society of Japan

Analytical Methods and Computer Calculations of XY Model with Impurity Spin

Takehiko Oguchi, Ikuo Ono

Published Jun 1, 1969• 11 citations
Publisher

Abstract

The one dimensional XY model of spin 1/2 is discussed by transforming the spin operators to the Fermi operators. This method can be applied to the XY model with an impurity spin. In that case a pair of localized exciton (or a kind of magnon) levels are obtained. The energy of states can be determined by distributing the excitons over two sets of exciton levels which depend on whether the number of excitons is even or odd. The energy levels of states are calculated by the computer for small rings of 4, 6 and 8 spins. The results can be perfectly understood by the theory.

DOI
10.1143/jpsj.26.1378

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The one dimensional XY model of spin 1/2 is discussed by transforming the spin operators to the Fermi operators. This method can be applied to the XY model with an impurity spin. In that case a pair of localized exciton (or a kind of magnon) levels are obtained. The energy of states can be determined by distributing the excitons over two sets of exciton levels which depend on whether the number of excitons is even or odd. The energy levels of states are calculated by the computer for small rings of 4, 6 and 8 spins. The results can be perfectly understood by the theory.
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.
The effect of impurities on the spectrum of one-magnon excitations of two quasi-one-dimensional spin systems—a double XY chain with an impurity atom and an analogous chain with an impurity link, in which two identical impurity atoms occupy neighboring nodes on different XY chains—is studied. The conditions for the existence of impurity levels and their number are determined for the system with one impurity, and for the system with an impurity link the explicit expressions for the energy and wave functions of localized states are also determined.
The influence of a magnetic impurity on the elementary excitation spectrum of a two-dimensional Ising ferromagnet is investigated in a low transverse magnetic field. It is shown that in the case of a “ferromagnetic” impurity in a matrix of spin 1/2, from one to three local energy levels can be split off from the lower edge of the continuous spectrum of one-magnon states where the conditions for their existence impose quite strict constraints on the spin-Hamiltonian parameters. Three distinct spin configurations can be realized in the presence of an “antiferromagnetic” impurity in the crystal in the ground state. Single-magnon states of a ferromagnet are considered for those values of the spin-Hamiltonian parameters when two of them are realized. It is shown that from one to six local levels above or below the continuous spectrum band can appear, and their distance from the band edge can substantially exceed its width. Quasilocal energy levels originating within the continuous spectrum of the single-magnon states are also investigated.
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.
The two-dimensional Heisenberg model at zero temperature with a quantum S=1/2 impurity spin, coupled to one site, is studied. The ground state properties of the model are calculated for large coupling, in order to study the impurity-host spin local singlet formation and related local suppression of the magnetization in the host lattice. Analytic results are obtained by developing perturbation theory around the exactly solvable two-body (impurity plus neighbor) problem and compared with results obtained by exact diagonalization of small clusters. We find that perfect screening of the impurity is only achieved for infinitely large interaction, while at intermediate coupling the local magnetization is suppressed, but nonzero.