Order, disorder, and tunable gaps in the spectrum of Andreev bound states in a multiterminal superconducting device

We consider the spectrum of Andreev bound states (ABSs) in an exemplary four-terminal superconducting structure where four chaotic cavities are connected by quantum point contacts to the terminals and to each other forming a ring. We nickname the resulting device 4T-ring. Such a tunable device can be realized in a 2D electron gas-superconductor or a graphene-based hybrid structure. We concentrate on the limit of a short structure and large conductance of the point contacts where there are many ABS in the device forming a quasicontinuous spectrum. The energies of the ABS can be tuned by changing the superconducting phases of the terminals. We observe the opening and closing of gaps in the spectrum upon changing the phases. This concerns the usual proximity gap that separates the levels from zero energy as well as less usual “smile” gaps that split the levels of the quasicontinuous spectrum. We demonstrate a remarkable crossover in the overall spectrum that occurs upon changing the ratio of conductances of the inner and outer point contacts. At big values of the ratio (closed limit), the levels exhibit a generic behavior expected for the spectrum of a disordered system manifesting level repulsion and Brownian “motion” upon changing the phases. At small values of the ratio (open limit), the levels are squeezed into narrow bunches separated by wide smile gaps. Each bunch consists of almost degenerate ABS formed by Andreev reflection between two adjacent terminals. We study in detail the properties of the spectrum in the limit of a small ratio, paying special attention to the crossings of bunches. We distinguish two types of crossings: (i) with a regular phase dependence of the levels and (ii) crossings where the Brownian motion of the levels leads to an apparently irregular phase dependence. We work out a perturbation theory that explains the observations both at a detailed level of random scattering in the device and at a phenomenological level of positively defined random matrices. The unusual properties of the spectrum originate from rather unobvious topological effects. The topology of the first kind is restricted to the semiclassical limit and related to the winding of the semiclassical Green function. It is responsible for the closing of the proximity gaps. The topology of the second kind comes about the discreteness of the number of modes in the point contacts and is responsible for the smile gaps. The topology of the third kind leads to the emergence of Weyl points in the spectrum and is not discussed in the context of this article.

Figure 1

Sketch of a possible experimental realization of a 4T-ring geometry using a 2D electron gas. Multiple gates (gray regions) are used to deplete the 2D gas and to form four chaotic cavities, which are connected to each other and build a ring structure (red blurry region). A gate inside the ring can be fabricated by air-bridge technique [35]. Each cavity is connected to the superconducting terminals (yellow regions) through ballistic contacts. ${\phi }_i$ ($i=0,1,2,\text{and}3$) indicates the superconducting phase in the terminal $i$. A challenge might be the realization of the central gate, that has to be contacted without disturbing the other connections.

Figure 2

Circuit-theory scheme of the 4T-ring. $G_k^{\mathrm{i}\left(\mathrm{o}\right)}$ denotes the conductance of the inner (outer) connector $k$. The nodes at the ends of the connectors and the superconducting terminals are characterized by matrix voltages ${\widehat{G}}_k$ and ${\widehat{G}}_k^{\mathrm{S}}$, respectively.

Figure 3

The scattering matrix of the 4T-ring (dimension ${\sum }_k{N}_k\times {\sum }_k{N}_k$) is composed of the random unitary matrices ${\widehat{s}}^{\left(i\right)}$ of the chaotic cavities. The dimension of such a matrix is $(N_i+M_i+M_{i-1})\times (N_i+M_i+M_{i-1})$.

Figure 4

Topological numbers in the 4T-ring.

Figure 5

Density of states at $\epsilon =0$ along the line $({\phi }_1,{\phi }_2,{\phi }_3)=(1,3,6)\phi$ in the open regime. The parameter $M/N$ takes values 0.05, 0.1, 0.2, and 0.5 as indicated by labels in the rectangular frames. The topological numbers of the gapped states are computed and given in the figure as $n_0{n}_1{n}_2{n}_3$.

Figure 6

Density of states (DOS) at $\epsilon =0$ along the line $({\phi }_1,{\phi }_2,{\phi }_3)=(1,3,6)\phi$ in the closed regime. The parameter $M/N$ takes values 1, 2, 5, and 50 for alternating curves of small and big thickness. The smaller values of the parameter corresponding to smaller peak DOS, so that the lowest thick curve corresponds to $M=N$. The topological numbers of the gapped states $n_k$ are given. As expected, no state with $n_4\equiv {\sum }_k{n}_k\ne 0$ occurs in this regime. The arrow indicates ($111-3$) state.

