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The gapless Bogoliubov-de Gennes (BdG) quasiparticles of a clean threedimensional spinless $p_x+ip_y$ superconductor provide an intriguing example ofa thermal Hall semimetal (ThSM) phase of Majorana-Weyl fermions in class D ofthe Altland-Zirnbauer symmetry classification; such a phase can support a largeanomalous thermal Hall conductivity and protected surface Majorana-Fermi arcsat zero energy. We study the effect of quenched disorder on such a topologicalphase with both numerical and analytical methods. Using the kernel polynomialmethod, we compute the average and typical density of states for the BdGquasiparticles; based on this, we construct the disordered phase diagram. Weshow for infinitesimal disorder, the ThSM is converted into a diffusive thermalHall metal (ThDM) due to rare statistical fluctuations. Consequently, the phasediagram of the disordered model only consists of ThDM and thermal insulatingphases. Nonetheless, there is a cross-over at finite energies from a ThSMregime to a ThDM regime, and we establish the scaling properties of the avoidedquantum critical point which marks this cross-over. Additionally, we show theexistence of two types of thermal insulators: (i) a trivial thermal bandinsulator (ThBI) [or BEC phase] supporting only exponentially localizedLifshitz states (at low energy), and (ii) a thermal Anderson insulator (AI) atlarge disorder strengths. We determine the nature of the two distinctlocalization transitions between these two types of insulators and ThDM.We alsodiscuss the experimental relevance of our results for three dimensional, timereversal symmetry breaking, triplet superconducting states.

Quantum phases of disordered three-dimensional Majorana-Weyl fermions