Physics

G. A. Bross

LOW-TEMPERATURE PROPERTIES OF AN ISOTROPIC FERROMAGNET IN THE CASE OF ARBITRARY SPIN

(Presented by Academician N. N. Bogolyubov on 16 V 1964)

In the last two years a number of attempts have been made \((^{1-4})\) to calculate the magnetization of a ferromagnetic dielectric in the case of arbitrary spin. All the calculations are based on the method of two-time temperature Green’s functions and differ from one another in the way the chain of equations for the Green’s functions is decoupled. Another difference appears when Bose operators are used instead of Pauli operators or spin operators. The first results of Tahir-Kheli and ter Haar \((^1)\) and of Tyablikov give only the zeroth approximation for the given problem, since in them the correlations of spin operators at different lattice sites are neglected. A similar decoupling procedure was used by Callen \((^4)\). Further calculations by Tahir-Kheli and ter Haar \((^2)\) were based on substituting, in place of Pauli operators, operators with Bose commutation relations, which may lead, as shown by Tyablikov \((^{5-7})\), in the case of spin \(s = 1/2\), to an incorrect result for the magnetization. In the present communication we briefly set out the results of certain calculations, which differ from Tyablikov’s calculations \((^3)\) by a better decoupling (for details see \((^{8,9})\)).

The Hamiltonian of our problem has the form

\[ \mathcal{H}=E_0-\left(\mu H-J(0)\langle S^z\rangle\right)\sum_f S_f^z -\frac{1}{2}\sum_{f,f'} I(f-f')S_f^-S_{f'}^+ -\frac{\varepsilon}{2}\sum_{f,f'} I(f-f')\left(S_f^z-\langle S^z\rangle\right)\left(S_{f'}^z-\langle S^z\rangle\right), \tag{1} \]

where \(H\) is the external field, which is assumed to be directed along the \(z\)-axis; \(\mu\) is the Bohr magneton; \(f\) and \(f'\) are numbers of lattice sites; \(I(f-f')\) is the exchange integral corresponding to the interaction of spins at the lattice sites \(f\) and \(f'\); \(N\) is the number of lattice sites; \(\varepsilon\) is a parameter, which we regard as small; it is introduced for the decoupling of the chain of equations for the Green’s functions. Below we use the notation:

\[ E_0=\frac{\varepsilon}{2}NJ(0)\langle S^z\rangle^2, \tag{2} \]

\[ J(p)=\sum_f I(f)\exp\left(-i(pf)\right); \tag{3} \]

\(S_f^{\pm}, S_f^z\) are spin operators satisfying the commutation relations

\[ [S_f^z,S_{f'}^{\pm}]=\pm S_f^{\pm}\Delta(f-f'); \tag{4} \]

\[ [S_f^+,S_{f'}^-]=2\{\langle S^z\rangle+\varepsilon(S_f^z-\langle S^z\rangle)\}\Delta(f-f'), \tag{5} \]

where \(\Delta(x)=1\) if \(x=0\), and \(\Delta(x)=0\) otherwise. Further, we use the relation

\[ S_f^-S_f^+=-\,[2\langle S^z\rangle-\varepsilon(2\langle S^z\rangle-1)](S_f^z-S) -\varepsilon[(S_f^z)^2-S^2], \tag{6} \]

which, together with conditions (4)—(5), becomes the generally known relation at \(\varepsilon=1\). To calculate the energy of the elementary excitations

(spin waves) up to terms of order \(\varepsilon^2\), it is necessary to introduce the Green’s functions

\[ G_1^n(g,f)=\left\langle\!\left\langle S_g^+ \mid S_f^- (S_f^z-\langle S^z\rangle)^n \right\rangle\!\right\rangle; \tag{7} \]

\[ G_2^n(k,g,f)=\left\langle\!\left\langle (S_k^z-\langle S^z\rangle)S_g^+ \mid S_f^- (S_f^z-\langle S^z\rangle)^n \right\rangle\!\right\rangle; \tag{8} \]

\[ G_3^n(l,k,g,f)=\left\langle\!\left\langle S_l^-S_k^+S_g^+ \mid S_f^- (S_f^z-\langle S^z\rangle)^n \right\rangle\!\right\rangle; \tag{9} \]

\[ G_4^n(l,k,g,f)=\left\langle\!\left\langle (S_l^z-\langle S^z\rangle)(S_k^z-\langle S^z\rangle)S_g^+ \mid S_f^- (S_f^z-\langle S^z\rangle)^n \right\rangle\!\right\rangle; \tag{10} \]

\[ G_5^n(m,l,k,g,f)=\left\langle\!\left\langle S_m^- (S_l^z-\langle S^z\rangle)S_k^+S_g^+ \mid S_f^- (S_f^z-\langle S^z\rangle)^n \right\rangle\!\right\rangle; \tag{11} \]

\[ G_6^n(r,m,l,k,g,f)=\left\langle\!\left\langle S_r^-S_m^-S_l^+S_k^+S_g^+ \mid S_f^- (S_f^z-\langle S^z\rangle)^n \right\rangle\!\right\rangle \tag{12} \]

and to calculate the functions \(G_2^n\) and \(G_3^n\) from the equations of motion up to terms of order \(\varepsilon\) inclusive, and the functions \(G_4^n—G_6^n\) up to terms of zeroth order in \(\varepsilon\). Therefore it is possible to terminate the chain of equations for the Green’s functions. To solve the equations of motion for \(G_2^n\), let us pass to Fourier transforms in time and space. For example:

\[ G_1^n(g,f)=\frac{1}{N}\sum_p \int_{-\infty}^{\infty} dE\,G_1^n(p;E)\exp\left[i(p,q-f)-iE(x_0-x_0')\right]. \tag{13} \]

