Маслов ИНТРОДУЦТИОН ТО ПХЫСИЦС ОФ СЕЦОНД-ОРДЕР МАГНЕТИЦ 2015

Jij < 0 for any two nearest atoms and negligibly small otherwise, the total energy is minimum when any two nearest moments are oppositely directed. Such a type of ordering is referred to as antiferromagnetic. It is reminiscent of a checkerboard order. In the general case, the sign of Jij may vary with the distance between the atoms i and j, resulting in rather complex magnetic textures, neither ferromagnetic nor antiferromagnetic.

It should be pointed out that the very possibility of one or another type of magnetic order to take place in a given solid strongly depends on its crystal structure. For example, the antiferromagnetic ordering cannot occur on a triangular lattice because of frustration. Besides, in anisotropic crystals different types of magnetic order may exist along different crystal axes.

10. Heisenberg model and Ising model

Up to now we have discussed the interaction between two local

magnetic moments and derived the operator ij of the corresponding

U

interaction energy, see Eqs. (9.1) and (9.2). We now turn to the macroscopic system of N 1 moments. With consideration for an external

magnetic field H the Hamiltonian is

where the sum over both i and j is from 1 to N, the prime implies that

i≠ j, and we took into account that U ji =U ij . For the sake of conven-

ience let us, first, introduce the operators σi = µi µB and, second, redefine the exchange integral Jij → 2Jij in Eq. (9.1), so that

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see § 4.

The Hamiltonian (10.3) is referred to as the Hamiltonian of the isotropic Heisenberg model. The term “isotropic” points to equal strength of interaction between the components of magnetic moments along x, y,

and z axes: Jij (σi σj ) = Jij σix σjx + Jij σiy σjy + Jij σiz σjz . In general, how-

ever, this is not the case because of the crystal field anisotropy, and the Hamiltonian has the form

This the Hamiltonian of the anisotropic Heisenberg model. For the special case of Jijxx = Jijyy = Jijzz it reduces to Eq. (10.3). For another limiting

case Jijxx , Jijyy << Jijzz ≡ Jij we have the Hamiltonian of the Ising model:

This is the Hamiltonian we shall study in what follows.

Contrary to the case of noninteracting magnetic moments, there is now the term that accounts for interaction of moments to one another.

As before, our objective is to find magnetization M as a function of T and H.Inasmuch as H = Hez , we have M = Mez , where

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Since all magnetic moments are alike, the value of σiz T does not depend on i, and hence

where one can take σiz T for any one of N moments. In general this

problem has no exact analytic solution, and one must either resort to numerical calculations or make use of one or another approximation.

11. Mean-field approximation in the Ising model

As mentioned above, in the general case the exact analytic solution of Ising model with interactions can not be found. However if we repre-

sent Hamiltonian (10.7) as the sum of terms depending only on the σiz , the problem will be solved. Further we shall use the approach that make

where σiz and σjz are the mean values of the magnetic moments,

σiz − σiz and σjz − σjz are the operators of magnetic moments fluc-

tuations (they define the difference between true moments and the mean value). One can see three different terms in the sum written above: terms that do not depend on fluctuations, terms that depend linearly on fluctuations, and terms that depend quadratically on fluctuations. In other words, we can present the interaction of two magnetic moments as a

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sum of three types of interactions (for each pair of magnetic moments): interaction of mean magnetic moments with each other, interaction of mean magnetic moments with fluctuations, and interaction of fluctua-

small, we neglect the terms in Hamiltonian that depend quadratically on fluctuations, but leave terms that depend linearly on fluctuations

σiz σjz ≈ σiz σjz + σiz (σjz − σjz )+ σjz (σiz − σiz )=

= σiz σjz + σjz σiz − σiz σjz .

(11.2)

This approach is called the mean-field approximation. Taking into account the approximation used (for each pair of magnetic moments) rewrite the Hamiltonian of the Ising model

Considering that Jij = J (Ri − Rj ), and consequently Jij = J ji , replacing i ↔ j , we require for the second term

that is equal to the first term. The third term is a constant

Therefore, we can write down the Hamiltonian of the Ising model in the following form

Considering that our system is homogeneous (i.e., it is free from intersti-

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(magnetic moment) to all other atoms (magnetic moments), the prime implies that Rl ≠ 0. This sum is the same for every atom, since we con-

sider that all atoms (magnetic moments) located periodically, i.e., they form the crystal. Thus, the interaction term takes the following form

ergy (sum of the energies of exchange interactions of randomly selected atom with all other atoms). Taking into account that J (Rl ) falls off ex-

ponentially as the Rl increases, denote the J (Rl )= J for the nearest neighbors and considering that J (Rl )= 0 for other atoms, we obtain ∑/ J (Rl )≈ zJ, where z is the number of nearest neighbors of each

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where θ is the some characteristic temperature (next, we shall omit the Boltzmann constant).

Thus, taking into account the notation introduced, we can write down the Hamiltonian of the Ising model in the following form

i=1

The main differences between this Hamiltonian and the Hamiltonian

i=1

of the constant E0′ (note that E0′ does not affect on the σz calculation)

and the field h substituted by the “effective field” heff = h + θσz .

Finally, in the mean-field approximation system of interacting magnetic moments can be approximately described as the system of nonin-

teracting magnetic moments placed into the “effective field” heff ≠ h.

12. Curie-Weiss equation and Curie-Weiss law

We now apply the mean-field approximation to the system of magnetic moments with Jij ≥ 0 (i, j) . In particular, we assume Jij = J

for the nearest neighbors and Jij = 0 for other atomic pairs. Magnetization of such system (projection on the z-axis) equals to

−1≤ x ≤1, and x = 0 in the case of µz = 0. Let us find x = x(T, H ).

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Recall that the Hamiltonian of the Ising model in the mean-field approximation is

i=1

N

Constant E0′ does not affect on the σz calculation, term −heff ∑σiz is

i=1

the operators sum of the interaction between each magnetic moment and “effective field”. In other words, in our approximation magnetic moments do not depend on each other, but they located in the “effective

field” heff . Therefore, the value σz for arbitrary magnetic moment can be obtained in terms of single magnetic moment located in the field

Recall that the heff = h + θσz ≡ h + θx . Therefore the Eq. (12.3) takes the form

Such dependence of x is referred to as Curie-Weiss equation.

Let us find the magnetization and magnetic susceptibility at high

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The latter temperature dependence of χ is referred to as Curie-Weiss law. Note that if the exchange interaction is absent ( Jij = 0 ), then

θ = 0 , so, we obtain the Curie law. The value θ is called the Curie temperature.

13. Ferromagnetic transition in the Ising model. Curie temperature. Order parameter

In the previous paragraph we have shown that at high temperatures

which something happens. Maybe it is a transition to ferromagnetic state? Let us study the Curie-Weiss equation in more detail.

Consider the h = 0 in the Curie-Weiss equation, thus the latter takes the following form

This equation can not be solved analytically. We shall try to solve it graphically. From the Fig. 13.1 and 13.2 one can see that at T > θ the Curie-Weiss equation has only one solution ( x = 0 ), but at T < θ it has three solutions ( x = 0; x ≠ 0 ).

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