The high-temperature expansion of the classical Ising model with S_z^2 term

We derive the high-temperature expansion of the Helmholtz free energy up to the order \beta^{17} of the one-dimensional spin-S Ising model, with single-ion anisotropy term, in the presence of a longitudinal magnetic field. We show that the values of some thermodynamical functions for the ferromagnetic models, in the presence of a weak magnetic field, are not small corrections to their values with h=0. This model with S=3 was applied by Kishine et al. [J.-i. Kishine et al., Phys. Rev. B, 2006, 74, 224419] to analyze experimental data of the single-chain magnet [Mn (saltmen)]_2 [Ni(pac)_2 (py)_2] (PF_6)_2 for T<40 K. We show that for T<35 K the thermodynamic functions of the large-spin limit model are poor approximations to their analogous spin-3 functions.


Introduction
The one-dimensional spin-1/2 Ising model with first-neighbor interaction, in the presence of a longitudinal magnetic field, was exactly solved in 1925 [1]. The Helmholtz free energy (HFE) of this model with S = 1/2 has a simple mathematical expression [2]. The exact expression of the HFE for the S = 1 ferromagnetic model in the presence of a longitudinal magnetic field was derived in 1976 by Krinsky and Furman [3] using the matrix density approach. (This work seems to have been neglected by the subsequent literature, though). More recently, the HFE of the S = 1 [4,5] and S = 3/2 [6] of the Ising model in the presence of an external magnetic field have been written as a set of coupled equations and solved numerically.
In 2007 Rojas et al. [7] published the high-temperature expansion of the HFE of the Ising model for [Mn(saltmen)] 2 [Ni(pac) 2 (py) 2 ](PF 6 ) 2 for temperature T 40 K. They concluded that the experimental data support the view that this SCM can be described, in this window of temperature, by the spin-3 of this model. We apply the method of cumulants described in [12] to calculate the high-temperature expansion of the HFE of any one-dimensional Hamiltonian which is invariant under space translation and satisfies the periodic space condition. In this approach, in order to obtain the exact coefficient that multiplies the term of order β n in the expansion we have to calculate a set of functions named H (n) 1,m , m = 1, 2, . . . , n. The interested reader will find a survey of the method in reference [13]. In section 2 we apply the results of reference [12] to calculate the high-temperature expansion of the HFE of the normalized one-dimensional spin-S Ising model with single-ion anisotropy term in the presence of an external longitudinal magnetic field up to order β 17 . The nice feature about this expansion is that it is faster to apply than the transfermatrix method [2], although the latter one provides exact curves in the whole interval of temperatures. In order to show the importance of calculating the high-temperature expansion of the spin-S Ising model in the presence of an external magnetic field, in section 3 we compare the behavior of certain thermodynamical functions of the ferromagnetic and of the AF Blume-Capel models in the presence of a weak longitudinal magnetic field. This comparison is made for various values of the spin, including the largespin limit (S → ∞). Section 4 presents the thermodynamics of the SCM (for spin-3 and the large-spin limit model) in the interval 11 K T 40 K. Finally, in section 5 we present our conclusions.

Thermodynamics of the spin-S Ising model with a single-ion anisotropy term
The Hamiltonian of the spin-S Ising model with a single-ion anisotropy term in the presence of a longitudinal magnetic field is [13] where S z i is the z component of the spin S operator with norm: || S|| 2 = S(S + 1), S = 1 2 , 1, 3 2 , 2, . . ., at the i -th site of the chain; J ′ is the exchange strength and it can have negative value (ferromagnetic model) or positive value (AF model). In references [14] and [13] we studied the exact thermodynamics of this model for S = 1 2 and 1 (with h = 0), respectively. If the large-spin limit (S → ∞) is applied directly to the Hamiltonian (1), all the thermodynamic functions of the large-spin limit model will diverge. In order to keep the functions finite in this limit, we study the normalized version of the Hamiltonian (1), that is, where s z i is the z component of the spin operator s that has norm 1. The s operator is defined as Making S → ∞ in the Hamiltonian (2) we obtain its large-spin limit.
The Hamiltonians (1) and (2) are identical when Its HFE, in the thermodynamic limit, is called W ′ S , where

