On the Role of Self-Adjointness in the Continuum Formulation of Topological Quantum Phases

TL;DR

This paper explores the importance of self-adjointness in the continuum description of topological quantum phases, demonstrating how bound states at edges can be derived and characterized using self-adjoint extension methods.

Contribution

It provides a pedagogical analysis of self-adjoint extensions in continuum topological models, linking boundary states to operator self-adjointness and symmetries.

Findings

Bound states at edges are derived using self-adjoint extension techniques.

Self-adjoint extensions are characterized by conserved local currents.

The role of symmetries in continuum topological models is elucidated.

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.