Singular continuous Cantor spectrum for magnetic quantum walks

Cedzich, C.; Fillman, Jake; Geib, T.; Werner, Albert H.

doi:10.1007/s11005-020-01257-1

“…By (19) we have to bound the scalar product (20) from below and |φ + 2y−1+j | and |φ − 2x−i | from above. Since φ − is left-compatible, we read off from (18) that it satisfies…”

Section: B Finite Unitary Restrictions and Resolvent Estimatesmentioning

confidence: 99%

“…In close analogy to the self-adjoint case, this quantum walk has purely singular continuous spectrum for all irrational frequencies and almost all offsets [30]. From a physics point of view this model describes the discrete evolution of a quantum mechanical particle on a two-dimensional lattice under the influence of a discrete homogeneous magnetic field [18,22].…”

Section: Introductionmentioning

confidence: 97%

Anderson localization for electric quantum walks and skew-shift CMV matrices

Cedzich,

Werner

2019

Preprint

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We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the Lyapunov exponent of this model. Making use of a connection between quantum walks and CMV matrices our result implies Anderson localization for CMV matrices with a particular choice of skew-shift Verblunsky coefficients as well as for quasi-periodic unitary band matrices.

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“…and the relations (39) are recovered. This is in perfect analogy to the DTQW scenario with θ = 0 studied in appendix C.…”

Section: Dirac Particle Analogymentioning

confidence: 96%

Complementarity in quantum walks

Grudka

¹

,

Kurzyński

²

,

Polak

³

et al. 2023

J. Phys. A: Math. Theor.

0

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The eigenbases of two quantum observables, $\{|a_i\rangle\}_{i=1}^D$ and $\{|b_j\rangle\}_{j=1}^D$, form Mutually Unbiassed Bases (MUB) if $|\langle a_i |b_j \rangle| = 1/\sqrt{D}$ for all $i$ and $j$. In realistic situations MUB are hard to obtain and one looks for Approximate MUB (AMUB), in which case the corresponding eigenbases obey $|\langle a_i |b_j \rangle| \leq c/\sqrt{D}$, where $c$ is some positive constant independent of $D$. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks on $d$-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter $q$. We solve the model analytically and observe that for prime $d$ the eigenvectors of two quantum walk evolution operators form AMUB. Namely, if $d$ is prime the corresponding eigenvectors of the evolution operators, that act in the $D$-dimensional Hilbert space ($D=2d$), obey $|\langle v_q|v'_{q'} \rangle| \leq \sqrt{2}/\sqrt{D}$ for $q\neq q'$ and for all $|v_q\rangle$ and $|v'_{q'}\rangle$. Finally, we show that the analogous AMUB relation still holds in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.

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“…In close analogy to the self-adjoint case, this quantum walk has purely singular continuous spectrum for all irrational frequencies and almost all offsets [30]. From a physics point of view this model describes the discrete evolution of a quantum mechanical particle on a two-dimensional lattice under the influence of a discrete homogeneous magnetic field [18,22].…”

Section: Rationalmentioning

confidence: 97%

Anderson Localization for Electric Quantum Walks and Skew-Shift CMV Matrices

Cedzich

¹

,

Werner

²

2021

Commun. Math. Phys.

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We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the Lyapunov exponent of this model. Making use of a connection between quantum walks and CMV matrices our result implies Anderson localization for CMV matrices with a particular choice of skew-shift Verblunsky coefficients as well as for quasi-periodic unitary band matrices.

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Singular continuous Cantor spectrum for magnetic quantum walks

Cited by 15 publications

References 76 publications

Anderson localization for electric quantum walks and skew-shift CMV matrices

Anderson localization for electric quantum walks and skew-shift CMV matrices

Complementarity in quantum walks

Anderson Localization for Electric Quantum Walks and Skew-Shift CMV Matrices

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