Figure 7

Number of ABS $N\left(\epsilon \right)$ vs energy at $\phi =0.3\pi$ (top) and $0.6\pi$ (bottom) on the line $({\phi }_1,{\phi }_2,{\phi }_3)=(1,3,6)\phi$ in the open regime. The parameter $M/N$ takes values ${10}^{-3}, {10}^{-2}$, 0.1, 0.2, and 0.5 for alternating curves of small and big thickness exhibiting a progressive transition from steplike to smooth shape. The most steplike thin curve corresponds to $M/N={10}^{-3}$. $N_{\mathrm{ABS}}=4M$.

Figure 8

Number of ABS $N\left(\epsilon \right)$ vs energy at $\phi =0.3\pi$ (left) and $0.6\pi$ (right) on the line $({\phi }_1,{\phi }_2,{\phi }_3)=(1,3,6)\phi$ in the closed regime. The parameter $M/N$ takes values 1, 2, 5, and 50 from the lower to the upper curve. The arrows indicate the formation of the smile gap at $N\left(\epsilon \right)=0.5N_{\mathrm{ABS}}$. Note that the lines do not cross with each other. $N_{\mathrm{ABS}}=2N$.

Figure 9

Energy spectrum of ABS in the 4T-ring for various ratios $r=M/N$ of inner and outer numbers of channels. The superconducting phase is swept on the line $({\phi }_1,{\phi }_2,{\phi }_3)=(1,3,6)\phi$. Only 400 positive ABS energies are shown in the left panels (a) to (d) and 200 levels in the right panels (e) to (g). The left panels show the ABS energies in the open regime, $r={10}^{-3}$ (a), ${10}^{-2}$ (b), ${10}^{-1}$ (c), and 0.2 (d). Here, the number of inner channels in each QPC differ slightly, $M_0=100,M_1=120,M_2=70,\text{and}{M}_3=110$. The average is $M=100$. Right panels show the ABS energies when the ratio is $r=1$ (e), 2 (f), and 5 (g). For these cases, $M_i$ are all the same and the number of outer channels is fixed to $N=100$ in each terminal. Capital latin letters denote the gapped states with distinct topological numbers, $\mathrm{A}:0000, \mathrm{B}:000-1, \mathrm{C}:001-1, \mathrm{D}:011-2, \mathrm{E}:011-3$.

Figure 10

Energy spectrum of ABS in the 4T-ring for various ratios $r=M/N$ of inner and outer numbers of channels. The superconducting phases satisfy $({\phi }_1,{\phi }_2,{\phi }_3)=(1,5,10)\phi$. The other parameters are the same as in Fig. 9. Capital latin letters denote the states with distinct topological numbers, $\mathrm{A}:0000, \mathrm{B}:000-1, \mathrm{C}:001-1, \mathrm{D}:011-1, \mathrm{E}:011-2, \mathrm{F}:011-3, \mathrm{G}:012-3, \mathrm{H}:012-4, \mathrm{I}:022-4, \mathrm{J}:022-5$.

Figure 11

Energy spectra of ABS in the 4T-ring for various ratios $r=M/N$ of inner and outer numbers of channels. The superconducting phases satify $({\phi }_1,{\phi }_2,{\phi }_3)=(1,1,2)\phi$. The other parameters are the same as in Fig. 9. The topological numbers of the two gapped states are given in the figure.

Figure 12

Sketch of three bunches that cross at three points surrounding a smile gap. The bunches have finite width, which is related to the finite width of the transmission distribution in the nodes. This finite width is given by the thick (red) curves bounding the bunches of Andreev levels. At each crossing point the number of Andreev levels in each crossing bunch must be conserved. Bundles indicated by the same colored circles consist of the same number of Andreev levels.

Figure 13

Stray levels induced inside the smile gaps by replacing a transmission eigenvalue at a single node with the value $T_{\mathrm{ext}}$ in the transmission distribution gap. We take equal numbers of internal modes $M_i=M=100$ and a ratio $M/N=0.001$. For (a), (c), and (d), $T_{\mathrm{ext}}=0$ and the eigenvalue is replaced in the nodes 0, 2, and 3, respectively. In (b), stray levels are plotted for a set of $T_{\mathrm{ext}}$ changing from 0 (red curves) to $T_{\mathrm{c}}$ (blue curve) in steps of $T_{\mathrm{c}}/8$. The eigenvalue is replaced in node 1.

Figure 14

Stray levels for extra transmission eigenvalues at all four nodes. At each node, one eigenvalue is replaced to $T_{\mathrm{ext}}=0$. The number of Andreev levels is the same as that in Fig. 13 ($4M=400$). No stray levels penetrate the proximity gaps marked in green, those survive in the closed limit.

