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

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