We study the effect of disorder in systems having a nontrivial Euler class. As these recently proposed multigap topological phases come about by braiding non-Abelian charged band nodes residing between different bands to induce stable pairs within isolated band subspaces, novel properties may be expected. Namely, a modified stability and critical phases under the unbraiding to metals can arise when the disorder preserves the underlying C2T or PT symmetry on average. Employing elaborate numerical computations, we verify the robustness of associated topology by evaluating the changes in the average densities of states and conductivities for different types of disorders. Upon performing a scaling analysis around the corresponding quantum critical points, we retrieve a universality for the localization length exponent of ν=1.4±0.1 for Euler-protected phases, relating to two-dimensional percolation models. We generically find that quenched disorder drives Euler semimetals into critical metallic phases. Finally, we show that magnetic disorder can also induce topological transitions to quantum anomalous Hall plaquettes with local Chern numbers determined by the initial value of the Euler invariant. Published by the American Physical Society 2024

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