Schwinger-boson mean-field theory of an anisotropic ferrimagnetic spin chain

Physics Letters A 374 (2010) 3514–3519

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Physics Letters A www.elsevier.com/locate/pla

Schwinger-boson mean-field theory of an anisotropic ferrimagnetic spin chain Yinxiang Li a,∗ , Bin Chen a,b a b

Department of Physics, Hangzhou Normal University, Hangzhou 310018, PR China Department of Physics, University of Shanghai for Science and Technology, Shanghai 200093, PR China

a r t i c l e

i n f o

Article history: Received 8 April 2010 Received in revised form 14 May 2010 Accepted 9 June 2010 Available online 12 June 2010 Communicated by A.R. Bishop

a b s t r a c t By using the Schwinger-boson mean-field theory, the Heisenberg ferrimagnetic spin chain with the single-ion anisotropy D is explored. Based on the effect of the single-ion anisotropy D, we obtain four branches of the low-lying excitation and calculate the anisotropy dependence of spin reduction and the longitudinal correlation at zero temperature. We also discuss the free energy, magnetic susceptibility and specific heat at finite temperature with different anisotropy D. © 2010 Elsevier B.V. All rights reserved.

Keywords: Schwinger-boson mean-field theory The single-ion anisotropy Heisenberg ferrimagnetic spin chain

1. Introduction In the recent decades, the low-dimensional magnetic systems have attracted much interest. Haldane [1] proposed that the excited state of integer-spin chain has an energy gap and the spin correlation decays exponentially, while the half-odd-integer spin chain is gapless and the spin correlation decays algebraically with distance. After that, enormous efforts have been devoted to verify the Haldane conjecture and to study the properties of Heisenberg ferrimagnetic spin chain. Brehmer et al. [2] studied the low-lying excited states and the low temperature properties of quantum antiferromagnetic spin chain in term of the spin-wave theory (SWT) and the quantum Monte Carlo method (QMC). By using the modified spin-wave theory (MSWT) and the densitymatrix renormalization-group method (DMRG), Yamamoto et al. [3–7] studied the thermodynamic properties of the Hersenberg Ferrimagnetic spin chain. Wu et al. [8] studied the ground state and thermodynamic properties of ferrimagnetic spin chain by Schwinger-boson mean-field theory (SBMFT) and pointed out that the long range ferrimagnetic order of the ground state is caused by the condensation of the Schwinger bosons. Su et al. [9] used Schwinger boson theory to study one-dimensional Heisenberg antiferromagnetic spin chain with single-ion anisotropy. The quantum magnetization plateaux of an anisotropic ferrimagnetic spin chain was studied by Sakai et al. [10]. In this Letter, we study the excited states and thermodynamic properties of Heisenberg

*

Corresponding author. Tel.: +86 571 28912996; fax: +86 571 28912996. E-mail address: [email protected] (Y. Li).

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ferrimagnetic spin chain with single-ion anisotropy D by using Schwinger-boson mean-field theory. This Letter is organized as follows. In the second section, we present Schwinger-boson mean-field theory for the anisotropy ferrimagnetic spin chain. In the third section, we obtain the ground state and related properties of this model at zero temperature. In the forth section, we give the thermodynamic properties such as the energy gaps, the free energy, the uniform susceptibility, the stagger susceptibility and the specific heat based on the effect of single-ion anisotropy D. Conclusion and discussion are made in the final section. 2. The Schwinger-boson mean-field theory The Hamiltonian of an anisotropic ferrimagnetic spin chain can be expressed as

H= J



si · S i +δ + D

i ,δ



S iz+δ

2

(1)

i

with J = 1, s = 12 , S = 1. δ is the index of the nearest neighbors. The spin operator siA can be represented by Schwinger Bosons ai ,↑ and ai ,↓ :

siA,+ = a+ a , i ,↑ i ,↓ siA,z =

siA,− = a+ a , i ,↓ i ,↑

 1 + ai ,↑ ai ,↑ − a+ i ,↓ ai ,↓ 2

(2)

with siA = 12 (a+ a + a+ a ) for each sublattice A and each subi ,↑ i ,↑ i ,↓ i ,↓ lattice B can be represented in a similar way. The Hamiltonian can be rewritten as

