[Paper Review] Band functions in the presence of magnetic steps

Peter D. Hislop, Nicolas Popoff+2 more|arXiv (Cornell University)|Jan 12, 2015

Spectral Theory in Mathematical Physics23 references4 citations

TL;DR

This paper analyzes band functions for 2D magnetic Schrödinger operators with piecewise constant magnetic fields, focusing on edge states induced by magnetic steps. Using asymptotic analysis of the fiber Hamiltonian's eigenvalues, it derives explicit asymptotic expansions for band functions at infinity, proving band splitting when the magnetic field ratio is rational, with quantitative dependence on the asymmetry of the effective double-well potential.

ABSTRACT

We complete the analysis of the band functions for two-dimensional magnetic Schrödinger operators with piecewise constant magnetic fields. The discontinuity of the magnetic field can create edge currents that ow along the discontinuity that have been described by physicists. Properties of these edge currents are directly related to the behavior of the band functions. The effective potential of the fiber operator is an asymmetric double well (eventually degenerated) and the analysis of the splitting of the bands incorporates the asymmetry. If the magnetic field vanishes, the reduced operator has essential spectrum and we provide an explicit description of the band functions located below the essential spectrum. For non degenerate magnetic steps, we provide an asymptotic expansion of the band functions at infinity. We prove that when the ratio of the two magnetic fields is rational, a splitting of the band functions occurs and has a natural order, predicted by numerical computations.

Motivation & Objective

• To complete the mathematical analysis of band functions for 2D magnetic Schrödinger operators with piecewise constant magnetic fields, particularly in previously unstudied configurations.
• To describe the spectral properties of the fiber Hamiltonian associated with the magnetic step, focusing on the behavior of band functions below the essential spectrum when one magnetic field vanishes.
• To establish asymptotic expansions of band functions at infinity for non-degenerate magnetic steps, capturing the influence of asymmetry in the effective double-well potential.
• To prove that rational ratios of magnetic fields lead to a splitting of band functions, with a natural order consistent with numerical predictions.

Proposed method

• Employing the direct fiber decomposition via Fourier transform in the y-direction, reducing the 2D problem to a family of 1D Schrödinger operators parameterized by quasimomentum k.
• Analyzing the effective potential V_b(x,k) = (k - a_b(x))², which forms an asymmetric double well due to the discontinuous magnetic field, and solving the associated eigenvalue problem.
• Using parabolic cylinder functions as special solutions to the Schrödinger equation in each half-plane, and matching solutions at x=0 via continuity and derivative conditions.
• Applying asymptotic analysis to the characteristic equation derived from the matching conditions, particularly for large k, to extract the leading-order behavior of band functions.
• Deriving explicit asymptotic expansions for the band functions ε⁺(k) and ε⁻(k) by analyzing the dominant terms in the discriminant of the characteristic equation.
• Using known asymptotic expansions of parabolic cylinder functions U(a,x) and V(a,x) for large |x| to estimate the behavior of solutions in the left and right regions.

Experimental results

Research questions

• RQ1 How do band functions behave for magnetic Schrödinger operators with a magnetic step when one magnetic field is zero?
• RQ2 What is the asymptotic behavior of the band functions as the quasimomentum k tends to infinity in the non-degenerate magnetic step case?
• RQ3 Does a rational ratio of the two magnetic fields lead to a splitting of the band functions, and if so, what is the order of this splitting?
• RQ4 How does the asymmetry of the effective double-well potential influence the splitting and asymptotic structure of the band functions?

Key findings

• For the case where one magnetic field vanishes (e.g., b₁=0, b₂=1), the band functions below the essential spectrum are explicitly described through the spectral analysis of the reduced operator.
• In the non-degenerate case with b₁ ≠ 0 and b₂ ≠ 0, the band functions admit asymptotic expansions as k → ∞, with ε⁺(k) ∼ C₁ k²ᵐ⁺¹ e⁻ᵏ²ᐟᵇ and ε⁻(k) ∼ C₂ k²ⁿ⁻³ e⁻ᵏ², where C₁ and C₂ are constants depending on m, n, and b.
• When the ratio b₁/b₂ is rational, a splitting of the band functions occurs, and the splitting has a natural order that matches numerical observations.
• The splitting is directly linked to the asymmetry of the effective double-well potential V_b(x,k), which breaks the symmetry of the system and leads to distinct asymptotic behaviors for the two lowest bands.
• The discriminant of the characteristic equation for the matching conditions is positive for large k, ensuring two distinct real roots corresponding to the two band functions ε⁺ and ε⁻.