On the role of self-adjointness in the continuum formulation of topological quantum phases

Abstract

Topological quantum phases of matter are characterized by an intimate relationship between the Hamiltonian dynamics away from the edges and the appearance of bound states localized at the edges of the system. Elucidating this correspondence in the continuum formulation of topological phases, even in the simplest case of a one-dimensional system, touches upon fundamental concepts and methods in quantum mechanics that are not commonly discussed in textbooks, in particular the self-adjoint extensions of a Hermitian operator. We show how such topological bound states can be derived in a prototypical one-dimensional system. Along the way, we provide a pedagogical exposition of the self-adjoint extension method as well as the role of symmetries in correctly formulating the continuum, field-theory description of topological matter with boundaries. Moreover, we show that self-adjoint extensions can be characterized generally in terms of a conserved local current associated with the self-adjoint operator.

Mechanical, electronic, optical, piezoelectric and ferroic properties of strained graphene and other strained monolayers and multilayers: an update

“…Indeed, a consistent treatment of boundaries within graphene strain engineering remains lacking. Nevertheless, multiple deformation fields are uniquely determined by boundaries and by sharp strain profiles: as it is the case in semiconductor heterojunctions, effective theories based on envelope wave functions call for supplemental boundary conditions of the form ψ = Mψ to retain hermiticity, and for self-adjoint extensions to preserve currents [72][73][74][75]. M is a matrix containing microscopic details and symmetries, and ψ is the electron/hole wavefunction at the boundary.…”

Kekulé-induced valley birefringence and skew scattering in graphene

“…For any effective theory that uses an envelope wavefunction, as is the case of the Dirac equation for graphene, the matching requires a supplemental boundary condition of the form Ψ = M Ψ in order to retain the hermiticity and preserve currents. Here M is a matrix containing the microscopic details and the symmetries of the problem [59][60][61][62][63][64] . Since we consider the Kekulé-Y bond modulation as a perturbation within the same graphene sheet, no major misalignment is expected and thus for small ∆ we can consider M as unitary throughout this work.…”

Kekulé Induced Valley Birefringence and Skew Scattering in Graphene

“…For any effective theory that uses an envelope wavefunction, as is the case of the Dirac equation for graphene, the matching requires a supplemental boundary condition of the form Ψ = M Ψ in order to retain the hermiticity and preserve currents. Here M is a matrix containing the microscopic details and the symmetries of the problem [58][59][60][61][62][63] . Since we consider the Kek-Y bond modulation as a perturbation within the same graphene sheet, no major misalignment is expected and thus for small ∆ we can consider M as unitary throughout this work.…”