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

144 C.W.J. Beenakker
The effective Hamiltonian Heff should not be confused with the Bogoliubov-de
Gennes Hamiltonian HBG, which contains the superconducting order parameter
in the off-diagonal blocks [cf. (2)]. The Hamiltonian HBG determines the
entire excitation spectrum (both the discrete part below ∆ and the continuous
part above ∆), while the effective Hamiltonian Heff determines only the
low-lying excitations Ep ≪ ∆.
The Hermitian matrix Heff (like HBG) is antisymmetric under the com-
bined operation of charge conjugation (C) and time inversion (T )[39]:
Heff = −σyH T �
0
effσy, σy =
i
�
−i
.
0
(37)
(An M × M unit matrix in each of the four blocks of σy is implicit.) The CT -
antisymmetry ensures that the eigenvalues lie symmetrically around E =0.
Only the positive eigenvalues are retained in the excitation spectrum, but the
presence of the negative eigenvalues is felt as a level repulsion near E =0.
6.2 Excitation Gap
In zero magnetic field the suppression of the density of states ρ(E) around
E = 0 extends over an energy range ET that may contain many level spacings
δ of the isolated billiard. The ratio g � ET /δ is the conductance of the point
contact in units of the conductance quantum e2 /h. Forg≫ 1 the excitation
gap Egap � gδ is a mesoscopic quantity, because it is intermediate between
the microscopic energy scale δ and the macroscopic energy scale ∆. One can
use perturbation theory in the small parameter 1/g to calculate ρ(E). The
analysis presented here follows the RMT of Melsen et al. [22]. An alternative
derivation [40], using the disorder-averaged Green function, is discussed in the
next sub-section.
In the presence of time-reversal symmetry the Hamiltonian H of the isolated
billiard is a real symmetric matrix. The appropriate RMT ensemble is
the GOE, with distribution [9]
�
P (H) ∝ exp − π2
�
Tr H2 . (38)
4Mδ2 The ensemble average 〈···〉is an average over H in the GOE at fixed coupling
matrix W . Because of the block structure of Heff, the ensemble averaged Green
function G(E) =〈(E −Heff) −1 〉 consists of four M × M blocks G11, G12, G21,
G22. By taking the trace of each block separately, one arrives at a 2×2 matrix
Green function
G =
� G11 G12
G21 G22
�
= δ
π
� Tr G11 Tr G12
Tr G21 Tr G22
(The factor δ/π is inserted for later convenience.)
�
. (39)