6.4 The spectrum
6.4.1 Extreme open limit
In the extreme open limit the number of modes towards the superconductors N is much larger than the number of internal modes M. For M/N æ 0 the lower boundary of transmission eigenvalues Tc æ 1, which means that all transmission eigenvalues are pinned toT = 1. The scattering matrix at each node becomes
and the matrices Eq. (6.11) - (6.14) are in this regime Xˆ = ˆW =0, Zˆ= ˆUOˆ2, Yˆ = ˆO1UˆT. The full normal region scattering matrix (6.6) becomes
Sˆe= ˆZYˆ = ˆUOˆUˆT. (6.18) and the determinant condition (6.1) reads
deth
e2i‰≠e|iψOeˆ{z≠iψOˆ}
eiÕˆ
i = 0, (6.19)
where we used that the commutator [ ˆU , eiψ] = 0. Uˆ is thus removed from Eq.
(6.19) by an appropriate unitary transformation. Õˆ is a diagonal matrix con-taining phase differences of neighboring terminals. We find for the determinant condition
Y3 j=0
e2i‰≠ei(Ïj≠Ïj+1) M e2i‰≠ei(Ïj+1≠Ïj) M = 0, (6.20) where we want to express a product over all neighboring phase differences.
In this sense the index 3 + 1 means 0. We find sets of degenerate solu-tions which depend only on a single phase difference and correspond to the solutions of 2-terminal Josephson junctions with ideal transmission T = 1:
Ej =±∆cos((Ïj+1≠Ïj)/2). In the following we use the notation Ïij = Ïi≠Ïj. 6.4.2 Numerical results
We calculate Andreev levels numerically by solving the eigenvalue problem described in section 6.3.2 for r = M/N < 1. This regime we call the open regime, since each node is strongly coupled to the neighboring superconductor and the coupling within the ring is weak. In the closed regime, which is not discussed in this work, but is considered in a not yet published paper, the coupling of the nodes inside the ring is stronger than the coupling to the superconductors. In the extreme closed regime M/N æ Œthe four nodes of the ring behave effectively like a single node.
Choosing the phase in the (0)-terminal to Ï0 = 0, we can characterize the phase-configuration of the superconductors by a vector in three-dimensional phase space: Ï˛ = (Ï1,Ï2,Ï3)T. Andreev levels are calcualted for two different phase-space trajectories: Fig. 6.3 is for (Ï1,Ï2,Ï2) = (1,3,6)Ï, Fig. 6.4 is for (Ï1,Ï2,Ï2) = (1,5,10)Ï, the parameter Ï is increased from0æfi, which corresponds to a straight line in phase-space. In each case we consider different values of r ranging from r = 0.001 to r ¥ 0.49. For r = 0.001 (Fig. 6.3 (a) and Fig. 6.4(b)) we find bunches of almost degenerate Andreev levels. Each bunch is attributed to a single phase-difference of neighboring superconductors, which interact only weakly among each other and with levels of other bunches.
Andreev levels in a bunch are strongly localized at the internal connector between the corresponding nodes. The number of levels within each bunch is equal to the number of modes M inside the particular connector. The bunches are separated by wide gaps, which close if two bunches cross at certain values of Ï. Also around E = 0 there are gaps, which are bounded by zero energy
6.4 The spectrum 101 crossings of single bunches.
Two types of crossing points are distinguished: Regular crossings have a very regular behavior as a function of Ï, whereas chaotic crossings are strongly fluctuating. Two examples for each type of crossing are marked in Fig. 6.3 (a).
The red crossing 2 and the yellow crossing 4 are of the regular type, whereas the blue crossing 1 and the green crossing 3 are of the chaotic type. These four crossings are plotted in more detail in Fig. 6.9. It turns out that regular crossings are due to perturbative corrections of first order and appear between bunches corresponding to neighboring internal connectors, whereas at chaotic crossings, the first order correction vanishes, and the leading contribution is of second order in the deviation from the extreme open limit (section6.4.1).
The green crossing 3 is an exception. It is of the chaotic type although it appears between bunches corresponding to neighboring internal connectors.
This happens because the first order correction, which would principally exist between these bunches, vanishes due to the special values ofÏ1, Ï2 and Ï3 at this crossing, which lead to a vanishing prefactor in front of the first order term.
These are very rare events and it happens only because of the special choice of the phase space trajectory in this example. The properties of crossings and their perturbative treatment is discussed in detail in section6.5.2.
By increasing the ratio r, the bunches become broader. For r = 0.01 the boundaries of the bunches and also the gaps between them are still well defined and no levels are penetrating the gaps. By further increasing the the ratio, some gaps begin to vanish due to overlapping bunches, others however are more stable and survive up to high values of r. Especially some gaps around E = 0 seem to be very stable. An explanation for this stability is given in section 6.4.4. At r= 0.49 almost all gaps have vanished, except some aroundE = 0 and one at Ï = fi, which still exists close to E = ∆. Outside of these gaps Andreev levels are continuously distributed and no separate bunches can be distinguished anymore. No qualitative difference is found for the two different phase space trajectories of Fig. 6.3 and Fig. 6.4.
