Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

TL;DR

This paper investigates magnetic Schrödinger operators with position-dependent magnetic fields, demonstrating the existence, localization, and stability of edge currents induced by Iwatsuka-type magnetic fields in two dimensions.

Contribution

It provides new results on edge current existence, localization, and stability for Schrödinger operators with specific magnetic field profiles, including cases with magnetic field jumps.

Findings

Edge currents flow along the magnetic barrier near x=0.

Edge states are localized within a region of size proportional to b_-^{-1/2}.

Edge currents are stable under various perturbations.

Abstract

We study purely magnetic Schr\"odinger operators in two-dimensions $(x,y)$ with magnetic fields $b(x)$ that depend only on the $x$-coordinate. The magnetic field $b(x)$ is assumed to be bounded, there are constants $0 < b_- < b_+ < \infty$ so that $b_- \leq b(x) \leq b_+$, and outside of a strip of small width $-\epsilon < x < \epsilon$, where $0 < \epsilon < b_-^{-1/2}$, we have $b(x) = b_\pm x$ for $\pm x > \epsilon$. The case of a jump in the magnetic field at $x=0$ corresponding to $\epsilon=0$ is also studied. We prove that the magnetic field creates an effective barrier near $x=0$ that causes edge currents to flow along it consistent with the classical interpretation. We prove lower bounds on edge currents carried by states with energy localized inside the energy bands of the Hamiltonian. We prove that these edge current-carrying states are well-localized in $x$ to a region of size $b_-^{-1/2}$, also consistent with the classical interpretation. We demonstrate that the edge currents are stable with respect to various magnetic and electric perturbations. For a family of perturbations compactly supported in the $y$-direction, we prove that the time asymptotic current exists and satisfies the same lower bound.