Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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Andreev Billiards 141 Equation (25) for the Andreev map has the same form as (22) for the Andreev billiard (with α → −i). In particular, both equations have roots that lie symmetrically around zero. A specific realization of the Andreev map is the Andreev kicked rotator. (See [30] for a different realization, based on the kicked Harper model.) The normal kicked rotator has Floquet operator [31] � F =exp i �τ0 × exp 4I0 � i �τ0 4I0 ∂2 ∂θ2 � � exp −i KI0 ∂ 2 ∂θ 2 � cos θ �τ0 � . (27) It describes a particle that moves freely along the unit circle (cos θ, sin θ) with moment of inertia I0 for half a period τ0, is then kicked with a strength K cos θ, and proceeds freely for another half period. Upon increasing K the classical dynamics varies from fully integrable (K = 0) to fully chaotic [K > ∼ 7, with Lyapunov exponent α ≈ ln(K/2)]. For K < 7 stable and unstable motion coexist (mixed phase space). If needed, a magnetic field can be introduced into the model as described in [32]. The transition from classical to quantum behavior is governed by the effective Planck constant heff ≡ �τ0/2πI0. For1/heff ≡ M an even integer, F can be represented by an M × M unitary symmetric matrix. The angular coordinate and momentum eigenvalues are θm =2πm/M and pm = �m, with m =1, 2,...M, so phase space has the topology of a torus. The NS interface is an annulus around the torus, either in the θ-direction or in the p-direction. (The two configurations give equivalent results.) The construction (23) produces a 2M × 2M Floquet operator F, which can be diagonalized efficiently in O(M 2 ln M) operations [rather than O(M 3 )] by combining the Lanczos technique with the fast-Fourier-transform algorithm [33]. 6 Random-Matrix Theory An ensemble of isolated chaotic billiards, constructed by varying the shape at constant area, corresponds to an ensemble of Hamiltonians H with a particular distribution function P (H). It is convenient to think of the Hamiltonian as a random M × M Hermitian matrix, eventually sending M to infinity. The basic postulate of random-matrix theory (RMT) [9] is that the distribution is invariant under the unitary transformation H → UHU † , with U an arbitrary unitary matrix. This implies a distribution of the form P (H) ∝ exp[−Tr V (H)] . (28) If V (H) ∝ H 2 , the ensemble is called Gaussian. This choice simplifies some of the calculations but is not essential, because the spectral correlations become
142 C.W.J. Beenakker largely independent of V in the limit M →∞. More generally, the ensemble of the form (28) is called the Wigner-Dyson ensemble, after the founding fathers of RMT. By computing the Jacobian from the space of matrix elements to the space of eigenvalues En(n =1, 2,...M), one obtains the eigenvalue probability distribution [9] P ({En}) ∝ � � β |Ei − Ej| e −V (Ek) . (29) i
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Lecture Notes in Physics Editorial
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W. Dieter Heiss (Ed.) Quantum Dots:
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Preface Nanoscale physics, nowadays
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Contents A Guide for the Reader ...
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A Guide for the Reader Quantum dots
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The Renormalization Group Approach
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RG for Interacting Fermions 5 where
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where � is a shorthand: � ≡ 3
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RG for Interacting Fermions 9 These
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RG for Interacting Fermions 11 is i
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RG for Interacting Fermions 13 the
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RG for Interacting Fermions 15 this
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RG for Interacting Fermions 17 equa
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� p � dθp = g 2π RG for Inter
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RG for Interacting Fermions 21 the
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References RG for Interacting Fermi
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Semiconductor Few-Electron Quantum
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1.2 Implementations Semiconductor F
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GaAs n-AlGaAs AlGaAs GaAs Semicondu
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ε1 ε0 e Semiconductor Few-Electro
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250 Ω twisted pair 20 MΩ powder
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10 5 0 -5 Semiconductor Few-Electro
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G ( e / h) 2 2 1 0 Semiconductor Fe
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V R (V) V R (V) Semiconductor Few-E
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V L (V) -1.084 -1.082 -1.080 -1.078
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B // Semiconductor Few-Electron Qua
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a RESERVOIR 200 nm Semiconductor Fe
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4.5 QPC Versus SET Semiconductor Fe
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a reservoir dot Semiconductor Few-E
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a b c V P (mV) 10 5 0 ∆I QPC E F
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Spin-down counts 200 100 a Semicond
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Page 155 and 156: 146 C.W.J. Beenakker Fig. 6. Ensemb
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Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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