6.4.3 Topological protection of the “Smile”-gaps
The scattering in each of the nodes is described by a scattering matrix of an asymmetric cavity. A consequence of this asymmetry is that transmission eigenvalues are not distributed in the whole interval [0,1], but there is a minimal transmission eigenvalueTc. Only in the regime of an infinite number of transport channelsTc is a hard boundary. For a finite number of channels mesoscopic fluctuations allow for single transmission eigenvalues to exist below
(a)
Figure 6.3: Andreev levels in the 4-T ring, numerically calculated from Eq.
(6.1) using the reduced scattering matrix approach. Andreev levels are plotted along a(Ï1,Ï2,Ï2) = (1,3,6)Ïphase-space trajectory for different values of the ratior =M/N ranging fromr= 0.001in (a) to r= 0.49in (f). For small values of r Andreev levels appear in quasi degenerate bunches, which are separated by well defined gaps. With increasing r the bunches get broader and the gaps begin to vanish due to overlap of bunches until the Andreev levels are distributed quasi-continuously and no bunches are distinguishable anymore. The crossings marked in (a) are discussed in detail in sec. 6.5.2.
6.4 The spectrum 103
Figure 6.4: Qualitatively no big difference exists between this phase space parametrization (Ï1,Ï2,Ï2) = (1,5,10)Ï and Fig. 6.3. Because of different phase differences between the superconductors, the crossings appear at different points, and the bunches have different shapes. Their qualitative behavior with increasingr corresponds to the one of Fig. 6.3.
Tc. However, the according scattering matrices are very improbable and such transmission eigenvalues are exponentially suppressed. It is known from the analytical calculations in section 4.4.4, that there is a correspondence between the existence of a gap in the transmission distribution of chaotic cavities and the appearance of gaps in the Andreev spectrum in proximity systems.
In the open regimeTc is close to1and there are bunches of almost degenerate Andreev levels, which are separated by large “Smile”-gaps. The bunches have finite width which is related to the finite value of Tc. There is a hard edge on one side with a high level density, where the bunch is confined by the curve of a level with ideal transmissionT = 1. Since transmission eigenvalues aboveT = 1 are not possible, not even mesoscopic fluctuations could break these edges. On the other side the levels lie less dense and the boundary of the bunch is defined by the curve of a level corresponding to Tc. Due to mesoscopic fluctuations this edge is no hard edge, but has a rather soft character. The different level densities at the two edges are related to different densities of transmission eigenvalues. At T = 1 the transmission distribution diverges, leading to a very dense distribution of Andreev levels, whereas at Tc the distribution remains finite. The exponential suppression of transmission eigenvalues belowTcdirectly translates into an exponential suppression of Andreev levels inside the gaps leading to an exponential protection of the “Smile”-gaps. The number of levels in each bunch is constant and equal to the number of transport modes in the corresponding internal connector. These properties are summarized in Fig. 6.5, where the crossings of three bunches, which surround a smile gap, are sketched.
The red lines indicate the finite width of the bunches. The number of levels in each bunch must be conserved at each crossing. If the two bunches have different numbers of levels, corresponding to different numbers of modes in the internal connectors, some levels have to cross straight through the crossing point in order to assure this.
The gap in the transmission distribution can be influenced in different ways:
Either by increasing the ratioM/N which leads to a decrease ofTc, or by adding artificially additional transmission eigenvalues to the spectrum. In both cases, the gap in the transmission spectrum is violated, which leads to a disturbance of the “Smile”-gaps in the Andreev spectrum. In the case of an increasing ratio M/N the gap in the transmission spectrum vanishes monotonously, and so do the gaps. The width of the bunches grows until the “Smile”-gaps vanish. It turns out that some “Smile”-gaps around zero energy are more stable than others and survive up to very high values of M/N. This stability can be understood from the spectrum of stray levels which are generated by adding
6.4 The spectrum 105 additional transmission eigenvalues out of the general distribution of a chaotic cavity. This leads to the generation of single Andreev levels inside the smile gaps, which are considered in detail in section6.4.4.
6.4.4 Stray levels Numerics
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 transmissionTc, 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 Text = 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 Text = 0. Text has a value betweenText = 0(red curve) andText =Tc (dark blue curve). ForText =Tc 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 Text = 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 Text = 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 Text 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
Se(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 byText = 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 fromText =Tc (blue) to Text = 0(red). All other
levels are plotted in black.