Y. Li, B. Chen / Physics Letters A 374 (2010) 3514–3519

H = −2



A+ A + i ,i +δ i ,i +δ

i =1,δ

 i =1,δ

  2 z

siA S iB+δ + D

Sj

(3)

j =1

λ A − (λ B + D  S zj )

E α2 = E α↓ =



where A i ,i +δ = 12 (ai ,↑ b i +δ,↓ − ai ,↓ b i +δ,↑ ). By taking the constraint  +  + + + A B i (ai ,↑ ai ,↑ + ai ,↓ ai ,↓ ) = 2s or j (b j ,↑ b j ,↑ + b j ,↓ b j ,↓ ) = 2S , we A B introduce two kinds of Lagrangian multiplier λi and λ j . At the mean field level, we take the average value of the bond operator  A i ,i +δ  = A , λiA  = λ A , λ Bj  = λ B . The mean field Hamiltonian reads

H =−



∗



  z



A (ai ,↑ b j ,↓ − ai ,↓ b j ,↑ ) +

i =1,δ

+D

+ E β2 = E β↑ = −





a+ a − 2s A + λ B i ,δ i ,δ

+ λA

i =1,δ A



b+ b − 2S B j ,δ j ,δ

B

+ 2zN A A + zNs S .



2N

=

2

1

β

− π2

H =−

∗

k,δ B

− 2N s λ + s λ

B



A

(5)

+ = sinh θ1 αk,↑ + cosh θ1 β− , k,↓

2

λ A + λ B + D  S zj  2

(6)

with θ given by

(7)

E β1 = E β↓ = −

+

λA

+ (λ B − D  S zj ) 2

+ z A∗ A

2

D z  2

z

z 2 

Sj − Sj

= =

z Λ1

,

2 zΛ1

,

2

1 cosh 2θ1 =  , 1 − η2 γk2

(9)

.

λ A − λ B + D  S zj  2

λ A − λ B − D  S zj  2

= =

z Λ2 2 zΛ2 2

, ,

Λ2 = Λ1 + Λ2 − Λ1 , (10)

sinh 2θ1 = 

2 − | z A γk | , 2

2 − | z A γk |2 ,

|ηγk | 1 − η2 γk2

sinh 2θ2 = 

,

|η γk | 1 − η 2 γk2

.

(11)

By minimizing the free energy (i.e. δ F /δ A = 0, δ F /δλ A = 0, δ F /δλ B = 0, δ F /δ S zj  = 0), the mean-field self-consistent equations can be represented as

S B − sA =

2

2

2

π

λ A − (λ B − D  S zj )

β E β,2

2

η

2

2

2



Λ1 η

λ A − (λ B − D  S zj )

+

2

1 cosh 2θ2 =  , 1 − η 2 γk2

we obtain the energy spectrum

λ A + (λ B − D  S zj )



Then, the angle of the Bogoliubov transformation is expressed as

+ = − sinh θ2 βk,↑ + cosh θ2 sinh θ2 α− , k,↓



β E α ,1

Λ η  4κ = 1 , β= , 2 2 z   z z Λ1 η Sj = − 1 .  A=

2D

+ bk,↑ = cosh θ2 βk,↑ − sinh θ2 α− , a+ k,↓ −k,↓

E α1 = E α↑ =



λ A + λ B − D  S zj 

B

+ ak,↑ = cosh θ1 αk,↑ + sinh θ1 β− , b+ k,↓ −k,↓

−2| zB γk | , λ A + λ B + D  S zj 

(8)

We take a transformation as

where  ikzδ is the number of the nearest neighbor sites and γk = 1 = cos k. The sum of k is retricted in the reduced first δe z Brillouin zone. By using the Bogoliubov transformation

tanh 2θ2 =



ln 2 sinh

2π

2

+ 2zN A A + zNs S ,

−2| z A γk | , λ A + λ B − D  S zj 

 

+ sA S B +

∗

tanh 2θ1 =

dk

2

2  A +  λ ak,δ ak,δ + λ B bk+,δ bk,δ − N D z S zj + A

− | z A γk |2 .

