Motivated by the experimental findings from the Mesoscopic-Physics group at LPA-ENS , we study theoretically smooth topological interfaces, i.e. interfaces between a topological and a normal insulator. In addition to the usual topologically protected chiral surface states, which do not depend on the specific form of the interface, several massive states appear if the interface width is larger than a particular intrinsic length (given by the bulk gap and the Fermi velocity). These states, first described by Volkov and Pankratov (VP) in the 1990ies , are intrinsically relativistic and can be related to Landau bands of relativistic fermions. We show that the gap variation can be interpreted precisely as a vector potential that is affected by an additional electric field in a relativistic manner . The electric field can thus be used not only to dope electronically these massive surface states, but they become even more accessible due to the reduction of the Landau gap in the presence of an electric field. The effect is at the origin of an oscillating resistance measured as a function of the electric field in high-frequency experiments at LPA-ENS . We finish with a short discussion of how this "Landau-level approach" can also be used in the framework of Weyl semimetals and the description of Fermi arcs that play the role of chiral Landau bands here .
 "Observation of Volkov-Pankratov states in topological HgTe heterojunctions
using high-frequency compressibility", A. Inhofer, S. Tchoumakov, B.A. Assaf, G. Fève, J.M. Berroir, V. Jouffrey, D. Carpentier, M. O. Goerbig, B. Plaçais, K. Bendias, D.M. Mahler, E. Bocquillon, R. Schlereth, C. Brüne, H. Buhmann, and L.W. Molenkamp, Phys. Rev. B 96, 195104 (2017).
 "Two-dimensional relativistic electrons in an inverted contact",
V. Volkov and O. Pankratov, JETP Lett. 42, 178 (1985).
 "Volkov-Pankratov states in topological heterojunctions", S. Tchoumakov, V. Jouffrey, A. Inhofer, E. Bocquillon, B. Plaçais, D. Carpentier, and M. O. Goerbig, Phys. Rev. B 96, 201302 (2017).
 "Magnetic description of the Fermi arc in type-I and type-II Weyl semimetals", S. Tchoumakov, M. Civelli, and M. O. Goerbig, Phys. Rev. B 95, 125306 (2017).
This seminar is organized in the framework of the PSI2 program "Topological phases of matter: from the quantum Hall effect to spin liquids" (Topmat)