PHYSICS

Jerzy Czerwonko

THERMODYNAMIC GREEN’S FUNCTIONS OF AN ISOTROPIC FERROMAGNET WITH ARBITRARY SPIN

(Presented by Academician N. N. Bogolyubov, May 28, 1962)

Let us consider a system of spins with exchange interaction, assuming that the exchange integral is a nonnegative function of the distance between the points at which the spins are localized. The thermodynamic Green’s functions of such systems were investigated by N. N. Bogolyubov and S. V. Tyablikov (¹), and also by Pu Fu-cho, S. V. Tyablikov, and T. Shiklosh (²–⁴) for spin \(1/2\).

In the present note we develop methods for solving the problem for higher spins. For this purpose we transform the exchange Hamiltonian, introducing the notation: \(S_f = S_f^x - iS_f^y\), \(S_f^+ = S_f^x + iS_f^y\), \(\sigma_f = S_f^z\), where \(S_f^x, S_f^y, S_f^z\) are the components of the spin operator of the atom localized at the point \(f\). In these operators the exchange Hamiltonian has the form

\[ H = -h\sum_f \sigma_f - \frac{1}{2}\sum_f \sum_{f \ne g} I\bigl(|f-g|\bigr)\bigl(\sigma_f \sigma_g + S_f^+ S_g\bigr), \qquad \sum_f 1 = N. \tag{1} \]

Here \(h\) is the product of the external magnetic field by the Bohr magneton; summation is over the whole lattice with the additional condition \(I(0)=0\).

We shall show that the one-particle density matrix will be diagonal if all \(\sigma_f\) are taken in diagonal form. This follows from the following propositions:

1) If \(\sigma_f |j\rangle = j |j\rangle\), the only nonvanishing elements of the operators \(S_f\) and \(S_f^+\) are the elements \(\langle j-1|S_f|j\rangle\) and \(\langle j|S_f^+|j-1\rangle\).

2) It follows from 1) that an arbitrary product of \(S_f, S_f^+, \sigma_f\) will be diagonal in the \(|j\rangle\)-representation when it contains as many operators \(S_f^+\) as \(S_f\). Only in this case can this product be diagonal and different from zero.

3) Only those terms of the expansion in a series of \(\exp(-H/\vartheta)\) in which the same number of \(S_f^+\) and \(S_f\) operators occurs will contribute to the one-particle density matrix. Here \(f\) denotes the coordinates of atoms over whose states the trace is taken. It follows from the form of Hamiltonian (1) that in all terms contributing to the one-particle density matrix, the operators \(S^+, S\) (associated with the atom over whose states the trace is not evaluated) enter in equal numbers—this is what ensures the diagonality of the one-particle density matrix. On the other hand, this indicates that for an isotropic ferromagnet with spin \(l\) there can exist only \(2l\) linearly independent one-particle moments. In a similar way one can prove that the mean value of an arbitrary product of \(S_f^+, S_f, \sigma_f\), where \(f\) runs through several values, will be nonzero only when in the product the numbers of \(S\)-operators “with a cross” and “without a cross” are equal. The same conclusions can be drawn for all mean values calculated with the aid of the density matrix \(\exp(-H/\vartheta)\), where the Hamiltonian \(H\) is a sum of terms in which there are as many \(S^+\) operators as \(S\) operators.

Following paper \((^1)\), we introduce the retarded and advanced Green functions:

\[ \langle\!\langle A(t)\mid B\rangle\!\rangle_{\mathrm{ret}} = \theta(t)\langle[A(t),B]\rangle, \qquad \langle\!\langle A(t)\mid B\rangle\!\rangle_{\mathrm{adv}} = -\theta(-t)\langle[A(t),B]\rangle, \qquad \theta(t)= \begin{cases} 1, & t>0,\\ 0, & t<0, \end{cases} \]

where \(\langle\ldots\rangle\) denotes averaging over the canonical ensemble; \(A(t), B(t)\) are operators in the Heisenberg representation; \([\ldots,\ldots]\) denotes the commutator. The equations for the Green functions
\(C^{1}_{fg}(t)\equiv \langle\!\langle S_f(t)\mid S_g^+\rangle\!\rangle\) and
\(C^{2}_{fg}(t)\equiv \langle\!\langle S_f^+(t)\mid S_g\rangle\!\rangle\)
do not depend on the spin of the ferromagnet and will be the same as in \((^1)\), if one takes into account the difference of units (we have \(\hbar=1\)). Eliminating the higher Green functions by the device used in \((^1)\), namely
\(\langle\!\langle S_f(t)\sigma_d(t)\mid S_g^+\rangle\!\rangle=\langle\sigma_d\rangle C^1_{fg}(t)\), etc., we obtain the solution

