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For a discussion of the localization properties of these systems we refer to F. Evers and A. D. Mirlin, arXiv:0707.4378 (unpublished). However, the quasi-one-dimensional systems studied in this work categorically show localization.
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This expectation holds only if the relevant electronic time scales exceed the "Ehrenfest time," the minimum time after which quantum effects appear in ballistic conductors; see Ref..
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This need not be immediately obvious if time-reversal symmetry is present (β=1), since Eq. 5 contains the matrix Gq, which cannot be expressed in terms of Tq. Still, the stochastic process defined by Eqs. 4 fully describes the evolution of the eigenvalues of Tq, which can be verified by considering the mean change of tr (Tq) m or of a product of such traces, which contains Gq in the combination Gq (T) n Gq† = (Tq) n+2 - (Tq) n+3 only, n=0,1,2,....
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This need not be immediately obvious if time-reversal symmetry is present (β=1), since Eq. 5 contains the matrix Gq, which cannot be expressed in terms of Tq. Still, the stochastic process defined by Eqs. 4 fully describes the evolution of the eigenvalues of Tq, which can be verified by considering the mean change of tr (Tq) m or of a product of such traces, which contains Gq in the combination Gq (T) n Gq† = (Tq) n+2 - (Tq) n+3 only, n=0,1,2,....
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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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The Moyal product between two phase space functions A and B is defined as (A B) (x) =exp (1 2 i x′ T I xδ) | A (x′) B (x) | x= x′, where I is the symplectic unit matrix.
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The Moyal product between two phase space functions A and B is defined as (AB) (x) =exp (1 2 i x′ T I xδ) | A (x′) B (x) | x= x′, where I is the symplectic unit matrix.
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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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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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PRLTAO 0031-9007 10.1103/PhysRevLett.75.2750
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N. Argaman, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.75.2750 75, 2750 (1995).
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(1995)
Phys. Rev. Lett.
, vol.75
, pp. 2750
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Argaman, N.1
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