RKKY approximation regime of validity The RKKY effective action applies 2nd order perturbation theory to the interaction of magnetic moments $\vec I_n$ with conduction electrons
$$H_{eS}= \frac12 \frac{J}{V} \sum_{k,k'} \vec I_n e^{i(k-k')r_n}c_{k\alpha}^\dagger \vec \sigma_{\alpha\beta} c_{k'\beta}$$
where $H_0 = \sum \epsilon_k c_{k\alpha}^\dagger c_{k\alpha}$ is the electron Hamiltonian. The result in 3d is
$$H_{RKKY} = \frac{4m^* J^2k_F^4}{(2\pi)^3}\sum_{n} \vec I_n \cdot \vec I_m F(2k_F r_{nm}),\qquad F(x) = \frac{x\cos x-\sin x}{x^4}$$
and it is often said to hold "for small $J$." But $J$ has dimension of energy times volume, so what does this mean? What is the dimensionless quantity that must be small for the RKKY approximation to be valid? Importantly, does it depend on the volume $V$ or lattice spacing $a$ (i.e. if we call the dimensionless quantity $d$, then is the regime $d<< 1$ or $d/N <<1$, where $N=V/a^3$?)?
 A: RKKY interaction is common in metals with rare-earth elements. There are two points for those system:

*

*metals: there exists itinerant electrons, e.g. s/p-orbitals, which can be described by tight-binding model:
$$H_0=-t\sum_{i,j}c_i^\dagger c_j+h.c.$$  diagonalization will gives the same $H_0$ as you said in the question. The physics picture of such term is itinerant electrons runs in the system with the time scale:
$$T_t\sim \frac{1}{t} \sim \frac{1}{W}$$
where $W$ is the band width of $H_0$.


*rare-earth elements: there exists local moment due to the flat band from f-orbitals. However, it is important to note that the behavior of "local moment" here is different from that in Mott insulator. The local moments in Mott insulator obtian the magnetic behaviors by "super-exchange interaction", which gives $H_{Mott}=\sum_{i,j}J_{i,j} S_i\cdot S_j$. On the other hand, the magnetic behaviors of local moment in this system(RKKY regime) coms from the following picture: itinerant electron which interacts with local moment at $r_i$ can carry some information of spin for local moment at $r_i$, then, this electron run forward to $r_j$, and interacts with with local moment at $r_j$. During such process, two local moments in $r_i$ and $r_j$ will talk with each other via the itinerant electrons. And how about the oscillation behavior in RKKY interaction?  This can be understood as itinearnt electron has not loss all the information of local moment at $r_i$ after interacting with local moment at $r_j$, and may deliver it to other local moments. The time scale of interaction between itinerant electron and local monument at $r_i$ or $r_j$ (you can understand as time scale of conveying information between them) is the inverse of coefficient of RKKY interaction as you said in the question:$$T_J \sim \frac{1}{m^* J^2 k_F^2}$$
Now, we obtain two time scales, we consider two limit:

*

*If $T_J \gg T_t$, this means itinerant electrons runs so slowly, and conveying information with local moments is so fast. Also, local moments are often much "heavier" than itinerant electrons, i.e. "heavier" here means the magnitude of spin $S$, local moments often have large $S$, which resulting in longer time of "spin flip". Thus, when itinerant electrons have enough time to exchange information with local moment at $r_i$, itinerant electron will prefer to follow the direction of local moment at $r_i$. And when it run to $r_j$, it will also follow the direction of local moment at $r_j$  (loosely speaking, itinerant electrons may loss the function of conveying information between two local moments).


*If $T_J \ll T_t$, this means itinerant electrons runs so fast, and conveying information with local moments is so slow. If so, itinerant electrons is possible to only gives part spin information to local moment, and carry the remaining spin information to next local moment. This is actually the picture of RKKY interaction. And you can you such dimensionless quantity
$$d=\frac{T_J}{T_t} \ll 1$$
And the relation of $a$ with $d$ is easy to obtain via analyze the relation between $T_J$ or $T_t$ with $a$.
