The magnetic properties of the spin-1 Heisenberg antiferromagnetic chain with single-ion anisotropy

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The magnetic properties of the spin-1 Heisenberg antiferromagnetic chain with single-ion anisotropy Gangsan Hu, Rengui Zhu n College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 14 July 2014 Received in revised form 24 November 2014 Accepted 24 November 2014

The magnetic properties of the spin-1 Heisenberg antiferromagnetic chain with exchange anisotropy and single-ion anisotropy are studied by the double-time Green's function method. The determinative equations for the critical temperature, the magnetization, and the zero-field susceptibility are derived analytically. The effects of the anisotropies on the magnetic properties are presented. & 2014 Published by Elsevier B.V.

Keywords: Single-ion anisotropy Green's function Heisenberg model Antiferromagnet

1. Introduction Since the early semiquantitative analytical predictions by Haldane [1] that properties for integer spin must differ qualitatively from those for half-integer spin, the interest in studying the integer spin systems has been triggered. So far, various Heisenberg models with spin magnitude S ¼1 have been theoretically investigated [2–16]. Already in the earlier stage of vigorous studies on the Haldane problem, Botet et al. [2] have indicated that the single-ion anisotropy generated by crystal fields plays an essential role in larger spin systems, so the effects of the single-ion anisotropy on the magnetic behavior of the magnetic systems have become an important content of research. It has been shown that the single-ion anisotropy suppresses the quantum and thermal spin fluctuations, and can have a fundamental influence on the ground state phases [2,8,11,15] and thermodynamic properties [5– 7,12,16] of the spin systems with spin greater than one-half. Because of the complexities caused by the single-ion anisotropy term, various theoretical methods have been devoted, such as quantum Monte Carlo simulation [3], coupled-cluster approximation [4], exact diagonalization [8], series expansion [11,12], double-time Green's funciton [6,9,13,14], density and transfer renormalization group [7,15], and modified spin wave theory [16]. Among the above methods, the double-time Green's function approach [17,18], which is applicable for all temperature regions and all dimensions, has got a great success in the research area of n

Corresponding author. E-mail address: [email protected] (R. Zhu).

quantum magnetism. When the single-ion anisotropy parameter is small, the Anderson–Callen decoupling approximation (ACA) [19] can be used to decouple the hierarchy of Green's function equation, and obtain reliable results. By using Green's function approach and the Anderson–Callen decoupling approximation, the magnetic properties of the one, two, and three dimensional (1D, 2D and 3D) ferromagnetic (FM) spin-1 Heisenberg models with single-ion anisotropy have been studied [9,13,20–22], as well as the 2D and 3D antiferromagnetic (AFM) models [14,23]. It is worth noting that in Ref. [6], a theoretical formulation of the secondorder double-time Green's function method for the 1D AFM model with single-ion anisotropy in a phase without long-range order was presented. In this paper, we will use Green's function approach and the ACA approximation to study the 1D spin-1 antiferromagnetic Heisenberg model with the easy-axis single-ion anisotropy in the antiferromagnetic phase, presenting the effects of the single-ion anisotropy on the magnetic properties. As mentioned above, the second-order Green's function method has been applied in Ref. [6], however, where only the isotropic case without long-range order was numerically calculated and discussed. Furthermore, the effects of the single-ion anisotropy which are important for the 1D spin-1 AFM model were not discussed in Ref. [6]. The importance of the effects is based on the following two points: firstly, it is known that the role of single-ion anisotropy in phase transitions is of particular importance when it has an opposite sign with respect to exchange interactions, while the AFM model with the easy-axis single-ion anisotropy just belongs to this case; secondly, in low dimensional spin systems, the intensity of the thermal and the

http://dx.doi.org/10.1016/j.physb.2014.11.095 0921-4526/& 2014 Published by Elsevier B.V.

Please cite this article as: G. Hu, R. Zhu, Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.095i

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G. Hu, R. Zhu / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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quantum fluctuations is relatively stronger than that in high dimensional systems, so studying the 1D systems will better reveal the suppression of the fluctuations from single-ion anisotropy. This paper is organized as follows: In Section 2, we present our 1D Heisenberg model and the formalism of Green's function approach. The basic self-consistent equations are obtained. In Section 3, we present our numerical results, investigating the effects of the single-ion anisotropy and exchange anisotropy on the critical temperature, staggered magnetization and zero-field susceptibility. In Section 4, a brief conclusion is given.

