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In this supplementary material, we investigate further the impurity-inducedfreezing mechanism in a doped system of 3D weakly coupled ladders resemblingBi(Cu$_{1-x}$Zn$_x$)$_2$ZnPO$_6$ using large scale Quantum Monte Carlosimulations.

Supplementary material for "Impurity-induced magnetic order in low dimensional spin gapped materials"