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The high-temperature expansion of the classical Ising model with S 2 z term The analogous functions for the normalized Hamiltonian (2) are Z s (J , h, D; β) = Tr e −βH s , and For arbitrary spin value S, the relation between the HFE's (5) and (6) is The Hamiltonian (1) and its normalized version (2) belong to a subclass of chain Hamiltonians that satisfy periodic conditions and that can be decomposed as H i,i+1 = P i,i+1 + Q i , in which the operator P i,i+1 depends on two sites (the i -th and the (i + 1)-th sites), the operator Q i depends only on the i -th site; moreover, It is simple to show that the functions H (n) 1,m calculated in the method presented in references [12,13] for the subclass of Hamiltonians satisfying the condition (8) can be written as where the g -trace (〈. . .〉 g ) in equations (9) and (11) where m i=1 n i = n and n i 0 with i = 1, 2, . . . , m.  [12,13].
The results (9)-(11) are valid for any 1D Hamiltonian that is invariant under space translation, satisfies the periodic space condition and the condition (8).
In this article we study a 1D model that satisfies the necessary conditions to apply the method of reference [12] plus the condition (8). All the terms in this Hamiltonian are commutative; hence, the gtraces in (9)-(11) can be replaced by the usual normalized traces [12].
We calculate the high-temperature expansion of W s (J , h, D; β), for arbitrary value of the spin S , up to order β 17 . The relation (7) can be applied to yield the high-temperature expansion of W ′ Here we present the high-temperature expansion of W s (J , h, D; β), for any spin value S, up to order β 2 , We should note that this high-temperature expansion is valid for positive, null or negative values of J , and for arbitrary values of S, h and D. The HFE of the large-spin limit model of Hamiltonian (2) is calculated from the high-temperature expansion of W s (J , h, D; β) by taking the limit S → ∞.
The authors maintain a website 1 in which the interested reader may find data files on the arbitrary finite spin-S and the large-spin limit (S → ∞) HFE's of the normalized Hamiltonian (2) up to order β 17 .
We have a few general comments on the function W s (J , h, D; β): i) the expansion (13) of the HFE is an even function of the external longitudinal magnetic field h; ii) the function W s (J , h, D; β) is even in the parameter J for an external magnetic field with null longitudinal component (h = 0); iii) even for h = 0, the HFE (13) is sensitive to the sign of the parameter D.
By direct comparison, we verify that our expansion of the HFE of the spin-S Ising model coincides for S = 1/2 and S = 1 with the high-temperature expansions of the exact results of references [14] and [3], respectively.
It is simple to understand why the second comment is valid for the exact expression of the HFE of the model (2). The partition function comes from the calculation of the traces of operators (H s ) n . In reference [19] we showed that for any spin-S only tr i s z

