Exploring $2d$ localization with a step dependent coin
- URL: http://arxiv.org/abs/2303.06769v2
- Date: Thu, 26 Oct 2023 05:44:08 GMT
- Title: Exploring $2d$ localization with a step dependent coin
- Authors: Kallol Sen
- Abstract summary: We generalize the coin operator of citeZahed023 to include a step dependent feature which induces localization in $2d$.
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.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We generalize the coin operator of \cite{Zahed_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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