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We introduce a unitary almost-Mathieu operator, which is obtained from atwo-dimensional quantum walk in a uniform magnetic field. We exhibit a versionof Aubry--Andr\'{e} duality for this model, which partitions the parameterspace into three regions: a supercritical region and a subcritical region thatare dual to one another, and a critical regime that is self-dual. In eachparameter region, we characterize the cocycle dynamics of the transfer matrixcocycle generated by the associated generalized eigenvalue equation. Inparticular, we show that supercritical, critical, and subcritical behavior alloccur in this model. Using Avila's global theory of one-frequency cocycles, weexactly compute the Lyapunov exponent on the spectrum in terms of the givenparameters. We also characterize the spectral type for each value of thecoupling constant, almost every frequency, and almost every phase. Namely, weshow that for almost every frequency and every phase the spectral type ispurely absolutely continuous in the subcritical region, pure point in thesupercritical region, and purely singular continuous in the critical region. Insome parameter regions, we refine the almost-sure results. In the critical casefor instance, we show that the spectrum is a Cantor set of zero Lebesguemeasure for arbitrary irrational frequency and that the spectrum is purelysingular continuous for all but countably many phases.

Almost Everything About the Unitary Almost Mathieu Operator