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PHYSICS
PU Fiz.-40
AN APPROXIMATE METHOD FOR CALCULATING THE MAGNETIZATION OF AN ISOTROPIC ANTIFERROMAGNET
(Presented by Academician N. N. Bogolyubov, 27 XI 1959)
In our work \((^1)\), with the aid of spectral representations of Green’s functions by the method of paper \((^2)\), we obtained formulas for the quantities of interest to us: the relative magnetization of each sublattice and the cosine \(\gamma\) of the angle between the sublattice magnetization vector and the direction of the magnetic field. We then investigated their solution for temperature \(\tau = 0\). In the present paper we study these equations for \(\tau \ne 0\). As was shown \((^1)\), because the vacuum is not determined by the condition \(\Phi = 0\), it is necessary to consider simultaneously Green’s functions of the type \(\ll h|l_h^+ \gg\), \(\ll l_h^+|l_h^+ \gg\). It is easy to see that \(\sigma\) and \(\gamma\) can be written in the form (see \((^1)\), equations (7), (8))
\[ \frac{1}{\sigma} = \frac{1}{(2\pi)^3} \int \frac{h\gamma-\sigma(2\gamma^2-1+\gamma^2Q/3)}{\varepsilon} \operatorname{cth}\frac{\varepsilon}{\tau}\,dv; \tag{1} \]
\[ \frac{1}{\gamma} = \frac{2}{h} \left\{ \sigma - \frac{1}{(2\pi)^3} \int \frac{Q}{3} \left[ \frac{h\gamma-\sigma(2\gamma^2-1+Q/3)} {h\gamma-\sigma(2\gamma^2-1)(1+Q/3)} \right]^{1/2} \operatorname{cth}\frac{\varepsilon}{\tau}\,dv \right\}, \tag{2} \]
where
\[ \varepsilon = \sqrt{ \bigl[h\gamma-\sigma(2\gamma^2-1+Q/3)\bigr] \bigl[h\gamma-\sigma(2\gamma^2-1)(1+Q/3)\bigr] }, \]
\(h=\mu H/|\bar J_{12}|\), \(\tau=\theta/|\bar J_{12}|\), \(Q=\cos x+\cos y+\cos z\), and the integrals are taken over the region \(-\pi < x, y, z < \pi\) \((dv=dx\,dy\,dz)\). We note that we restrict ourselves to taking into account interactions only between nearest neighbors.
To investigate the expansion at low temperatures, let us expand \(\operatorname{cth}\frac{\varepsilon}{\tau}\) in a series in powers of \(e^{-2\varepsilon/\tau}\) and calculate the integrals by the saddle-point method. We shall then use the iteration method with the solutions \(\sigma, \gamma\) for \(\tau=0\) as the zeroth approximation; as a result we obtain:
Case A. \(h=0\). In this case \(\gamma=0\), i.e., the spins of the two sublattices are oriented parallel,
\[ \sigma = \sigma_0 \left\{ 1 - \sigma_0 a_1\left(\frac{\tau}{\sigma_0}\right)^2 - \sigma_0(a_2+\sigma_0a_1^2)\left(\frac{\tau}{\sigma_0}\right)^4 - \right. \]
\[ \left. - \sigma_0(a_3+4a_1a_2\sigma_0+2\sigma_0^2a_1^3) \left(\frac{\tau}{\sigma_0}\right)^6 -\ldots \right\}, \tag{3} \]
where
\[ \sigma_0=\frac{1}{J}=0.865,\quad a_1=\frac{3^{3/2}\zeta(2)}{2\pi^2} = \frac{3^{3/2}}{12},\quad a_2= \frac{9\sqrt{3}}{4\pi^2}\zeta(4) = \frac{\sqrt{3}}{40}\pi^2,\ldots \]
and
\[ J= \frac{1}{(2\pi)^3} \int \frac{dv}{\sqrt{1-(Q/3)^2}}. \tag{4} \]
We note that spin-wave theory gives
\[ \sigma = 2-J-a_1\tau^2-a_2\tau^4-a_3\tau^6-\ldots , \tag{5} \]
where \(a_1,\ a_2,\ldots\) are determined from (4). The coefficients of \(\tau^2\) in both formulas coincide, but starting with \(\tau^4\) they begin to diverge.