The ferromagnetic and anti-ferromagnetic models in the presence of a weak magnetic field
The exact expression of the HFE of the Hamiltonian (2) and its thermodynamic functions are unknown for an arbitrary value of spin s. Our work has been that of calculating the high-temperature expansion of thermodynamic functions of one-dimensional models. In section 2 we mentioned that the HFE's of the Hamiltonians (1) and (2) in a magnetic field with null longitudinal component (h = 0), is an even function of J . As a consequence, for h = 0, several thermodynamic functions are the same for the ferromagnetic (J < 0) and AF (J > 0) models, namely: the specific heat, the internal energy, the entropy and the mean value of the square of the z component of the spin (S z ) 2 . For h = 0, we also have that the 1 http://www.proac.uff.br/mtt.
In order to verify how important the presence of a longitudinal magnetic field is to the thermodynamic properties of the ferro and AF models, in this section we examine how the thermodynamic functions of the ferromagnetic and AF models differ from their respective values at h = 0 when they are in the presence of a weak magnetic field for different values of spin.
Throughout this section we consider J = 1 (AF model) or J = −1 (ferromagnetic model). Again, the parameters are in units of |J | and the expansions are in powers of (|J |β).
The previous discussion exemplifies the fact that the high-temperature expansion of the specific heat of the AF model in the presence of a weak longitudinal magnetic field can be approximated by the corresponding expression derived from the results of reference [13] (where we have h = 0). Figure 1 shows us that for arbitrary spin-s we can improve this approximation by recognizing that C s (1, h/|J |, D/|J |; |J |β) ≈ C s (1, 0, D/|J |; |J |β) + ∆C 1/2 (1, h/|J |, D/|J |; |J |β), at least in the region in which h/|J | ≪ 1 and |J |β 1.6. It is important to recall that the exact expression of C 1/2 (J , h, D; β) is known [14].
For the spin-s ferromagnetic model (2)

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The high-temperature expansion of the classical Ising model with S 2 z term It is simple to obtain the high-temperature expansion of the magnetic susceptibility per site, χ s (J , h, D; β), from the series (13) for the HFE χ s (β) = −∂ 2 W s (β)/∂h 2 . At h = 0, the magnetic susceptibility of the ferromagnetic and AF models are distinct. We define a percentage difference analogous to equation (15) to present in figure 2  We point out that to derive the high-temperature expansion of χ s (J , h, D; β) from the HFE, we need the dependence of the HFE on h. That is not the case of the previous paper on the spin-S Ising model [7].
In [19] we studied the thermodynamics of the unitary spin-s X X Z model with a single-ion anisotropy term in the presence of a magnetic field in the z direction. We obtained that the specific heat, the magnetization and the magnetic susceptibility of this model with s = 3 are well approximated by their respective large-spin limit versions, in the temperature range of |J |β 1. For the non-normalized S = 3 X X Z model this range corresponds to |J |β 0.083 (see section 2.2 of reference [19]).
In this section, we compare the Hamiltonian (1) having spin-3 with its large-spin limit version for T 40 K. By "large-spin limit model" we mean that the z component of its spin vector S varies continuously, namely, S z i = 2 3 cos(θ i ), in which θ i ∈ [0, π], and i ∈ {1, 2, . . . , N }. In order to relate our analysis to the aforementioned SCM we use the same parameter values as in [9], namely, in the spin-3 and large-spin limit Hamiltonians. One is reminded that k is the Boltzmann constant. Taking the value (18a) for J ′ , for T ∼ 40 K we have |J ′ |β ∼ 0.04. Note that the temperature region characterized by |J |β 0.04 is contained in the temperature region in which the spin-3 X X Z model behaves very much like its large-spin limit version.
Along this section, we take h ′ /k = 0.25 K, which is a weak magnetic field (h ′ /|D ′ | = 0.1). Let F S (h; β) be a thermodynamic function. Its percentage difference of the S = 3 and the large-spin limit models is defined as In figure 3 we plot the percentage difference (19) of the specific heat, the z component of the mag-  Due to the ratio |J ′ |/|D ′ | value of 0.64 for this SCM, the single-ion anisotropy term gives the main contribution to its thermodynamics. Since the crystal field D ′ is negative (see equation (18b)), the states with S z i = ±3 and ±2 are the most probable. The modulus of the amount of energy, in units of k, for the single-ion anisotropy term to change from states with S z i = ±2 to states with S z i = ±3, and vice-versa, is 12.5 K. This value is close to one-third of 40 K. We conclude that for T 40 K (that is, |J ′ |β 0.04), the discretized nature of the spin-3 is still an important feature of the model. This result is very different from that of reference [19] for the spin-3 X X Z model with a single-ion anisotropy term in the presence of a magnetic field in the z direction.
Our expansion of the HFE with S = 3 can be easily used to fit the experimental data of this SCM. It can also be applied to determine the best fit of the parameters J ′ and D ′ .