2

z

k

A

2

    1 1 − sA + λA − S B + λB



+ + + Azγk ak+,↑ b+ −k,↓ − ak,↓ b −k,↑    +D S zj bk+,↑ bk,↑ − b+ −k,↓ b −k,↓



λ A + (λ B + D  S zj )

+ ln 2 sinh

∗



2



A zγk (ak,↑ b−k,↓ − ak,↓ b−k,↑ )

k



      β E β,1 β E α ,2 + ln 2 sinh + ln 2 sinh

(4)

In momentum space, it can be transformed into



− | z A γk |2 ,

λ A − (λ B + D  S zj )

π

F

j =1,δ

∗

2

+ (λ B + D  S zj )

The free energy can be written as

z 2 

− Sj S j b+ b − b+ b j ,↑ j ,↑ j ,↓ j ,↓

j

2

λA

2

+

  − a+ b+ A i ,↓ j ,↑

a+ b+ i ,↑ j ,↓

3515

− π2

dk  2π

coth

 



κ −Λ2 + Λ1 1 − η2 γk2

    − coth κ Λ2 + Λ1 1 − η2 γk2      + coth κ −Λ2 + Λ1 1 − η 2 γk2     − coth κ Λ2 + Λ1 1 − η 2 γk2 ,



(12)

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Y. Li, B. Chen / Physics Letters A 374 (2010) 3514–3519 π

2 B

A

S +s +1=

dk  2π

− π2

coth

π



 

2

 2

κ −Λ2 + Λ1 1 − η2 γk



1

z

=

2 z

dk  2π

− π2



1 1 − η 2 γk2

,

− Λ1 η

2

− π2

dk

2 k

2

2π

coth

 × −1 − 



κ −Λ2 + Λ1 1 − η2 γk2

B

     + coth κ Λ2 + Λ1 1 − η2 γk2 1−    2



+ coth κ −Λ2 + Λ1 1 − η 2 γk

  2



+ coth κ Λ2 + Λ1 1 − η 2 γk

(14)

1−η

1+ 

2

2 k

−π



γ

2

D

η −1 −1 η

1 − η2 2

(18)

,

1 − η 2 γ 2 1

−1 − η 2 γk2

2N

=−

dk 2π

.

A

sz =

−π 2

dk 2π



1 1 − η2 γk2

γ



1 1 − η 2 γk2

(19)

.

2N

(20)

1 2N

β  (sinh θ1 )2nk,↓ k=0,T →0k .

(21)



S B − sA



2

1 2

+



τ A = s A + s zA = s A −

S B − sA 2



1 2



1

(22)

1 − η2

and the spin reduction on the site A is

−

1 1 − η2

 .

(23)

The zero temperature free energy per unit cell is π

N



2 1 − η2

The spin reduction on the site B is

F

2

1

+ 

2 k

β  (cosh θ1 )2nk,↓ k=0,T →0k .



