Point-Like Rashba Interactions as Singular Self-Adjoint Extensions of the Schrödinger Operator in One Dimension

Vladimi L Kulinskii,Dmitry Yu Panchenko

doi:10.3389/fphy.2019.00044

Abstract

We consider singular self-adjoint extensions for the Schr\{o}dinger operator of spin-$1/2$ particle in one dimension. The corresponding boundary conditions at a singular point are obtained. There are boundary conditions with the spin-flip mechanism, i.e. for these point-like interactions the spin operator does not commute with the Hamiltonian. One of these extensions is the analog of zero-range $\delta$-potential. The other one is the analog of so called $\delta^{(1)}$-interaction. We show that in physical terms such contact interactions can be identified as the point-like analogues of Rashba Hamiltonian (spin-momentum coupling) due to material heterogeneity of different types. The dependence of the transmissivity of some simple devices on the strength of the Rashba coupling parameter is discussed. Additionally, we show how these boundary conditions can be obtained in the non-relativistic limit of Dirac Hamiltonian.

Highlights

Point-like interactions can be described as the singular extensions of the Hamiltonian and are very useful quantum mechanical models because of their analytical tractability [1,2,3,4,5]
They are equivalent to some boundary conditions imposed on a wave function at the singular points and represent the limiting cases of field inhomogeneities
The main result of the paper is that those extensions of the Schrödinger operator which are physically constructed on the basis of the inhomogeneous distribution of the electric field potential φ(x) can be augmented with the spin-flip mechanism

Summary

INTRODUCTION

Point-like interactions can be described as the singular extensions of the Hamiltonian and are very useful quantum mechanical models because of their analytical tractability [1,2,3,4,5]. In accordance with the spin-momentum nature of the r-couplings the physical reason of such factorization is that X3 contact interaction does not include spatial inhomogeneity in electric field potential φ This is quite consistent with the difference between X2 and X3 from the point of view of breaking the gauge symmetry [14, 17]. These results demonstrate that spin-flip mechanism even at small values of r-coupling can reach high probabilities with increasing the energy of incident particle This directly follows from the boundary conditions (19) and (22) since the effects depend on both r and the momentum. More intriguing problem here is the inclusion of the correlation effects due to spin statistics and investigation of phases with magnetic (dis)order in dependence on the intensity of point-like interactions This way of research may be useful for modeling 1-dimensional magnetic systems [18]

Spin-Flip Contact Interaction in 3D

PHYSICAL ORIGIN OF THE SPIN-FLIP BOUNDARY CONDITIONS

Findings

CONCLUSION