Multiple hysteresis behaviors in spin models: Effect of anisotropy in the exchange interaction

Author’s Accepted Manuscript Multiple hysteresis behaviors in spin models: effect of anisotropy in the exchange interaction Ümit Akıncı, Yusuf Yuksel

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To appear in: Physica B: Physics of Condensed Matter Received date: 30 June 2017 Revised date: 3 September 2017 Accepted date: 6 September 2017 Cite this article as: Ümit Akıncı and Yusuf Yuksel, Multiple hysteresis behaviors in spin models: effect of anisotropy in the exchange interaction, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.09.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Multiple hysteresis behaviors in spin models: effect of anisotropy in the exchange interaction ¨ Umit Akıncı, Yusuf Yuksel ˙ Department of Physics, Dokuz Eyl¨ ul University, Tr-35160 Izmir, Turkey

Abstract Hysteresis characteristics of the general spin-S (S > 1) Heisenberg model have been studied within the effective field approximation. Particular emphasis has been paid on the large negative valued crystal field region. Since spin-S Ising model displays maximum number of 2S windowed hysteresis loops in this region, this investigation gives the answer of the question: what is the effect of the anisotropy in the exchange interaction on this multiple hysteresis behavior? As an answer, rising number of windows in the hysteresis loops have been observed. Keywords: Multiple hysteresis, general spin Heisenberg model, effective field theory 1. Introduction Higher spin models (S > 1) are important to explain the behaviors of the real magnetic materials, since S −1/2 models are highly idealized. One furter idealization is Ising model. This model is highly anisotropic limit of the more general model, namely Heisenberg model. Although it is very important to investigate higher spin models, there is a downward trend with the rising spin value in the theoretical literature. This fact is understandable since solutions of the higher spin models needs rising computational time for simulations and cope with mathematical difficulties for approximation methods. Ising model including the crystal field or the single-ion anisotropy was introduced as a S − 1 Blume - Capel (BC) model [1, 2]. Later on, it was generalized to the higher spin problems [3]. After this mean field approximation ¨ Email address: [email protected] (Umit Akıncı, Yusuf Yuksel) Preprint submitted to Physica B

September 11, 2017

(MFA) solution, general spin-S model has been solved by some other methods, such as cluster variational method [4, 5], effective field theory (EFT) [6, 7, 8, 9, 10] pair approximation [11] and Monte Carlo (MC) simulation [12]. In a recent work we have demonstrated that, spin-S BC model could exhibit multiple hysteresis behaviors for low temperatures and large negative values of the crystal fields [13]. The literature review about the higher spin Ising model can also be found in this work. This multiple hysteresis behavior appears in some physical systems such as F e3 O4 /M n3 O3 superlattices [14], F eGa/CoF eB nanowires [15]. Besides, on the theoretical side, similar multiple hysteresis behaviors were reported on nanotube and nanowire geometry within the EFT formulations [16, 17] and MC simulations [18, 19]. Though not as much as Ising model, higher spin Heisenberg model has been investigated within several methods. For instance, S − 1 Heisenberg model has been investigated within the two-spin cluster mean field method [20], pair approximation [21]; S − 2 Heisenberg model has been investigated within the Green’s function technique [22]. Besides, half integer Heisenberg models have been worked out such as S − 3/2 three leg Heisenberg tube by density matrix renormalization group method [23, 24], S − 3/2 Heisenberg antiferromagnet with tensor network optimization algorithms [25] and S−5/2 Heisenberg model within the series expansions method [26]. There are also mixed spin Heisenberg models present. For instance, mixed S1/2 − 3/2 Heisenberg model within the Oguchi approximation [27], Green’s function method [28]; S1 − 3/2 Heisenberg model within the coupled cluster method [29], numerical density-matrix renormalization group calculations [30] and mixed S2 − 5/2 within the Green’s function method [31, 32, 33]. As seen from this short literature summary, there has been less attention paid on the hysteresis behaviors. The aim of this work is to determine the hysteresis properties of the Spin-S Heisenberg model and to obtain some general results about the multiple hysteresis behaviors. EFT has been used in order to investigate hysteresis behaviors of the spin-S Heisenberg model. First attempts of the constructing formulation for Ising model can be found in Ref. [34] and decoupling approximation has been constructed in [35]. A useful review about these formulations can be found in Ref. [36]. On the other hand, EFT for spin-S Heisenberg model requires a little more effort. The formulation constructed here is based on the EFT with two spin cluster for S − 1/2 Heisenberg model [37]. The aim of the paper is to obtain general results about the multiple 2

