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Volumn 78, Issue 7, 2008, Pages

Anderson localization from classical trajectories

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EID: 49249112663     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.78.075304     Document Type: Article
Times cited : (15)

References (71)
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    • We prefer to use a formulation with continuous momenta p and p′ instead of one with discrete momenta, as used in most of the semiclassical literature. For each set trajectory that contributes to the conductance, there exists a large number (∼ gc) of sets of deformed trajectories that have different momenta upon entrance and exit. The difference between a momentum sum and a momentum integral disappears if gc 1.
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    • In fact, the phase space of individual dots supports a small set of fluctuations that decay on time scales longer than the generic tf: the probability Π (x, x̄ ′, t) to propagate from a phase space point x to the time reversed of a closeby point x′ relaxes on scales ∼ λ-1 ln (EF tf /us), where λ is the dominant Lyapunov exponent of the system and s and u are the locally most stable and unstable coordinate of the point x′ in a coordinate system that has x as its center. Since the phase space resolution of the quantum theory is limited by us∼, these long-time memory effects decay on time scales of the order of the Ehrenfest time τE ∼ λ-1 ln (S/us). Thus, the decoupled dots have relaxed into a fully ergodic configuration on time scales τE < τD.
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