Applications of the potential algebras of the two-dimensional Dirac-like operators

TL;DR

This paper explores how potential algebras can analyze 2D Dirac-like quantum systems, leading to solvable models, energy calculations for fullerenes, and connections to supersymmetry.

Contribution

It introduces a framework using potential algebras for constructing solvable 2D Dirac models and links shape-invariance to N=4 nonlinear supersymmetry.

Findings

Derived integrals of motion forming specific algebraic structures.

Constructed solvable models for (2+1)D Dirac equations.

Linked shape-invariance to N=4 nonlinear supersymmetry.

Abstract

Potential algebras can be used effectively in the analysis of the quantum systems. In the article, we focus on the systems described by a separable, 2x2 matrix Hamiltonian of the first order in derivatives. We find integrals of motion of the Hamiltonian that close centrally extended so(3), so(2,1) or oscillator algebra. The algebraic framework is used in construction of physically interesting solvable models described by (2+1) dimensional Dirac equation. It is applied in description of open-cage fullerenes where the energies and wave functions of low-energy charge-carriers are computed. The potential algebras are also used in construction of shape-invariant, one-dimensional Dirac operators. We show that shape-invariance of the first-order operators is associated with the N=4 nonlinear supersymmetry which is represented by both local and nonlocal supercharges. The relation to the shape-invariant non-relativistic systems is discussed as well.