RKKY Interaction

The Ruderman–Kittel–Kasuya–Yosida interaction is the exchange coupling between spatially separated local moments generated by the spin response of itinerant electrons. One moment polarizes the electron sea; a second moment samples that polarization. In weak coupling, the pair interaction is second order in the local moment–electron exchange and is determined by the host’s static spin susceptibility.

RKKY exchange is long ranged and oscillatory in an ordinary metal. For a three-dimensional spherical Fermi surface,

$$J_{\mathrm{RKKY}}(R) \sim \frac{ \cos(2k_{\mathrm F}R) }{ R^3 }$$

up to an amplitude and phase fixed by conventions. The oscillation is not decorative: changing moment separation or carrier density can reverse the preferred alignment.

The central result is $J_{ij}=-\mathcal J^2\chi_s(\mathbf R_i-\mathbf R_j)$ when the pair energy is written as $J_{ij}\mathbf S_i\cdot\mathbf S_j$ and the host responds to a source $-\mathbf h\cdot\mathbf s$. The sign of the spatial response, rather than the sign of the local exchange alone, selects parallel or antiparallel alignment. The derivation below fixes the units and signs before connecting the result to ordering and measurements.

Required background. Fermi Surface supplies host geometry, while Susceptibilities supplies the response conventions used to derive the mediated interaction.

Helpful background. Exchange Interactions provides the mechanism comparison, and the Kondo Model Preview supplies the local moment–electron coupling.

Indirect exchange mediated by conduction electrons

Published RKKY formulas differ by signs and factors because they use different exchange Hamiltonians, spin normalizations, Fourier transforms, and response conventions. We use the following definitions throughout.

Dimensionless spins

A local spin $\mathbf S_i$ is a dimensionless angular momentum in units of $\hbar$. The continuum electron spin density has units of inverse volume (inverse $d$-dimensional volume in dimension $d$):

$$\mathbf s(\mathbf r) = \frac12 \psi^\dagger_\alpha(\mathbf r) \boldsymbol\sigma_{\alpha\beta} \psi_\beta(\mathbf r).$$

Thus $\int d^dr\,\mathbf s(\mathbf r)$ is dimensionless, the physical angular-momentum density is $\hbar\mathbf s$, and the electron magnetic-moment density is $-g\mu_{\mathrm B}\mathbf s$. A lattice-site spin operator is dimensionless.

Local exchange

The moment–electron interaction is

$$H_K = \mathcal J \sum_i \mathbf S_i\cdot \mathbf s(\mathbf R_i).$$

For this plus-sign convention, $\mathcal J>0$ is antiferromagnetic. In a continuum point-contact model, $\mathcal J$ has units of energy times volume because $\mathbf s(\mathbf r)$ is a density. A lattice-normalized $J_K$ instead has units of energy.

Response source

Define the electron spin source by

$$H_h = - \int d^dr\, h^a(\mathbf r)s^a(\mathbf r),$$

and the static response by

$$\delta \left\langle s^a(\mathbf r) \right\rangle = \int d^dr'\, \chi_s^{ab}( \mathbf r-\mathbf r' ) h^b(\mathbf r').$$

The source $h$ has energy units. Consequently $\chi_s(\mathbf q)$ has units $1/(\text{energy}\times\text{volume})$, while $\chi_s(\mathbf R)$ has units $1/(\text{energy}\times\text{volume}^2)$. The uniform equilibrium susceptibility is positive; it allows spin populations to equilibrate and is not the finite-frequency response of a strictly conserved total spin.

Pair Hamiltonian

Write the effective pair interaction as

$$H_{\mathrm{pair}} = \sum_{i<j} J_{ij}\, \mathbf S_i\cdot\mathbf S_j.$$

$J_{ij}>0$ favors an antiferromagnetic pair and $J_{ij}<0$ favors a ferromagnetic pair. Sources using $H=-\sum I_{ij}\mathbf S_i\cdot\mathbf S_j$ have $I_{ij}=-J_{ij}$.

Fourier transform

For a translationally invariant continuum,

$$\chi_s^{ab}(\mathbf R) = \int \frac{d^dq}{(2\pi)^d} e^{i\mathbf q\cdot\mathbf R} \chi_s^{ab}(\mathbf q).$$

Conventions for Quantum Matter gives the corresponding cell-normalized lattice transform and basis-position alternatives.

