Dyson-Maleev theory of ferrimagnetic spin chain with bond alternation in longitudinal magnetic field

Physica B 575 (2019) 411709

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Dyson-Maleev theory of ferrimagnetic spin chain with bond alternation in longitudinal magnetic field Yuge Chen a,b , Yinxiang Li c ,∗, Lijun Tian b , Bin Chen c,d a

Department Department Department d Department b c

of of of of

Physics, Physics, Physics, Physics,

ARTICLE

Fudan University, Shanghai 200433, PR China University of Shanghai, Shanghai 200444, PR China University of Shanghai for Science and Technology, Shanghai 200093, PR China Hangzhou Normal University, Hangzhou 310018, PR China

INFO

Keywords: Dyson–Maleev mean-field theory Ferrimagnetic spin chain Bond alternation Longitudinal magnetic field

ABSTRACT By using the method of Dyson–Maleev mean-field theory, we study the effect of bond alternation and longitudinal magnetic field in the model of ferrimagnetic spin chain. Based on the numerical results of self-consistent equations, properties of the excited spectrums, magnon internal energy, static susceptibility, magnetization plateau and specific heat are obtained with different bond alternation parameters 𝛿 and external magnetic field 𝐵. This method provides qualitative consistent results to describe the phase from ferrimagnetic state to bond altered state.

1. Introduction In the past three decades, the field of low-dimensional magnetic system has attracted many theoretical and experimental researchers. The appearance of various exotic quantum phenomena boosts the development of this field, such as Luttinger liquid behavior, magnetic plateaux, spin-Peierls states. Since the discovery of cuprates high-temperature superconductor [1], the relation between antiferromagnetic (AF) fluctuation and unconventional superconductor is also interesting and mysterious. For one-dimensional antiferromagnetic spin chain, Haldane [2] made a conjecture that the ground state of an integer-𝑆 Heisenberg AF spin chain has a finite spectral gap, while the spectrum of half-integer AF spin chain is gapless. This statement pushed the rapid development of one-dimensional spin chain in theory and experiment. The material of mixed spin chain, which is consisted of two different spin 𝑆 and 𝑠 (𝑆 ≠ 𝑠), was synthesized in the lab such as 𝑁𝑖𝐶𝑢(𝑝𝑏𝑎)(𝐷2 𝑂)3 ⋅ 𝐷2 𝑂 [3]. The ferrimagnetic spin materials arise the enthusiasm of theoreticians. Different mean-field methods have been proposed to deal with low-dimensional spin system such as Dyson–Maleev mean-field (DMMF) [4–9], bond operator (BO) [10– 15], Schwinger-boson mean-field (SBMF) [16–18] and modified spin wave (MSW) [19,20]. Besides above mean-field methods, the methods of numerical calculation, such as DMRG [19], QMC [21] and ED [19], also be applied to study the physical properties of spin chain and ladders. Due to the synthesization of real low-dimensional spin material, the effects of dimerization, anisotropic interaction and frustration also be considered in the modified spin model. Sachdev [10]

introduced bond operator formalism to understand the dimerized state in two-dimensional spin-1/2 antiferromagnets. One of us [8] used the Dyson–Maleev mean-field theory to study the model of ferrimagnetic spin chain. In this paper, we apply the method of Dyson–Maleev meanfield to deal with the effect of bond alternation and external magnetic field in the ferrimagnetic spin chain. The paper is organized as follows. In the second section, we present the Dyson–Maleev mean-field approximation for ferrimagnetic spin chain with bond alternation under longitudinal magnetic field. In the third section, we study numerical results of the excited spectrums, magnon internal energy, specific heat at finite temperature based on the effect of bond alternation. Subsequently, we consider the effect of longitudinal magnetic field in ferrimagnetic spin chain model. We also discuss the corresponding numerical results about several physical properties as static susceptibility, magnetization and specific heat. Summary and conclusion are made in the final section.

