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Tangent Fermions: Dirac or Majorana Fermions on a Lattice Without Fermion Doubling
Authors:C. W. J. Beenakker  A. Donís Vela  G. Lemut  M. J. Pacholski  J. Tworzydło
Affiliation:1. Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, Leiden, 2300 RA The Netherlands;2. Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, Leiden, 2300 RA The Netherlands

Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany;3. Faculty of Physics, University of Warsaw, ul. Pasteura 5, Warszawa, 02–093 Poland

Abstract:Methods 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) are reviewed. The method of tangent fermions, pioneered by Richard Stacey, is singled out as being uniquely suited for this purpose. Tangent fermions propagate on a 2 + 1 ${2bm {+}1}$ dimensional space-time lattice with a tangent dispersion: tan 2 ( ε / 2 ) = tan 2 ( k x / 2 ) + tan 2 ( k y / 2 ) ${text{tan}^2 (bm {varepsilon }/2) bm {=} text{tan}^2 (k_x/2) bm {+}text{tan}^2 (k_y/2)}$ in dimensionless units. They avoid the fermion doubling lattice artefact that will spoil the topological protection, 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. Applications that are discussed 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.
Keywords:fermion doubling  lattice fermions  topological insulators  topological superconductors
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