with a small number of nodes provides already a suitable description, for which this method is a considerable simplification. In chapters 3 and 4 a chaotic cavity is considered, which is described by a single normal node, coupled to two superconducting reservoirs. In order to simulate a system, where the spatial dependence in the normal region is important, in section4.4.6the normal metal is discretized using three nodes which are coupled to two superconductors. A finer subdivision is not necessary for our purposes.
2.3 Random Matrix Theory
A completely different, phenomenological theory of transport in mesoscopic systems is given by the so-called Random Matrix Theory (RMT) [Bee97].
The idea behind this method is the fact that in chaotic systems the matrices, determining their properties, are unknown, because they depend on microscopic details, like for example the exact shape of the boundaries or the distribution of impurities in a diffusive system. These matrices are either the Hamiltonian in case of a closed system, or the scattering matrix in case of an open system.
On the other hand, it is known that these microscopic details should not be important if the systems are large enough, similar as for quasiclassical Green’s functions. Thus the idea of RMT is to start from an ensemble of random matrices and to use as input only the fundamental symmetries of the system [Dys62; Zir11]. These symmetries are time-reversal symmetry (TRS), which can be broken by a magentic field and spin-rotation symmetry (SRS), which is broken by spin-orbit interaction.
2.3.1 General properties
Starting from a matrix with randomly distributed elements, RMT makes predictions on how the eigenvalues and eigenvectors, or in the case of scattering matrices, the transmission eigenvalues, of these matrices are distributed. In most cases, the main interest concerns the eigenvalues (transmission eigenvalues in the case of scattering matrices), because they determine the values of physical observables. In general, they are correlated and their distribution is described by correlation functions. Depending on the particular observable, it might be sufficient to know the average distribution of eigenvalues. Observables for which this is the case are called linear statistics, they are just a sum of functions of single eigenvalues, not necessarily a linear function like the name might suggest. The correlation functions give access not only to the average values of observables, but determine all their statistical properties and can also be used
to calculate their fluctuations.
One of the main assumptions of RMT is that correlations of eigenvalues are purely “geometrical”[Bee97], which means that they are a purely mathemat-ical effect, which arises due to the parametrization of a matrix in terms of eigenvalues and eigenvectors. Physical properties do not generate correlations between eigenvalues themselves. This is the reason why the distribution of eigenvalue spacings is a universal function, which is known as the Wigner-Dyson-distribution, and why many systems which are describable by RMT show universal behavior in all kinds of respects. More details on this topic can be found in [Bee97].
2.3.2 Random Matrix Theory of random Hamiltonians
The original problem which led to the development of RMT was the need for a description of energy levels in heavy nuclei, about whom very little knowledge exists, besides the fundamental symmetries that their Hamiltonians must fulfill.
Using these symmetries as the only input led Wigner and Dyson to study the ensemble
P(H)≥e≠—Tr(V(H)), (2.30) which is known as the Wigner-Dyson-ensemble [Bee97]. Later it turned out that this ensemble is not only applicable to atomic nuclei, but to chaotic systems in general [Efe82; Efe83]. Such systems are called “non-integrable”. It doesn’t matter whether the chaotic character in these systems is due to impurity scattering in a diffusive system, or due to chaotic boundary scattering in a ballistic system. The symmetry index — is 1 in the presence of TRS and SRS, which is the case considered throughout this work. The Hamiltonian is a real, symmetric matrix in this case. V is a general function, which must have the property of preventing eigenvalues from escaping to infinity. Its exact functional form is related to microscopic details of the system. ForV(H)≥H2 the matrix elements are independently distributed, which makes some calculations easier.
The probability distribution of energy eigenvalues related to this ensemble is given by [Bee97]
P({En})≥exp
≠—✓ X
i<j
≠ln(|Ei≠Ej|) +X
i
V(Ei)◆
, (2.31)
which has the form of a Gibbs distribution at inverse temperature— for particles in an external potentialV, including a logarithmic repulsion term. A logarithmic
2.3 Random Matrix Theory 17 repulsion is realized in nature by two line charges, which repell each other electrostatically. This analogy leads to the fact that the repulsion of energy eigenvalues is commonly compared with a Coulomb gas of charged particles.
This analogy is sketched in Fig. 2.2. According to Eq. (2.31) no correlations arise due to the functionV, containing information on the microscopic details of the system. This independence of correlations of this function, is what is meant with “geometrical correlations” in section 2.3.1. In fact, it can be shown that this logarithmic repulsion of energy levels is true only for energy separations smaller than the Thouless energy of the system, setting a limit to the applicability of RMT [Alt86].
2.3.3 Random Matrix Theory of quantum transport
In order to be able to describe transport through open systems, a similar method of random matrices was developed for scattering matrices of mesoscopic systems. This theory is known as RMT of quantum transport. Instead of ensembles of random Hamiltonians it uses ensembles of random scattering matrices to describe transport through chaotic systems. Maybe the most important difference compared to the RMT of random Hamiltonians is that in the case of scattering matrices the interest concerns not directly the eigenvalues of the random matrix, but rather the transmission eigenvalues of the matrix.