Physical Mechanism

Hold one local moment $\mathbf S_1$ fixed. Its exchange term has the form of an electron-spin source:

$$h_1^a(\mathbf r) = - \mathcal J S_1^a \delta( \mathbf r-\mathbf R_1 ).$$

The induced conduction-electron spin density is therefore

$$\delta \left\langle s^a(\mathbf r) \right\rangle_1 = - \mathcal J \chi_s^{ab}( \mathbf r-\mathbf R_1 ) S_1^b.$$

A second moment at $\mathbf R_2$ couples to this induced polarization:

$$\begin{aligned} \Delta E_{12} &= \mathcal J S_2^a \delta \left\langle s^a(\mathbf R_2) \right\rangle_1 \\ &= - \mathcal J^2 S_2^a \chi_s^{ab}( \mathbf R_2-\mathbf R_1 ) S_1^b. \end{aligned}$$

This argument already gives the tensor structure, sign, and dependence on the host. It also explains why RKKY is not a direct overlap integral between the two localized orbitals.

Second-Order Derivation

For an initially unpolarized host, the electron free-energy change in a weak static source is

$$\Delta F_e[h]={}-\frac12\int d^dr\,d^dr' \times h^a(\mathbf r)\chi_s^{ab}(\mathbf r-\mathbf r') \times h^b(\mathbf r')+O(h^3).$$

All local moments together generate

$$h^a(\mathbf r) = - \mathcal J \sum_i S_i^a \delta( \mathbf r-\mathbf R_i ).$$

Substitution gives

$$\Delta F_e^{(2)} = - \frac{\mathcal J^2}{2} \sum_{i,j} S_i^a \chi_s^{ab}( \mathbf R_i-\mathbf R_j ) S_j^b.$$

If the host already has a moment, a first-order term $\mathcal J\sum_i\mathbf S_i\cdot\langle\mathbf s(\mathbf R_i)\rangle_0$ must also be included. The quadratic term uses the connected response about that state.

The $i=j$ terms are local self-energy corrections. For a rotationally invariant host and fixed spin length they are constants; with spin anisotropy they can instead renormalize single-ion terms. Combining the two ordered cross terms for each unordered pair yields

$$H_{\mathrm{RKKY}} = - \mathcal J^2 \sum_{i<j} S_i^a \chi_s^{ab}( \mathbf R_i-\mathbf R_j ) S_j^b.$$

For a spin-rotation-invariant host,

$$\chi_s^{ab}(\mathbf R) = \delta^{ab}\chi_s(\mathbf R),$$

so

$$J_{\mathrm{RKKY}}(\mathbf R) = - \mathcal J^2 \chi_s(\mathbf R).$$

The factor $1/2$ has disappeared because the pair $(i,j)$ and $(j,i)$ contribute equally. Losing this pair-counting step is a common factor-of-two error.

Perturbative meaning

The same result follows from second-order degenerate perturbation theory or from integrating out electrons in an imaginary-time functional integral. The static susceptibility is

$$\chi_s^{ab}(\mathbf R) = \int_0^\beta d\tau\, \left\langle T_\tau s^a(\mathbf R,\tau) s^b(\mathbf 0,0) \right\rangle_c.$$

Here $\beta=1/(k_{\mathrm B}T)$, imaginary time has inverse-energy units, and $s(\tau)=e^{\tau H_e}s e^{-\tau H_e}$. The connected correlator subtracts the product of equilibrium means. The static approximation also assumes the local moments evolve slowly compared with the electronic response; integrating out the host generally gives a frequency-dependent interaction. Effective Hamiltonians in Many-Body Systems develops the general projection logic and its error controls.

