Geometry-independent tight-binding method for massless Dirac fermions in two dimensions

Abstract

The Nielsen-Ninomiya theorem, dubbed 'fermion-doubling', poses a problem for the naive discretization of a single (massless) Dirac cone on a two-dimensional surface. The inevitable appearance of an additional, unphysical fermionic mode can, for example, be circumvented by introducing an extra dimension to spatially separate Dirac cones. In this work, we propose a geometry-independent protocol based on a tight-binding model for a three-dimensional topological insulator on a cubic lattice. The low-energy theory, below the bulk gap, corresponds to a Dirac cone on its two-dimensional surface which can have an arbitrary geometry. We introduce a method where only a thin shell of the topological insulator needs to be simulated. Depending on the setup, we propose to gap out the states on the undesired surfaces either by breaking the time-reversal symmetry or by introducing a superconducting pairing. We show that it is enough to have a thickness of the topological-insulator shell of three to nine lattice constants. This leads to an effectively two-dimensional scaling with minimal and fixed shell thickness. We test the idea by comparing the spectrum and probability distribution to analytical results for both a proximitized Dirac mode and a Dirac mode on a sphere which exhibits a nontrivial spin-connection. The protocol yields a tight-binding model on a cubic lattice simulating Dirac cones on arbitrary surfaces with only a small overhead due to the finite thickness of the shell.

Tangent Fermions: Dirac or Majorana Fermions on a Lattice Without Fermion Doubling

“…One can then think of the Dirac cone on the top surface as being doubled on the bottom surface, but if the surfaces are widely separated there will effectively be only a single species of massless excitations on each surface. Since it is computationally costly to work with a 3D lattice, [4] a fully 2D formulation is preferable. In what follows we will review the options developed by particle physicists, with one key criterion in mind: If we add disorder to the Dirac Hamiltonian, as is unavoidable in a real material, will the Dirac cone remain gapless?…”