Dyson–Maleev mean-field theory of ferrimagnetic spin chain

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Dyson–Maleev mean-field theory of ferrimagnetic spin chain

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Yuge Chen a , Yinxiang Li b , Lijun Tian a , Bin Chen b a b

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College of Sciences, Shanghai University, Shanghai 200444, PR China Tin Ka-Ping College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China

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a r t i c l e

i n f o

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a b s t r a c t

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Article history: Received 28 May 2017 Received in revised form 30 June 2017 Accepted 3 July 2017 Available online xxxx Communicated by M. Wu

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Keywords: Dyson–Maleev mean-field theory Ferrimagnetic spin chain

Based on the method of Dyson–Maleev mean-field theory, we study the Heisenberg ferrimagnetic spin chain at zero temperature. The energy spectrum exhibits the coexistence of a nearly gapless branch and a gapful branch. Compared with numerical results of other methods, the energy gap, h = 1.32, of Dyson–Maleev mean-field theory is reasonable and only a little bit lower. At finite temperature, the properties of energy gap, magnetization, uniform susceptibility, internal energy, specific heat are calculated respectively. The trends of these thermodynamic properties in this method are consistent with the existed results of other methods. © 2017 Elsevier B.V. All rights reserved.

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1. Introduction

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In the past decades, the low-dimensional quantum physics, especially the one-dimensional magnets, has attracted much attention from the theorists and experimentalists due to the exotic quantum phenomenons. In the field of one-dimensional magnets, Haldane [1] discovered the fundamental difference between the integer and half-integer spin chains which pushes the development of low-dimensional quantum magnets significantly. Subsequently, the discovery of cuprates [2] high temperature superconductors which are intimately correlated with the strong magnetic fluctuation also has made a new boost for this field. One of the specific spin models in one-dimensional magnets is the mixed spin chain which contains two different kinds of spins S and s (S > s). The material of ferrimagnetic spin chain can be synthesized in the experiment as the molecular magnet NiC u ( pba)( D 2 O )3 · D 2 O [3]. From the theoretical aspect, the powerful bosonization method has been explored extensively to understand the unique physics of one-dimensional field. These different methods of many-body theory include Schwinger-boson mean-field (SBMF) [4], extended self-consistent mean-field (SCMF) [5], modified spin wave (MSW) [6], Green Function (GF) [7] and bond operator (BO) [8]. Various numerical methods such as QMC [9], DMRG [6] and ED [6] are also used to calculate the ground state and correlated effects of low-dimensional spin model. Based on the most popular bosonic representation of spin operators as Holstein–Primakoff transformation [10], Dyson–Maleev representation [11–14] made a correction

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E-mail addresses: [email protected] (Y. Li), [email protected] (B. Chen). http://dx.doi.org/10.1016/j.physleta.2017.07.004 0375-9601/© 2017 Elsevier B.V. All rights reserved.

to project out the unphysical boson state in the formalism which goes beyond the linear spin-wave theory. In this paper, we use the Dyson–Maleev mean-field theory to study the ground state and thermodynamic properties of Heisenberg spin-1/2 and spin-1 ferrimagnetic chain. This paper is organized as follows. In the second section, we present the Dyson–Maleev mean-field theory for isotropic ferrimagnetic spin chain. In the third section, we obtain the ground state energy and excited spectrum at zero temperature and numerical results of energy gap, magnetization, uniform susceptibility, internal energy and specific heat at finite temperature. Summary is made in the final section. 2. Dyson–Maleev mean-field formalism for spin-1/2 and spin-1 ferrimagnetic chain

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The Hamiltonian of spin-1/2 and spin-1 ferrimagnetic spin chain is

H=

1  2



(s+ s− + s− s+ ) + i j i j



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siz s zj ,

(1)



where si = 1/2 and s j = 1. We only consider the spin interaction between the nearest neighbor sites < i j >. This simplest system is a fundamental model which represents the generic type of ferrimagnets. The spin wave theory, especially the Holstein– Primakoff transformation [10], is one of the powerful tools in the field of magnets. This transformation has been used for ferromagnetic state [10] and antiferromagnetic N e´ el state [15]. Compared with the exact Holstein–Primakoff formalism, Dyson–Maleev