Case B. \(h \simeq 0\) \((0<h<1)\). For simplicity we shall restrict ourselves to only the first two terms of the expansion in temperature:
\[ \sigma=\sigma_0+\sigma_1 h^2\ln h+O(h^2)+[\sigma'_0+(h^2\ln h)]\tau^2+\ldots; \tag{6} \]
\[ \gamma=\gamma_1 h+\gamma_2 h^3\ln h+O(h^3)+[\gamma'_1 h+O(h^3\ln h)]\tau^2+\ldots, \tag{7} \]
where \(\sigma_0,\ \sigma_1,\ \gamma_1,\ \gamma_2\) are determined by formulas (10), (11) of work \(^{(1)}\), and
\[ \sigma'_0=-\frac{3^{3/2}\zeta(2)}{4\pi^2}=-\frac{3^{3/2}}{12}, \qquad \gamma'_1=\frac{3^{3/2}}{\pi^2}\zeta(2)\gamma_1^2\left(1-\frac{1}{\sigma_0^2}\right) =\frac{\sqrt{3}}{2}\gamma_1^2\left(1-\frac{1}{\sigma_0^2}\right). \]
Case C. \(h\simeq 2\) and \(h<2\) \((0<\eta\equiv 2-h<2)\). As was noted in \(^{(1)}\), at \(\tau=0\), when \(h\) approaches the value \(h=2\), a transition occurs to a state with ferromagnetic spin arrangement. The same situation also takes place for \(\tau\ne 0\). For this region we have:
\[ \sigma=1+\sigma_1\eta+\sigma_2\eta^{3/2}+O(\eta^2\ln\eta)+[\sigma'_0+O(\eta)]\tau^{3/2}+\ldots; \tag{8} \]
\[ \gamma=1+\gamma_1\eta+\gamma_2\eta^{3/2}+O(\eta^2\ln\eta)+O(\eta)\tau^{3/2}+\ldots, \tag{9} \]
where \(\sigma_1,\ \sigma_2,\ \gamma_1,\ \gamma_2\) are determined by formulas (13), (14) of work \(^{(1)}\), and
\[ \sigma'_0=-\frac{1}{4}\left(\frac{3}{\pi}\right)^{3/2}\zeta\left(\frac{3}{2}\right). \]
Let us note that \(\gamma_1<0\), and therefore at sufficiently low temperatures \(\gamma\) cannot exceed 1. When \(\gamma=1\), the substance passes into the ferromagnetic state (parallel orientation of the spins in both sublattices).
Case D. \(h>2,\ \gamma=1\). The substance behaves like a ferromagnet.
\[ \sigma=1-a_1\tau^{3/2}-a_2\tau^{5/2}-\frac{a_1^2}{2}\tau^{6/2}-a_3\tau^{7/2}-\ldots, \tag{10} \]
where
\[ a_1=\frac{1}{4}\left(\frac{3}{\pi}\right)^{3/2}\varphi\left(\frac{3}{2},x\right),\qquad a_2=\frac{9}{64}\left(\frac{3}{\pi}\right)^{3/2}\varphi\left(\frac{5}{2},x\right), \]
\[ a_3=\frac{33}{128}\frac{1}{6}\times\frac{3^{7/2}}{\pi^{3/2}}\varphi\left(\frac{7}{2},x\right) \quad\text{and}\quad \varphi(s,x)=\sum_{n=1}^{\infty}\frac{e^{-nx}}{n^s}, \qquad x=\frac{2}{\tau}(h-2\sigma)\simeq\frac{2}{\tau}(h-2). \]
When \(h\to 2\) \((h\ge 2)\), then \(x\to 0\), \(\varphi(s,x)=\zeta(s)\), and
\[ \sigma=1-\frac{1}{4}\left(\frac{3}{\pi}\right)^{3/2}\times \zeta\left(\frac{3}{2}\right)\tau^{3/2}-\ldots, \]
which coincides with the result of the limiting transition \(h\to 2\) \((h\le 2)\) for case C. It is important to note that here, in contrast to work \(^{(3)}\), at \(h=2\) \(\sigma\) does not undergo a jump
\[ \Delta\sigma=\frac{1}{3}\left(\frac{3}{\pi}\right)^{3/2}\zeta\left(\frac{3}{2}\right)^{(3)} \]
and that the quantity \(\sigma\) in equation (8) does not exceed the value \(\sigma=1\). Finally, for the magnetization \(M\) and the susceptibility \(\chi\) of the sample we obtain the following expressions, for example, for case B:
\[ M=2N\sigma\gamma=2N\{\sigma_0\gamma_1 h+(\sigma_0\gamma_2+\sigma_1\gamma_1)h^3\ln h+O(h^3)+ \]
\[ +[\sigma'_0\gamma'_1 h+O(h^3\ln h)]\tau^2+\ldots\}; \tag{11} \]
\[ \chi=\frac{\partial M}{\partial H}=\frac{2N\mu}{|J_{12}|}\{\sigma_0\gamma_1+3(\sigma_0\gamma_2+\sigma_1\gamma_1)h^2\ln h+O(h^3)+ \]
\[ +[\sigma'_0\gamma'_1+O(h^2\ln h)]\tau^2+\ldots\}; \tag{12} \]
it is not difficult to obtain formulas for \(M\) and \(\chi\) in the other cases as well.
