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Summary
This summary is machine-generated.

We studied quantum hopping in one dimension and found a localization transition. This transition occurs when hopping decreases slowly, changing how quantum states decay.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Statistical Physics

Background:

  • Understanding quantum transport in disordered systems is crucial.
  • Random matrix theory provides a framework for studying localization phenomena.
  • Sparse, long-range interactions introduce complex behaviors not captured by standard models.

Purpose of the Study:

  • Introduce and analyze a novel banded random matrix model for sparse, long-range quantum hopping in one dimension.
  • Determine the phase diagram and critical behavior of this model.
  • Investigate the nature of quantum localization and state decay under varying hopping rates.

Main Methods:

  • Analytic arguments and derivations.
  • Numerical simulations of the quantum hopping model.
  • Mapping the model to a long-range epidemics model for comparative analysis.

Main Results:

  • Established the phase diagram of the banded random matrix model.
  • Identified a genuine localization transition occurring when hopping rate decreases slower than ℓ^{-2}.
  • Observed a transition in the decay of localized states from exponential to stretched exponential and finally to a logarithmic form exp(-Cln^{κ}ℓ).

Conclusions:

  • The study reveals a rich phase diagram for sparse, long-range quantum hopping.
  • Localization transitions and mobility edges are present under specific conditions.
  • The decay of localized states exhibits distinct behaviors depending on the hopping rate, highlighting the impact of long-range interactions.