Susceptibility of Band Electrons

For normalized full Bloch states $|n\mathbf k\rangle$, define the one-particle spin vertices $s^a_{\mathbf q}=e^{-i\mathbf q\cdot\mathbf r}\sigma^a/2$. Write their matrix elements as

$$M^a_{nm}(\mathbf k,\mathbf q) =\langle n\mathbf k|s^a_{\mathbf q}|m,\mathbf k+\mathbf q\rangle.$$

Since $(s^b_{\mathbf q})^\dagger=s^b_{-\mathbf q}$, the return vertex is $[M^b_{nm}(\mathbf k,\mathbf q)]^*$. This complex conjugate retains the opposite momentum transfer and the reversed initial and final states. The noninteracting response is

$$\chi_s^{ab}(\mathbf q) = \frac{1}{V} \sum_{\mathbf k} \sum_{n,m} \frac{ f(\epsilon_{n\mathbf k}) - f(\epsilon_{m,\mathbf k+\mathbf q}) }{ \epsilon_{m,\mathbf k+\mathbf q} - \epsilon_{n\mathbf k} } \times M^a_{nm}[M^b_{nm}]^*.$$

Both vertices have the displayed $(\mathbf k,\mathbf q)$ arguments. Degenerate denominators are understood as derivatives or regulated limits. The band and orbital matrix elements matter: two hosts with similar densities of states can have different RKKY tensors and ordering vectors.

Free-electron Lindhard benchmark

For a spin-degenerate parabolic band,

$$\epsilon_{\mathbf k} = \frac{\hbar^2k^2}{2m},$$

the response of one spin component is

$$\chi_s(q) = \frac12 \int \frac{d^3k}{(2\pi)^3} \frac{ f(\epsilon_{\mathbf k}) - f(\epsilon_{\mathbf k+\mathbf q}) }{ \epsilon_{\mathbf k+\mathbf q} - \epsilon_{\mathbf k} }.$$

At zero temperature, let $\rho_0=mk_{\mathrm F}/(2\pi^2\hbar^2)$ be the density of states per spin per volume at the Fermi energy and define

$$y = \frac{q}{2k_{\mathrm F}}.$$

Then

$$\chi_s(q) = \frac{\rho_0}{2} \left[ \frac12 + \frac{1-y^2}{4y} \ln \left| \frac{1+y}{1-y} \right| \right].$$

The bracket tends to $1$ as $q\to0$, so

$$\chi_s(0) = \frac{\rho_0}{2}.$$

The function is continuous at $q=2k_{\mathrm F}$, where its derivative is nonanalytic. This Kohn anomaly, rather than a divergence of $\chi_s$ itself in three dimensions, produces the long-distance $2k_{\mathrm F}$ oscillation.

Three-Dimensional Range Function

For the continuum contact exchange and conventions above, the real-space spin susceptibility at $R>0$ is

$$\chi_s(R) = \frac{m} {32\pi^3\hbar^2R^4} \left[ \sin x - x\cos x \right].$$

Here $x=2k_{\mathrm F}R$.

Therefore

$$J_{\mathrm{RKKY}}(R) = \frac{ \mathcal J^2mk_{\mathrm F}^4 }{ 2\pi^3\hbar^2 } \frac{ x\cos x-\sin x }{ x^4 }.$$

The prefactor assumes a point contact, a continuum field normalized by

$$\{ \psi_\sigma(\mathbf r), \psi_{\sigma'}^\dagger(\mathbf r') \} = \delta_{\sigma\sigma'} \delta(\mathbf r-\mathbf r'),$$

and a per-spin density of states. A lattice exchange, extended impurity orbital, or different spin normalization changes the amplitude and short-distance behavior.

At $k_{\mathrm F}R\gg1$,

$$J_{\mathrm{RKKY}}(R) \sim \frac{ \mathcal J^2mk_{\mathrm F} }{ 16\pi^3\hbar^2 } \frac{ \cos(2k_{\mathrm F}R) }{ R^3 }.$$

The oscillation period is

$$\Delta R_{\mathrm{period}} = \frac{\pi}{k_{\mathrm F}},$$

and successive asymptotic zero crossings are separated by

$$\Delta R_{\mathrm{zero}} = \frac{\pi}{2k_{\mathrm F}}.$$

The formula is exact to this perturbative order for the stated continuum model at $R>0$, but its small-$R$ divergence is not a reliable atomic-scale material prediction. Finite orbital extent, lattice-scale band structure, direct exchange, and superexchange then matter.

