No abstract

Recent years have brought an explosion of activities in the research of topological aspects of condensed-matter systems. Topologically nontrivial phases of matter are typically accompanied by protected surface states or exotic degenerate excitations such as Majorana end states 1,2 or Haldane's localized spinons. 3,4 Topologically protected degeneracies can, however, also appear in the bulk. An intriguing example is provided by Weyl semimetals, where topologically protected electronic band degeneracies and exotic surface states emerge even in the absence of interactions. 5-7 Here we demonstrate experimentally and theoretically that Weyl degeneracies appear naturally in an interacting quantum dot system, for specific values of the external magnetic field. These magnetic Weyl points are robust against spin-orbit coupling unavoidably present in most quantum dot devices. 8,9 Our transport experiments through an InAs double dot device placed in magnetic field reveal the presence of a pair of Weyl points, exhibiting a robust ground state degeneracy and a corresponding protected Kondo effect.Mathematical tools borrowed from topology find more and more applications in contemporary condensedmatter physics. In Weyl semimetals, 7,10 for example, the electronic band structure exhibits isolated degeneracy points, 5 where two bands touch. In three-dimensional systems, these degeneracy points can be protected by topology -and classified by a suitably chosen Chern number: continuous perturbations may displace these Weyl points in momentum space, but cannot break their degeneracy. Weyl point related degeneracies of electronic states in molecules termed conical intersections are also thought to play a fundamental role in various phenomena in photochemistry. 11 They have also been predicted to appear in the context of multi-terminal Josephson junctions 12 and that of photonics, 13 and have also been engineered and demonstrated in coupled superconducting qubits. 14,15 The simplest example of a Weyl point arises when a spin-1/2 electron is placed in a homogeneous magnetic field (see Fig. 1a-c). In this example, the parameter space is spanned by the magnetic-field vector B = (B x , B y , B z ), and the two energy eigenstates are degenerate at B = 0. We can associate a nonzero topological charge to this degeneracy point: the ground state Chern number C(S) = 1 evaluated on an arbitrary closed surface S surrounding the degeneracy point (see Methods and Supplementary Information for details). This nonzero Chern number promotes this B = 0 degeneracy point to a Weyl point, and underlines the robustness of its (Kramers) degeneracy against perturbations.Let us now turn to the case of two coupled interacting spins, and investigate the fate of Weyl points in the presence of -possibly strong -spin-orbit in-↑ ↑ ↑

Gate-controlled supercurrent (GCS) in superconducting nanobridges has recently attracted attention as a means to create superconducting switches. Despite the clear advantages for applications, the microscopic mechanism of this effect is still under debate. In this work, we realize GCS for the first time in a highly crystalline superconductor epitaxially grown on an InAs nanowire. We show that the supercurrent in the epitaxial Al layer can be switched to the normal state by applying ≃±23 V on a bottom gate insulated from the nanowire by a crystalline hBN layer. Our extensive study of the temperature and magnetic field dependencies suggests that the electric field is unlikely to be the origin of GCS in our device. Though hot electron injection alone cannot explain our experimental findings, a very recent non-equilibrium phonons based picture is compatible with most of our results.