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method projects out the unphysical state which usually appears in the truncation of asymptotic square-root series. Here, we use the Dyson–Maleev representation to study the ground state and thermodynamic properties of the mixed spin chain. For the onedimensional ferrimagnetic spin system, the Dyson–Maleev representation [11–14] of spin operators can be written as follow,

s+ = (2s1 ) i

(2s1 ai − a+ a a ), s− = (2s1 ) a+ , siz = s1 − a+ a, i i i i i i i

s+ = (2s2 ) j

(2s2 b+j − b+j b+j b j ), s−j = (2s2 ) b j ,

− 12 − 12

s zj

1 2

(2)

2 1

[ai , a+ ] = 1, [b j , b+j ] = 1, [ai , b j ] = 0, [a+ , b+j ] = 0. i i

2

2

zM = −

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H=

[ 2(ai b j − a+ a a b )+ i i i j

2

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2

ij

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+

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+ + +

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+ +

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H=



√

ψ Wψ + √

− (−

a b c d

W =

2

4

where



2

+

k

√ a = (−

√

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b=[

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(5)

In the momentum space, the quadratic Hamiltonian can be obtained as

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√

× √

−

√

2 [ 22 (1 − 4



√ zP +

2

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M+

2 1 2



c=[

2 2

4

√

2

zP M +

2

4

zQ M −

1 2

2

z+M z (6)

) z,

 ,ψ =

√

uk v k+

 ,

M + 1) z ,

(2 − Q ) − M ]zγk ,

2

(1 − P ) − M ]zγk , 2 √

d = (−

2

4

M+

1 2

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4

(a

+ d)2

 1 

− 4bc √

4N

2

4

 E+  2T

− 3 4 2 (a + d) + (

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,

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2T

+ coth √

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z=

 E+ 

k

+ coth √

− coth

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2T

P ) − M ] z2 γk2

z Q + 2M z +



×

2T

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+ coth

 E −  2T

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√ ×

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2

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 E − 

2 (1 − 2

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Q ) − M ] z2 γk2

k

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2T P ) − M) + (

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 (a + d)2 − 4bc

2 (2 − 4



Q ) − M ) z2 γk2

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,

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+

− < ai b j >< ai b j > .

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+ +

+ +

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+

a+ a b + b = < a+ b+ > a i b j + a+ b+ < ai b j > i i j j i j i j

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2 [ 42 (2 − 2

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2T

where N is the lattice number of A(B) sublattice.

− < a+ b+ >< b+j b j >, i j

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√

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(8)

ai b j b j b j = < ai b j > b j b j + ai b j < b j b j >

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2T

+ coth

 E − 

+

a >< ai b j >, − < a+ i i

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coth

 E+ 

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ai ai ai b j = < ai ai > ai b j + ai ai < ai b j >

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k

4N

2

(4)

2

+

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(2a+ b+ − a+ b+ b+ b )] i j i j j j

(7)

, (a + d)2 − 4bc √   E+   E −  2 1  coth + coth zM = −

2

In the mean field approximation, we suppose that < a+ a >= P , i i < b+j b j >= Q , and < a+ b+ >=< ai b j >= M. Those remaining i j four-operator terms can be approximated as

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2

ij

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2

 1 1 (− + b+j b j + a+ a − a+ a b+ b ). + i i i i j j

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√

1√

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4N

(3)

By substituting the above boson operators into formula (1), the Hamiltonian can be rewritten as

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1 

×

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The parameters P , Q and M are obtained by numerically solving the following self-consistent equations:

√

where s1 = 1/2 and s2 = 1. ai and b j are the boson operators defined on the lattice sites i ∈ A and j ∈ B respectively. The operators ai and b j satisfy the following bosonic communication relation

(a + d)2 − 4bc ],

2

+

= −s2 + b j b j ,



[(a − d) +

 E = [−(a − d) + (a + d)2 − 4bc ]. −

√

1 2

1

E+ =

) z,

here z = 2 is the number of nearest neighbors for one dimensional chain and γk = cos(k). The energy spectrums are