+ + The equations of motion for S Ai (t) and SBj (t) can be written as

i

d + z + + + z S Ai = [S Ai , H] = J ∑ (aS Ai SBj − S Ai SBj ) dt j

z + + z + + D (S Ai S Ai + S Ai S Ai) + hS Ai ,

i

(8)

d + SBj = [SBj+, H] dt z + = J ∑ (aSBjz S Ai − SBj+S Ai ) + D (SBj+SBjz + SBjz SBj+) − hSBj+. i

Using Eqs. (8) and (9) we can obtain the equations of motion for Green's functions:

2. The model and Green's function approach The Hamiltonian of the 1D spin-1 antiferromagnetic Heisenberg model with the exchange anisotropy and single-ion anisotropy under the staggered magnetic field can be described as

⎡ z z z 2 x x y y H = J ∑ [a (S Ai SBj + S Ai SBj ) + S Ai SBj ] − D ⎢∑ (S Ai ) + ⎢ ⎣ i 〈ij〉

d z + − + − 〈〈S Ai 〉〉 = 2δt δii 〈S Ai 〉 + 〈〈 [S Ai 〉〉 (t); S Ai , H]; S Ai ′ ′ ′ dt z z + + z − = 2δt δii 〈S Ai 〉 + J ∑ 〈〈aS Ai 〉〉 SBj − S Ai SBj; S Ai ′ ′ j

z + + z − + − + D〈〈S Ai 〉〉 + h〈〈S Ai 〉〉, S Ai + S Ai S Ai; S Ai ; S Ai ′ ′

⎤

j

⎥⎦

i

(10)

d + − + − 〈〈S Ai (t); SBj 〉〉 = 〈〈 [S Ai , H]; SBj 〉〉 dt z + + z − = J ∑ 〈〈aS Ai SBj′ − S Ai SBj′; SBj 〉〉 j′

(1)

j

where 〈i, j〉 denotes that the summation is over the nearestneighbor spins i and j. The parameters a and D denote the exchange anisotropy and the single-ion anisotropy respectively. Increasing D and decreasing a both lead to stronger anisotropy. Here we only consider the easy-axis case with 0 < a < 1 and D > 0. In this parameter region, the ground-state phase is antiferromagnetic [2], so that we write the Hamiltonian (1) in the two-sublattice formulation as usually done in Green's function approach. A and B denote the two sublattices. This two-sublattice treatment was also used in linked-cluster series expansion approach in Ref. [12]. J is the exchange coupling constant between neighboring spins. h is the staggered magnetic field, which makes the system to be in the antiferromagnetic phase even in the isotropic case with a ¼1 and D¼ 0. In the following, we apply the spin raising and lowering operators S ±j = S jx ± iS jy to simplify the above Hamiltonian, which can be rewritten as

⎡ ⎤ ⎡ a + − − + z z⎥ z 2 H = J ∑ ⎢ (S Ai SBj + S Ai SBj) + S Ai SBj − D ⎢∑ (S Ai ) + ⎢⎣ i ⎥⎦ ⎢2 〈ij〉 ⎣ z − h ∑ S Ai + h ∑ SBjz. i

i

∑ (SBjz )2⎥

z − h ∑ S Ai + h ∑ SBjz. i

(9)

⎤ ⎥⎦

d − − 〈〈SBj+ (t); S Ai 〉〉 = 〈〈 [SBj+, H]; S Ai 〉〉 dt z + − = J ∑ 〈〈aSBjz S Ai − SBj+S Ai 〉〉 ; S Ai ′ ′ i′

− − + D〈〈SBj+SBjz + SBjz SBj+; S Ai 〉〉 − h〈〈SBj+; S Ai 〉〉,

i

(12)

d − − 〈〈SBj+ (t); SBj 〉〉 = 2δt δ jj 〈SBjz 〉 + 〈〈 [SBj+, H]; SBj 〉〉 ′ ′ ′ dt z + − = 2δt δ jj 〈SBjz 〉 + J ∑ 〈〈aSBjz S Ai − SBj+S Ai 〉〉 ; SBj ′ ′ i