hysteresis behaviors of the higher spin valued Heisenberg model. On the basis of obtained results in [13] for spin-S Ising model, the answer of the following question will be given in this work: What is the effect of anisotropy in the exchange interaction on these multiple hysteresis loops? For this aim, the paper is organized as follows: In Sec. 2 we briefly present the model and formulation. The results and discussions are presented in Sec. 3, and finally Sec. 4 contains our conclusions 2. Model and Formulation The Hamiltonian of the spin-S anisotropic Heisenberg model can be given as, H = −J

      Δ Six Sjx + Siy Sjy + Siz Sjz − D (Siz )2 − H Siz ,

i

(1)

i

where Six , Siy , Siz denote the x, y, z components of the Pauli spin operators at a site i, respectively, J stands for the exchange interactions between the nearest neighbor spins, Δ is the anisotropy in the exchange interaction, D is the crystal field and H is the longitudinal external magnetic field. The first summation in Eq. (1) is over the nearest-neighbor pairs of spins and the other summations are over all the lattice sites. In a typical EFT solution, a finite cluster is constructed and within this cluster all spin-spin interactions are treated exactly. Spin-spin interactions between the spins within this cluster and outside of the cluster are treated approximately. For the aim of including the anisotropy in the exchange interaction, two spin cluster is constructed. The Hamiltonian of this two spin cluster within the axial approximation [38] is given by H

(2)

= −J

[Δ (S1x S2x

+

S1y S2y )

+

S1z S2z ]

−D

2  k=1

(Skz )2

−

2 

(hk + H) Skz . (2)

k=1

Here hk (k = 1, 2) is the local field acting on the site k, which mimics the spin-spin interaction of the site k with the nearest neighbors, which is located at the outside of the two spin cluster. hk = J

zk  δ=1

3

Sk,δ ,

(3)

where zk is the number of nearest neighbor interactions and Sk,δ is the nearest neighbors of k, which are outside of the two spin cluster. In this finite cluster, EFT equations can be constructing by starting with generalized Callen-Suzuki [39] identities which are given as    z n (2) (S ) exp −βH T r 2 n k (Skz )  = , (4) T r2 exp (−βH(2) ) where, T r2 stands for the partial trace over the lattice sites 1 and 2, β = 1/(kB T ), where kB is the Boltzmann constant and T is the temperature. Here, number of 2S quantities represented by Skz , (Skz )2 , (Skz )3 which are named as magnetic dipol moment, quadrupol moment, octupole moment etc. exist in the model. The matrix representation of Eq. (2) can be constructed by using standard basis set i.e. {|φi }, i = 1, 2, . . . , (2S + 1)2 where an element of the set can be represented as |φi  = |s1 s2 , such that s1 , s2 are eigenvalues of the operators S1z and S2z , respectively. In order to obtain the matrix representations of the exponentials of H(2) in Eq. (4), matrix representation of H(2) in standard basis has to be diagonalized. Let the diagonal matrix elements are given by,



(5) rα = φα H(2) φα , (n)

Tk,α = φα |(Skz )n | φα  ,

(6)

where the basis set ({φα }) makes Eq. (2) diagonal. Then, with these definitions, Eq. (4) can be written as (2S+1)2

 (Skz )n  =



(n)

Tk,α exp (−βrα ) 

α=1

.

(2S+1)2



(7)

exp (−βrα )

α=1

For the aim of using differential operator technique [40], let us write Eq. (7) in closed form as (Skz )n  = Fkn (h1 , h2 ) . (8) Note that, dependency of Fkn on other Hamiltonian parameters other than the local fields hk has not been shown explicitly. The differential operator technique starts with identity exp (a∇)F (x) = F (x + a) , 4

(9)

where ∇ = ∂/∂x is differential operator, F is any function and a is any constant. In this way one can obtain polynomial form of Eq. (8) as (Skz )n  = exp (h1 ∇1 + h2 ∇2 ) Fkn (x1 , x2 ) |x1 =0,x2 =0 ,