Conclusions
The method of reference [12] permits one to calculate the high-temperature expansion of the HFE of any chain Hamiltonian with interaction between first neighbors which is invariant under space translation and satisfies a periodic space condition. If the Hamiltonian also satisfies the condition (8), we show that some of the terms that contribute to the function H (n) 1,m have already been calculated in a lower order of n, that is, H (n−1) 1,m−1 . A set of important 1D Hamiltonians satisfy the condition (8). This contribution from lower order of n to this auxiliary function permits us to compute the HFE of the one-dimension Ising model with single-ion anisotropy term in the presence of a longitudinal magnetic field up to order β 17 , for arbitrary values of spin S and of the other parameters J , h and D. Upon performing the numerical analysis of the thermodynamics of spin models (1) and (2) one must know the value of the spin beforehand.
We discuss the thermodynamics of the normalized Hamiltonian (see equation (2)), but equation (7) relates the HFE of the Hamiltonian (1) and its normalized version (equation (2)).
Some thermodynamic functions are insensitive to the sign of J in the absence of a magnetic field; for instance, the specific heat of the ferromagnetic (J < 0) and AF (J > 0) spin-S Ising models with single-ion anisotropy term are identical with h = 0. In the absence of a magnetic field, the ferromagnetic model favors parallel neighboring spins while in the AF model the anti-parallel pairs are more probable. The

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The high-temperature expansion of the classical Ising model with S 2 z term effect of the presence of an external magnetic field in both models is that of favoring the alignment of spins at each site to the field direction. In the ferromagnetic model such alignment is favored by the coupling between neighboring spins. On the other hand, in the AF model there is a competition between the coupling of neighboring spins (favoring anti-parallel alignment) and the Zeeman term (forcing all the spins in the chain to align with the external magnetic field). As a consequence of this competition, there is a perturbative effect in the AF model due to the presence of a weak magnetic field; in such regime, their thermodynamic functions F s (1, h/|J |, h/|J |; |J |β) for the spin-s model can be approximated by F s (1, h/|J |, D/|J |; |J |β) ≈ F s (1, 0, D/|J |; |J |β) + ∆F 1/2 (1, h/|J |, h/|J |; |J |β). The exact expression of the HFE of the spin-1/2 Ising model with a single-ion anisotropy term in the presence of a longitudinal magnetic field is known [14]. In the ferromagnetic model, the effect of the Zeeman coupling cannot be treated anymore as a perturbation to the interaction between first neighbors and to the single-ion anisotropy term for h/|J | 0.04.
Kishine et al. [9] applied the spin-3 Hamiltonian (1) to analyse the low energy dynamics of the [Mn(saltmen)] 2 [Ni(pac) 2 (py) 2 ](PF 6 ) 2 SCM for temperatures T < 40 K. Although S = 3 could be considered a high spin value in the region of T ∼ 40 K (|J ′ |β ∼ 0.04), the negative value of D ′ favors the states with S z i = ±2 and ±3. The modulus of the amount of energy required by the single-ion anisotropy term to have S z varied from ±2 ⇋ ±3 and vice-versa is about one-third of the thermal energy available for T 40 K. Our results show that the large-spin limit model, with the spin-3 replaced by the classical vector in the Hamiltonian, yields a poor approximation to the behavior of this SCM for T 18 K and for the set of parameters values (18a) and (18b). Finally, it is very important to point out that our high-temperature expansion of the HFE of the spin-s Ising model can be applied to fit experimental data of new materials with one-dimensional behavior and strong anisotropy axis. This expansion leaves the spin value of the material as one of the parameters to be determined by the best fit. The expansion is valid for positive and negative values of J and D in the presence of a longitudinal magnetic field that does not have to be a weak field.