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We examine the stability of magnetic order in a classical Heisenberg modelwith quenched random exchange couplings. This system represents the spindegrees of freedom in high-$T_\textrm{c}$ compounds with immobile dopants.Starting from a replica representation of the nonlinear $\sigma$-model, weperform a renormalization-group analysis. The importance of cumulants of thedisorder distribution to arbitrarily high orders necessitates a functionalrenormalization scheme. From the renormalization flow equations we determinethe magnetic correlation length numerically as a function of the impurityconcentration and of temperature. From our analysis follows thattwo-dimensional layers can be magnetically ordered for arbitrarily strong butsufficiently diluted defects. We further consider the dimensional crossover ina stack of weakly coupled layers. The resulting phase diagram is compared withexperimental data for La$_{2-x}$Sr$_x$CuO$_4$.Comment: 12 pages, 5 figure

physical review. b, condensed matter

Bond-disordered spin systems: Theory and application to doped high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>compounds

Bond-disordered spin systems: Theory and application to doped high-Tccompounds