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.

10 pages, 7 figures

References in corpus (9)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Topological Insulators with Inversion Symmetry
• The fate of topological-insulator surface states under strong disorder
• Finite difference method for transport properties of massless Dirac fermions
• Majorana state on the surface of a disordered 3D topological insulator
• Finite-difference method for transport of two-dimensional massless Dirac fermions in a ribbon geometry
• Floquet simulators for topological surface states in isolation
• Finite temperature effects on Majorana bound states in chiral $p$-wave superconductors

Cited by in corpus (3)

• Topologically protected Casimir effect for lattice fermions
• Four-band effective square lattice model for Bernal-stacked bilayer graphene
• Spin-polarized scanning tunneling microscopy measurement scheme for determining the quantum geometric tensor