(10)

where ∇1 and ∇2 are differential operators with respect to x1 and x2 . By writing definitions of local fields from Eq. (3) in Eq. (10) we can get equations as  z  z2 1 z n (Sk )  = exp (JS1,δ1 ∇1 ) exp (JS2,δ2 ∇2 ) Fkn (x1 , x2 ) |x1 =0,x2 =0 . δ1 =1

δ2 =2

(11) In order to get the polynomial forms of the terms in the thermal averages, we can use approximated van der Waerden identity [41]. This identity is given by

 Sk,δk Θk,δk = exp (JSk,δk ∇k ) = cosh (Jηk,δk ∇k ) + sinh (Jηk,δk ∇k ) , (12) ηk,δk 2  2 where k = 1, 2 and ηk,δ = Sk,δ k k This approximation simplifies the solution in a considerable manner with n n the assumptions Sk2n = Sk2  and Sk2n+1 = Sk Sk2  with integer n. Thus, the number of 2S equations in Eq. (11) reduces to two and they are given as,  z z  1 2 Sk  = Θ1,δ1 Θ2,δ2 Fk1 (x1 , x2 ) |x1 =0,x2 =0 , (13) δ1 =1δ2 =1



Sk2



 =

z1 z2

 Θ1,δ1 Θ2,δ2

Fk2 (x1 , x2 ) |x1 =0,x2 =0 .

(14)

δ1 =1δ2 =1 2 By decoupling the terms Sk,δk and Sk,δ in the expanded forms of Eqs. k (13) and (14) and using the translationaly invariance property of the lattice we obtain the equations:

1 (S1  + S2 ) = Φz11 Φz22 G1 (x1 , x2 ) |x1 =0,x2 =0 , 2 1  2  2  η2 = S1 + S2 = Φz11 Φz22 G2 (x1 , x2 ) |x1 =0,x2 =0 , 2 m=

5

(15) (16)

where the operator and functions are defined as

 m Φk = cosh (Jη∇k ) + sinh (Jη∇k ) , η

(17)

1 (F1i + F2i ) , i = 1, 2. (18) 2 By solving the system of nonlinear equations given by Eqs. (15) and (16) EFT results for the spin-S anisotropic Heisenberg model can be obtained. Linearization of Eqs. (15) and (16) in m will yield equation system for the second order critical temperature. Expanding operators in Eqs. (15) and (16) gives  3 m m2 m z1 z2 , (19) Φ1 Φ 2  X 0 + X 1 + X 2 2 + O η η η3 where X0 = [cosh (Jη∇1 )]z1 [cosh (Jη∇2 )]z2 Gi =

X1 = z1 [cosh (Jη∇1 )]z1 −1 [cosh (Jη∇2 )]z2 sinh (Jη∇1 ) +z2 [cosh (Jη∇1 )]z1 [cosh (Jη∇2 )]z2 −1 sinh (Jη∇2 ) X2 = z1 z2 [cosh (Jη∇1 )]

z1 −1

[cosh (Jη∇2 )]

z2 −1

(20)

sinh (Jη∇1 ) sinh (Jη∇2 ).

Applying Eq. (19) to Eqs. (15) and (16) near the critical temperature (i.e. m  0) yields critical temperature equations as, η 2 = X0 G2 (x1 , x2 ) |x1 =0,x2 =0 η = X1 G1 (x1 , x2 ) |x1 =0,x2 =0 .

(21)

Two unknown (η and critical temperature) can be determined by solving equation system Eq. (21). 3. Results and Discussion For the numerical calculations, following scaled (dimensionless) quantities have been used d = D/J, t = kB T /J, h = H/J.

(22)

The hysteresis loops can be obtained for a given parameter set (d, t) by calculating m according to the procedure given above, and by sweeping the longitudinal magnetic field from −h0 to h0 and then in reverse direction (i.e. h0 → −h0 ). We study on simple cubic lattice (i.e. z = 6) within this work. 6

14

Δ=0.0

14

Δ=0.0

12

Δ=1.0

12

Δ=1.0

10

10

8

8

tc

tc

3.1. Phase diagrams Phase diagrams in (d, t) plane are similar for Ising model (Δ = 0.0) and results for isotropic Heisenberg model (Δ = 1.0) with integer spin S can be seen in Fig. (1) (a). Second order critical line ends with tricritical point, which is the junction of the second and first order transitions. On the other hand, for the half integer spin model there is a qualitative difference between the Ising and isotropic Heisenberg cases. As seen in Fig. (1) (b), Δ = 0.0 curve extends to the large negative values of the crystal field, while Δ = 1.0 does not. In other words for half integer S values, Ising model is ordered at large negative values of the crystal field and low temperatures, while isotropic Heisenberg model is not. Note that, in the Ising case (i.e. Δ = 0.0) the phase diagrams for the S − 2 BC model have already obtained within MFA [42], EFT [43, 44, 45] and also for the S − 5/2 BC model within EFT [46]. Our Δ = 0.0 curves are consistent with the literature.

