Local Magnetization in the Impure Spin 1/2 Anisotropic Ising-Heisenberg Chains
A theory of the Friedel-type oscillations of the local magnetization in the impure antiferromagnetic spin 1/2 chains is developed using the Green function equations of motion in the pseudo-fermion representation. For the isotropic XY (XX) chain, the problem is solved exactly, while the Ising-Heisenberg model is investigated numerically within a temperature-dependent Hartree-Fock approximation. It is shown that the Hartree-Fock self consistency equations for the uniformly magnetized XXZ chain can be recovered as a particular case of the formalism developed in the present work. Comparison with the earlier perturbation theory treatment in a free-fermion approximation reveals that the magnetic field dependence of the perturbation of the local magnetization is sensitive to the formation of the localized states and the exact form of the energy dispersion law of the quasi-particles. In particular it is shown that the perturbations of the local magnetization in the impure spin 1/2 chains disappear in the absence of the external magnetic field. Using the exact solution for the XY chain it is shown that unless the localized energy levels are formed outside the pseudo-fermion energy band the singularity of the local magnetization existing in the pure chain disappears at an arbitrary distance from the single impurity spin. For the ferromagnetic chain with the ferromagnetically coupled impurity the solution of the Hartree-Fock equations at low temperatures agrees reasonably with the results of the linear spin-wave theory. If the impurity is antiferromagnetically coupled, then, in contrast with the results of the spin -wave theory, the Hartree-Fock approximation agrees with the exact result for the zero-field ground state spin defect at the impurity site. Unlike the previous methods, the technique developed in this work permits investigation of the whole temperature range and predicts the correct Curie-Weiss behavior at sufficiently large temperatures.
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- Physics: General