Disorder inevitably exists in realistic samples, manifesting itself in various exotic properties for the topological states. In this paper, we summarize and briefly review work completed over the last few years, including our own, regarding recent developments in several topics about disorder effects in topological states. For weak disorder, the robustness of topological states is demonstrated, especially for both quantum spin Hall states with Z2 = 1 and size induced nontrivial topological insulators with Z2 = 0. For moderate disorder, by increasing the randomness of both the impurity distribution and the impurity induced potential, the topological insulator states can be created from normal metallic or insulating states. These phenomena and their mechanisms are summarized. For strong disorder, the disorder causes a metal-insulator transition. Due to their topological nature, the phase diagrams are much richer in topological state systems. Finally, the trends in these areas of disorder research are discussed. PACS numbers: 73.61.-r, 71.23.-b, 73.43.-f I. INTRODUCTION Topological states, including gapped topological insulators and gapless topological semimetals, have become a focus of the condensed matter research 1-8 . In condensed matter systems, the first well-known insulating topological state is quantum Hall effect (QHE) under a strong magnetic field 9 . Subsequently, the quantum anomalous Hall effect (QAHE), a topological state similar as the QHE but without magnetic field, was proposed in 1988 10 and observed in 2013 11 . In 2005, Kane et al made a great step in the topological states research. They proposed the two-dimensional Z 2 topological insulator -quantum spin Hall effect (QSHE) in graphene, which extends the topological state into the class of systems protected by time reversal symmetry 12,13 . Soon afterwards, the concept of symmetry protected topological states is broaden into three-dimension 14,15 and other discrete symmetries 7,16 . Besides the insulating systems, the topological states can also exist in gapless systems 17,18 . More recently, topological semimetals, containing both Weyl semimetals and Dirac semimetals, were experimentally verified 19,20 , soon after they were predicted in the corresponding materials 21,22 . The topological states are different from normal metallic and insulating states because of the existence of nontrivial topological order, which originates from the global properties of all electrons below the Fermi energy 23 and can be characterized by various types of topological invariant numbers 2,13,23 . Due to the nontrivial topological order, corresponding gapless states emerge on the surface, leading to numerous exotic properties in topological systems. These properties have been reviewed in references 1-8 .Experimentally, disorder is ubiquitous because of the defects in manufacturing processes and usually plays a dominant role in the transport properties of the samples being studied. Due to their unique electric structures, the response of topological states to disord...
