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\title{RKKY interaction by Vadim Cheianov}
\begin{document}
\maketitle
The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is the electron-mediated long-distance exchange coupling between localized magnetic moments in metals. It was first proposed by M.A. Ruderman and C. Kittel \cite{1} as a mechanism of the unexpectedly strong broadening of the nuclear spin resonance in tin, thallium and silver. Later on, in the works by T. Kasuya \cite{2} and K. Yosida \cite{3} it was applied to the localized spins of atoms with a partially filled inner d-shell.
\textbf{The model.} The RKKY interaction is mediated by the electrons partially occupying the conduction band with a single-particle dispersion relation
$E(k)$.
The conduction electrons experience a local spin-conserving interaction with localized spins $\vec S_1, \dots, \vec S_N$
positioned at points $R_1,\dots, R_N$ inside the metal. The
Hamiltonian of the system is
$$
H=
\sum_{\sigma=\uparrow\downarrow}\sum_\mathbf k
E(\mathbf k) c^\dagger_{s \mathbf k} c_{s \mathbf k} +
\sum_{i=1}^N \sum_{s, s'=\uparrow\downarrow}
\sum_{\mathbf k, \mathbf k'} e^{i \mathbf R_i\cdot (\mathbf k' - \mathbf k)}
\Delta_{\mathbf k \mathbf k'} \vec S_i \cdot
( c_{s\mathbf k}^\dagger \vec{\sigma_{ss'}} c_{s'\mathbf k'} )
$$
where $ c^\dagger_{s\mathbf k} (c_{s\mathbf k}) $
is the creation (annihilation) operator of an electron
with spin $s$ and the Bloch momentum $\mathbf k$,
$\vec \sigma$ is the vector whose
components are the three Pauli matrices, and
$\Delta_{\mathbf k\mathbf k'}$ is the system-specific
form-factor defining the interaction between the
localized spin and the conduction
electrons.
The long distance RKKY interaction is dominated by the
conduction electrons close to the \cref{Fermi surface}
and therefore only depends on $\Delta=\Delta_{k_F k_F}$.
Regarding $\Delta$
as a small parameter, one can find the interaction
between localized spin using the second-order perturbation theory. In a
three-dimensional system the perturbative expression for the
interaction between two localized spins is given by
$$
H_{ij}=\frac{\vec S_i\cdot \vec S_j}{4}
\frac{|\Delta|^2 m_*}{(2\pi)^3 R_{ij}^4 \hbar^2}\left[2k_F R_{ij}
\cos (2k_F R_{ij}) - \sin(2k_F R_{ij}) \right].
$$
In this expression $k_F$
is the Fermi momentum of conduction electrons and it is assumed
that with a reasonable accuracy
$E(\mathbf k)= \hbar^2 k^2/2m_*$
for
$k \le k_F$
(for the discussion of systems with non-spherical Fermi surfaces see \cite{4}).
A peculiar property of the RKKY exchange, is that it experiences
oscillations with the period
$\pi/k_F$ as a function of
distance between localized spins.
These are known as \cref{Friedel oscillations} and play
an important role in applications such as the
\cref{giant magnetoresistance} (GMR) and \cref[spin glasses]{spin glass}.
\begin{thebibliography}{1}
\bibitem{1} M.A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).
\bibitem{2} T. Kasuya, Prog. Theor. Phys. 16, 45 (1956).
\bibitem{3} K. Yosida, Phys. Rev. 106, 893 (1957).
\bibitem{4} D. I. Golosov and M. I. Kaganov, J. Phys.: Condens. Matter 5, 1481-1492 (1993).
\end{thebibliography}
\end{document}