Beyond the quadratic saddle: quantum transmission through non-quadratic saddle-point potentials in a strong magnetic field

For an electron in a two-dimensional electron gas under a strong perpendicular magnetic field moving through a saddle-point (constriction) potential, Fertig and Halperin (Phys. Rev. B 36, 7969 (1987)) obtained an exact, closed-form transmission coefficient — but only for a purely quadratic, separable saddle, where the Hamiltonian splits into a confining harmonic oscillator (cyclotron motion) and an inverted harmonic oscillator (guiding-center motion), and where the oscillator/Landau quantum number is conserved. They also showed that the WKB approximation agrees with the exact result only when the classical turning points are well separated, breaking down near the saddle energy. The open problem is to extend this exact transmission theory beyond the quadratic idealization: to non-quadratic saddle geometries in which Landau levels and transverse modes mix, to the associated level splittings of realistic double-well landscapes connected by saddle barriers, and to approximations uniformly valid between the deep-tunneling (WKB) regime and the exact saddle-energy regime. No canonical transformation decoupling the guiding-center and cyclotron degrees of freedom is known for non-separable saddles, and no exact non-quadratic transmission coefficient is known; non-quadratic effects have been treated only perturbatively, numerically, or within adiabatic or gauge-field mode-mixing frameworks, and the general problem remains open.

Background

In a two-dimensional electron gas (2DEG) — a semiconductor layer in which electrons are free to move in a plane — a perpendicular magnetic field $B$ quantizes the cyclotron motion into Landau levels, separated by the cyclotron energy $\hbar\omega_c$ ($\omega_c=eB/mc$ is the cyclotron frequency). The electron coordinate separates into a rapidly rotating cyclotron coordinate and a slowly drifting guiding-center coordinate; the characteristic length scale of the cyclotron orbit is the magnetic length $l_0=(\hbar c/eB)^{1/2}$. A quantum point contact (QPC) is a short, gate-defined constriction between two reservoirs in a 2DEG, and in the quantum Hall regime (high $B$, where transport is carried by chiral edge states) smooth saddle points of the electrostatic potential control which states connect across the constriction. This physics underlies the network (Chalker–Coddington) description of quantum Hall plateau transitions — a percolation-type model in which the plateau-to-plateau transition is governed by scattering at the saddle points of a smooth random potential, each saddle contributing the elementary scattering amplitude.

Fertig and Halperin (1987) modeled the constriction by the quadratic saddle-point potential $V_{\mathrm{SP}}(x,y)=-U_xx^2+U_yy^2+V_0$, with curvatures $U_x,U_y$ and saddle energy $V_0$. By two successive Bogoliubov-type canonical transformations of the ladder operators — unitary changes of variables mixing coordinates and momenta, of the kind used to diagonalize coupled oscillators — they showed that the Hamiltonian splits into two commuting one-dimensional Hamiltonians: a harmonic oscillator of scale $E_2$ describing the cyclotron coordinate, and an inverted harmonic oscillator (a parabolic barrier top, the Hamiltonian $H_1=E_1(P^2-X^2)$) of scale $E_1$ describing the guiding-center coordinate. Because the two parts commute, the oscillator quantum number $n$ is conserved, and the two-dimensional transmission problem reduces exactly to one-dimensional scattering of the guiding center off the inverted harmonic oscillator. Writing $E_G=E-(n+\tfrac12)E_2$ for the guiding-center part of the total energy $E$ and $\epsilon=(E_G-V_0)/E_1$, the exact transmission probability is

$$T(\epsilon)=\frac{1}{1+\exp(-\pi\epsilon)}.$$

In the same paper the WKB (Wentzel–Kramers–Brillouin) semiclassical approximation — an expansion in powers of the small parameter $l_0$ (or equivalently $1/|\epsilon|$), valid when classical turning points are well separated — gives, on the tunneling side of the barrier, $T_{\mathrm{WKB}}=16\theta^2/(4\theta^2+1)^2$ with $\theta=\exp(\pi X_0^2/2)$ and $X_0^2=-\epsilon$; the above-barrier result follows from the reflection–transmission symmetry $R(\epsilon)=T(-\epsilon)$. This agrees with the exact result for $|\epsilon|\gg1$ but fails near the saddle energy: at $\epsilon=0$ WKB gives $16/25$ where symmetry requires $T(0)=1/2$. The exactness of the result rests entirely on the quadratic, separable form of $V_{\mathrm{SP}}$: for that form, and only for it, the guiding-center and cyclotron degrees of freedom decouple and no Landau-level mixing occurs.

The 1987 paper and its limitations are the origin of the present problem. Fertig and Halperin's treatment is specialized to the strictly quadratic saddle (the transmission is diagonal in the oscillator quantum number $n$, so each subband is handled separately, but superpositions do not mix); it notes implications for energy-level splitting in double-well potentials whose wells are connected by saddle barriers; and it leaves open the nonsemiclassical regime $\epsilon\approx0$, where WKB fails and only the exact quadratic answer is known.

Problem Statement

Extend the Fertig–Halperin exact transmission theory for an electron in a perpendicular magnetic field beyond the strictly quadratic, separable saddle-point potential $V_{\mathrm{SP}}(x,y)=-U_xx^2+U_yy^2+V_0$, while preserving its two achievements: an explicit transmission coefficient and controlled accuracy in the nonsemiclassical regime, where controlled accuracy means an error bound stated explicitly and holding uniformly over the energy range from the deep-tunneling regime to the barrier top. Concretely, for a smooth saddle-point potential of general (non-quadratic) form, i.e. any saddle whose curvature varies spatially so that the cyclotron/Landau levels couple (mix) with the transverse modes, determine the quantum transmission coefficient as a function of energy, magnetic field, and the parameters of the potential, in the regime where the magnetic length $l_0=(\hbar c/eB)^{1/2}$ (the cyclotron-orbit size set by the field $B$) is small compared with the scale of variation of the potential. The objective comprises three interconnected results: (1) the multi-level transmission matrix (or a controlled approximation to it) for non-quadratic saddle geometries, reducing exactly to $T(\epsilon)=1/(1+e^{-\pi\epsilon})$ per subband in the quadratic limit, where $\epsilon=(E_G-V_0)/E_1$ is the dimensionless guiding-center energy — $E_G=E-(n+\tfrac12)E_2$ being the guiding-center part of the total energy $E$ at oscillator quantum number $n$, measured from the saddle energy $V_0$ in units of the barrier scale $E_1$; (2) the implications for energy-level splitting in double-well potentials whose wells are connected by realistic (non-strictly-quadratic, possibly multiple) saddle barriers, generalizing the tunneling-splitting estimates that the quadratic-saddle transmission supports, in the form of an explicit level-splitting formula with a stated error bound in the saddle-transmission limit; (3) the quantification of the nonsemiclassical corrections to WKB near the saddle energy ($\epsilon\approx0$, where WKB gives $16/25$ instead of $1/2$ even in the quadratic case), in the form of approximations uniformly valid between the deep-tunneling regime $|\epsilon|\gg1$ and the barrier top; the distinct object required here is the explicit nonsemiclassical correction to the WKB transmission near $\epsilon\approx0$ — the deviation from $T_{\mathrm{WKB}}$ and its matching to the deep-tunneling and over-barrier limits — which the uniformly bounded coefficient of (1) does not by itself supply. The common scientific objective uniting these results is a quantitative theory, with explicitly controlled error, of quantum transmission through realistic saddle-point constrictions in a strong magnetic field — the element from which conductance of quantum point contacts, quantum Hall plateau-transition scattering amplitudes, and tunneling splittings in smooth disorder landscapes are built.

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