Surface spectra of Weyl semimetals through self-adjoint extensions

TL;DR

This paper uses self-adjoint extensions to analyze surface states in Weyl semimetals, providing a comprehensive method to determine boundary conditions and surface spectra, revealing how boundary choices influence surface phenomena like Fermi arcs.

Contribution

It introduces a systematic approach using self-adjoint extensions to characterize boundary conditions and surface spectra in Weyl semimetals, including effects of symmetries and dispersion shapes.

Findings

Surface spectra depend on boundary conditions.

Fermi arc shapes are influenced by boundary choices.

Bound states and dispersions can exhibit Mexican-hat features.

Abstract

We apply the method of self-adjoint extensions of Hermitian operators to the low-energy, continuum Hamiltonians of Weyl semimetals in bounded geometries and derive the spectrum of the surface states on the boundary. This allows for the full characterization of boundary conditions and the surface spectra on surfaces both normal to the Weyl node separation as well as parallel to it. We show that the boundary conditions for quadratic bulk dispersions are, in general, specified by a $\mathbb{U}(2)$ matrix relating the wavefunction and its derivatives normal to the surface. We give a general procedure to obtain the surface spectra from these boundary conditions and derive them in specific cases of bulk dispersion. We consider the role of global symmetries in the boundary conditions and their effect on the surface spectrum. We point out several interesting features of the surface spectra for different choices of boundary conditions, such as a Mexican-hat shaped dispersion on the surface normal to Weyl node separation. We find that the existence of bound states, Fermi arcs, and the shape of their dispersion, depend on the choice of boundary conditions. This illustrates the importance of the physics at and near the boundaries in the general statement of bulk-boundary correspondence.