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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 3: Quantum Dynamics

MP 3.7: Talk

Monday, March 18, 2024, 17:30–17:50, HL 001

Almost everything about the unitary almost Mathieu operator — •Christopher Cedzich¹, Jake Fillman², and Darren Ong³ — ¹Heinrich Heine Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany — ²Texas State University, San Marcos, TX 78666, USA — ³Xiamen University Malaysia, Jalan Sunsuria, Bandar Sunsuria, 43900 Selangor, Malaysia

We introduce the unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry-André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. We exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results.

Keywords: Quantum Walks; Almost Mathieu Operator; Spectral Theory; CMV matrices