2. Dyson-Maleev representation for spin-1/2 and spin-1 ferrimagnetic chain with bond alternation The Hamiltonian of spin-1/2 and spin-1 ferrimagnetic spin chain with bond alternation in the longitudinal magnetic field is ∑ ∑ ∑ 𝐻 =𝐽 [(1 + 𝛿)𝑠𝑖 𝑠𝑗 + (1 − 𝛿)𝑠𝑗 𝑠𝑖+1 ] + 𝐵 𝑠𝑧𝑖 + 𝐵 𝑠𝑧𝑗 (1) 𝑖,𝑗

∗ Corresponding author. E-mail addresses: [email protected] (Y. Li), [email protected] (B. Chen).

https://doi.org/10.1016/j.physb.2019.411709 Received 2 April 2019; Received in revised form 4 August 2019; Accepted 16 September 2019 Available online 17 September 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.

𝑖

𝑗

Physica B: Physics of Condensed Matter 575 (2019) 411709

Y. Chen et al.

where 𝑠𝑖 = 1∕2, 𝑠𝑗 = 1 and 𝐽 = 1. 𝛿 is the bond alternation parameter and 𝐵 is the magnetic field. We only consider the local spin interaction between the nearest neighbor (NN) sites ⟨𝑖𝑗⟩. This modified spin system is a fundamental model with bond alternation in the real material which might be related with spin-Peierls material 𝐶𝑢𝐺𝑒𝑂3 [22]. Compared with Holstein–Primakoff transformation [23], Dyson–Maleev method projects out the unphysical state by truncating the asymptotic square-root series. In this paper, we use the Dyson–Maleev representation to study the ground state and thermodynamic properties of the mixed spin chain with bond alternation 𝛿. The Dyson–Maleev representation of spin operators can be shown as, 1

1

−2 − + 2 + 𝑧 𝑠+ (2𝑠1 𝑎𝑖 − 𝑎+ 𝑖 =(2𝑠1 ) 𝑖 𝑎𝑖 𝑎𝑖 ), 𝑠𝑖 = (2𝑠1 ) 𝑎𝑖 , 𝑠𝑖 = 𝑠1 − 𝑎𝑖 𝑎𝑖 , 1

1

−2 + + − 𝑧 + 2 (2𝑠2 𝑏+ 𝑠+ 𝑗 − 𝑏𝑗 𝑏𝑗 𝑏𝑗 ), 𝑠𝑗 = (2𝑠2 ) 𝑏𝑗 , 𝑠𝑗 = −𝑠2 + 𝑏𝑗 𝑏𝑗 , 𝑗 =(2𝑠2 )

(2)

where 𝑠1 = 1∕2 and 𝑠2 = 1. 𝑎𝑖 and 𝑏𝑗 are the boson operators defined on the lattice sites 𝑖 ∈ 𝐴 and 𝑗 ∈ 𝐵 respectively. The bosonic operators 𝑎𝑖 and 𝑏𝑗 satisfy the anticommute relation. After algebraic operation, the Hamiltonian can be rewritten as √ ∑√ 2 + + 1 + + (2𝑎𝑖 𝑏𝑗 − 𝑎+ 𝐻 = (1 + 𝛿) [ 2(𝑎𝑖 𝑏𝑗 − 𝑎+ 𝑎 𝑎 𝑏 ) + 𝑖 𝑏𝑗 𝑏𝑗 𝑏𝑗 )] 𝑖 𝑖 𝑖 𝑗 2 2 𝑖𝑗 ∑ 1 1 + + + (1 + 𝛿) (− + 𝑏+ 𝑏 + 𝑎+ 𝑖 𝑎𝑖 − 𝑎𝑖 𝑎𝑖 𝑏 𝑗 𝑏 𝑗 ) 2 2 𝑗 𝑗 𝑖𝑗 ∑√ 1 + (1 − 𝛿) [ 2(𝑎𝑖+1 𝑏𝑗 − 𝑎+ 𝑎 𝑎 𝑏) 𝑖+1 𝑖+1 𝑖+1 𝑗 2 (3) 𝑖𝑗 √ 2 + + (2𝑎𝑖+1 𝑏𝑗 − 𝑎+ 𝑏+ 𝑏+ 𝑏 )] + 𝑖+1 𝑗 𝑗 𝑗 2 ∑ 1 1 + (1 − 𝛿) (− + 𝑏+ 𝑏 + 𝑎+ 𝑎 − 𝑎+ 𝑎 𝑏+ 𝑏 ) 𝑖+1 𝑖+1 𝑖+1 𝑖+1 𝑗 𝑗 2 2 𝑗 𝑗 𝑖𝑗

Fig. 1. Energy spectrums with different bond alternation parameters 𝛿 = 0 (black), 𝛿 = 0.6 (red), 𝛿 = 0.8 (green) and 𝛿 = 1.0 (blue) at zero temperature. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

+ + 𝐵(1∕2 − 𝑎+ 𝑖 𝑎𝑖 ) + 𝐵(−1 + 𝑏𝑗 𝑏𝑗 ).

