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We address the coexistence of massless and massive topological edge states at the interface between two materials with different topological phases. We modify the well-known Bernevig-Hughes-Zhang model to introduce a smooth function describing the band inversion and the band bending due to electrostatic effects between the bulk of the quantum well and the vacuum. Within this minimal model we identify distinct parameter sets that can lead to the coexistence of the two types of edge states and that determine their number and characteristics. We propose several experimental setups that could demonstrate their presence in two-dimensional topological systems, as well as ways to regulate or tune the contribution of the massive edge states to the conductance of associated electronic devices. Our results suggest that such states may also be present in novel two-dimensional van der Waals topological materials. ; The work of T.L.v.d.B. and D.B. was supported by the Spanish Ministerio de Economía y Competitividad (MINECO) through Project No. FIS2014-55987-P, by Spanish Ministerio de Ciecia, Innovation y Universidades (MICINN) through Project No. FIS2017-82804-P, and by the Transnational Common Laboratory Quantum-ChemPhys. M.R.C. acknowledges funding from the Spanish Government through Project No. MAT2017-88377-C2-2-R (AEI/FEDER UE) and the Generalitat Valenciana through Grant No. Cidegent2018004.

In topological systems, a modulation in the gap onset near interfaces can lead to the appearance of massive edge states, as were first described by Volkov and Pankratov. In this work, we study graphene nanoribbons in the presence of intrinsic spin-orbit coupling smoothly modulated near the system edges. We show that this space modulation leads to the appearance of Volkov-Pankratov states, in addition to the topologically protected ones. We obtain this result by means of two complementary methods, one based on the effective low-energy Dirac equation description and the other on a fully numerical tight-binding approach, finding excellent agreement between the two. We then show how transport measurements might reveal the presence of Volkov-Pankratov states, and discuss possible graphenelike structures in which such states might be observed. ; The work of T.B. and D.B. was supported by the Spanish Ministerio de Ciencia, Innovation y Universidades (MICINN) through Project No. FIS2017-82804-P, and by the Transnational Common Laboratory Quantum-ChemPhys. M.R.C. acknowledges funding from the Spanish Government through Project No. MAT2017-88377-C2-2-R (AEI/FEDER) and the Generalitat Valenciana through Grant No. Cidegent2018004.