Low-Temperature Magnetism of Metals in the Dynamic Spin-Fluctuation Theory

Abstract

The low-temperature behavior of magnetization in ferromagnetic metals is studied by means of the dynamic spin-fluctuation theory. An expression is derived for the dynamic magnetic susceptibility, which extends the one in the random phase approximation. The transverse susceptibility is shown to have the spin-wave poles, which yield the asymptotic T ^3/2 law for magnetization. An explicit expression for the coefficient in the T ^3/2 law is derived on the basis of the multiband Hubbard Hamiltonian and real band structure. The temperature dependence of magnetization is demonstrated by the example of Fe and disordered Fe-Ni Invar.

Appendix

1.1 Transverse Spin Correlator

In the momentum-“frequency” representation, the energy of the field is written as

$$ F_{0}(V) \,=\, \frac{1}{\tilde{u}}\! {\sum}_{\mathbf{q}m}\!\! \left( \frac{1}{2} V^{-}_{\mathbf{q}m} V^{+}_{-\mathbf{q}-m} \!\,+\, \frac {1}{2} V^{+}_{\mathbf{q}m} V^{-}_{-\mathbf{q}-m} \!\,+\,\! V^{z}_{\mathbf{q}m} V^{z}_{-\mathbf{q}-m}\right). $$

(21)

Here, $V^{\alpha }_{\mathbf {q}m}$, α=−,+,z, are the coefficients in the expansion of the matrix V _{q m} in terms of the spin matrices $\sigma ^{\pm } = \frac {1}{2} (\sigma ^{x} \pm \mathrm {i} \sigma ^{y})$ and σ ^z:

$$\begin{array}{@{}rcl@{}} V_{\mathbf{q}m} = V^{-}_{\mathbf{q}m} \sigma^{+} + V^{+}_{\mathbf{q}m} \sigma^{-} + V^{z}_{\mathbf{q}m}\sigma^{z}, \end{array} $$

and the Fourier transformations of the field are given by the formulae

$$\begin{array}{@{}rcl@{}}V^{\alpha}_{\mathbf{q}}(\tau) &=& \frac{1}{N_{\mathrm{a}}} {\sum}_{j} V^{\alpha}_{j} (\tau)\, \mathrm{e}^{-\mathrm{i} \mathbf{q} \mathbf{R}_{j}},\\ V^{\alpha}_{\mathbf{q}m} &=& T {\int}_{0}^{1/T} V^{\alpha}_{\mathbf{q}}(\tau)\, \mathrm{e}^{\mathrm{i} \omega_{m} \tau}\, \mathrm{d} \tau. \end{array} $$

The free energy F ₁(V) can be written as

$$ F_{1}(V) = - T \ln {\text{Tr}} \exp(-H(V)/T), $$

(22)

where the Hamiltonian has the form

$$H(V) \,=\, H_{0} + 2 {\sum}_{\mathbf{q}m} \left(V^{-}_{\mathbf{q}m} s^{+}_{-\mathbf{q}-m} \,+\, V^{+}_{\mathbf{q}m} s^{-}_{-\mathbf{q}-m} \,+\, V^{z}_{\mathbf{q}m} s^{z}_{-\mathbf{q}-m}\right). $$

The Fourier transformations of the spin operator are given by

$$s^{\alpha}_{\mathbf{q}} = {\sum}_{j} s^{\alpha}_{j}\, \mathrm{e}^{-\mathrm{i} \mathbf{q} \mathbf{R}_{j}}, \qquad s^{\alpha}_{\mathbf{q}m} = T {\int}_{0}^{1/T} s^{\alpha}_{\mathbf{q}}(\tau)\, \mathrm{e}^{\mathrm{i} \omega_{m} \tau}\, \mathrm{d} \tau. $$

Now, we consider the second derivative of the free energy averaged over the fluctuating field:

$$\begin{array}{@{}rcl@{}} &&\left\langle \frac{\partial^{2} F_{1}(V)}{\partial V^{-}_{\mathbf{q}m} \partial V^{+}_{-\mathbf{q}-m}} \right\rangle\\ &&\quad= Q^{-1} \int \frac{\partial^{2} F_{1}(V)}{\partial V^{-}_{\mathbf{q}m} \partial V^{+}_{-\mathbf{q}-m}}\, \mathrm{e}^{-(F_{0}(V)+F_{1}(V) )/T} \mathrm{D} V, \end{array} $$

where Q ⁻¹ is the normalizing factor. Integrating by parts, we come to the equality

$$\begin{array}{@{}rcl@{}} &&\left\langle \frac{\partial^{2} F_{0}(V)}{\partial V^{-}_{\mathbf{q}m} \partial V^{+}_{-\mathbf{q}-m}}\right\rangle - \frac {1}{T} \left\langle \frac{\partial F_{0}(V)}{\partial V^{-}_{\mathbf{q}m}} \frac{\partial F_{0}(V)}{\partial V^{+}_{-\mathbf{q}-m}}\right\rangle \\ &&\quad= \left\langle \frac{\partial^{2} F_{1}(V)}{\partial V^{-}_{\mathbf{q}m} \partial V^{+}_{-\mathbf{q}-m}}\right\rangle - \frac{1}{T}\left\langle \frac{\partial F_{1}(V)}{\partial V^{-}_{\mathbf{q}m}} \frac{\partial F_{1}(V)}{\partial V^{+}_{-\mathbf{q}-m}}\right\rangle, \end{array} $$

(23)

which is independent of a particular form of the functions F ₀(V) and F ₁(V). Substituting the explicit expressions (21) and (22) to (23), we obtain

$$\left\langle s^{-}_{\mathbf{q}m} s^{+}_{-\mathbf{q}-m}\right\rangle = \frac{1}{4} \left(\frac{1}{\tilde{u}^{2}} \left\langle V^{-}_{\mathbf{q}m} V^{+}_{-\mathbf{q}-m} \right\rangle - \frac{T}{\tilde{u}}\right). $$