https://doi.org/10.1140/epjp/s13360-024-05923-4
Regular Article
Exploring 2d localization with a step-dependent coin
ICTP South American Institute for Fundamental Research, IFT-UNESP (1° andar), Rua Dr. Bento Teobaldo Ferraz 271, Bloco 2 - Barra Funda, 01140-070, São Paulo, SP, Brazil
Received:
26
October
2023
Accepted:
12
December
2024
Published online:
31
December
2024
We generalize the coin operator of Zahed and Sen (2023), to include a step-dependent feature which induces localization in 2d. This is evident from the probability distributions which can be further used to categorize the localized walks. Localization is also evident from the entropic measures. We compute and compare three distinct measures (a) Shannon entropy in the position and coin space, (b) entanglement entropy between position and spin space and (c) Quantum Relative Entropy which is a POVM of density operators of the step-dependent and step-independent coins. Shannon entropy and entanglement entropy are periodic and bounded functions of the time steps. The zeros of Shannon and entanglement entropies signify a complete localization of the wave function. The Quantum Relative Entropy and Quantum Information Variance exhibit a similar periodic feature with a zero minima where the step-dependent and step-independent walks coincide. Finally, we compute the numerical localization length (inverse of the Lyapunov exponent) for the step-dependent coin as a function of energy and compare with an approximate perturbative computation, where we put the step-dependent coin as a perturbation in the background of a step-independent coin. In both the instances, we find that the localization length peaks at approximately the same positions in the momentum space.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2024
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