\[ C^1(E,k)=\frac{i\sigma}{\pi}\{E+h+\sigma[I'(0)-I'(k)]\}^{-1} = C^2(-E,k), \tag{2} \]

in which
\(I'(k)=\sum_g I(|g|)\exp i(g,k)\);
\(\sigma=\langle\sigma_d\rangle\);
\(C^\nu(E,k)\) is defined by means of \(C^\nu_{fg}(t)\), \(\nu=1,2\):

\[ C^\nu(E,k)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dt\, \sum_g \exp[i(k,g)+itE]\,C^\nu_{f+g,f}(t). \tag{3} \]

Calculating the spectral functions and the mean values of the operators by the method developed in \((^1)\), we obtain equations for \(\sigma\):

\[ \langle S_g^+S_g+S_gS_g^+\rangle =2\sigma\,\frac{v}{(2\pi)^3}\int d^3k\, \operatorname{cth}\frac{L+\sigma\varepsilon(k)}{\tau}, \tag{4} \]

where
\(\varepsilon(k)=1-I'(k)/I'(0)\),
\(L=h/I'(0)\),
\(\tau=2\vartheta/I'(0)\),
\(v=V/N\), and \(V\) is the volume of the system. For a ferromagnet with spin \(1/2\) the operator under the averaging sign is equal to unity, and we obtain the result of paper \((^1)\) for \(\hbar=1\). It is useful to note that from equation (4) one can obtain the critical temperature for arbitrary spin. This is connected with the fact that for \(L=0\) at the Curie point the one-particle density matrix possesses spherical symmetry, i.e. all spin projections are equiprobable. Since
\((S_g^+S_g+S_gS_g^+)|j\rangle=2[l(l+1)-j^2]|j\rangle\), where
\(\sigma_g|j\rangle=j|j\rangle\) and \(l\) is the maximal value of \(j\), calculation of the mean value reduces to simple summation. Hence, taking (4) into account, we have

\[ \tau_c^{-1}=\frac{3}{2l(l+1)}\frac{v}{(2\pi)^3} \int \frac{d^3k}{\varepsilon(k)}. \tag{5} \]

To consider higher spins we take the Green functions

\[ \langle\!\langle P_f(t)\mid S_g^+\rangle\!\rangle\equiv Q^1_{fg}(t), \qquad \langle\!\langle P_f^+(t)\mid S_g\rangle\!\rangle\equiv Q^2_{fg}(t), \]

where \(P_f=S_f\sigma_f+\sigma_fS_f\) and \(\langle P_f\rangle=0\), which follows from the diagonality of the one-particle density matrix. The equation for \(Q^1\) will be

\[ i\dot Q^1_{fg}(t) = i\varphi\,\Delta(f-g)\,\delta(t)-hQ^1_{fg}(t)- \tag{6} \]

\[ -\sum_d I(|f-d|) \left\{ \langle\!\langle P_f(t)\sigma_d(t)\mid S_g^+\rangle\!\rangle +\frac12\langle\!\langle \varphi_f(t)S_d(t)\mid S_g^+\rangle\!\rangle -\langle\!\langle S_f^2(t)S_d^+(t)\mid S_g^+\rangle\!\rangle \right\}, \]

and the equation for \(Q^2\) can be obtained from the equation for \(Q^1\) by changing the sign in one of the parts of the equation, replacing at the same time the operators standing in the Green functions by their Hermitian conjugates. In equation (6)
\(\varphi=\langle\varphi_f\rangle\);
\(\varphi_f=-4\sigma_f^2+S_f^+S_f+S_fS_f^+\);
\(\Delta(f-g)\) is the Kronecker \(\delta\)-function, and \(\delta(t)\) is the Dirac \(\delta\)-function. Decoupling the equations for \(Q^1\) and \(Q^2\) by means of the approximation given by the formulas

