Seminarium Fizyki Materii Skondensowanej

I briefly review different methods, which were used to discretize the Hamiltonian of a topological insulator or topological superconductor, without giving up on the topological protection of the massless excitations (respectively, Dirac fermions or Majorana fermions). The method of tangent fermions is singled out as being uniquely suited for this purpose. Tangent fermions avoid the fermion doubling lattice artefacts, while preserving the fundamental symmetries of the Dirac Hamiltonian. Although the discretized Hamiltonian is nonlocal, as required by the fermion-doubling no-go theorem, it is possible to transform the wave equation into a generalized eigenproblem that is local in space and time. The applications of this method and the examples of its usage that I intend to present include Klein tunneling of Dirac fermions through a potential barrier, the absence of localization by disorder, the anomalous quantum Hall effect in a magnetic field, and the thermal metal of Majorana fermions.