interact much more strongly between themselves than with any other nearby atoms [@Levitov1990DelocalizationInteraction]. Suppose that atoms $i,j$ (with $\mathcal L_{ij}=1$) are identified as the most strongly interacting pair, by a prescription given below. Then, we can re-write $\mathcal M$ as $\mathcal M=\mathcal M_{\text{pair}}+(\mathcal M-\mathcal M_{\text{pair}})$, where the only non-zero elements of $\mathcal M_{\text{pair}}$ involve atoms $i,j$. This effective $2\times 2$ matrix reads [ $$\begin{array}{c}
\mathcal M_{\text{pair}} ={\langle \omega \rangle}_{ij} \mathbb{I}
+
\left(\begin{array}{cc}
\delta\omega_{ij} & - G^{\text{near}}_{ij}\\\\
-  G^{\text{near}}_{ij} & -\delta\omega_{ij}
\end{array}
\right),
\end{array}$$ ]{} where ${\langle \omega \rangle}_{ij}= (\omega_i +\omega_j)/2$ and $\delta\omega_{ij}= (\omega_i -\omega_j)/2$, and where we have included the coherent near-field interaction in $\mathcal M_{\text{pair}}$. The remaining far-field interactions between atoms $i$ and $j$, as well as all the other atoms, are included in $(\mathcal M-\mathcal M_{\text{pair}})$. The large near-field interaction motivates diagonalizing $\mathcal M_{\text{pair}}$ first, while treating $(\mathcal M-\mathcal M_{\text{pair}})$ as a perturbation.

![**Renormalization group analysis.** a) Representative optical response of two identical atoms separated by a distance $\rho_{12} \ll 1$. Here, we plot the absorption spectrum (blue curve), which consists of two well-separated Lorentzians. The positions of the resonances are given by $\sim \pm G_{12}^{\text{near}}$, where $G_{12}^{\text{near}}\propto 1/\rho_{12}^3$ is the near-field component of the Green’s function. To 