Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

Abstract

We study purely magnetic Schrödinger 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

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

“…The basic model consists of a transverse magnetic field that is constant in each half plane so that it is equal to b + for x > 0 and b − for x < 0, with b − = b + . In [14], two of us studied the generalized Iwatsuka model for which 0 < b − < b + < ∞. In this paper, we study the case for which b + = b > 0 for x > 0, and b − = −b < 0 for x < 0.…”

“…This immediately implies the eigenvalue estimate 14) and the difference of the two eigenvalues is bounded as…”

“…In a wider context, these models (see also [14]) indicate how the magnetic field can confine the electrons along the magnetic discontinuity to create a net edge current. In the present article, we prove the existence of quantum states for which the expectation of the current operator is bounded from below and we give an explicit lower bound.…”

The Distribution of Superconductivity Near a Magnetic Barrier

“…The case 0 < a < 1. This case is studied in [HS15,Iwa85]. The eigenvalue µ a (ξ) is simple and is a decreasing function of ξ.…”

Absence of embedded eigenvalues for translationally invariant magnetic Laplacians

“…In [5], the author is mainly concerned by proving the absolute continuity of the spectrum. Note that this issue is closely connected to the existence of edge currents (quantified by Mourre estimates), as explained for instance in [4], where positive magnetic fields are considered. The reader might also want to consider -the physical considerations in [11], -the paper [3] considering the dispersion curves associated with non-smooth magnetic fields, -the contribution [13] generalizing Iwatsuka's result by adding a translationnaly invariant electric potential, -the paper [14] devoted to dimension three and fields having cylindrical and longitudinal symmetries, -or [7] where various estimates of the band functions are established for increasing, positive, and bounded magnetic fields, and applied to the estimate of quantum currents.…”