Application of Mean-Field Jordan Wigner Transformation to Antiferromagnet System

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1 Commun. Theor. Phys. Beijing, China pp c Chinese Physical Society Vol. 50, o. 1, July 15, 008 Application of Mean-Field Jordan Wigner Transformation to Antiferromagnet System LI Jia-Liang, 1,3 LEI Shu-Guo, and JIAG Yu-Chi 1 1 Department of Physics, Changshu Institute of Technology Changshu 15500, China Department of Applied Physics, anjing University of Technology, anjing 10009, China 3 Jiangsu Laboratory of Advanced Functional Materials, Changshu 15500, China Received August 4, 007; Revised ovember 1, 007 Abstract By using the mean-field Jordan Wigner transformation analysis, this paper studies the one-dimensional spin-1/ XYZ antiferromagnetic chain in the transverse field with uniform long-range interactions among the z- components of the spins. The thermodynamic quantities, such as Helmholtz free energy, the internal energy, the specific heat, and the isothermal susceptibility, are obtained. Under degenerating condition, our results agree with numerical results of the other literatures PACS numbers: Jm, Cx, s Key words: XYZ antiferromagnetic chain, mean-field Jordan Wigner transformation, long-range interactions 1 Introduction In recent years, the low-dimensional magnetic system, especially oxygen superconductor materials with stronger antiferromagnetic coupling, has received extensive attention from researchers in a wide range of scientific and engineering fields. So thermodynamics properties of the one-dimensional antiferromagnetic system are always one of the most activating research fields [1 3] in theory and experimentation of condensed matter physics. Studies in recent ten years also have shown that the Heisenberg chains with nonlocal quantum spin system are also used in such frontier physical fields as quantum dots and nuclear spin. [4,5] It is well nown that the Heisenberg models are frequently used to describe one-dimensional quantum spin chain. From the perspective of interaction anisotropy, there are three types of Heisenberg models named XXX, XXZ, and XYZ models respectively. They provide a good platform for studying the thermodynamic properties of low-dimensional quantum magnetic systems. Theoretically, the physical properties of spin-1/ XY chain can be studied analytically by the general Jordan Wigner transformation, which represents the spin operator to spinless Fermi operator with no interactions in the quadratic form and can be diagonalized exactly. However, the same operation is inapplicable to the isotropic spin-1/ XXX Heisenberg chain and the anisotropic spin-1/ XYZ chain because they are transformed to the interacting spinless Fermi system. Some approximate techniques are developed and applied to the model. [6] The nearest-neighbor interaction is mainly considered in the Heisenberg s model. This is because that although the nearest-neighbor interaction is not the wholly spin interaction, it is the most dominating one and its solution can also provide primary cognition and deduction for other problems. However, the effect of long range interaction as a real spin one on properties of system thermodynamics especially on phase transition of system is essential. One-dimensional finite range interaction and finite spin system have been proved by means of mathematics to have no phase transition [7] in history. The thermodynamics properties of isotropy ferromagnetic S-1/ XXZ model with uniform long range interaction are discussed by using Jordan Wigner transform and Gaussion integral transform in Refs. [8] and [9]. The phase transition of the one-dimensional antiferromagnetic Ising model with the exponential decaying long range interaction is discussed by the real space renormalization group renormalizing in Ref. [10]. The studies mentioned above show that effect of long range interaction on system thermodynamic properties is remarable, and it was also presented in Ref. [10] that the effect of long range interaction on antiferromagnetic chain is different from that on ferromagnetic chain. This paper applies a method of mean-field Jordan Wigner transformation to investigate the anisotropy S- 1/ XYZ antiferromagnetic chain with uniform long range interaction. We have found that the trend of thermodynamics quantities varying with temperature for antiferromagnetic agrees well with the result by using traditional thermodynamics integral method. Our discussion of phase transition problem is also agreeable with the viewpoint in Ref. [10]. By comparison, we have applied our method to study the ferromagnetic system, and obtained numerical result which is consistent with that in Ref. [8]. But as our focal point, the effect of anisotropy and self-consistent field on system thermodynamics properties has not been reported so far. The project supported by the Open Fund of Jiangsu Laboratory of Advanced Functional Materials under Grant o. 06KFJJ004