6

6

4

4

2

2

S-2 (a)

0 -5

-4

-3

-2

-1

S-5/2 (b)

0 0

-5

d

-4

-3

-2

-1

0

d

Figure 1: Variation of the critical temperature of the spin-S anisotropic Heisenberg model with crystal field. Fig. 1 (a) contains integer spins while (b) contains half integer spin model. Both figures contain Ising Δ = 0.0 and isotropic Heisenberg Δ = 1.0 cases.

For the large negative values of the crystal field and in Ising case, integer 7

spin models could not develop magnetic ground state, while half integer spin models have magnetic ground state for the large negative values of d (compare curves labeled as Δ = 0.0 in Figs. (1) (a) and (b)). This is because of the fact that, for the large negative values of the crystal field, only the minimum value of si makes the expectation value of the Hamiltonian given in Eq. (1) minimum. This value is nothing but s = 0 for the integer S, and s = ±1/2 for half integer S. Thus, there has to be a special value of crystal field that makes the ground state changes from s = S to s = 0 for integer model and s = 1/2 for the half integer model. It has been shown that in ground state, and for S −3/2 and S −5/2 BC models this transition occurs at D/J = −z/2 where z is the coordination number [46, 47]. 3.2. Hysteresis behaviors Ground state properties of the models are the key point about the multiple hysteresis behaviors. As demonstrated in Ref. [48], S − 1 BC model has double hysteresis behavior for low temperature and large negative values of crystal field. Later on this conclusion is generalized to the spin-S Ising model [13]: in general spin-S Ising model could display 2S windowed hysteresis behavior at large negative values of the crystal field. The mechanism behind these facts is simple: rising external longitudinal magnetic field in positive direction (in ground state) cause transitions s = 0 → s = 1 . . . → s = S for integer Ising models and s = 1/2 → s = 3/2 . . . → s = S for half integer Ising models. All these transitions are history dependent i.e. multiple hysteresis behavior appears. This plateau behavior of the magnetization for S − 1, S − 3/2 and S − 5/2 models have already been obtained theoretically on Bethe lattice [49]. Oigination of multiple hysteresis loops with changing d at low temperatures can be seen in Fig. 2. Firstly, as seen in Figs. 2 (a)-(d) rising d in negative direction causes to origination of multiple hysteresis loops. Maximum number of windows in these multiple hysteresis loops are 2S for the Ising case (see Figs. 2 (a) and (c)). In the isotropic Heisenberg case this situation is a little different. Rising d in negative direction causes also origination of multiple hysteresis loops from the ferromagnetic loops, but number of windows are two more from the loops of Ising model (see Figs. 2 (b) and (d) for d = −6.0 loops). This is the sign of additional more accessible magnetization value for the isotropic case other than m = 0, 1, . . . , S for integer model or m = 1/2, 3/2, . . . , S for the half integer model, in the ground state.