In the mean field approximation, we define the variables as ⟨𝑎+ 𝑖 𝑎𝑖 ⟩ = 𝑃 , + + ⟨𝑏+ 𝑗 𝑏𝑗 ⟩ = 𝑄, and ⟨𝑎𝑖 𝑏𝑗 ⟩ = ⟨𝑎𝑖 𝑏𝑗 ⟩ = 𝑀. The above four-operator interaction terms can be treated as + + + 𝑎+ 𝑖 𝑎𝑖 𝑎𝑖 𝑏𝑗 =⟨𝑎𝑖 𝑎𝑖 ⟩𝑎𝑖 𝑏𝑗 + 𝑎𝑖 𝑎𝑖 ⟨𝑎𝑖 𝑏𝑗 ⟩ − ⟨𝑎𝑖 𝑎𝑖 ⟩⟨𝑎𝑖 𝑏𝑗 ⟩, + + 𝑎+ 𝑖 𝑏𝑗 𝑏𝑗 𝑏𝑗 + 𝑎𝑖 𝑎𝑖 𝑏 + 𝑗 𝑏𝑗

+ + + + + + + + =⟨𝑎+ 𝑖 𝑏𝑗 ⟩𝑏𝑗 𝑏𝑗 + 𝑎𝑖 𝑏𝑗 ⟨𝑏𝑗 𝑏𝑗 ⟩ − ⟨𝑎𝑖 𝑏𝑗 ⟩⟨𝑏𝑗 𝑏𝑗 ⟩, + + + + + + =⟨𝑎𝑖 𝑏𝑗 ⟩𝑎𝑖 𝑏𝑗 + 𝑎𝑖 𝑏𝑗 ⟨𝑎𝑖 𝑏𝑗 ⟩ − ⟨𝑎𝑖 𝑏𝑗 ⟩⟨𝑎𝑖 𝑏𝑗 ⟩.

Fig. 2. Magnon internal energy 𝑈 with different bond alternation parameters 𝛿 = 0 (black), 𝛿 = 0.6 (red), 𝛿 = 0.8 (green) and 𝛿 = 1.0 (blue) at low temperature. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(4)

In the momentum space, the quadratic Hamiltonian can be obtained as √ √ ∑ 2 2 1 𝐻= 𝜓 +𝑊 𝜓 + 𝑧𝑃 𝑀 + 𝑧𝑄𝑀 − 𝑧 2 4 2 𝑘 √ 2 1 3 + 𝑀 2 𝑧 − (− 𝑀 + )𝑧 − 𝐵, (5) 4 2 2 where ( ) ( ) 𝑎𝑘 𝑎 𝑏 𝑊 = ,𝜓 = , + 𝑏𝑘 𝑐 𝑑

self-consistent equations with different bond alternation parameters 𝛿, the variables as 𝑃 , 𝑄 and 𝑀 can be obtained,

