Quantum tunneling and its absence in deep wells and strong magnetic fields

TL;DR

We study quantum tunneling between two deep potential wells in two dimensions under a strong perpendicular magnetic field by formulating the magnetic double-well Hamiltonians $h_\lambda$ and $H_\lambda$, with ground energy $e_\lambda$ and hopping coefficient $\rho_\lambda$ that governs the eigenvalue splitting $\Delta_\lambda$. For radially symmetric wells, $\Delta_\lambda$ is nonzero and exponentially small, with $\Delta_\lambda$ tied to $\rho_\lambda$. The paper demonstrates that in a non-radial perturbation family $\mathcal{F}_\lambda$ one can realize $\Delta_\lambda=0$ (no tunneling) and can tune the magnetic ground state symmetry between even and odd, while also proving that typical deep wells yield a quantitative lower bound on $|\rho_\lambda|$ and $\Delta_\lambda$. The results connect to flat-band crystal design and rely on complex-analytic continuation and perturbative decompositions to bound tunneling effects, highlighting regimes where tunneling cannot be suppressed and where tunable counterexamples exist.

Abstract

We present new results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to anti-symmetric. However, for typical double wells in a certain regime, tunneling is not suppressed, and we provide a lower bound for the eigenvalue splitting.

Figures (2)

• Figure 1: Inversion-symmetric potential of the type used in the proof of Theorem 2. Large circles are the supports, $B_0^L$ and $B_0^R$, of $v_0^L(x)$, centered at $x^L_0=(-d_1,0)$, and $v_0^R(x)$, centered at $x^R_0=(+d_1,0)$. For $\alpha=L,R$, we add four potentials ("sophons"), which are inversion symmetrically placed relative to $x_0^\alpha$ . The sophons are supported on the small red discs, $B_\mu^\alpha$, which are centered at $x_\mu^\alpha$; $\mu=1,2,3,4$, the vertical components of which determine the phase $\theta$ in [\ref{['eq:rho-asymp']}]; the amplitudes, support radii and centers of the sophon potentials are specially tuned $\lambda-$ dependent functions.
• Figure 2: Single-well potential for flat band crystal.