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arXiv:2610.26637

Average lower bound on magnetic double-well tunneling with asymptotically large well separation

Charles L. Fefferman, Say-Yeon Kwon, Jacob Shapiro, Michael I. Weinstein

math-phcond-mat.mes-hallmath.APquant-phSubmitted Oct 19, 2026• Announced Oct 20, 2026
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Abstract

We study tunneling between two identical, smooth, compactly supported potential wells in a constant perpendicular magnetic field, with the coupling and magnetic field fixed and the well separation tending to infinity. For a sufficiently small positive magnetic field, we prove lower bounds on the hopping coefficient and on the splitting of the two lowest eigenvalues outside a set of separations of density zero. If the well centers are separated by $2d_1$ and the magnetic field strength is $bλ$, the lower bounds have the form $e^{-bλd_1^2-Cd_1^{1+δ}}$ for every fixed $δ\in(0,1)$. No radial or reflection symmetry of the single-well potential is assumed. The proof combines the explicit Landau resolvent kernel with a complex analytic estimate for logarithmic averages of the hopping coefficient.

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12 pages, 1 figure

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The tunneling rate ${t}_{v}∕\ensuremath{\hbar}$ of a vortex between two pinning sites (of strength V‾\overline{V}V separated by ddd) is computed using the Bogoliubov expansion of vortex wave-functions overlap. For BCS vortices, tunneling is suppressed beyond a few Fermi wavelengths. For Bose condensates, ${t}_{v}=\overline{V}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}{n}_{s}{d}^{2}∕2)$, where ns{n}_{s}ns​ is the boson density. The analogy between vortex hopping in a superconducting film and two-dimensional electrons in a perpendicular magnetic field is exploited. We derive the variable range hopping temperature, below which vortex tunneling contributes to magnetoresistance. Using the ``quantum Hall insulator'' analogy we argue that the Hall conductivity (rather than the inverse Hall resistivity) measures the effective carrier density in domains of mobile vortices. Details of vortex wave functions and overlap calculations, and a general derivation of the Magnus coefficient for any wave function on the sphere, are provided in appendixes.
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