√

( ( +) ( − )) 2 1 ∑ 𝐸 𝐸 𝑧𝑀 = − 𝑐𝑜𝑡ℎ + 𝑐𝑜𝑡ℎ 2 4𝑁 𝑘 2𝑇 2𝑇

√

√ √ 2 [ 42 (2 2 √

− 𝑄) − 𝑀]𝑧2 𝛾𝑘2 , √ (𝑎 + 𝑑)2 − 4𝑏𝑐

√

( ( +) ( − )) 2 [ 2 (1 − 𝑃 ) − 𝑀]𝑧2 𝛾 2 𝑘 1 ∑ 𝐸 𝐸 4 2 𝑐𝑜𝑡ℎ + 𝑐𝑜𝑡ℎ , √ 4𝑁 𝑘 2𝑇 2𝑇 (𝑎 + 𝑑)2 − 4𝑏𝑐 √ √ 2 2 𝑧𝑄 + 2𝑀𝑧 + 𝑧= 4 4 [( ( +) ( − )) ∑ 𝐸 1 𝐸 − −𝑐𝑜𝑡ℎ + 𝑐𝑜𝑡ℎ 4𝑁 𝑘 2𝑇 2𝑇 √ ( ( ) ( )) + 2 𝐸 𝐸− × + 𝑐𝑜𝑡ℎ + 𝑐𝑜𝑡ℎ 4 2𝑇 2𝑇 √ √ [ √2 ] ] 3 2 − 4 (𝑎 + 𝑑) + ( 2 (1 − 𝑃 ) − 𝑀) + ( 42 (2 − 𝑄) − 𝑀) 𝑧2 𝛾𝑘2 × , √ (𝑎 + 𝑑)2 − 4𝑏𝑐

2 𝑧𝑀 = − 4 √ 2 𝑧𝑃 + 2

√ 2 𝑎 =(− 𝑀 + 1)𝑧 − 𝐵, 2 √ 2 𝑏 =[ (2 − 𝑄) − 𝑀]𝑧𝛾𝑘 , 4 √ 2 𝑐 =[ (1 − 𝑃 ) − 𝑀]𝑧𝛾𝑘+ , 2 √ 2 1 𝑑 =(− 𝑀 + )𝑧 + 𝐵, 4 2 here 𝑧 = 2 is the number of NN sites in one-dimensional system and 𝛾𝑘 = 𝑐𝑜𝑠(𝑘) + 𝑖𝛿𝑠𝑖𝑛(𝑘). The energy spectrums are √ 1 𝐸 + = [(𝑎 − 𝑑) + (𝑎 + 𝑑)2 − 4𝑏𝑐], 2 (6) √ 1 𝐸 − = [−(𝑎 − 𝑑) + (𝑎 + 𝑑)2 − 4𝑏𝑐]. 2 In order to minimize the energy, we could obtain the self-consistent 𝜕𝐻 equations by 𝜕𝐻 = 𝜕𝐻 = 𝜕𝑀 = 0. By numerically solving the following 𝜕𝑃 𝜕𝑄

(7)

where 𝑁 is the lattice number of A or B sublattice and 𝛾𝑘2 = 𝑐𝑜𝑠2 (𝑘) + 𝛿 2 𝑠𝑖𝑛2 (𝑘). 2

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effect in the Heisenberg ferrimagnetic spin model by numerically solving the above self-consistent equations. The two branches of energy spectrums with different bond alternation parameters 𝛿, (𝛿 = 0, 𝛿 = 0.6, 𝛿 = 0.8, 𝛿 = 1.0), are shown in Fig. 1. In the absence of bond alternation 𝛿 = 0, the system is Heisenberg ferrimagnetic spin-(1/2,1) chain. The energy spectrums of this state are consisted of a gapless branch and another gapful branch. With adding the magnitude of bond alternation, the lower branch varies from gapless to gapful. The gap between these two branches becomes larger and the dispersions of these branches are narrower. When the bond alternation is less than 0.6 (𝛿 < 0.6), the lower branch of energy spectrums is nearly gapless. This result is qualitatively consistent with exact-diagonalization estimate [24]. As the model approaches the decoupled-dimer limit (𝛿 = 1.0) in our paper, the gap between ferromagnetic and antiferromagnetic bands is 1.5 and the energy spectrums are dispersionless. These properties agree with above cited paper. The small gap (𝛥 = 0.3) in decoupleddimer limit mainly depends on the parameters (P, Q, M) by solving self-consistent equations. To retain the original data of our meanfield approximation, we have not artificially adding the constrain of − = 0 to make the lower branch of energy spectrums gapless. These 𝐸𝑘=0 results are qualitatively consistent with the transition from Heisenberg ferrimagnetic state to one-bond state [24–26]. In Fig. 2, the variation of 𝐸𝑘𝑖 ∑ ) with various parameters magnon internal energy (𝑈 = 𝑖,𝑘 𝐸𝑖

Fig. 3. The specific heat 𝐶 with different bond alternation parameters 𝛿 = 0 (black), 𝛿 = 0.6 (red), 𝛿 = 0.8 (green) and 𝛿 = 1.0 (blue) at low temperature. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

