Random Coulomb antiferromagnets: from diluted spin liquids to Euclidean random matrices

We study a disordered classical Heisenberg magnet with uniformly antiferromagnetic interactions which are frustrated on account of their long-range Coulomb form, {\em i.e.} $J(r)\sim -A\ln r$ in $d=2$ and $J(r)\sim A/r$ in $d=3$. This arises naturally as the $T\rightarrow 0$ limit of the emergent interactions between vacancy-induced degrees of freedom in a class of diluted Coulomb spin liquids (including the classical Heisenberg antiferromagnets on checkerboard, SCGO and pyrochlore lattices) and presents a novel variant of a disordered long-range spin Hamiltonian. Using detailed analytical and numerical studies we establish that this model exhibits a very broad paramagnetic regime that extends to very large values of $A$ in both $d=2$ and $d=3$. In $d=2$, using the lattice-Green function based finite-size regularization of the Coulomb potential (which corresponds naturally to the underlying low-temperature limit of the emergent interactions between orphan-spins), we only find evidence that freezing into a glassy state occurs in the limit of strong coupling, $A=\infty$, while no such transition seems to exist at all in $d=3$. We also demonstrate the presence and importance of screening for such a magnet. We analyse the spectrum of the Euclidean random matrices describing a Gaussian version of this problem, and identify a corresponding quantum mechanical scattering problem.