− − + D〈〈SBj+SBjz + SBjz SBj+; SBj 〉〉 − h〈〈SBj+; SBj 〉〉, ′ ′

i

(13)

d + − 2 + (t); (S Ai ) S 〉〉 〈〈S Ai ′ Ai′ dt

j

(2)

j

i

(11)

+ − 2 + + − 2 + (t), (S Ai ) S ] 〉 + 〈〈 [S Ai , H]; (S Ai ) S 〉〉 = δt δii 〈 [S Ai ′ ′ Ai′ ′ Ai′ + − 2 + (t), (S Ai ) S Ai′] 〉 = δt δii 〈 [S Ai ′ ′ z + + z − 2 + ) S 〉〉 + J ∑ 〈〈aS Ai SBj − S Ai SBj; (S Ai ′ Ai′

∑ (SBjz )2⎥ j

z + + z − + − + D〈〈S Ai S Ai + S Ai S Ai; SBj 〉〉 + h〈〈S Ai ; SBj 〉〉,

In order to calculate the magnetic properties of this model, we introduce the retarded Green's functions, which are defined as

z + + z − 2 + ) S 〉〉 + D〈〈S Ai S Ai + S Ai S Ai; (S Ai ′ Ai′ + − 2 + + h〈〈S Ai; (S Ai ) S Ai′〉〉. ′

(14)

In order to solve the system of the equations generated by Eqs. (10)–(14), we need to break the higher-order Green's functions. We apply the random phase approximation (RPA) [17] for the exchange coupling terms:

+ − + − 〈〈S Ai (t); S Ai 〉〉 = − iθ (t) 〈 [S Ai (t), S Ai ] 〉, ′ ′

(3)

+ − + − 〈〈S Ai (t); SBj 〉〉 = − iθ (t) 〈 [S Ai (t), SBj ] 〉,

(4)

+ + 〈〈SBj+ (t); S Ai 〉〉 = − iθ (t) 〈 [SBj+ (t), S Ai ] 〉,

(5)

and apply the Anderson–Callen approximation (ACA) [19] for the single-ion anisotropy term:

− − 〈〈SBj+ (t); SBj 〉〉 = − iθ (t) 〈 [SBj+ (t), SBj ] 〉, ′ ′

(6)

〈〈Si+Siz + Siz Si+; S −j 〉〉

+ 〈〈S Ai (t);

− (S Ai

′

+ )2S Ai 〉〉 ′

=−

+ iθ (t) 〈 [S Ai (t),

− (S Ai

′

+ )2S Ai ] 〉. ′

(7)

+ z − + − 〈S Ai Sl ; SBj 〉 = 〈Slz 〉〈〈S Ai ; SBj 〉〉,

⎡ ⎛ ⎞⎤ 1 = ⎢2〈Siz 〉 − 〈Siz 〉 ⎜〈Si+Si− 〉 + 〈Si− Si+〉⎟ ⎥ 〈〈Si+; S −j 〉〉. 2 ⎝ ⎠⎦ ⎣ 2S

Please cite this article as: G. Hu, R. Zhu, Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.095i

(15)

(16)

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G. Hu, R. Zhu / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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For S ¼1, using the following relationships:

fk (ω) =

Si− Si+ = 2 − Siz − (Siz )2 ,

(17)

Si+Si− = 2 + Siz − (Siz )2 ,

(18)

3

z 2 ⎡ ω1 − ω k ⎤ 2 + m − 3〈 (S Ai ) 〉 ω1 + ω k ⎢ ⎥, − ωk ω ω ω + ωk ⎦ − ⎣ k

(31)

with z 2 ω1 = h + 2Jm + 2Dm〈 (S Ai ) 〉,

(32)

Eq. (16) can be rewritten to be

ω12 − 4J 2 m2a2 cos2k .

ωk =

〈〈Si+Siz + Siz Si+; S −j 〉〉 = 〈Siz 〉〈 (Siz )2〉〈〈Si+; S −j 〉〉.

