Formalism of general continuum models with boundary conditions, propagation of bound states from nontrivial to trivial topological classes, and the general surface-state structure near one node of a Weyl semimetal

TL;DR

This paper develops a comprehensive formalism for continuum models with boundary conditions across various dimensions, explores topological surface states, and reveals how bound states propagate between topological classes and dimensions.

Contribution

It introduces a symmetry-incorporating formalism for continuum models with boundary conditions, deriving general forms and analyzing bound states across 1D, 2D, and 3D systems, including topological and trivial classes.

Findings

Derived the most general boundary conditions for continuum models in various dimensions.

Established a bulk-boundary correspondence for Weyl points relating surface states to Chern numbers.

Identified propagation of bound states from topologically nontrivial to trivial classes across dimensions.

Abstract

We present the {\em (symmetry-incorporating) formalism of general continuum models with boundary conditions} and apply it to the model with the minimal number of degrees of freedom necessary to have a well-defined boundary: a model with a two-component wave function and a linear-in-momentum Hamiltonian. We derive the most general forms (class A) of both the Hamiltonian and boundary condition in 1D (insulator), 2D (quantum anomalous Hall insulator), and 3D (Weyl node) and analytically calculate and explore the corresponding general bound/edge/surface-state structures. In 1D, one bound state exists in the half of the $\text{U}(1)$ parameter space of possible boundary conditions. Considering several dimensions simultaneously ties the models together and uncovers important relations between them. We formulate a version of bulk-boundary correspondence that fully characterizes the vicinity of a Weyl point: the chirality of the surface-state spectrum along a path enclosing the projected Weyl point is equal to the Chern number of the Weyl point. We demonstrate how symmetries are naturally incorporated into the formalism, by deriving the most general form of the model with chiral symmetry (class AIII). We show that the (perhaps unexpected) existence of persistent bound states in the topologically trivial 1D class-A model is not accidental and has at least two topological explanations, by relating it to the topologically nontrivial 2D class-A (by viewing it as an effective 2D quantum anomalous Hall system) and 1D class-AIII (via deviation from the cases of chiral symmetry in the parameter space) models. We identify this as a systematic "propagation effect", whereby bound states from topologically nontrivial classes, where they are protected and guaranteed to exist, propagate to the related, "adjacent" in dimension or symmetry, topologically trivial classes.