3. Numerical results and discussion

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By solving the above self-consistent equations at zero temperature, we obtain the energy spectrums of system which exhibit the coexistence of a nearly gapless branch and a gapful branch as shown in Fig. 1. The energy gap of excitation spectrums is h = 1.32. In order to compare with other numerical results completely, we list a Table which contains the energy gaps estimated by other methods as shown in Table 1. The excitation gap, h = 1.32, of our Dyson–Maleev mean-field theory agrees quite well with the results given by SBMF [4], SCMF [5], MSW [6], DMRG [6], BO [8], QMC [9] and GF [7]. To be precisely, our result is only a little bit lower than these methods mentioned above. This phenomenon is mainly caused by the higher orders of interaction term which are usually ignored in the mean field approximation. The gap of lower excitation branch E k−=0 is 0.06 while the gap of higher excitation branch E k+=0 is 1.38. Compared with the methods of SBMF [4] and QMC [9], our result shows that the lower excitation branch exists a small gap. This phenomenon also occurs in the methods of the extended self-consistent mean-field theory [5], E k−=0 = 0.107, and bond operator theory [8], E k−=0 = 0.06. By adding a constraint E k−=0 = 0, we could remove this small trivial gap in the calculation. However, we keep the original data and retain this existed gap based on the Goldstone theorem in field theory. It is also important to mention that the appearance of this gap is also general in the theory of self-consistent mean-field [5] and bond operator [8]. In order to study the thermodynamic properties of this system, we solve the self-consistent equations numerically at finite temperature. As shown in the Fig. 2, we obtain the energy gaps of two excitation branches at different temperatures. With the increase of the temperature, the energy of gapful branch increases comparatively obvious while the variation of lower branch is slight. Undoubtedly, the

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Fig. 2. Red cycle line and black square line represent the energy gaps of excitation spectrum E + and E − versus T respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 1. The blue and yellow lines represent the excitation spectrums of E + and E − for isotropic ferrimagnetic chain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1 Energy gaps are estimated by various methods.

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Method



Method



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SBMF [4] GF [7] DMRG [6] This work

1.778 1.76 1.759 1.32

SCMF [5] MSW [6] QMC [9] BO [8]

1.764 1.676 1.767 1.64

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gap between these two branches is enhanced and the bandwidth of excitation spectrums is wider. Magnetization plateau is an exotic and prevalent phenomenon in the one-dimensional spin chain and strongly alternating spin ladder [16]. We study the magnetization properties of this model by applying the uniform magnetic field to the ferrimagnetic spin chain as shown in the Fig. 3. Due to the existence of a small gap in the lower branch at zero temperature, the system has a trivial plateau at m = 0 which means that the gap is closed by the external magnetic field H . In our calculation, we also find that the system has a small gap between the nearly gapless ferromagnetic branch and the gapful antiferromagnetic branch. The system has to increase the magnetic field by a finite number to overcome the gap which is the reason why the existence of nontrivial quantum plateau at m = 0.5. Before the system reaches the saturated magnetic field, the magnetization of ferrimagnetic spin chain exists a plateau which approximately equals 0.08 at m = 0.5. The T times uniform susceptibility per unit site

MF T χuni

N g 2

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Fig. 3. The magnetization of ferrimagnetic spin chain under the uniform magnetic field.

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versus tem-

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perature is plotted as shown in the Fig. 4. At the low temperature region, the uniform susceptibility is increasing with the increase of temperature. In the Fig. 5, we plot the temperature dependence of magnon internal energy. The power-law behavior of magnon internal energy, U ( T ) ∼ T 2 , is qualitatively consistent with the result of bond operator theory [8]. Fig. 6 shows the trend of specific heat with the variation of temperature. With the increase of temperature, the specific heat increases linearly which agrees with that of SCMF [5], BO [8] and SBMF [17] quite well. Only the estimated value of bond operator theory is a little bit larger than our theory. From the above figures and analyses, both the ground state energy spectrum and thermodynamic properties of this model can be described correctly and reasonably by the Dyson–Maleev mean-field theory compared with other existed methods.

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4. Summary

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In this paper, we study the energy spectrum in the ground state and thermodynamic properties of Heisenberg ferrimagnetic spin chain by using the Dyson–Maleev mean-field theory. The excita-

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Fig. 4. T times uniform susceptibility per unit site versus T.

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tion spectrum contains the coexistence of a nearly gapless branch and a gapful branch. The excitation gap, h = 1.32, is estimated reasonable compared with other established methods. Under a uniform magnetic field, the quantum plateau of magnetization in the ferrimagnetic spin chain is observed at m = 0.5. At finite temperature, the thermodynamic properties as internal energy and specific heat agree quite well with the results of SBMF [17], SCMF [5] and BO [8]. Compared with various existed methods, the Dyson–Maleev mean-field theory for ferrimagnetic spin chain is also valid and reasonable.

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Acknowledgements

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The work is supported by NSFC (No. 10774035) and Qianjiang RenCai Program of Zhejiang Province (No. 2007R0010).

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Fig. 5. The magnon internal energy per unit site versus T.

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Fig. 6. The specific heat per unit site versus T. Our results are compared with those from SBMF [17], SCMF [5] and BO [8].

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