Exact solutions for topological surface states of three-dimensional lattice models

Abstract

In this work, we employ a generalized transfer matrix method that provides exact analytical and numerical solutions for lattice versions of topological models with surface termination in one direction. We construct a generalized eigenvalue equation, equivalent to the conventional transfer matrix, which neither suffers from nor requires singular (non-invertible) inter-layer hopping matrices. This contrasts with some previous approaches that use the transfer matrix technique to obtain analytical expressions for surface states, which only focus on singular cases. We then apply this formalism to derive, with exactness, the topological surface states and Fermi arc states in two prototypical topological models: the 3D Bernevig-Hughes-Zhang model and a lattice model exhibiting Weyl semimetal behavior. Our results show that the surface states and bulk bands, across the projected 2D Brillouin zone, agree perfectly with those obtained through direct numerical diagonalization of the corresponding Hamiltonians in a slab geometry. This highlights that the generalized transfer matrix method is not only a powerful tool but also a highly efficient alternative to fully numerical methods for investigating surface physics and interfaces in topological systems, particularly when it is required to go beyond low-energy effective descriptions.

13 pages, 4 figures

References in corpus (17)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Progress and prospects in magnetic topological materials
• Weyl Semimetals, Fermi Arcs and Chiral Anomalies (A Short Review)
• Topological Surface States and Dirac point tuning in ternary Bi2Te2Se class of topological insulators
• Real space finite difference method for conductance calculations
• 2024 roadmap on 2D topological insulators
• Ferrimagnetic Regulation of Weyl Fermions in a Noncentrosymmetric Magnetic Weyl Semimetal
• Spin- and angle-resolved photoemission on topological materials
• Highly Tunable Spin-Orbit Torque and Anisotropic Magnetoresistance in a Topological Insulator Thin Film Attached to Ferromagnetic Layer
• Dirac Fermions and Topological Phases in Magnetic Topological Insulator Films
• Spontaneous Formation of Exceptional Points at the Onset of Magnetism