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Communications in Theoretical Physics

Magnetic Properties of One-Dimensional Ferromagnetic Mixed-Spin Model within Tyablikov Decoupling Approximation

Yuan Chen, Xiang Ying, Chuang-Chuang Song

Published Oct 1, 2010• 4 citations
Publisher

Abstract

In this paper, we apply the two-time Green's function method, and provide a simple way to study the magnetic properties of one-dimensional spin-(S,s) Heisenberg ferromagnets. The magnetic susceptibility and correlation functions are obtained by using the Tyablikov decoupling approximation. Our results show that the magnetic susceptibility and correlation length are a monotonically decreasing function of temperature regardless of the mixed spins. It is found that in the case of S=s, our results of one-dimensional mixed-spin model is reduced to be those of the isotropic ferromagnetic Heisenberg chain in the whole temperature region. Our results for the susceptibility are in agreement with those obtained by other theoretical approaches.

DOI
10.1088/0253-6102/54/4/30

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In this paper, we apply the two-time Green's function method, and provide a simple way to study the magnetic properties of one-dimensional spin-(S,s) Heisenberg ferromagnets. The magnetic susceptibility and correlation functions are obtained by using the Tyablikov decoupling approximation. Our results show that the magnetic susceptibility and correlation length are a monotonically decreasing function of temperature regardless of the mixed spins. It is found that in the case of S=s, our results of one-dimensional mixed-spin model is reduced to be those of the isotropic ferromagnetic Heisenberg chain in the whole temperature region. Our results for the susceptibility are in agreement with those obtained by other theoretical approaches.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 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 cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn 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/ kB, we find that the specific heat goes as T1/2 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 c1 = -0.55407 and c2 = 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.
The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈S0z{\mathit{S}}_{0}^{\mathit{z}}S0z​Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉 and 〈S0+{\mathit{S}}_{0}^{+}S0+​${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈Sz{\mathit{S}}^{\mathit{z}}Sz〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.
The spin-1/2 quantum Heisenberg ferromagnet with anisotropic spin exchange interactions in one dimension is investigated by means of the Green's function method and the spin-wave theory. In this paper we use the Holstein-Primakoff transformation to get the approximately Hamiltonian. We show that the magnetic field and anisotropy can have a rather drastic effect on the correlation functions and susceptibilities in the low-temperature region. Our results show that inverse correlation lengths kx and ky are independent of the temperature in the weak magnetic field. kx is less dependent on the anisotropy than ky. These results are consistent with the theoretical and experimental results.
We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional XY\mathrm{XY}XY models with S=1,S=1,S=1, using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.
We use the Quantum Monte Carlo Stochastic Series Expansion (SSE) algorithm to investigate the properties of the one-dimensional Heisenberg ferromagnet.We consider the model for two spin values s = 1 2 and s = 1 and study the dependence of the specific heat, magnetisation, magnetic susceptibility and correlation functions on temperature and magnetic field.Obtained results are compared to those derived by analytical methods (Bethe ansatz, Green's function, Random Phase Approximation).In particular, the surprising prediction of a double-peak structure in the specific heat at low temperatures and small magnetic fields can be confirmed.
The Green's function formalism proposed by Kondo and Yamaji for low-dimensional spin systems with S =1/2 has been extended to the cases of S >1/2. We apply this formalism to the study of one- and two-dimensional Heisenberg ferromagnets over the whole temperature range. It is shown that our theory can reproduce the correct results obtained by the high-temperature expansion method. On the other hand, the results at low temperatures are similar to those of the modified spin-wave theory, which is considered to predict the low-temperature properties of such systems rather correctly. The gross behaviors of the calculated thermodynamic quantities of ferromagnetic Heisenberg chains agree with those obtained by the exact diagonalization method.