Top: the three-dimensional response is largest at zero momentum and has a logarithmic slope singularity at $q=2k_{\mathrm F}$. Bottom: multiplying $J(R)$ by $x^3/A$, where $x=2k_{\mathrm F}R$ and $A=\mathcal J^2mk_{\mathrm F}^4/(2\pi^3\hbar^2)$, gives $\cos x-\sin x/x$. The dashed cosine is its large-distance limit. The rescaling exposes the oscillations without confusing their envelope with $R^{-1}$ decay.

Fermi-Surface Origin

The asymptotic interaction is controlled by low-energy particle–hole excitations connecting points on the Fermi surface. For a smooth, convex Fermi surface, stationary-phase points have group velocities parallel or antiparallel to the separation $\mathbf R$. The phase is $(\mathbf k_+-\mathbf k_-)\cdot\mathbf R$, where the two stationary points have normals parallel and antiparallel to $\mathbf R$. Their spanning vector need not itself be parallel to $\mathbf R$ on an anisotropic surface; its projection along the separation fixes the oscillation period.

For a noninteracting isotropic metal with a smooth ordinary Fermi surface in spatial dimension $d$,

$$J(R) \sim \frac{ \cos( 2k_{\mathrm F}R+\phi_d ) }{ R^d },$$

where the phase $\phi_d$ depends on convention and dimension.

Host	Generic long-distance behavior	Important qualification
One-dimensional metal	$R^{-1}$ oscillation	especially sensitive to interactions and boundaries
Two-dimensional parabolic metal	$R^{-2}$ oscillation	exact form uses Bessel functions
Three-dimensional parabolic metal	$R^{-3}$ oscillation	exact spherical range function given above
Nonspherical Fermi surface	direction-dependent periods and amplitudes	caliper vectors replace a single $2k_{\mathrm F}$
Nested or locally flat surface	enhanced response and potentially slower decay	curvature and matrix elements decide the power
Undoped Dirac system	nonmetallic power law	sublattice, valley, and symmetry structure can fix signs

The familiar rule “decay as $R^{-d}$” is not universal. It assumes an ordinary Fermi surface with nonzero local curvature and a pointlike coupling. Roth, Zeiger, and Kaplan showed that cylindrical or nearly parallel Fermi-surface regions can alter the decay and strengthen directionality.

Momentum-space ordering tendency

For local moments on a lattice, write

$$H_{\mathrm{RKKY}} = \frac{1}{2N} \sum_{\mathbf q} J(\mathbf q) \mathbf S_{\mathbf q} \cdot \mathbf S_{-\mathbf q}.$$

Use the cell-normalized lattice susceptibility $\chi_{\mathrm{lat}}(\mathbf q)$ and $\mathbf S_{\mathbf q}=\sum_j e^{-i\mathbf q\cdot\mathbf R_j}\mathbf S_j$. Removing the local self term gives the pair-only transform

$$\begin{aligned} J(\mathbf q)&=-J_K^2[\chi_{\mathrm{lat}}(\mathbf q)-\chi_{\mathrm{loc}}],\\ \chi_{\mathrm{loc}}&=\frac1N\sum_{\mathbf q}\chi_{\mathrm{lat}}(\mathbf q). \end{aligned}$$

This ensures $J_{ii}=0$. The subtraction shifts every scalar eigenvalue by the same constant, so it leaves the minimizing wavevector unchanged for fixed-length spins. It matters when calculating the pair sum or a mean-field transition temperature: a spin does not act on itself through an ordinary pair molecular field.

For a one-site Bravais lattice with isotropic exchange, a classical spiral at a minimizing $\mathbf Q$ can satisfy every local length constraint. With several inequivalent sites or anisotropic couplings, minimizing the Fourier matrix gives a quadratic tendency or relaxed-constraint lower bound; its eigenvector must also be checked against the fixed length of each spin. Multiple wavevectors may be needed.

The spherical three-dimensional gas has its maximum continuum susceptibility at $q=0$, although the far-field oscillation comes from the nonanalyticity at $2k_{\mathrm F}$. A lattice of moments samples and sums reciprocal-space copies of the continuum response, with orbital form factors; one should not identify its ordering wavevector with $2k_{\mathrm F}$ automatically.