2 44 LI Jia-Liang, LEI Shu-Guo, and JIAG Yu-Chi Vol. 50 Model and Partition Function The Hamiltonian of the one-dimensional S-1/ antiferromagnetic chain model with uniform long-range interactions among the z-components of the spins in the external field is H = J [1 + γsj x Sj+1 x + 1 γs y j Sy j+1 ] + I j,=1 S z j S z + h Sj z, 1 where Sj α α = x, y, z is the spin operator of S = 1/ on site j, is the number of system spins, J is the nearest-neighbor exchange coupling, γ is the anisotropy parameter, I is the long range interaction parameter in z-direction, h is the transverse field parameter. Using the spin raising and lowering operators S x j = 1 S+ j + S j, Sy j = 1 i S+ j S j, and Jordan Wigner transformation, [11] S + j = [ exp ] c l c l c j, j 1 iπ l=1 S j = c j [exp iπ j 1 c l c l l=1 ], S z j = c + j c j 1, 3 equation 1 can be replaced with [ 1 H = J c j c j+1 + c j+1 c j + γ c j c j+1 + c j+1c j ] + I j,=1 c j c j j c j c jc c + h I h I. 4 The mean-field theory is applied to four-operator term in Eq. 4. Using the self-consistent field, D = c j c j, and under the mean field approximation, equation 4 can be written as H = J [c j c j+1 + c j+1 c j + γc j c j+1 + c j+1c j ] I ID h c j c j Introducing Fourier transformation, c j = 1 expijc, = πn n=1 n = 1,, 3,...,, h I ID. 5 equation 5 is transformed into H = [ Ac c + ib ] c c + c c h I ID. 6 With Bogoliubov transformation, c = U α + iv α, c = U α iv α, U K = U K, V K = V K. The Hamilton function of the system can be expressed as H = [ A + B α α 1 A + B + 1 ] A h I ID. 7 Partition function is expressed as [ β Z = exp h I + βid ] [ Tr exp [ β A + B α α 1 A + B + 1 ]] A, lnz = β h I + βid + ln1 + e βε + 1 βε 1 βa, 8 where A = J cos + h I + ID, B = Jγ sin, and ε = A + B. 3 Thermodynamics Quantities of System 3.1 Helmholtz Free Energy per Site From Eq. 8, we can obtain F = T ln1 + e βε A 1 ε h 1 ID. 9 Imposing the condition F/ D = 0, the self-consistent field D can be obtained as follows: D = 1 A βε tanh + 1 ε Internal Energy per Site From Eq. 8, we can also obtain U = 1 βε ε tanh + 1 A Specific Heat per Site h I ID. 11 According to Eq. 11, the specific heat per site can be expressed as:

3 o. 1 Application of Mean-Field Jordan Wigner Transformation to Antiferromagnet System 45 where C = I [ A βε T tanh ε + A 4 sech βε 1 ] I D β + I T D β = 1/ A/4sech βε / 1 + I/ B /ε 3 tanhβε / + βa /ε sech βε /. ε 4 sech βε + I T D D β, Magnetization per Site From Eq. 8, we can obtain 3.5 Susceptibility per Site M Z = 1 From Eq. 13, we can obtain χ = A ε 1 + e βε + 1 A = 1 A tanh ε ε βε. 13 1/ βa /4ε sech βε / + 1/ B /ε 3 tanhβε / 1 1/ IβA /ε sech βε / 1/ IB /ε 3 tanhβε /. 14 All the functions and variables i.e., A, B, ε in the above-mentioned equations satisfy Eq umerical Result and Discussion 4.1 Thermodynamics Quantities as a Function of Temperature We have solved Eqs through self-consistent method and discussed how the thermodynamics quantities vary with temperature. Figures 1a 1c show the dependence of the thermodynamic quantities on temperature in a transverse field when h/j = 0.0, γ = 0.0, and I/J = 0.8, 1.0, 1. respectively. It is also shown that with the increase of long range interaction, not only phase transition does not occur, but also the variation trends of the free energy, the internal energy and the specific heat are not changed with the increase of long range interaction. This is because that in the numerical computation by using the self-consistent approximate method which we mentioned above, D 0.5 and D/ β 0. We have found that the results are reasonable through Eqs. 9, 11, and 1 and are in good agreement with the viewpoint in Ref. [10]. Fig. 1 The free energy F a, internal energy U b and specific heat C c versus the temperature T/J with γ = 0, h/j = 0 and for different long-range interactions I/J = 0.8,1.0,1.. In order to compare the effect of long range interaction on antiferromagnetic chain with that on ferromagnetic chain, we compute the effect of long range interaction on system thermodynamic quantities in Figs. a c when h/j = 0.0, γ = 0.0, and I/J = 1.6, 1.8, respectively. The first two parts in Hamiltonian function Eq. 1 can be considered as perturbation because the value of I/J is larger. Furthermore, h/j = 0 but I/J is negative, so equation 1 describes the standard Ising ferromagnetic chain. We have solved Eqs with the help of the self-consistent approximation method, and our numerical results show that for ferromagnetic, firstly long range interaction has great effect on thermodynamics quantities, and secondly apparent phase transition occurs. The results are in good accordance with the conclusion drawn by different theoretical methods in Ref. [8].

