Stray levels

For the numerical calculation of stray levels, we consider the open limit of small ratioM/N, where Andreev levels come in almost degenerate bunches, which are separated by wide “Smile”-gaps. The minimal transmissionT_c, which is determined by the ratio M/N, is close to 1 in this regime. Andreev levels are strongly located at internal connectors. Instead of increasing the ratio M/N, which shifts all transmission eigenvalues to smaller values, we break the gap in the transmission distribution by artificially decreasing a single transmission eigenvalue and keeping the others constant. Because of the correspondence of transmission gap and “Smile”-gap this leads to a disturbance of the “Smile”-gaps by single Andreev levels. It thus allows us to study the closing of a “Smile”-gap at the level of single Andreev levels related to this transmission eigenvalue.

While single Andreev levels penetrate into the “Smile”-gaps, all other levels remain in bunches of almost degenerate Andreev levels corresponding to the particular ratio M/N. For Fig. 6.6 we consider equal numbers of internal modes M =M₀₁ = M₁₂ =M₂₃ = M₃₀ = 100 and a ratio M/N = 1/1000. A single transmission eigenvalue at a single node ((a) node 0, (b) node 1, (c) node 2, (d) node 3) is replaced by T_ext = 0, while the transmission distributions at all other nodes are not changed. Fig. 6.6 (b) demonstrates the evolution of a single stray level, which penetrates the gap as Text is decreased from Tc to T_ext = 0. T_ext has a value betweenT_ext = 0(red curve) andT_ext =T_c (dark blue curve). ForT_ext =T_c the dark blue curve is already absorbed by the bunches and hardly visible. We find stray levels existing out of the quasi-degenerate bunches, which penetrate into the “Smile”-gaps. For T_ext = 0 one channel is fully reflected, leading to a constant E = ∆ level, while a second stray level with a complicatedÏ-dependence breaks the topological protection of the

“Smile”-gaps. Its complicatedÏ-dependence is related to three superconducting phases appearing in the equation for this level. Its shape also depends on the unitary matrix entering the scattering matrix at the particular node. It is calculated analytically in the next section. Fig. 6.7 shows stray levels for T_ext = 0 at all nodes. We find that none of the stray levels penetrates the three gaps, which are marked in blue in Fig. 6.7. Even a different random unitary matrixU, which leads to a different shape of theText = 0-curve (red),

Figure 6.5: Sketch of three bunches which cross at three points and surround a “Smile”-gap. The bunches have finite width, which is related to a gap in the transmission distribution of a chaotic cavity. This finite width is symbolized by the red boundaries of the bunches of Andreev levels. At each crossing point the number of Andreev levels in each bunch is conserved.

cannot change this. These gaps are thus very stable, and do not vanish even if the ratio M/N is decreased (see Fig. 6.3 and 6.4).

Analytics

In this section we derive an analytic expression for the stray levels in the extreme open regime M/N æ0. In this limit all transmission eigenvalues at all nodes are exactly T = 1 and Andreev levels exist in degenerate bunches.

We replace a single transmission eigenvalue at node (0)by T_ext and calculate the stray level related to this transmission eigenvalue. The calculation for a replacement at a different node is straightforward. Scattering matrices at nodes (i”= 0) are given by

S_e^(i”=0) =✓ 0 1 1 0

◆

, (6.21)

where we have chosen the unitary matrices U^(i”=0) = 1, which can be done without any loss of generality, because there is ideal transmission in all channels and phases of holes cancel those of electrons. The scattering matrix at node

6.4 The spectrum 107

Figure 6.6: Comparison of stray levels appearing in the spectrum if one transmission eigenvalue at a single node ((a) node 0, (b) node 1, (c) node 2, (d) node 3) is replaced byT_ext = 0(red curves). We consider equal numbers of

internal modesM₀₁=M₁₂=m₂₃=M₃₀= 100 and a ratioM/N = 0.001. In (b) stray levels are plotted fromT_ext =T_c (blue) to T_ext = 0(red). All other

levels are plotted in black.

(0) is given by

where matrices A and B are:

A=✓Ô1≠T_ext

Figure 6.7: Stray levels for an additional transmission eigenvalueT_ext= 0 at all nodes. No stray levels penetrate the low-energy gaps marked in blue. These gaps are very stable if the ratioM/N is decreased.

The unitary matrix U⁽⁰⁾ can be chosen as

U⁽⁰⁾ = 0 BB

@ 1M≠1

cos(–)e^≠i— sin(–)e^≠i“

≠sin(–)e^i“ cos(–)e^i—

1M≠1

1 CC

A, (6.24)

–, — and “ being uniformly distributed in the interval [0,2fi]. This choice is possible because it doesn’t matter which and how many of the T = 1-channels are mixed with the Text channel by the unitary matrix U⁽⁰⁾, since they are all of equal transmission. We thus mix it only with a single one of them, the strength of the mixing is described by the angle –. In this regime the node(2) is not important for stray levels related to Text and we can consider a simple scattering problem with scattering at a single node (0), whose scattering is described by the matrix (6.22), which is connected to three superconducting reservoirs. For the description of Andreev reflection at these superconductors

6.4 The spectrum 109

where it is important to note that there are two subblocks of phaseÏ0 because of2M channels leading to this superconductor, whereas only M channels lead to the superconductors of phases Ï₁ and Ï₂ respectively. The determinant det[e^2i‰≠ÏSˆ e^(0)úψ^úSe⁽⁰⁾]can be calculated analytically for a generalText, however it becomes a quite lengthy expression. We restrict ourselves to the extremal case T_ext = 0, where it becomes This expression and thus the stray level depends only on –. — and “ drop out in the calculation. As expected from numerics there is a double root ate^2i‰ = 1, which generates a constant level at E =∆. The other two roots can be found by solving a quadratic equation. The corresponding energies are given by

E

where the only random parameter is –. In the following we consider the (Ï0,Ï1,Ï2,Ï3) = (0,1,3,6)Ï parametrization. Fig. 6.8 (a) shows the analytic solution of the stray level from – = 0 (purple) to – = fi/2 (red) in steps of fi/16. For Fig. 6.8 (b) we choose – = 0.229fi (blue curve) which leads to a maximal overlap with the numerical data (red/black).

π/2 π π/4 3π/4 00

0.2 0.4 0.6 0.8 1

φ

E / Δ

(a) (b)

Figure 6.8: (a) Analytical solution for the stray level with T_ext = 0 for different values of the parameter – from – = 0 to – = fi/2. (b) Fit of the analytic expression of the stray level (blue) to the numerically calculated curve (red). Good agreement is found for–= 0.229fi.