Material Corrections

Several bands and orbitals

In a crystal, $\chi_s^{ab}$ can carry band, orbital, sublattice, and basis-position indices. Intraband terms emphasize the Fermi surface; interband terms can remain important near avoided crossings or in spin–orbit-coupled systems. The exchange between moments on orbitals $\alpha$ and $\beta$ is more accurately written

$$J_{\alpha\beta}^{ab}(\mathbf R) = - J_{K,\alpha} \chi_{\alpha\beta}^{ab}(\mathbf R) J_{K,\beta}.$$

Orbital form factors can suppress a geometrically available spanning vector. A density-of-states-only estimate cannot predict this.

Spin–orbit coupling

Without spin-rotation symmetry, the response is a tensor and the pair Hamiltonian can be decomposed as

$$H_{ij} = J_H \mathbf S_i\cdot\mathbf S_j + \mathbf D_{ij}\cdot ( \mathbf S_i\times\mathbf S_j ) + S_i^a \Gamma_{ij}^{ab} S_j^b.$$

The antisymmetric susceptibility produces the Dzyaloshinskii–Moriya term, while its symmetric traceless part produces anisotropic exchange. Inversion and other bond symmetries can force some components to vanish. Exchange Interactions derives the tensor decomposition and bond-symmetry constraints.

Finite temperature

Thermal smearing rounds the Kohn anomaly. In the clean three-dimensional far field, at $k_{\mathrm B}T\ll E_{\mathrm F}$, linearizing $k(E)$ about the Fermi energy gives

$$J(R,T)\simeq J_{\mathrm{asym}}(R,0)\,\frac{z}{\sinh z},$$ $$z=\frac{2\pi R k_{\mathrm B}T}{\hbar v_{\mathrm F}}.$$

The factor follows by averaging the oscillatory phase over the Fermi edge:

$$\int_{-\infty}^{\infty}d\epsilon\, \frac{e^{2i\epsilon R/(\hbar v_{\mathrm F})}} {4k_{\mathrm B}T\cosh^2[\epsilon/(2k_{\mathrm B}T)]} =\frac{z}{\sinh z}.$$

The asymptotic phase and slowly varying prefactor are evaluated at the chemical potential; replacing it by $E_{\mathrm F}$ is the leading low-temperature approximation. For $z\ll1$, the reduction is $1-z^2/6+\cdots$; for $z\gg1$, it is $2ze^{-z}$. An order-of-magnitude length $\hbar v_{\mathrm F}/(k_{\mathrm B}T)$ is useful, but omitting $2\pi$ can seriously underestimate damping in a numerical example. This formula is a clean-host far-field result, not a universal cutoff for diffusive or interacting hosts.

Elastic disorder

In a weakly disordered metal, the disorder-averaged RKKY interaction is commonly damped on the mean-free-path scale:

$$\overline{ J(R) } \sim J_{\mathrm{clean}}(R) e^{-R/\ell}.$$

Sample-specific fluctuations need not vanish on exactly the same scale. Random impurity positions and oscillatory signs can therefore produce a broad distribution of couplings even when the spatial average is small.

Electron interactions

The most general weak local-moment coupling samples the interacting host susceptibility. Fermi-liquid parameters, paramagnons, vertex corrections, and collective modes can enhance or redirect the exchange:

$$J_{\mathrm{eff}}(\mathbf q) = - J_K^2 \chi_s^{\mathrm{host}}(\mathbf q).$$

Random Phase Approximation gives one controlled resummation only when its assumptions apply. Near an instability, truncating the moment–electron coupling at second order can fail even if an enhanced susceptibility looks tempting.

Gaps and nonmetallic hosts

A band gap, superconducting gap, pseudogap, or finite-size level spacing changes the long-distance response. For a fully gapped short-range host with no relevant gapless modes, the static spin response is generally exponentially short ranged. Nodes in a superconducting gap, boundary states or other gapless modes can retain longer-ranged terms. Interband exchange in insulators is often classified as Bloembergen–Rowland exchange or another virtual-excitation mechanism rather than metallic RKKY. Naming the mediator and its low-energy spectrum is more useful than stretching the acronym.

From Pair Couplings to Magnetic Order

RKKY exchange can organize a lattice of local moments into ferro-, antiferro-, helical, or incommensurate order. The ordering vector is not obtained by inspecting one pair. The full Fourier-space pair matrix identifies the leading quadratic tendency, subject to the local spin-length constraints described above.

