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.
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This is analogous to a nontrivial knot that cannot be undone without cutting. Locally, the knot has the same geometry as a loop; however, there is a global twist that distinguishes it from a simple loop. Technically, the twist in the wavefunction is expressed in terms of Berry's phases; see Ref. 3.
The transpose operation is defined as .
An absolutely continuous function on an interval is defined as one for which there exists an integrable function that satisfies for any (see, for example, p. 22 in Ref. 6). Then, according to the fundamental theorem of calculus, almost everywhere in I.
Note that the states do not belong to since .