Figure 15

Analytical solution given by Eqs. (32) and (33). (a) Stray level with $T_{\mathrm{ext}}=0$ at node 0 for a set values of the parameter $\alpha$ ranging from 0 to $\pi /2$ in steps of $\pi /16$. (b) Fit of the analytic expression of the stray level (blue) to the numerically calculated curve (red) in the case of $\alpha =0.229\pi$.

Figure 16

Numerics: two types of bunch crossings in the open limit. The Andreev levels and assumed parameters are referred from Fig. 9. The parameters for the number of levels in the bunches is $M_0=100, M_1=120$, and $M_3=110$. The ratio is $M/N={10}^{-3}$. (a) and (b) illustrate a regular and an irregular crossing, respectively. Red lines show excess levels, their number being $M_3-M_0$ in (a) and $M_1-M_0$ in (b). The dashed blue lines indicate the reference curves of the bunches, $cos({\phi }_{10}/2)$ and $cos({\phi }_{03}/2)$ in (a) and $cos({\phi }_{10}/2)$ and $cos({\phi }_{12}/2)$ in (b).

Figure 17

Comparison of the first-order perturbation results in Eq. (48) with full numerical results for two ratios $r={10}^{-3}$ [(a) and (c)] and $r={10}^{-4}$ [(b) and (d)] in the vicinity of one regular crossing at $\phi =2\pi /7\equiv {\phi }_{\mathrm{c}}$. (a) and (b) demonstrate the Andreev levels (thin black curves) and analytical results by the perturbation with $g=\langle g_i\rangle =r$ (thick solid green line) and $g=0,4r$ (thick dashed green line). The dotted line in (a) and (b) indicates the half-sum of the two energies for $g_i=0, {\overline{\epsilon }}_{g_i=0}=({\epsilon }_{g_i=0,+}+{\epsilon }_{g_i=0,-})/2$, namely, the first two terms in Eq. (48). (c) and (d) are corrected results by substracting the half-sum from Eq. (48). The blue solid curve in (c) and (d) indicates the points of minimum separation of two anticrossing levels for various $g_i$. This curve implies asymmetric deviation with respect to $\phi -{\phi }_{\mathrm{c}}$. The dotted lines indicate zero.

Figure 18

Parametric dependence of the eigenvalues of $H_{2\mathrm{nd}}\left(c\right)$ given by Eq. (52). Transmission and reflection matrices in the expression are computed from a random realization of the scattering matrices. We set $M=100$ (matrix size is $100\times 100$). The ratio is $M/N={10}^{-3}$. (a) The eigenvalues of $H_{2\mathrm{nd}}\left(c\right)$. In $0<c<2$, they are all positive. (b) The velocities vs the eigenvalues at $c=0.1$ (top), 0.7 (middle), and 2 (bottom panel). In the bottom panel, we also give a linear fit.

Figure 19

Fine structure of the bunch ($M=100, M/N={10}^3$). (a) The ABS energy shifts from $cos({\phi }_{10}/2)$ in a narrow interval centered at $\phi =0.73\pi$ [see Fig. 9 for a bigger plot]. (b) The level velocities $v_i={\langle (i{e}^{-i2\chi }\sqrt{1-{(\epsilon /\mathrm{\Delta })}^2}/2)\partial S/\partial \phi \rangle }_i$ at $\phi =0.73\pi$ vs the energy shifts ${\epsilon }_i$. The fitting line is given by $v=a(\epsilon /\mathrm{\Delta })$ ($a\simeq 1.53412$). The lower panel shows the velocities upon subtracting the linear fit. (c) ABS energies upon subtracting ${\epsilon }_i(\phi =0.73\pi )\{exp\left(a(\phi -0.73\pi )\right)-1\}$ from each curve. The remaining dependence is mostly irregular.

Figure 20

Irregular crossing. We plot the eigenvalues and the velocities of $H_{\mathrm{eff}}\left(\xi \right)$ given by Eq. (61). The parameters are the same as in Fig. 18. The matrix size of $H_{\mathrm{eff}}\left(\xi \right)$ is $200\times 200$ ($M_0=M_1=100$). (a) Eigenvalues of $H_{\mathrm{eff}}\left(\xi \right)$. The left panel shows all eigenvalues. Dashed red lines in the left panel correspond to $H=\pm \xi /2$. In the right panel, we plot three groups of selected eigenvalues with indexes $i=1\sim 6$ (black), $81\sim 86$ (red), and $101\sim 106$ (blue). (b) Velocities of the corresponding eigenvalues (a). (c) The velocites vs the eigenvalues $\xi =0$ (top), 0.01 (middle), and 0.02 (bottom).