π

1−η

2

k

1

τ B = S B − S zB = S B +



1

The average value of the spin on the site A is expressed in a similar way



(16)

  1 − = − S B − sA −  2 1 − η2



1  + bk,↑ bk,↑ − bk+,↓ bk,↓



S zB =

 k

1+ 

dk 2π

2

1

 Λ 2 = Λ1 1 − η 2 , 2Λ 1

γ + 2 k

By solving these self-consistent equations numerically, the excited energy spectrums at different anisotropy (i.e. D = 0, D = 0.5, D = 1.0, D = 1.5) can be plotted respectively as Fig. 1. From these figures, it can be observed clearly that the energy levels are transformed from degeneracy to non-degeneracy and the energy gaps increase with increasing anisotropy D around k = 0. However, the energy level of the antiferromagnetic branch E α ,1 and E α ,2 get together while k tends to π /2 or −π /2. When the anisotropy D = 0, our numerical results are found in good agreement with Wu et al. [8]. We obtain the average value of the spin on the site B at T = 0k



1

A

+

We suppose that an infinitesimal external stagger magnetic field is downward at site B and upward at site A. The βk,↓ branch has the lowest energy and the bosons condense at the βk,↓ branch at T = 0k. Thus, the self-consistent equations are reduced as



 Λ1 1 − η 2 γk2

π

3. Properties at zero temperature



1−η

2

2

(15)



|γk |2



− π2



1 − η2 γk2

 

2π

−π

S +s +1=



 

dk

−

2

|γk |2  Λ1 1 − η2 γk2



 

1

2

π





2π

− π2

 D z  z

2 Sj −1 2 =

1 − η 2 γk2

π

2 z



dk

=

    + coth κ −Λ2 + Λ1 1 − η 2 γk2     |γk |2  − coth κ Λ2 + Λ1 1 − η 2 γk2 , Λ1 1 − η 2 γk2

π

(17)

,

1

2

π

2



2

2



2π

− π2

   coth κ −Λ2 + Λ1 1 − η2 γk2

− coth κ Λ2 + Λ1 1 − η γ

D

dk

S B + sA + 1 −



 

1 − η 2 γk2

π

(13) 2

1

2

1 − η2 γk2

    + coth κ −Λ2 + Λ1 1 − η 2 γk2      − coth κ Λ2 + Λ1 1 − η 2 γk2

π



2π

− π2



    − coth κ Λ2 + Λ1 1 − η2 γk2

2

dk

+

2 = − π2

dk  4π





4Λ1 1 − η2 γk2 + 4Λ1 1 − η 2 γk2



  z + Λ21 η2 − 2s A + 1 (Λ1 + Λ2 ) 2    2   − 2S B + 1 Λ1 − Λ2 + 2s A S B + 2D S zj − S zj .

(24)

Y. Li, B. Chen / Physics Letters A 374 (2010) 3514–3519

3517

Fig. 1. The dotted, solid, dashed and dotdashed lines respectively represent the excited energy spectrum of E α ,1 , E β,1 , E α ,2 and E β,2 .

In the Schwinger-boson mean-field theory, the longitudinal correlations between two sites can be expressed as follow









siA,z S Bj,z − siA,z S Bj,z



 π2 2   dk  −ikR i , j  = − sinh 2θ1 e  ,   2π − π2



siA,z s Aj,z



−



siA,z



s Aj,z



 π2 2   dk  −ikR i , j  = − cosh 2θ1 e  ,   2π − π2









S iB,z S Bj,z − S iB,z S Bj,z



 π2 2   dk  −ikR i , j  = − cosh 2θ1 e  .   2π

(25)

−π 2

4. Effect of the anisotropy D at finite temperatures





Λ 2 = Λ1 1 − η − 2

2 1 − η2 T 2

Λ1 η 2

.

(26)

By solving the self-consistent Eqs. (20)–(22), the energy gaps at different anisotropy D are plotted as Fig. 2. From above figures, we clearly observe that the energy gaps increase with larger anisotropy D as the single-ion anisotropy D tends to enlarge the gaps. This result is similar with Su et al. [9]. When T = 0k and D = 0, these energy gaps agree with the results of Wu et al. [8]. The free energy versus temperature is also calculated and plotted in Fig. 3. As can be seen, the free energy decreases more slowly with larger anisotropy D. We have to divide the part of fluctuation per unit cell F M F / N + 2s A S B by 2 and add the classic ground energy per cell −2s A S B because the degrees of freedom of the Schwinger-bosons are overcounted by a factor 2, as argued by Arovas and Auerbach [11,12]. The mean-field static uniform magnetic susceptibility per unit cell is π

