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In commonly employed models for 2D topological insulators, bulk gaplessstates are well known to form at the band inversion points where the degeneracyof the states is protected by symmetries. It is thus sometimes quite temptingto consider this feature, the occurrence of gapless states, a result of theband inversion process under protection of the symmetries. Similarly, the bandinversion process might even be perceived as necessary to induce 2D topologicalphase transitions. To clarify these misleading perspectives, we propose asimple model with a flexible Chern number to demonstrate that the bulk gaplessstates emerge at the phase boundary of topological phase transitions, despitethe absence of band inversion process. Furthermore, the bulk gapless states donot need to occur at the special $k$-points protected by symmetries. Given thesignificance of these fundamental \textit{conceptual} issues and theirwide-spread influence, our clarification should generate strong generalinterests and significant impacts. Furthermore, the simplicity and flexibilityof our general model with an arbitrary Chern number should prove useful in awide range of future studies of topological states of matter.

Non-necessity of band inversion process in 2D topological insulators for bulk gapless states and topological phase transitions