(33)

(19) Then through the spectral theorem [24],

Then Eqs. (10)–(14) become

〈S −j Si+ (t) 〉 =

d + − − + − i 〈〈S Ai (t); S Ai 〉〉 = 2δt δii m + Jm ∑ (a〈〈SBj+; S Ai 〉〉 + 〈〈S Ai ; S Ai 〉〉) ′ ′ ′ ′ dt j z 2 + − + − + Dm〈 (S Ai ) 〉〈〈S Ai ; S Ai 〉〉 + h〈〈S Ai ; S Ai 〉〉, ′ ′

i

+ i

− SBj 〉〉

+

+ h〈〈S Ai ;

e βω k − 1

(20)

− SBj 〉〉,

(21)

d − + − − 〈〈SBj+ (t); S Ai 〉〉 = − Jm ∑ (a〈〈S Ai ; S Ai 〉〉 − 〈〈SBj+; S Ai 〉〉) ′ dt i′ (22)

d − + − − 〈〈SBj+ (t); SBj 〉〉 = − 2δt δii m − Jm ∑ (a〈〈S Ai 〉〉 − 〈〈SBj+; SBj 〉〉) ; SBj ′ ′ ′ ′ dt i

i N

∑∫ k

ei [k· (i − j) − ωt],

∞

−∞

e βω k − 1

(34)

dω 2π

ei [k· (i − j) − ωt],

(35)

we can obtain the correlation functions: − + 〈S Ai S Ai〉 = mΦ,

(36)

z 2 − 2 + + 〈 (S Ai ) S AiS Ai〉 = (2 + m − 3〈 (S Ai ) 〉) Φ,

(37)

here β = 1/k B T , and

Φ=

(23)

ω1 + ω k 1 ⎛⎜ 1 − ∑ N ⎜⎝ k e βω k − 1 ω k

∑ k

1 e−βω k

ω1 − ω k ⎞ ⎟⎟. − 1 ωk ⎠

(38)

For S ¼1, Eqs. (36) and (37) can also be written as

d + (t ); (S − ) 2S + 〉〉 = 2δ t δ ii [2 + m − 3〈 (S z ) 2〉] 〈〈S Ai Ai Ai ′ dt ′ Ai′ + ; (S − ) 2S + 〉〉 + a〈〈S + ; (S − ) 2S + 〉〉) + Jm ∑ (〈〈S Bj Ai Ai′ Ai Ai Ai′

′

j

dω 2π

fk (ω + i0+) − fk (ω − i0+)

− − − Dm〈 (SBjz )2〉〈〈SBj+; SBj 〉〉 − h〈〈SBj+; SBj 〉〉, ′ ′

i

−∞

k

〈 (S −j )2S +j Si+ (t) 〉 =

− − − Dm〈 (SBjz )2〉〈〈SBj+; S Ai 〉〉 − h〈〈SBj+; S Ai 〉〉,

i

∞

∑∫

gk (ω + i0+) − gk (ω − i0+)

d + − − + − 〈〈S Ai (t); SBj 〉〉 = Jm ∑ (a〈〈SBj+′; SBj 〉〉 + 〈〈S Ai ; SBj 〉〉) dt j′ z 2 + Dm〈 (S Ai ) 〉〈〈S Ai ;

i N

z 2 2 − m − 〈 (S Ai ) 〉 = mΦ,

(39)

z 2 z 2 2〈 (S Ai ) 〉 − 2m = (2 + m − 3〈 (S Ai ) 〉) Φ.

(40)

′

z ) 2〉〈〈S + ; (S − ) 2S + 〉〉 + h〈〈S + ; (S − ) 2S + 〉〉 . + Dm〈 (S Ai Ai Ai Ai′ Ai Ai Ai′

′

′

(24)

z Here the values of the magnetizations 〈S Ai 〉 and 〈SBjz 〉 are considered to be independent of their sites i and j. We have set z 〈S Ai 〉 = − 〈SBjz 〉 = m for sublattice symmetry. For S ¼1, the relations

Solving Eqs. (39) and (40), we can obtain the magnetization and the z-component self-correlation function:

z 3 z z 4 z 2 and (S Ai (S Ai ) = S Ai ) = (S Ai ) have been used in Eq. (24). After Fourier transforming these equations with respect to the space and time variable,

(41)

+ − 〈〈S Ai 〉〉 = (t); S Ai ′

+ − 〈〈S Ai 〉〉 = (t); SBj

〈〈SBj+ (t);

− 〉〉 S Ai

1 N

∫−∞

1 N

∫−∞

1 = N

1 − 〈〈SBj+ (t); SBj 〉〉 = ′ N

∞

∞

∞

∫−∞ ∞

∫−∞

dω 2π

∑ eik· (i − i ′) − iωt gk1(ω),

dω 2π

∑ eik· (i − j) − iωt gk2 (ω),

dω 2π dω 2π

k

k

(25)