These are the eigenvalues of the transmission matrix, multiplied with its hermitian conjugate. The transmission matrix is a submatrix of the original random scattering matrix. Another difference is that the random matrices are unitary instead of hermitian. For TRS and SRS present in the system, a further constraint besides unitarity is that the matrix has to be symmetric. More details on additional properties for the other symmetry classes can be found in [Bee97]. Since the volume of the space of unitary matrices with respect to an appropriately defined Haar measure is finite [Zyc94], no confining potential V is needed in order to define an ensemble of random scattering matrices. A generalization of the Wigner-Dyson ensemble to scattering matrices is
P(S)≥e≠—Tr[V(tt†)], (2.32) t being the transmission matrix. This ensemble leads to a logarithmic trans-mission eigenvalue repulsion, similar to Eq. (2.31). Microscopic details of the system are related to the functionV and generate no correlations.
There are two physical systems for which the statistics of transmission eigenvalues are known: A chaotic cavity and a diffusive connector. Both cases
Figure 2.2: Schematic representation of line charges at positionsEi interacting logarithmically and being forced into a finite region by a confining potential V. Taken from [Bee97].
are discussed in the following. The chaotic cavity in more detail, since this is the system relevant for this work.
Chaotic cavity
The ensemble of scattering matrices for a chaotic cavity can be derived from the Wigner-Dyson-Ensemble of Hamiltonians by coupling the discrete energy levels of the isolated cavity to the transport modes entering and leaving the cavity. Depending on the transmission properties of the leads, one finds different ensembles of scattering matrices. For point contacts with perfect transmission the ensemble of scattering matrices is known as the circular ensemble. This name results from the fact that in this ensemble scattering phase-shifts are uniformly distributed in the interval [0,2fi], which corresponds to a uniform distribution of phase factors on the unit circle in the complex plane. This ensemble is described by a constant probability distribution in the space of unitary matrices:
P(S) = const. (2.33)
It corresponds to the ensemble (2.32) withV = 0. The transmission eigenvalues of the circular ensemble are distributed according to
P({Tn})≥ Y
n<m
|Tn≠Tm|—Y
k
Tk—(|N2≠N1|+1≠2/—)/2, (2.34)
2.3 Random Matrix Theory 19 and thus have a logarithmic repulsion. N1 andN2 are the numbers of transport channels in the two leads. A generalization to non-ideal contacts is given by the so-called Poisson-Kernel [Bee97]
P(S) =|det(1≠S¯†S)|≠—(N1+N2≠1+2/—), (2.35) where S¯is a sub-unitary matrix given by
S¯=✓ r1 0
0 r2
◆ .
r1 and r2 are the reflection matrices of the two leads respectively. The circular ensemble (2.33) follows from Eq. (2.35) forS¯= 0. The validity of Eq. (2.35) is shown in [Bro95], where the transfer matrix method is used to derive the distribution of the full scattering matrix of three scatterers in series. The scattering matrices of the leads are constant and the central scattering matrix of the cavity is assumed to be distributed according to the circular ensemble (2.33). The ensembles (2.33) and (2.35) are very useful if the the energy dependence of S(E) is not needed or in regimes where it is so weak that it can be neglected. In problems where the energy dependence ofS(E) is essential, these ensembles can not be used, but instead one has to use the underlying Hamiltonian ensemble of the cavity. The Hamiltonian contains information on correlations of scattering matrices at different energies. The energy-dependent scattering matrix can be constructed from the Hamiltonian via
S(E) = 1≠2fiiW†⇥
E≠H+ifiW W†⇤≠1
W, (2.36)
where W(E)is a M ◊N coupling matrix describing the coupling of M levels inside the cavity to N = N1 + N2 modes in the leads. The N non-zero eigenvalues ofW W† are given by [Bee97]
wn = M”s
fi2Tn
(2≠Tn±2p
1≠Tn). (2.37)
Tnare the channel transmissions in the leads,”sis the level-spacing of the cavity.
The Hamiltonian H is distributed according to the Wigner-Dyson ensemble (2.30) with V(H) = fi2/(4M”s2)H2, which is thus called Gaussian ensemble.
The sign in (2.37) is arbitrary, the final results do not depend on it.
Disordered wire
The second system for which the distribution of transmission eigenvalues is known, is the disordered wire. There are two ways to approach this problem.
The first is by considering a system of many ballistically coupled chaotic cavities in series, where for each the ensemble of scattering matrices is known.
A second approach is to consider many weakly scattering segments in series and to treat the effect of an additional segment perturbatively. This leads to a differential equation for the probability distribution P, known as the Dorokhov-Mello-Pereyra-Kumar equation [Dor82; Mel88]: results at large length scales. It turns out that the eigenvalue repulsion is non-logarithmic, thus the ensemble of scattering matrices cannot be of the form of Eq. (2.32).