𝑒𝑥𝑝[ 𝜅 𝑘𝑇 ]−1 𝑏

𝛿 is displayed at low temperature. With the increasing of temperature, the internal energy of system is enhanced apparently. The system with larger bond alternation is lower than ferrimagnetic state at nearly zero temperature. When the temperature is lifted, the internal energy with larger bond alternation exceeds the normal bond state. Obviously, there exists a critical point, near 𝑇𝑐 = 0.15, about the transition of internal energy with bond alternation. The temperature dependence ) with bond alternation 𝛿 are plotted in of specific heat (𝐶 = 𝜕𝑈 𝜕𝑇 Fig. 3. In the temperature region [0.2, 0.4], the trend of specific heat behaves nearly linear even altering the bond alternation parameter. In the lower temperature region, the specific heat with larger bond alternation increases more slowly near zero temperature and then is enhanced higher. There exists a transition point between the system with different bond alternations. Fig. 4 shows the bond alternation effect ∑ on the T times uniform static susceptibility 𝑇 𝜒𝑢𝑛𝑖 ∕𝑁𝑔 2 = 𝑖,𝑘 𝑛𝑖𝑘 (𝑛𝑖𝑘 + 1) under small longitudinal magnetic field (𝐵 = 0.1), where 𝑛𝑖𝑘 is the Bose–Einstein statistics. The static susceptibility becomes smaller with continually adding the bond alternation parameter 𝛿. The line with smaller parameter 𝛿 is increased slower. In the low-dimensional magnetic system, the magnetic plateau is a prevalent phenomenon in the spin-alternating chain and ladder [27]. Based on the famous Lieb– Schulz–Mattis theorem [28], system exists a nontrivial magnetization plateau at 𝑚 = 0.5 in spin-(1/2,1) ferrimagnetic chain. The single-ion anisotropy induced phase transition of magnetization plateau is also observed by Chen [9] and Saika [29]. We discuss the magnetization 𝜕 𝑙𝑛𝑇 𝑟𝑒−𝛽𝐻 (𝛽 = 𝜅 1𝑇 ) in the longitudinal magnetic field 𝐵. With 𝑚 = 𝛽1 𝜕𝐵 𝑏 the enhancement of bond alternation, the lower spectrum changes from gapless to gapful. The system needs additional external magnetic field to close the gap. Thus, there exist a trivial magnetization plateau at 𝑚 = 0 as plotted in Fig. 5. When the magnetic field is enhanced further, the nontrivial quantum plateau occurs at 𝑚 = 0.5 and the length of nontrivial plateau with larger bond alternation is longer. This phenomenon is caused by the gap of two spectrums becomes larger. The narrowing of spectrum dispersion makes that the line between nontrivial plateau (𝑚 = 0.5) and trivial plateau (𝑚 = 0) is sharper. With adding bond alternation parameter, the nontrivial magnetization plateau is longer at zero temperature. This trend can be easily understood by former existed theory and the variation of energy spectrum. When the state varies from ferrimagnetic plateau (m=0.5) to fully polarized plateau (m=1.5), the dispersion in this Luttinger Liquid (LL) state is same as

Fig. 4. The static susceptibility with different bond alternation parameters 𝛿 = 0 (black), 𝛿 = 0.6 (red), 𝛿 = 0.8 (green) and 𝛿 = 1.0 (blue) in weak longitudinal magnetic field (B=0.1) at low temperature. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. The magnetization plateau 𝑀 with different bond alternation parameters 𝛿 = 0 (black), 𝛿 = 0.6 (red), 𝛿 = 0.8 (green) and 𝛿 = 1.0 (blue) in longitudinal magnetic field at zero temperature. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Properties with bond alternation 𝜹 in longitudinal magnetic field 𝑩

Bond alternation is the ubiquitous phenomenon in the real transition metal compound. In this part, we consider the bond alternation 3

