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We study disorder induced topological phase transitions in magnetically doped(Bi, Sb)$_2$Te$_3$ thin films, by using large scale transport simulations ofthe conductance through a disordered region coupled to reservoirs in thequantum spin Hall regime. Besides the disorder strength, the rich phase diagramalso strongly depends on the magnetic exchange field, the Fermi level, and theinitial topological state in the undoped and clean limit of the films. In aninitially trivial system at non-zero exchange field, varying the disorderstrength can induce a sequence of transitions from a normal insulating, to aquantum anomalous Hall, then a spin-Chern insulating, and finally an Andersoninsulating state. While for a system with topology initially, a similarsequence, but only starting from the quantum anomalous Hall state, can beinduced. Varying the Fermi level we find a similarly rich phase diagram,including transitions from the quantum anomalous Hall to the spin-Cherninsulating state via a state that behaves as a mixture of a quantum anomalousHall and a metallic state, akin to recent experimental reports.

Topological Phase Transitions Induced by Disorder in Magnetically Doped (Bi, Sb)$_2$Te$_3$ Thin Films