4 46 LI Jia-Liang, LEI Shu-Guo, and JIAG Yu-Chi Vol. 50 Fig. The internal energy U a, magnetization M Z b, and specific heat C c versus the temperature T/J with γ = 0, h/j = 0 and for different long-range interactions I/J = 1.6, 1.8. Figures 3a 3c show the dependence of the thermodynamic quantities on temperature when h/j = 0.0, I/J = 1., and γ = 0., 0.5, 0.8 respectively. The numerical results show that no phase transition occurs either and the thermodynamic quantities vary with temperature apparently as anisotropy variables are increased. This is because that the first two parts in Hamilton function become larger with increasing anisotropy variables. Fig. 3 The internal energy U a, specific heat C b and susceptibility χ c versus the temperature T/J with I/J = 1., h/j = 0 and for different anisotropic γ = 0.,0.5,0.8. Figures 4a 4c show the dependence of the thermodynamic quantities on temperature when I/J = 1. and h/j = 0., 0.5, 0.8 respectively. We have found that the trend of the specific heat varying with temperature by using mean-field Jordan Wigner transformation method agrees well with the result obtained by traditional thermodynamics Bethe integral equation method. [1] Fig. 4 The free energy F a, specific heat C b, and susceptibility χ c versus the temperature T/J with γ = 0.5, I/J = 1. and for different external field h/j = 0.,0.5,0.8.

5 o. 1 Application of Mean-Field Jordan Wigner Transformation to Antiferromagnet System The Magnetization as a Function of External Magnetic Field Figures 5a 5c show the dependence of the magnetization on external magnetic field. Although all variables are different, our numerical results show that with the increase of external magnetic field, the spin direction of system trends to be consistent and reaches a definite value, which agrees with the theory of thermodynamics. Fig. 5 Function of external field h/j. a For T/J = 0.0, γ = 0.0, and I/J = 0.3,0.6,0.9; b For T/J = 0.0, I/J = 0.5, and γ = 0., 0.5, 0.8; c For I/J = 0., γ = 0., and T/J = 0.05, 0.15, Summary In summary, using mean-field Jordan Wigner transform method, this paper discusses the thermodynamics properties of the system based on the S-1/ Heisenberg XYZ antiferromagnetic model with uniform long range interaction in z-direction in external magnetic field. We have placed great emphasis on studies on the effect of the competition between the anisotropic parameter and the long-range interaction parameter upon the system. Our numerical results on degenerating condition agree well with those in other papers, while the conclusion drawn under normal condition has not been reported so far. Acnowledgments We would lie to than professor Tong Pei-Qing for many important discussions and suggestions. References [1]. Laflorencie and H. Rieger, Eur. Phys. J. B [] H.H. Fu, K.L. Yao, and Z.L. Liu, Phys. Lett. A [3] H. Yoshizawa, G. Shirane, H. Shiba, and K. Hiraawa, Phys. Rev. B [4] A. Imamoglu and D.P. Divencenzo, Phys. Rev. Lett [5] S.B. Zheng and G.C. Guo, Phys. Rev. Lett [6] M. Shiroishi and M. Taahashi, Phys. Rev. Lett [7] R.B. Griffiths, Phase Transition and Critical Phenomenon, Vol. 1, Academic, London 197 pp [8] L.L. Goncalves, L.P.S. Coutinho, and J.P.de Lima, Physica A [9] Ping Lou, Phys. Rev. B [10] Zhang Jin-Shan, J. Beijing ormal University in Chinese. [11] E. Fradin, In Field Theories of Condensed Matter Systems, Addison-Wesley, Redwood City, 1991 Chap. 4. [1] A. Klumper, Eur. Phys. J. B

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