For a scalar Bravais-lattice model in the present sign convention, the Curie–Weiss temperature is

$$\Theta_{\mathrm{CW}} = - \frac{ S(S+1) }{ 3k_{\mathrm B} } \sum_j J_{ij}.$$

A negative $\Theta_{\mathrm{CW}}$ indicates a positive net exchange sum, but it need not equal the ordering temperature or reveal the ordering wavevector. Frustration, low dimension, anisotropy, disorder, and fluctuations can strongly separate those scales.

Mean-field ordering occurs schematically when

$$k_{\mathrm B}T_{\mathrm{MF}} = - \frac{ S(S+1) }{ 3 } \lambda_{\min},$$

For equivalent isotropic spins, $\lambda_{\min}<0$ is the minimum eigenvalue of the pair-only exchange matrix, with $J_{ii}=0$. Inequivalent spins and crystal-field anisotropy require the local susceptibility matrix instead of one common $S(S+1)/3$. This estimate ignores critical fluctuations and does not establish the true transition temperature in low-dimensional or frustrated systems.

Dilute random moments

If local moments occupy random positions, the factor $2k_{\mathrm F}R_{ij}$ varies from pair to pair. The resulting mixed ferro- and antiferromagnetic bonds can generate frustration and glassy freezing. Oscillatory RKKY exchange is an important ingredient in canonical metallic spin glasses such as dilute CuMn or AuFe, but random signs alone do not prove a thermodynamic spin-glass phase. Aging, memory, nonlinear susceptibility, and frequency-dependent freezing must be tested.

Rare-earth metals and intermetallics

Localized $4f$ moments can couple through more extended $s$, $p$, and $d$ bands. Fermi-surface geometry then helps select long-period helices or commensurate order. Crystal-field anisotropy and several exchange channels must be included before comparing with neutron diffraction or spin-wave spectra.

Metallic multilayers and surfaces

Magnetic layers separated by a nonmagnetic metal can exhibit oscillatory interlayer exchange as spacer thickness changes. Surface-state-mediated interactions can be strongly directional and can be mapped by engineered adatom structures. These settings share the susceptibility and Fermi-surface logic of RKKY, although interfaces, quantum-well states, extended contacts, and multiple reflections modify the point-impurity range function.

Competition with Kondo Screening

The same antiferromagnetic local exchange can generate two qualitatively different tendencies:

• second-order RKKY coupling correlates distinct local moments;
• repeated local spin flips generate Kondo screening.

The schematic RKKY scale is algebraic,

$$k_{\mathrm B}T_{\mathrm{RKKY}} \sim C J_K^2 \rho_0,$$

whereas the single-impurity Kondo scale is nonanalytic in weak coupling. Kondo Lattices compares the two scales; Heavy Fermions discusses how coherence appears in measured material properties.

RKKY perturbation theory is most credible when the running local exchange remains weak at the energy and distance scales being probed. Once moments are substantially screened, treating a bare $J_K^2\chi_0$ pair coupling and independent single-impurity Kondo effects as additive is uncontrolled.

Experimental Inference

Pair spectroscopy

For two spin-$1/2$ moments with

$$H = J_{12} \mathbf S_1\cdot\mathbf S_2,$$

the singlet and triplet energies are

$$E_s = - \frac{3J_{12}}{4}, \qquad E_t = \frac{J_{12}}{4}.$$

Thus

$$E_t-E_s = J_{12}.$$

Inelastic tunnelling, neutron spectroscopy, or field-dependent level crossings can determine the sign and magnitude, provided single-ion anisotropy and other pair interactions are included.

Carrier-density tuning

Gate voltage, chemical doping, or pressure can change Fermi-surface dimensions. A sign reversal correlated with a change in a spanning vector is strong evidence for carrier-mediated oscillatory exchange. The comparison must use the measured or calculated multiband Fermi surface, not merely a spherical $k_{\mathrm F}$ inferred from total density.

Ordering and diffuse scattering

Neutron or resonant x-ray scattering measures the ordering wavevector and diffuse correlations above the transition. A successful RKKY model should reproduce:

1. the location and symmetry of response maxima;
2. the ordered structure;
3. spin-wave energies and intensities;
4. carrier-density and pressure trends;
5. the influence of disorder and crystal-field anisotropy.