If we introduce the mass operator \(M_1^n(p;E)\), then the equation for the first Green’s function takes the form

\[ \{E-E(p)-M_1^n(p;E)\}G_1^n(p;E)=I_1^n(p), \tag{14} \]

where

\[ I_1^n(p)=-\frac{1}{2\pi}\left\langle [S_f^+,S_f^-(S_f^z-\langle S^z\rangle)^n]\right\rangle; \tag{15} \]

\[ E(p)=\mu H+[J(0)-J(p)]\langle S^z\rangle. \tag{16} \]

The mass operator \(M_1^0(p;E(p))\) can be determined in exactly the same way as in the case \(S=\frac12\) \((^{5,7})\). Neglecting terms of order \(T^{7/2}\) and higher, we obtain for it the expression

\[ M_1^0(p;E(p))=\varepsilon [J(0)-J(p)]p_S^0+\varepsilon B_1+\varepsilon^2 B_2+i\varepsilon^2 B_2', \tag{17} \]

where

\[ p_S^0=\frac{1}{N}\sum_q N_S^0(p)=\frac{1}{N}\sum_p \left\{ e^{\beta \widetilde{E}_1^0}-1\right\}^{-1}; \tag{18} \]

\[ \varepsilon B_1=\frac{\varepsilon}{N}\sum_q \{J(p)+J(q)-J(p-q)-J(0)\}N_S^0(q); \tag{19} \]

\[ \varepsilon^2 B_2=\frac{\varepsilon^2}{2N^2}\sum_{q,q'} P\, \frac{[J(p+q-q')+J(q')-J(p-q')-J(q'-q)]^2} {E(p)+E(q)-E(q'-q)-E(p-q')}\,N_S^0(q); \tag{20} \]

\[ \varepsilon^2 B_2'=\frac{\varepsilon^2\pi}{2N^2}\sum_{q,q'} [J(p+q-q')+J(q')-J(p-q') \]

\[ {}-J(q'-q)]^2\, \delta [E(p)+E(q)-E(p-q')-E(q'-q)]\,N_S^0(q); \tag{21} \]

\(P\) is the principal value. The expressions for \(B_1\), \(B_2\), \(B_2'\) are close to the expressions obtained by Tyablikov \((^{5,7})\) for the case of spin \(S=\frac12\). For this reason we do not write out explicit expressions for them in the low-temperature case. Corrections to the spin-wave energy due to the interaction between them are deter-

are the real part of the mass operator

\[ \widetilde{E}_{1}^{0}=\widetilde{E}_{1}^{0}(p)=E(p)+M_{1}^{\prime\,0}(p;\widetilde{E}_{1}^{0})= \]
\[ =\mu H+[J(0)-J(p)]S+\varepsilon B_{1}+\varepsilon^{2}B_{2}+O(T^{7/2}), \tag{22} \]

whereas the imaginary part determines their mean lifetime. Calculation of the magnetization by the method proposed by Tyablikov \({}^{(3)}\) for large values of the spin \(s\gg 1/2\) requires considerable computations. Therefore we shall construct an approximate solution, using equation (6). With its help one can show that, in the case \(s>1/2\), the magnetization is determined by the expression:

\[ \langle S^{z}\rangle=S-P_{S}^{0}+O(T^{7/2}), \tag{23} \]

whereas in the case \(s=1/2\)

\[ \langle S^{z}\rangle=\frac{1}{2}-P_{1/2}^{0}+2(P_{1/2}^{0})^{2}+O(T^{7/2}). \tag{24} \]

It is easy to show that at low temperatures the first nontrivial correction to the spin-wave approximation (in the expression for \(\langle S^{z}\rangle\)) is proportional to \(T^{3}\) in the case \(s=1/2\) and proportional to \(T^{4}\) in the case \(s>1/2\). The same result was obtained by other methods by Tahir-Kheli and ter Haar \({}^{(2)}\) and by Callen \({}^{(4)}\).

The author expresses deep gratitude to Prof. S. V. Tyablikov for suggesting that this investigation be carried out and for numerous useful discussions of the results of the work.

Federal Republic of Germany

Received
7 V 1964

REFERENCES

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\({}^{3}\) S. V. Tyablikov, Fiz. met. i metalloved., 16, 321 (1963).
\({}^{4}\) H. B. Callen, Phys. Rev., 130, 890 (1963).
\({}^{5}\) S. V. Tyablikov, DAN, 149, 573 (1963).
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