(26)

m=

z 2 〈 (S Ai ) 〉=

∑

k 3 (ω),

k

k

1 N

∑ k

2ω1 βω k2

1 + − 2 + 〈〈S Ai (t); (S Ai ) S 〉〉 = ′ Ai′ N

∫−∞

dω 2π

∑ eik· (i − i ′) − iωt fk (ω), k

(28)

(29)

we can obtain the Fourier-transform solutions for Green's functions:

gk1 (ω) =

ω1 − ω k ⎤ m ⎡ ω1 + ω k − ⎢ ⎥, ωk ⎣ ω − ωk ω + ωk ⎦

,

4 , 3Φ

ω1 = 2Jm + ∞

(42)

(43)

(27) m=

∑ eik· (j− j′) − iωt gk1(ω),

2Φ 2 + 4Φ + 4 . 3Φ 2 + 6Φ + 4

When the staggered magnetic field h → 0 and the temperature T → TN , the magnetization will be very small, that leads to k B T > > ω k . Then, Eqs. (38), (41) and (32) can be approximated respectively to be

Φ=

eik· (j − i) − iωt g

4(1 + Φ) , 3Φ 2 + 6Φ + 4

(30)

(44) 4 Dm. 3

(45)

Then we can estimate the critical temperature, which is obtained from the limiting of Eqs. (43) and (44) as the magnetization m approaches zero. Combining Eq. (33) with Eqs. (43)–(45), we obtain the expression of TN:

TN =

2 ⎛ 4a2 ⎞2 2J ⎛ 4 a2 ⎞ ⎜4 + 2D˜ − ⎟ −⎜ ⎟ , 3k B ⎝ 2 + D˜ ⎠ ⎝ 2 + D˜ ⎠

Please cite this article as: G. Hu, R. Zhu, Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.095i

(46)

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G. Hu, R. Zhu / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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where

4D D˜ = . 3J

(47)

When a ¼1 and D¼0, the critical temperature TN is reduced to be zero, which agrees with the result of the isotropic case, implying that the long-range order does not exist. When T > TN , according to the definition of the zero field susceptibility χ = limh → 0 (m /h), using Eqs. (33), (43)–(45), we get

χ=

⎤−1 ⎡ 2 1 ⎢ ⎛ 3k B T ⎞ 4a2 4a2 ˜ − 4⎥ . − + D + 2 ⎜ ⎟ ⎥ J ⎢ ⎝ 2J ⎠ 2 + D˜ 2 + D˜ ⎦ ⎣

(48)

In isotropic case with a¼ 1 and D¼ 0, we get TN ¼0 from Eq. (46) and χ ∼ T −2 as T → 0 from Eq. (48).

3. Results and discussions

Fig. 2. The critical temperature versus the single-ion anisotropy parameter D when a is taken as 0.1, 0.5 and 1 (respectively for the solid line, long dash line and short dash line). Here J = k B = 1.

In Figs. 1 and 2, we show the variations of the critical temperature TN with the exchange anisotropy parameter a and the single-ion anisotropy parameter D respectively. We can see that TN increases with decrease of a and increase of D, which means that the exchange anisotropy and the single-ion anisotropy both tend to suppress the thermal and quantum spin fluctuations. We note that the behavior curve of TN versus the single-ion anisotropy parameter D in Fig. 2 is qualitatively similar to the results of the same model on 3D body-centered-cubic (bcc) and simple-cubic (sc) lattices [12]. A distinct difference is the result of the isotropic case (a ¼1 and D ¼0): the 3D lattice gives a nonzero TN close to 4.0 for bcc and 2.7 for sc [12], while here the 1D linear chain gives TN ¼ 0. This can be explained theoretically by Mermin– Wagner theorem [25], which claims that the isotropic low-dimensional (1D and 2D) quantum spin systems with short-range exchange couplings cannot have long-range order at any finite temperature. It also reflects the fact that the magnetic properties of lower dimensional systems are more affected by quantum and thermal fluctuations than the higher dimensional systems. In Figs. 3 and 4, we show the temperature dependence of the zero-field susceptibility χ in the temperature region T > TN (the vertical solid line corresponds to TN). It is shown that, when a and D are given, χ always decreases with increasing T. It is interesting that for T fixed, the susceptibility raises with the increasing D and decreasing a. This is because that the stronger the anisotropy becomes, the more the quantum spin fluctuations are suppressed. We note that in Ref. [7], the authors took a Hamiltonian with the