For the investigation of the expansion at high temperatures we shall restrict ourselves only to the case of a pure antiferromagnetism \((\gamma=0)\). In this case, instead of (1) and (2), we have:
\[ \frac{1}{\sigma} = \frac{1}{(2\pi)^3}\int \frac{2}{\sqrt{1-(Q/3)^2}} \operatorname{cth}\frac{\sigma}{\tau} \sqrt{1-\left(\frac{Q}{3}\right)^2}\,dv . \tag{13} \]
Since \(\sigma\) tends to zero as \(\tau\) tends to the Néel point, one may expand
\(\operatorname{cth}\frac{\sigma}{\tau}\sqrt{1-\left(\frac{Q}{3}\right)^2}\)
in powers of \(\sigma/\tau\). Solving equation (11) approximately near the Néel point, we obtain
\[ \frac{\sigma}{\tau} = \zeta\left\{1+\frac{1}{36}\zeta^2-\frac{6.5}{1000}\zeta^4+\cdots\right\}, \tag{14} \]
where
\[ \zeta = \sqrt{\frac{3(1-K\tau)}{\tau}}, \qquad K=\frac{1}{(2\pi)^3}\int\frac{dv}{1-Q/3}. \]
The Néel temperature \(\tau_N\) is determined, obviously, as \(\tau_N=1/K=0.66\). Near the Néel point, \(\sigma\) varies proportionally to \(\sqrt{\tau_N-\tau}\) (as in a ferromagnet near the Curie point \((^{4,5})\)); this agrees with the result of the Néel molecular-field method \((^6)\). An indication of such a dependence of \(\sigma\) on temperature is also found in \((^7)\).
It is of interest to study the boundary separating the regions of the antiferromagnetic and ferromagnetic states. The equation for the boundary is obtained by substituting \(\gamma=1\) in (1) and (2):
\[ \frac{1}{\sigma} = \frac{1}{(2\pi)^3}\int \operatorname{cth}\frac{h-\sigma(1+Q/3)}{\tau}\,dv; \tag{15} \]
\[ \frac{h}{2} = \sigma-\frac{1}{(2\pi)^3}\int \frac{Q}{3}\operatorname{cth}\frac{h-\sigma(1+Q/3)}{\tau}\,dv . \tag{16} \]
In the region of low temperatures the equation for the boundary has the form
\[ h=2-A_1\tau^{3/2}-A_2\tau^{5/2}-A_3\tau^{6/2}-A_4\tau^{7/2}-A_5\tau^{8/2}-\cdots, \tag{17} \]
where
\[ A_1=\left(\frac{3}{\pi}\right)^{3/2}\zeta\left(\frac{3}{2}\right), \qquad A_2=\frac{3}{16}\left(\frac{3}{\pi}\right)^{3/2}\zeta\left(\frac{5}{2}\right), \qquad A_3=\frac{9}{4}\frac{1}{\pi^3}\zeta^2\left(\frac{3}{2}\right), \]
\[ A_4=\frac{135}{1024}\left(\frac{3}{\pi}\right)^{3/2}\zeta\left(\frac{7}{2}\right), \qquad A_5=\frac{3}{16}\left(\frac{3}{\pi}\right)^3 \zeta\left(\frac{3}{2}\right)\zeta\left(\frac{5}{2}\right). \]
In conclusion the author expresses deep gratitude to S. V. Tyablikov for suggesting the topic and for constant valuable advice, and also to T. Shiklosh for discussion of the work.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
11 XI 1959
REFERENCES
- Pu Fu-cho, DAN, 130, No. 6 (1959).
- N. N. Bogolyubov, S. V. Tyablikov, DAN, 126, 53 (1959).
- S. V. Tyablikov, Fiz. met. i metalloved., 2, 193 (1956).
- S. V. Tyablikov, Ukr. matem. zhurn., 11, 3 (1959).
- Pu Fu-cho, Dokl. Vyssh. shkoly, ser. fiz.-matem., No. 1 (1959).
- L. Néel, Ann. de Phys., 3, 134 (1948).
- S. V. Vonsovskii, in: Antiferromagnetism, IL, 1956, p. 69; note to article \((^6)\).