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

摘要

We consider the spectrum of Andreev bound states (ABSs) in an exemplary four-terminal superconductingstructure where four chaotic cavities are connected by quantum point contacts to the terminals and to each otherforming a ring.We nickname the resulting device 4T-ring. Such a tunable device can be realized in a 2D electrongas-superconductor or a graphene-based hybrid structure. We concentrate on the limit of a short structure andlarge conductance of the point contacts where there are many ABS in the device forming a quasicontinuousspectrum. The energies of the ABS can be tuned by changing the superconducting phases of the terminals. Weobserve the opening and closing of gaps in the spectrum upon changing the phases. This concerns the usualproximity gap that separates the levels from zero energy as well as less usual “smile” gaps that split the levelsof the quasicontinuous spectrum. We demonstrate a remarkable crossover in the overall spectrum that occursupon changing the ratio of conductances of the inner and outer point contacts. At big values of the ratio (closedlimit), the levels exhibit a generic behavior expected for the spectrum of a disordered system manifesting levelrepulsion and Brownian “motion” upon changing the phases. At small values of the ratio (open limit), the levelsare squeezed into narrow bunches separated by wide smile gaps. Each bunch consists of almost degenerate ABSformed by Andreev reflection between two adjacent terminals. We study in detail the properties of the spectrumin the limit of a small ratio, paying special attention to the crossings of bunches. We distinguish two types ofcrossings: (ⅰ) with a regular phase dependence of the levels and (ⅱ) crossings where the Brownian motion ofthe levels leads to an apparently irregular phase dependence. We work out a perturbation theory that explainsthe observations both at a detailed level of random scattering in the device and at a phenomenological levelof positively defined random matrices. The unusual properties of the spectrum originate from rather unobvioustopological effects. The topology of the first kind is restricted to the semiclassical limit and related to the windingof the semiclassical Green function. It is responsible for the closing of the proximity gaps. The topology of thesecond kind comes about the discreteness of the number of modes in the point contacts and is responsible forthe smile gaps. The topology of the third kind leads to the emergence of Weyl points in the spectrum and is notdiscussed in the context of this article.