single-ion anisotropy term different from ours by a minus sign. They calculated the zero-field susceptibility for the model in a different phase from ours (according to Ref. [2], that is the X–Y phase). In the following, we use our solutions of Green's functions to discuss the temperature-dependent behaviors of the magnetization affected by the anisotropies. In Figs. 5 and 6, we can see that when the parameters a, D, h are given, the magnetization m decreases from the maximum as the temperature increases. The variations of the exchange anisotropy parameter a and the singleion anisotropy parameter D lead to different behaviors of the magnetization. Figs. 5 and 6 show that the magnetization curves both move right with a decreasing and D increasing respectively. Both the above behaviors imply that the anisotropy can enhance the magnetization and long-range order. These behaviors qualitatively coincide with the higher dimensional systems. When the temperature approaches the critical temperature TN, the curves corresponding to different parameters are almost overlapping and the magnetization approaches zero. These behaviors qualitatively coincide with the results of higher dimensional systems [12,14], but present saddle shape in intermediate temperature region, this is because we did not take the staggered field h to be zero.

Fig. 1. The critical temperature versus the exchange anisotropy when the D is taken as 0.1, 0.5 and 1 (respectively for the solid line, long dash line and short dash line). Here J = k B = 1.

Fig. 3. The zero-field susceptibility versus the temperature when a is taken as 0.1, 0.5 and 1 (respectively for the solid line, long dash line and short dash line). Here J = k B = 1 and D¼ 0.5.

Please cite this article as: G. Hu, R. Zhu, Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.095i

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G. Hu, R. Zhu / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Fig. 4. The zero-field susceptibility versus temperature when D is taken as 0.1, 0.5 and 1 (respectively for the solid line, long dash line and short dash line). Here J = k B = 1 and a¼ 0.5.

5

parameter region 0 < a < 1 and D > 0. Green's function approach with RPA and ACA approximations has been used to study the effects of the single-ion anisotropy on the magnetic properties of the model. We have analytically derived the equations for the critical temperature TN and the zero-field susceptibility χ, as well as the determinative self-consistent equations for the staggered magnetization m. Through numerical calculations, we have plotted the variation curves of these quantities with the variations of the anisotropy parameters and temperature. Our results show that making the anisotropies stronger (i.e, decreasing a or increasing D) leads to the raising of TN, χ and m, which is in agreement with higher dimensional models [11,12,14]. Furthermore, our results unambiguously confirm the fact that the exchange anisotropy and the single-ion anisotropy both tend to suppress the thermal and quantum spin fluctuations, and the intensity of suppression is relatively strengthened by reducing the dimensions. Also, our results show the validity of the double-time Green's function approach with RPA and ACA approximations for the present model (1). Finally, we point out that our research contents are different from the previous works on the 1D model [3,5–7,15], and our results are helpful supplements for them. Some of them did not consider the effects of the single-ion anisotropy, while others considered the properties of the phases different from ours. Furthermore, we note that in Ref. [16], the authors considered the model without the exchange anisotropy, otherwise the same as ours, however, their study is restricted in low temperature region and did not give the results about the transition temperature TN nor the staggered magnetization m.

Acknowledgement

Fig. 5. The magnetization versus the temperature when a is taken as 0.1, 0.5 and 1 (respectively for the solid line, long dash line and short dash line). Here J = k B = 1 and D = h = 0.5.

This work was supported by the Natural Science Foundations of the Department of Science and Technology (under Grant no. 1208085QA10) of Anhui Province, China.

References

Fig. 6. The magnetization versus the temperature when D is taken as 0.1, 0.5, 1 (respectively for the solid line, long dash line and short dash line). Here J = k B = 1 and a = h = 0.5.

4. Conclusion In this paper, we have considered the magnetic properties of the 1D spin-1 antiferromagnetic Heisenberg model (1) with the exchange anisotropy a and the single-ion anisotropy D, in the

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Please cite this article as: G. Hu, R. Zhu, Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.095i

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