Green's-Function Formalism of the One-Dimensional Heisenberg Spin System
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The one-dimensional Heisenberg model with S=1/2 is treated with the use of the twotime Green's functions. The hierarchy of the equations of motion of the Green's functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, <S^z>, is set to zero, because the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions C_n≡4<S_0^zS_n^z>. The Green's function is essentially of the form representing undamped spin waves, whose spectrum depends on c_1, c_2 and one more parameter. They are determined by the requirement that c_1 and c_2 should be self-consistent and that c_0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by c_n=(J/4k_BT)^n. In the case of ferromagnetic coupling, the correlation function c_n at T=0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T=0 also agrees with the correct one. At T≪J/k_B, we find that the specific heat goes as T^<1/2> and the susceptibility goes as T^<-2>. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c_1=-0.55407 and c_2=0.16100 at T=0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
- 理論物理学刊行会の論文
- 1972-03-25
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