(0) is given by
where matrices A and B are:
A=✓Ô1≠Text
Figure 6.7: Stray levels for an additional transmission eigenvalueText= 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(–)ei“ cos(–)ei—
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[e2i‰≠ÏSˆ e(0)úψúSe(0)]can be calculated analytically for a generalText, however it becomes a quite lengthy expression. We restrict ourselves to the extremal case Text = 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 ate2i‰ = 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 Text = 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.
6.5 Perturbative description of crossings in the open limit
In this section we consider the open regime M/N π1, where Andreev levels appear in almost degenerate bunches (see for example Fig. 6.3 (a) and Fig. 6.4 (a)). We investigate in detail the properties of the Andreev spectrum close to the crossing points of two bunches. For this purpose we treat the deviations from the extreme open limit, which was described in section 6.4.1, in a perturbative way. The phase space parametrization is again fixed to (Ï1,Ï2,Ï3) = (1,3,6)Ï.
We distinguish two types of crossing points and relate the difference between both to the different role of perturbative corrections at different crossings.
6.5.1 Numerical results for crossings in the open limit
In the numerical analysis of the Andreev spectrum close to crossing points between bunches of different internal connectors we find two different types of crossings. The first type we call a “regular crossing”. It is characterized by a very regular behavior of Andreev levels as a function of phase without strong fluctuations. Two examples for this type of crossing are marked in Fig. 6.3 (a) in red and yellow. The second type of crossing we call “chaotic crossing”.
Close to the crossing point Andreev levels have a very irregular behavior and fluctuate strongly as a function of phase. Two examples of chaotic crossings are marked in Fig. 6.3 (a) in blue and green. These four examples are plotted in detail in Fig. 6.9. Here we discuss qualitative differences in the numerical results for the two types. Quantitative differences are calculated in Sec. 6.5.3.
6.5 Perturbative description of crossings in the open limit 111
398 399 400 401 402 0.3
398 399 400 401 402 0.3
Figure 6.9: We distinguish two different types of crossings: The first type we call “regular” crossing, the second type are “chaotic” crossings. All crossings considered here are marked in Fig. 6.3 (a) with colors and numbers. The parameters are the same like those of Fig. 6.3 (a): N = 100000,M = 100. (a) Regular crossing “2” between theÏ10 and theÏ03 bunches (red). (b) Regular crossing “4” between theÏ23and theÏ03bunches (yellow). (c) Irregular crossing
“1” between the Ï21 and the Ï03 bunches (blue). (d) Irregular crossing “3”
between theÏ21 and the Ï23 bunches (green). The red dashed lines correspond to the degenerate bunches of the extreme open limitM/N æ0.
One of the most obvious differences concerns the region where the two bunches overlap. Whereas this region is for regular crossings just a point, corresponding approximately to the point where the two bunches cross in the limitM/N æ 0, irregular crossings have an overlap in a region of finite size.
Before and after this region the width of the bunches seems to be roughly constant. At regular crossings the bunches increase in width already relatively far from the actual crossing point.
Another difference concerns the gaps between the two bunches. Whereas
for regular crossings the gap seems to increase more or less linearly with the distance from the critical phase (∆Egap ≥ |Ï≠Ïc|), the irregular crossings seem to have a more complicated functional dependence (∆Egap≥|Ï≠Ïc|x).
This non-linear functional dependence leads to the typical “smile” shape of the gaps.
A third point becomes clear by considering the positions of the red dashed lines, which correspond to the degenerate levels in the extreme open limit M/N æ 0, with respect to the numerical results. For regular crossings the position of the numerical results with respect to these reference lines changes at the crossing point: If the numerical results are above the red line before the crossing point, they are below it after the crossing or vice versa. At chaotic crossings the numerical results are always situated above the red lines. A quantitative explanation for the shape of regular crossings within a first order model is given in Sec. 6.5.3.
6.5.2 Perturbation theory
For M/N π 1 but still finite all transmission eigenvalues are close to but slightly below 1. This allows to treat deviations from the levels calculated in section 6.4.1 perturbatively in the small parameters Ô
Á = Ô1≠T. We focus on the points in phase space where in 0th order (sec. 6.4.1) two of the M-degenerate bunches cross, leading to an2M-degeneracy at these points. In order to reach a separation of states with negative energies from those with positive energies, we rewrite Eq. (6.1) as an eigenvalue equation for an effective Hamiltonian
Hˆeff(˛Ï)|ŒÍ=i1≠S(ˆ Ï)˛
1 + ˆS(˛Ï)|ŒÍ= tan(‰)|ŒÍ, (6.27) where Sˆ is the product of all scattering matrices appearing in Eq. (6.1):
1 + ˆS(˛Ï)|ŒÍ= tan(‰)|ŒÍ, (6.27) where Sˆ is the product of all scattering matrices appearing in Eq. (6.1):