\[ \langle\!\langle P_f(t)\sigma_d(t)\mid S_g^+\rangle\!\rangle = \sigma Q^1_{fg}(t), \qquad \langle\!\langle \varphi_f(t)S_d(t)\mid S_g^+\rangle\!\rangle = \varphi C^1_{dg}(t), \]

\[ \langle\!\langle S_f^2(t)S_d^+(t)\mid S_g^+\rangle\!\rangle=0 \]

and by analogous formulas for the Green function of the conjugate operators, using transformation (3) and solution (2), we obtain

\[ Q^1(E,k)=\frac{i\varphi}{2\pi}\{E+h+\sigma[I'(0)-I'(k)]\}^{-1} =Q^2(-E,k). \tag{7} \]

With the aid of the methods developed in (1), we have

\[ \langle S_g^+P_g\rangle= \frac{\varphi v}{(2\pi)^3}\int d^3k\,[e^{-\omega(k)/\vartheta}-1]^{-1}, \qquad \langle S_gP_g^+\rangle= -\frac{\varphi v}{(2\pi)^3}\int d^3k\,[e^{-\omega(k)/\vartheta}-1]^{-1}, \]

\[ \omega(k)=h+\sigma[I'(0)-I'(k)], \]

whence

\[ \langle S_g^+P_g+S_gP_g^+\rangle = -\frac{\varphi v}{(2\pi)^3}\int d^3k\, \operatorname{cth}\frac{L+\sigma\varepsilon(k)}{\tau}. \tag{8} \]

Equations (4) and (8) are suitable for arbitrary spin. For spin 1 they determine two independent one-particle moments. If, in addition to \(\sigma\), one takes the mean number of atoms with zero spin projection in the \(z\) direction, referred to the number of atoms in the lattice—\(n\), equations (4) and (8) can be reduced to the form

\[ n=\frac{1}{3}\left[(4-3\sigma^2)^{1/2}-1\right], \qquad 2+[4-3\sigma^2]^{1/2} = \frac{3\sigma v}{(2\pi)^3}\int d^3k\, \operatorname{cth}\frac{L+\sigma\varepsilon(k)}{\tau}. \tag{9} \]

Let us now solve equations (9) in the case of low temperatures. Expanding \(\operatorname{cth}\xi\) in a series in powers of \(e^{-\xi}\), and evaluating the integrals by the saddle-point method, we find for a simple cubic lattice, as in work \((^2)\):

\[ \sigma=1-\sum_{j\geq 3}A_j\tau^{j/2}, \qquad n=\sum_{j\geq 3}B_j\tau^{j/2}, \]

where

\[ A_3=B_3=\frac{1}{2}\left(\frac{3}{4\pi}\right)^{3/2}Z_{3/2}; \qquad A_4=B_4=0; \]

\[ A_5=B_5=\frac{3\pi}{4}\left(\frac{3}{4\pi}\right)^{5/2}Z_{5/2}; \qquad A_6=\frac{3}{8}\left(\frac{3}{4\pi}\right)^3Z_{3/2}^2; \qquad B_6=-\frac{1}{8}\left(\frac{3}{4\pi}\right)^3Z_{3/2}^2; \]

\[ A_7=B_7=\frac{33\pi^2}{32}\left(\frac{3}{4\pi}\right)^{7/2}Z_{7/2}; \qquad A_8=\frac{3\pi}{2}\left(\frac{3}{4\pi}\right)^4Z_{3/2}Z_{5/2}; \qquad B_8=0; \]

\[ A_9=B_9=\frac{1}{64}\left(\frac{3}{4\pi}\right)^{1/2} \left[9Z_{3/2}^{\,3}+281\pi^3Z_{9/2}\right]; \]

\[ A_{10}=B_{10}=\frac{5\pi^2}{32}\left(\frac{3}{4\pi}\right)^5 \left[\frac{33}{2}Z_{3/2}Z_{7/2}+9Z_{5/2}^{\,2}\right]. \]

The expression \(Z_p\) is given by the sum

\[ \sum_{m=1}^{\infty}m^{-p}\exp\left(-\frac{Lm}{v}\right). \]

A comparison of these results with the results of Dyson \((^5)\) and S. V. Tyablikov \((^2)\) can be represented by the formulas (Tyablikov’s result is taken for \(\hbar=1\)):

\[ A_k=A_{k,T}=A_{k,D}\quad \text{for } k=3,5,7; \]

\[ A_6=\frac{3}{8}A_{6,T}; \qquad A_{6,D}=0; \qquad A_8=0.5\,A_{8,T}=0.3\,A_{8,D}\quad \text{for } l=\frac{1}{2}; \]

\[ A_8=0.6\,A_{8,D}\quad \text{for } l=1. \]

The terms \(A_9\) and \(A_{10}\) were not calculated by Dyson or Tyablikov.

