A solvable model of Landau quantization breakdown

Abstract

Physics of two-dimensional electron gases under perpendicular magnetic field often displays three distinct stages when increasing the field amplitude: a low field regime with classical magnetotransport, followed at intermediate field by a Shubnikov-de Haas phase where the transport coefficients present quantum oscillations, and, ultimately, the emergence at high field of the quantum Hall effect with perfect quantization of the Hall resistance. A rigorous demonstration of this general paradigm is still limited by the difficulty in solving models of quantum Hall bars with macroscopic lateral dimensions and smooth disorder. We propose here the exact solution of a simple model exhibiting similarly two sharp transitions that are triggered by the competition of cyclotron motion and potential-induced drift. As a function of increasing magnetic field, one observes indeed three distinct phases showing respectively fully broken, partially smeared, or perfect Landau level quantization. This model is based on a nonrotationally invariant, inverted two-dimensional harmonic potential, from which a full quantum solution is obtained using 4D phase space quantization. The developed formalism unifies all three possible regimes under a single analytical theory, as well as arbitrary quadratic potentials, for all magnetic field values.

Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids

“…Putting γ = µ = 0 in the arguments of hypergeometric functions in Equation ( 108), one obtains λ = (ω i − ω f )/ω i . Then, it is easy to verify that formulas (109) and (110) coincide with the instantaneous jump expressions (42) for the coefficients u ± . More precise estimations of the accuracy of this approximation are given in Appendix D. The ratio E f /E i versus parameter κ for different negative values of the final frequency ω f (shown nearby the respective lines) in the case of exponentially varying frequency on the time semi-axis (101).…”

“…Hence, formula Γ(x) = Γ(1 + x)/x ≈ 1/x (valid for |x| 1) leads immediately to the sudden jump relations (42). Analyzing the maximum of ratio | ω/ω 2 | as function of time for the Epstein-Eckart profile (131) with ω f > 0, we obtain the condition of the adiabatic approximation…”

“…Operators (4) describe nothing but the coordinates of the center of a circle, which the particle rotates around with the cyclotron frequency Ω = eB/(mc). The importance of these integrals of motion was emphasized by many authors during decades [4,[31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Equivalent integrals of motion, obtained by the multiplication of x c and y c by mΩ, were considered under the name "pseudomomentum" in papers [36,46].…”

“…One can question this result, because the limit ω f → 0 is not justified in the initial Equations ( 35) and (42). However, the exact solution to Equation (3) with ω(t) = 0 at t > 0, satisfying the initial conditions (28), has the form (this solution was used in reference [64] in connection with the concept of "quantum sling")…”

“…Formulas ( 53), ( 56) and ( 57) can be simplified in the special case of the sudden jump of magnetic field, when coefficients u ± are real: see Equation (42). Then, for any sign of the final frequency ω f , we obtain…”