Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians

Abstract

Dirac-like Hamiltonians, linear in momentum $k$, describe the low-energy physics of a large set of novel materials, including graphene, topological insulators, and Weyl fermions. We show here that the inclusion of a minimal $k^2$ Wilson's mass correction improves the models and allows for systematic derivations of appropriate boundary conditions for the envelope functions on finite systems. Considering only Wilson's masses allowed by symmetry, we show that the $k^2$ corrections are equivalent to Berry-Mondragon's discontinuous boundary conditions. This allows for simple numerical implementations of regularized Dirac models on a lattice, while properly accounting for the desired boundary condition. We apply our results on graphene nanoribbons (zigzag and armchair), and on a PbSe monolayer (topological crystalline insulator). For graphene, we find generalized Brey-Fertig boundary conditions, which correctly describe the small gap seen on \textit{ab initio} data for the metallic armchair nanoribbon. On PbSe, we show how our approach can be used to find spin-orbital-coupled boundary conditions. Overall, our discussions are set on a generic model that can be easily generalized for any Dirac-like Hamiltonian.

arXiv comment14 pages, 6 figures

References in corpus (33)

• The electronic properties of graphene
• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Energy Gaps in Graphene Nanoribbons
• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs
• Quantum Spin Hall Effect
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Higher-Order Topological Insulators
• Topological Crystalline Insulators in the SnTe Material Class
• Electronic States of Graphene Nanoribbons
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Kwant: a software package for quantum transport
• Reflection symmetric second-order topological insulators and superconductors
• Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Boundary conditions for Dirac fermions on a terminated honeycomb lattice
• Energy levels of triangular and hexagonal graphene quantum dots: a comparative study between the tight-binding and the Dirac approach
• Symmetry of boundary conditions of the Dirac equation for electrons in carbon nanotubes
• Crystal Field Effect Induced Topological Crystalline Insulators In Monolayer IV-VI Semiconductors
• Magnetically Defined Qubits on 3D Topological Insulators
• Finite difference method for transport properties of massless Dirac fermions
• Two-Dimensional Lattice Model for the Surface States of Topological Insulators
• Qsymm: Algorithmic symmetry finding and symmetric Hamiltonian generation
• Topological Crystalline Insulator and Quantum Anomalous Hall States in IV-VI based Monolayers and their Quantum Wells
• Finite-difference method for transport of two-dimensional massless Dirac fermions in a ribbon geometry
• Confinement and Fermion Doubling Problem in Dirac-like Hamiltonians
• Paradoxical extension of the edge states across the topological phase transition due to emergent approximate chiral symmetry in a quantum anomalous Hall system
• Topological Nonsymmorphic Ribbons out of Symmorphic Bulk

Cited by in corpus (9)

• Absence of equilibrium edge currents in theoretical models of topological insulators
• Tangent fermions: Dirac or Majorana fermions on a lattice without fermion doubling
• Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice
• Fabry-Pérot resonant vortices and magnetoconductance in topological insulator constrictions with magnetic barriers
• Casimir effect for fermion condensate in conical rings
• Resonant helical multi-edge transport in Sierpiński carpets
• Exact lattice-model calculation of boundary modes for Weyl semimetals and graphene
• Single-cone Dirac edge states on a lattice
• Symmetry of electron bands in graphene: (nearly) free electron vs. tight-binding