A solution for temperatures \(0\leq (t_c-t)t_c^{-1}\ll 1\) can be obtained by expanding \(\operatorname{cth}\xi\) in a series in \(\xi\). We find

\[ \sigma=\sum_{m\geq 0}a_m v^{2m+1}, \qquad n=\frac{1}{3}+\sum_{m\geq 1}b_m v^{2m}, \]

where

\[ v=\left(1-\tau/\tau_c\right)^{1/2}, \qquad a_0=2\left[\frac{3}{4}+\frac{1}{\tau_c}\right]^{-1/2}, \]

\[ a_1=-\tau_c^{-1}\left(\frac{3}{4}+\frac{1}{\tau_c}\right)^{-3/2} -\left[\frac{9}{16}-\frac{4c}{15\tau_c^3}\right] \left(\frac{3}{4}+\frac{1}{\tau_c}\right)^{-5/2}, \]

\[ b_1=-\frac{3}{4}a_1^2, \qquad b_2=-\left[\frac{3}{2}a_1a_2+\frac{27}{32}a_1^4\right], \]

and \(c\) is equal to \(3/2\), \(11/8\), \(11/9\) for the simple, body-centered, and face-centered cubic lattices, respectively. In contrast to the solution for spin \(1/2\), this solution cannot be represented in the form of an expansion in powers of \([\tau^{-1}-\tau_c^{-1}]^{1/2}\). It should be noted that the term \(a_0\) for cubic lattices is very close to \(4\sqrt{2}/3\), which corresponds to the result of the self-consistent-field theory for \(l=1\).

In the region above the Curie point, in the presence of an external field, we have

\[ \sigma=\sum_{m\geq 0} C_m\tau^{-m}, \qquad n=\sum_{m\geq 0} D_m\tau^{-m}, \]

where

\[ C_0=4v\,(3+v^2)^{-1}, \]

\[ C_1=8(1-v^2)(3-v^2)(3+v^2)^{-1}(5-2v^2)^{-1}; \]

\[ D_0=(1-v^2)(3+v^2)^{-1}, \]

\[ D_1=64v(1-v^2)(3-v^2)(3+v^2)^{-2}(5-2v^2)^{-1} \]

for \(v=\operatorname{th}(L/\tau)\).

An analogous consideration of the problem of spins greater than unity presents no fundamental difficulties.

One should mention one attempt to study higher spins in the work of Kawasaki and Mori \((^6)\). In addition to the functions \(C^1\) and \(C^2\), they also introduce the three-time Green function \(\langle\!\langle S_f^+(t)S_d^+(t)\mid S_g^2\rangle\!\rangle\). It can be shown that introducing the two-time function \(\langle\!\langle [S_f^+(t)]^2\mid S_g^2\rangle\!\rangle\) instead of \(Q^2\) leads, in the superposition approximation, to serious shortcomings of the solutions near absolute zero and the critical point, since in \((^6)\) double integrations over \(k\) enter into the equation for \(\sigma\); however, as is clear from this note, this is not a necessary condition for a satisfactory solution of the problem.

Institute of Theoretical Physics
University of Wrocław
Wrocław, Poland

Received
24 V 1962

REFERENCES

\(^1\) N. N. Bogolyubov, S. V. Tyablikov, DAN, 126, 53 (1959).
\(^2\) S. V. Tyablikov, Ukr. Math. J., 11, 287 (1959).
\(^3\) Pu Fu-cho, S. V. Tyablikov, T. Shiklosh, Acta Phys. Acad. Sci. Hung., 11, 323 (1960).
\(^4\) Pu Fu-cho, Scientific Reports of Higher Schools, 2, 141 (1959).
\(^5\) F. Dyson, Phys. Rev., 102, 1217, 1230 (1956).
\(^6\) K. Kawasaki, H. Mori, Progr. Theor. Phys., 25, 1045 (1961).