Bond-disordered spin systems: Theory and application to doped high-$T_c$ compounds

We examine the stability of magnetic order in a classical Heisenberg model with quenched random exchange couplings. This system represents the spin degrees of freedom in high-$T_c$ compounds with immobile dopants. Starting from a replica representation of the nonlinear-$\sigma$ model, we perform a renormalization-group analysis. The importance of cumulants of the disorder distribution to arbitrarily high orders necessitates a functional renormalization scheme. From the renormalization flow equations we determine the magnetic correlation length numerically as a function of the impurity concentration and temperature. From our analysis it follows that two-dimensional layers can be magnetically ordered for arbitrarily strong but sufficiently diluted defects. We further consider the dimensional crossover in a stack of weakly coupled layers. The resulting phase diagram is compared with experimental data for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4.$