For irrational magnetic flux, the spectrum of the two-dimensional Hadamard magnetic quantum walk is a zero-measure Cantor set and the walk has no pure point spectrum, so the spectrum is purely singular continuous.
Ballistic Transport for Limit-Periodic Jacobi Matrices with Applications to Quantum Many-Body Problems
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abstract
We study Jacobi matrices that are uniformly approximated by periodic operators. We show that if the rate of approximation is sufficiently rapid, then the associated quantum dynamics are ballistic in a rather strong sense; namely, the (normalized) Heisenberg evolution of the position operator converges strongly to a self-adjoint operator that is injective on the space of absolutely summable sequences. In particular, this means that all transport exponents corresponding to well-localized initial states are equal to one. Our result may be applied to a class of quantum many-body problems. Specifically, we establish a lower bound on the Lieb--Robinson velocity for an isotropic XY spin chain on the integers with limit-periodic couplings.
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Singular continuous Cantor spectrum for magnetic quantum walks
For irrational magnetic flux, the spectrum of the two-dimensional Hadamard magnetic quantum walk is a zero-measure Cantor set and the walk has no pure point spectrum, so the spectrum is purely singular continuous.