Duality between existence condition and construction procedure for interface states in a class of narrow-gap heterostructures

TL;DR

This paper reformulates the theory of interface states in narrow-gap heterostructures using category theory, revealing a duality between the existence criterion and localization procedure through commutative diagrams.

Contribution

It introduces a categorical duality framework that separates and relates the existence and localization problems of interface states in heterostructures.

Findings

The existence condition and localization of interface states are dual problems.

Commutative diagrams encode the criteria and procedures for interface states.

Categorical duality provides a new perspective on heterostructure interface theory.

Abstract

The problem for interface solutions in the quantum theory of heterostructures comprising narrow-gap semiconductors is reformulated in the language of commutative diagrams. By this way the theory of interface states in such heterostructures is naturally factorized in the two subproblems: (i) Criterion for the existence of interface states, (ii) Their localization in the common energy gap; the solution of both of them being presented by the requirement for commutativity of the relevant diagrams. It is shown that these two problems are dual in the sense of categorical duality, the passing from the one commutative diagram to the other being realized by a (contravariant) functor Op.