8

In order to visualize the effect of the anisotropy in the exchange interaction on these multiple hysteresis loops we depict hysteresis loops for several values of Δ in Fig. 3. As seen in Figs. 3 (a) and (b) rising Δ (means lowering of the anisotropy in the exchange interaction) shrinks the loops (compare loops labeled as Δ = 0.0 and Δ = 0.4 in Figs. 3 (a) and (b)). This shrinking behavior give place to splitting behavior for the nearest loops to the center, while Δ continuing to rise (compare loops labeled as Δ = 0.4 and Δ = 0.8 in Figs. 3 (a) and (b)). Note that, one qualitative difference between the integer and half integer model is, rising Δ could destroy the central loop which is absent in the integer model. This is consistent with the phase diagrams given in Fig. 1 (b), i.e. lowering of the anisotropy in the exchange interaction could develop paramagnetic phase from the ferromagnetic phase for large negative values of the crystal field. Since all these multiple behaviors are tightly bound with the plateau behavior of the magnetization, let us see these plateau behavior in (d, h) plane with low temperatures for both integer (Figs. 4 (a) and (b)) and half integer (Figs. 4 (c) and (d)) models. As seen in Fig. 4, for the large negative values of the crystal field, in the Ising case (Figs. 4 (a) and (c)) rising magnetic field can induce transitions m = 0 → 1 → 2 for integer model and m = 1/2 → 3/2 → 5/2 for the half integer model. But when the anisotropy in the exchange interaction decreases (meaning that rising Δ) some intermediate magnetization values appear between m = 0 and m = 1 in the integer model and m = 1/2 and m = 3/2 in the half integer model (see Figs. 4 (b) and (d)). This explains the splitting behavior of the loops located in the vicinity of the center for the anisotropic Heisenberg model when the value of the Δ rises. 4. Conclusion Hysteresis characteristics of the general Spin-S (S > 1) anisotropic Heisenberg model have been studied within the effective field approximation. This work can be considered as the generalization of Ref. [13]. In that work it has been demonstrated that, spin-S BC model has 2S windowed hysteresis behavior for large negative values of the crystal field and low temperature. This work pursues this investigation for the anisotropic Heisenberg model. It has been demonstrated that, changing anisotropy in the exchange interaction changes these multiple hysteresis behaviors and loops that are near the center of the hysteresis division into two. We hope that the results obtained 9

in this work may be beneficial form both theoretical and experimental points of view.

10

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13

3

3

d=0.0 2

1

d=0.0 2

d=-2.0 d=-4.0

1

d=-4.0 d=-6.0

m

m

d=-6.0

d=-2.0

0

-1

-1

Δ=0.0 t=0.3

-2

0

Δ=1.0 t=0.3

-2

S-2

S-2 (a)

-3 -20

-10

0

10

(b)

-3 -20

20

-10

h

d=0.0 2

d=-2.0 d=-4.0

1

d=-2.0 d=-4.0 d=-6.0

m

m

d=-6.0 0

-1

0

-1

Δ=0.0 t=0.3

-2

Δ=1.0 t=0.3

-2

S-5/2 -3 -20

20

3

d=0.0

1

10

h

3

2

0

S-5/2

(c) -10

0

10

-3 -20

20

h

(d) -10

0

10

20

h

Figure 2: Hysteresis loops of S − 2 and S − 5/2 BC models for various values of d and t = 0.3. While Δ = 0.0 corresponds to the Ising case, Δ = 1.0 corresponds to the isotropic Heisenberg model.

14

3

3

Δ=0.0 2

1

Δ=0.0 2

Δ=0.4 Δ=0.8

1

Δ=0.8 Δ=1.0

m

m

Δ=1.0

Δ=0.4

0

-1

-1

d=-6.0 t=0.3

-2

0

d=-6.0 t=0.3

-2

S-2 -3 -20

S-5/2 (a)

-10

0

10

-3 -20

20

h

(b) -10

0

10

20

h

Figure 3: Hysteresis loops of the S-2 and S-5/2 anisotropic Heisenberg models for selected values of Δ and d = −6.0 and t = 0.3.

15

20

20 3

3

2.5

15

2.5

15

10

S-2 Δ=0.0

5

1.5

2

h

h

2 10

S-2 Δ=1.0

1 5 0.5

(a)

0 -8

-7

-6

-5

-4

-3

-2

-1

0

1 0.5

0

(b)

0

1

-8

-7

-6

-5

-4

d

-3

-2

-1

0

0

1

d

20

20 3

3

2.5

15

2.5

15

10

S-5/2 Δ=0.0

5

1.5

2

h

2

h

1.5

10

S-5/2 Δ=1.0

1 5 0.5

(c)

0 -8

-7

-6

-5

-4

-3

d

-2

-1

0

1

0

1.5 1 0.5

(d)

0 -8

-7

-6

-5

-4

-3

-2

-1

0

0

1

d

Figure 4: Magnetization values of the integer and half integer models in (d, h) plane for selected values of t = 0.3. While Δ =160.0 corresponds to the Ising case, Δ = 1.0 corresponds to the isotropic Heisenberg model.