Материал: Маслов ИНТРОДУЦТИОН ТО ПХЫСИЦС ОФ СЕЦОНД-ОРДЕР МАГНЕТИЦ 2015

thermodynamic average σ1z σ2 z T is called the correlation function. In

a crystal with regular atomic arrangement it depends on a relative position of two atoms:

To begin with consider the case T > θ where we expect the qualitative modifications to the mean field approximation. Note that for a sys-

configurations. In one half of those configurations the moment µ1 is

directed “up” (that is, the eigenvalue σ1z of the operator σ1z equals to +1), while in another half it is directed “down” (σ1z = –1), and hence the

where the symbols ...+ and ...− stand for the thermodynamic averag-

es over 2N-1 configurations with the fixed values of σ1z = +1 and σ1z = –1 respectively. Since in zero magnetic field the Hamiltonian and the value

By virtue of the fact that σ1z = +1 in all configurations involved, from Eq. (25.4) one has

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where the symbol Tr+ implies summation over the diagonal matrix elements calculated for the states with σ1z = +1 only.

Let us now rewrite the Hamiltonian of the Ising model, Eq. (10.7), separating out the terms that account for interactions of the magnetic moment µ2 with all other moments:

(the external magnetic field H = 0). Next we apply the mean field approximation to the first part of Eq. (25.6), leaving the second one intact:

In doing so, we neglect only the interactions between the fluctuations of the magnetic moment µ2 and fluctuations of other moments, while the

interactions between the fluctuations of all remaining moments are tak-

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Designating

(2)

we see that H can be viewed as the Hamiltonian of the single magnetic moment µ2 placed in an effective magnetic field H / = H / ez where H / =h′/ µB , see Eq. (25.10). By analogy with an isolated magnet-

ic moment in the field H = Hez , see § 4, we come up with the expression

Eq. (25.4), and taking into account Eqs. (25.5), (25.11) and (25.12), we arrive at the following expression:

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Writing down similar expressions for g(R1 − Rj ) at j ≠ 2, we obtain a

set of N coupled equations for correlation functions g(R1 − Ri ) with

i = 2, 3, ..., N .

The system of nonlinear equations (25.13) has no exact analytical solution. Let us find an approximate expression for g(R1 − R2 ) in the case that the distance R1 − R2 between the magnetic moments greatly ex-

ceeds the interatomic spacing a. In this case, at least one of two strong inequalities, R1 − Rj >> a and Rj − R2 >> a , should hold for any j, so

that either g(R1 − Rj ) <<1 (because there is no long range magnetic or-

der at T > θ) or J (Rj − R2 ) is exponentially small (see § 8), or combina-

tion of both. As a consequence, the argument of hyperbolic tangent in the right-hand side of Eq. (25.13) is much less than unity, and one can restrict himself to the first term in a Taylor series tanh(z) = z + … :

It only remains for us to supplement Eq. (25.15) with a boundary condition at R = 0 . Since g(R = 0) = σiz σiz T =1 for any T, one can approximately write

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Expanding the functions g(R) and J (R) into the Fourier series,

where β = 1/T. We also present without proof the expression for g(q) at T < θ:

where x is the temperature dependent order parameter. One can see that at T → θ (that is, at x → 0) Eq. (25.20) transforms into Eq. (25.19).

sum over q in Eq. (25.17) comes from the region of small q. Expanding

J (q) in powers of qR and restricting ourselves to quadratic terms we have

that we set kB = 1), and the second term in Eq. (25.21) equals to zero from the symmetry considerations, we arrive at

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