Agreement with only a Curie–Weiss temperature is underdetermined.

Exercises

1. Sign from the induced polarization

In the convention of this page, suppose $\chi_s(R)>0$ for two moments separated by $R$. What is the sign of $J_{\mathrm{RKKY}}(R)$, and which pair alignment is favored?

Solution

The pair coupling is

$$J_{\mathrm{RKKY}}(R) = - \mathcal J^2 \chi_s(R).$$

Since $\mathcal J^2>0$ and $\chi_s(R)>0$,

$$J_{\mathrm{RKKY}}(R) < 0.$$

With $H=J_{\mathrm{RKKY}}\mathbf S_1\cdot\mathbf S_2$, a negative coefficient favors parallel, ferromagnetic alignment. If a source uses $H=-I\mathbf S_1\cdot\mathbf S_2$, it will quote $I>0$ for the same physical result.

2. Far-field limit of the 3D range function

Starting from

$$J(R) = A \frac{ x\cos x-\sin x }{ x^4 }, \qquad x=2k_{\mathrm F}R,$$

find the leading large-$R$ behavior.

Solution

For $x\gg1$, the term $x\cos x$ is one power of $x$ larger than $\sin x$. Hence

$$\frac{ x\cos x-\sin x }{ x^4 } = \frac{\cos x}{x^3} + O(x^{-4}).$$

Since $x=2k_{\mathrm F}R$,

$$J(R) \sim \frac{A} {(2k_{\mathrm F})^3} \frac{ \cos(2k_{\mathrm F}R) }{ R^3 }.$$

The subleading sine term shifts finite-distance zero crossings but not the asymptotic decay power.

3. Ordering wavevector from a susceptibility matrix

A material has two candidate wavevectors. The largest eigenvalue of its static spin susceptibility is $2.1\,\mathrm{eV^{-1}}$ at $\mathbf q=0$ and $3.4\,\mathrm{eV^{-1}}$ at $\mathbf Q$. Assume equivalent local spins, the same scalar $J_K$ at every site, and a site-independent isotropic on-site subtraction. Which wavevector has the stronger quadratic tendency?

Solution

The corresponding exchange eigenvalues are

$$\lambda_J(\mathbf q) = - J_K^2 [\lambda_\chi(\mathbf q)-\chi_{\mathrm{loc}}].$$

The larger susceptibility eigenvalue gives the more negative exchange eigenvalue. Therefore

$$\lambda_J(\mathbf Q) < \lambda_J(\mathbf 0),$$

and the leading quadratic tendency is at $\mathbf Q$. The eigenvector at $\mathbf Q$, not the eigenvalue alone, determines the orbital, sublattice, and spin pattern. The mode must also satisfy local spin-length constraints; nonlinear terms and fluctuations decide the final phase and transition order.

4. Thermal and disorder cutoffs

Take $v_{\mathrm F}=1.0\times10^6\,\mathrm{m\,s^{-1}}$, $T=50\,\mathrm K$, mean free path $\ell=20\,\mathrm{nm}$, and moment separation $R=100\,\mathrm{nm}$. Estimate the clean-host factor $z/\sinh z$ and, separately, the disorder-average factor $e^{-R/\ell}$. Assume the low-temperature far-field approximations apply; do not infer a full disordered finite-temperature result by simply multiplying these estimates.

Solution

The thermal length is

$$\begin{aligned} \ell_T &= \frac{ (1.055\times10^{-34}) (1.0\times10^6) }{ (1.381\times10^{-23}) (50) } \mathrm m \\ &\simeq 1.53\times10^{-7}\,\mathrm m \\ &\simeq 153\,\mathrm{nm}. \end{aligned}$$

Here $z=2\pi R/\ell_T\simeq4.11$, so $z/\sinh z\simeq0.135$. The clean-host oscillation amplitude is already reduced by about a factor of 7.4; using $R/\ell_T$ alone would miss this.

The disorder factor is

$$e^{-R/\ell} = e^{-5} \simeq 6.74\times10^{-3}.$$

The separate disorder-average estimate gives the stronger reduction in this example. A particular sample can still have fluctuating couplings not represented by the simple average.

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