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even dispersionless at one-bond state 𝛿 = 1.0. These results could qualitatively describe the state from Heisenberg ferrimagnetic state to one-bond state between NN sites. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] J.G. Bednorz, K.A. Muller, Z. Phys. B 64 (1986) 189. [2] F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [3] M. Hagiwara, K. Minami, Y. Narumi, K. Tatani, K. Kindo, J. Phys. Soc. Japan 67 (1998) 2209. [4] F. Dyson, Phys. Rev. 102 (1956) 1217. [5] F. Dyson, Phys. Rev. 102 (1956) 1230. [6] S.V. Maleev, Sov. Phys.—JETP 6 (1958) 776. [7] A.F. Albuquerque, A.S.T. Pires, M.E. Gouvea, Phys. Rev. B 72 (2005) 174423. [8] Y.G. Chen, Y.X. Li, L.J. Tian, B. Chen, Phys. Lett. A 381 (2017) 2872. [9] Y.G. Chen, Y.X. Li, L.J. Tian, B. Chen, Chin. Phys. B 27 (2018) 127501. [10] S. Sachdev, R.N. Bhatt, Phys. Rev. B 41 (1990) 9323. [11] H.T. Wang, J.L. Shen, Z.B. Su, Phys. Rev. B 56 (1997) 14435. [12] H.T. Wang, H.Q. Lin, J.L. Shen, Phys. Rev. B 61 (2000) 4019. [13] B. Kumar, Phys. Rev. B 82 (2010) 054404. [14] Y.X. Li, B. Chen, Phys. Lett. A 379 (2015) 408. [15] Y.G. Chen, Y.X. Li, L.J. Tian, B. Chen, Physica B 520 (2017) 65. [16] C.J. Wu, B. Chen, X. Dai, Y. Lu, Z.B. Su, Phys. Rev. B 60 (1999) 1057. [17] Y.X. Li, B. Chen, Phys. Lett. A 374 (2010) 3514. [18] Y.X. Li, B. Chen, Chin. Phys. B 24 (2015) 027502. [19] S. Yamamoto, T. Fukui, K. Maisinger, U. Schllöck, J. Phys.: Condens. Matter 10 (1998) 11033. [20] Yuan Chen, You Wu, Solid State Commun. 159 (2013) 49. [21] S. Brehmer, H.J. Mikeska, S. Yamamoto, J. Phys.: Condens. Matter 9 (1997) 3921. [22] M. Hase, I. Terasaki, K. Uchinokur, Phys. Rev. Lett. 70 (1993) 3651. [23] T. Holstein, H. Primakoff, Phys. Rev. 58 (1940) 1098. [24] S. Yamamoto, S. Brehmer, H.J. Mikeska, Phys. Rev. B 57 (1998) 13610. [25] N. Avalishvili, G.I. Japaridze, D. Nozadze, S. Mahdavifar, Bull. Georg. Natl. Acad. Sci. 6 (2012) 53. [26] R.S. Lapa, A.S.T. Pires, Eur. Phys. J. B 86 (2013) 392. [27] M.H. Qin, G.Q. Zhang, K.F. Wang, X.S. Gao, J.M. Liu, J. Appl. Phys. 109 (2011) 07E103. [28] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 7 (1997) 1984. [29] T. Sakai, K. Okamoto, Phys. Rev. B 65 (2002) 214403. [30] W.M. da Silva, R.R. Montenegro-Filho, Phys. Rev. B 96 (2017) 214419.

Fig. 6. The specific heat 𝐶 with different bond alternation parameters 𝛿 = 0 (black), 𝛿 = 0.6 (red), 𝛿 = 0.8 (green) and 𝛿 = 1.0 (blue) in weak longitudinal magnetic field (B=0.3) at low temperature. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

quantum Monte Carlo (QMC) data [30]. When the bond alternation is considered, the nontrivial quantum plateau at m=0.5 and dispersion in LL state are also qualitatively reasonable. We also calculate the specific heat with various bond alternation in low longitudinal magnetic field (𝐵 = 0.3) as shown in Fig. 6. In the temperature region [0.2, 0.4], the trend of specific heat with external magnetic field is similar with that without 𝐵. Near zero temperature, the line equals zero which is closely related with the lower energy spectrum is lifted by external magnetic field. Meanwhile, the region around the transition point of specific heat with bond alternation is narrowed. From above figures and analysis, our method can qualitatively describe the phase from ferrimagnetic state to bond alternation state. 4. Results and conclusions In this paper, we use the Dyson–Maleev mean-field approximation to study the bond alternation effect on Heisenberg ferrimagnetic spin chain with longitudinal magnetic field. By numerically solving the selfconsistent equations, we discuss the energy spectrums, magnon internal energy, specific heat, static susceptibility and magnetization plateau. Especially, the low branch of energy spectrums becomes narrower and

4