The SBMFT is not good to descibe the thermodynamic properties of the ferrimagnetic spin chain at intermediate and high temperature, as pointed out by Wu et al. [8]. So the properties are shown when T < 0.5 in our Letter. At low temperature, there is no boson condensation. We obtain the following equation from Eq. (15),

T χuni N g2

2 = − π2

dk  nα ,1 (nα ,1 + 1) + nβ,1 (nβ,1 + 1) 4π

 + nα ,2 (nα ,2 + 1) + nβ,2 (nβ,2 + 1) ,

as Fig. 4.

(27)

3518

Y. Li, B. Chen / Physics Letters A 374 (2010) 3514–3519

Fig. 2. The dotted, solid, dashed and dotdashed lines respectively represent the energy gap of E α ,1 , E β,1 , E α ,2 and E β,2 .

The mean-field static stagger magnetic susceptibility per unit cell is π

T χstag N g2

2

dk

=

4π

− π2







nα ,1 (nα ,1 + 1) + nβ,1 (nβ,1 + 1) (cosh 2θ1 )2

+ 2(sinh 2θ1 )2

1 + nα ,1 + nβ,1

β( E α ,1 + E β,1 )

 + nα ,2 (nα ,2 + 1) + nβ,2 (nβ,2 + 1) (cosh 2θ2 )2  1 + nα ,2 + nβ,2 , + 2(sinh 2θ2 )2 β( E α ,2 + E β,2 ) 

(28)

as Fig. 5. We multiply the mean-field susceptibility with the factor 2/3 due to the same argument as Arovas and Auerbach [11,12]. In Figs. 4 and 5, the effects of single-ion anisotropy on the uniform susceptibility and stagger susceptibility are shown clearly at low temperature. The effects are more obvious with decreasing temperature. Both the uniform susceptibility and stagger susceptibility are increasing with decreasing temperature. However, the stagger

Fig. 3. The free energy F versus T with different anisotropy D.

susceptibility is increasing more rapidly than the uniform susceptibility. Moreover, both the uniform susceptibility and stagger sus-

Y. Li, B. Chen / Physics Letters A 374 (2010) 3514–3519

Fig. 4. The red, green, blue, gray, pink, and black lines respectively represent T times the uniform susceptibilities per unit site when D = 0, 0.001, 0.005, 0.01, 0.0572, 0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

3519

Fig. 6. The specific heat versus T with different anisotropy D.

From all the above figures, it is obviously that all the thermodynamic properties have a good agreement with the results of Wu. et al. [8] while the single-ion anisotropy D = 0. 5. Conclusion and discussion By using the Schwinger-boson mean field theory, we have obtained the ground state and the thermodynamic properties of the anisotropy ferrimagnetic spin chain. Wu et al. [8] pointed out that the SBMFT results describe the thermodynamic properties well at tow temperature. Thus, we give these thermodynamic properties when T < 0.5. The relationships between the the thermodynamic properties of an anisotropy ferrimagnetic spin chain and the anisotropy D are shown clearly in the former figures. Acknowledgement

Fig. 5. The red, green, blue, gray, pink, and black lines respectively represent T times the stagger susceptibilities per unit site when D = 0, 0.001, 0.005, 0.01, 0.0572, 0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

ceptibility exist different crossing points with different anisotropy at lower temperature. The specific heat versus temperature is calculated and plotted in Fig. 6. The effects of single-ion anisotropy on the specific heat are clearly at low temperature, as shown in Fig. 6. It is found that the effects are not obvious with increasing temperature. The specific heat with smaller anisotropy increases more rapidly. The specific heat with larger anisotropy parameter is lower at low temperature.

The work is supported by NSFC (10574035) (10774035). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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