Vacancy-induced spin textures and their interactions in a classical spin liquid

Motivated by experiments on the archetypal frustrated magnet SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ (SCGO), we study the classical Heisenberg model on the pyrochlore slab (Kagom\'e bilayer) lattice with site-dilution $x=1-p$. This allows us to address generic aspects of the physics of non-magnetic vacancies in a classical spin liquid. We explicitly demonstrate that the pure ($x=0$) system remains a spin-liquid down to the lowest temperatures, with an unusual {\em non-monotonic} temperature dependence of the susceptibility, which even turns diamagnetic for the apical spins between the two kagome layers. For $x> 0$ but small, the low temperature magnetic response of the system is most naturally described in terms of the properties of spatially extended spin textures that cloak an "orphan" $S=3/2$ Cr$^{3+}$ spin in direct proximity to a pair of missing sites belonging to the same triangular simplex. In the $T \rightarrow 0$ limit, these orphan-texture complexes each carry a net magnetization that is exactly half the magnetic moment of an individual spin of the undiluted system. Furthermore, we demonstrate that they interact via an entropic {\em temperature dependent} {\em pair-wise exchange interaction} $J_{eff}(T,\vec{r}) \sim T {\mathcal J} (\vec{r} \sqrt{T})$ that has a logarithmic form at short-distances and decays exponentially beyond a thermal correlation length $\xi(T) \sim 1/\sqrt{T}$. The sign of $J_{eff}$ depends on whether the two orphan spins belong to the same Kagome layer or not. We provide a detailed analytical account of these properties using an effective field theory approach specifically tailored for the problem at hand. These results are in quantitative agreement with large-scale Monte Carlo numerics.