Bott periodicity for the topological classification of gapped states of matter with reflection symmetry

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-two and four period-eight cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme the classification in arbitrary dimensions $d \ge 1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)]. The use of a scattering-matrix inspired method allows us to address the second descendant \bZ_2 phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.Comment: 18 pages, 7 figure

Long-Range Superharmonic Josephson Current

We consider a long superconductor-ferromagnet-superconductor junction with one spin-active region. It is shown that an \textit{odd} number of Cooper pairs cannot have a long-range propagation when there is \textit{only one} spin-active region. When temperature is much lower than the Thouless energy, the coherent transport of \textit{two} Cooper pairs becomes dominant process and the \textit{superharmonic} current-phase relation is obtained ($I\propto\sin2\phi$).Comment: 4 pages, 3 figure

Bulk-and-edge to corner correspondence

We show that two-dimensional band insulators, with vanishing bulk polarization, obey bulk-and-edge to corner charge correspondence stating that the knowledge of the bulk and the two corresponding ribbon band structures uniquely determines the fractional part of the corner charge irrespective of the corner termination. Moreover, physical observables related to macroscopic charge density of a terminated crystal can be obtained by representing the crystal as collection of polarized edge regions with polarizations $\vec P^\text{edge}_\alpha$, where the integer $\alpha$ enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed "Wannier cut", which allows us to compute $\vec P^\text{edge}_\alpha$. We find that $\vec P^\text{edge}_\alpha$ consists of two pieces: the bulk piece expressed via quadrupole tensor of the bulk Wannier functions' charge density, and the edge piece corresponding to the Wannier edge polarization---the polarization of the edge subsystem obtained by Wannier cut. For a crystal with $n$ edges, out of $2n$ independent components of $\vec P^\text{edge}_\alpha$, only $2n-1$ are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.Comment: published version. v2: application to Benalcazar-Bernevig-Hughes mode

Long-Range Interaction of Spin-Qubits via Ferromagnets

We propose a mechanism of coherent coupling between distant spin qubits interacting dipolarly with a ferromagnet. We derive an effective two-spin interaction Hamiltonian and estimate the coupling strength. We discuss the mechanisms of decoherence induced solely by the coupling to the ferromagnet and show that there is a regime where it is negligible. Finally, we present a sequence for the implementation of the entangling CNOT gate and estimate the corresponding operation time to be a few tens of nanoseconds. A particularly promising application of our proposal is to atomistic spin-qubits such as silicon-based qubits and NV-centers in diamond to which existing coupling schemes do not apply.Comment: 6 pages, 7 pages of appendi

Towards a realistic transport modeling in a superconducting nanowire with Majorana fermions

Motivated by recent experiments searching for Majorana fermions (MFs) in hybrid semiconducting-superconducting nanostructures, we consider a realistic tight-binding model and analyze its transport behavior numerically. In particular, we take into account the presence of a superconducting contact, used in real experiments to extract the current, which is usually not included in theoretical calculations. We show that important features emerge that are absent in simpler models, such as the shift in energy of the proximity gap signal, and the enhanced visibility of the topological gap for increased spin-orbit interaction. We find oscillations of the zero bias peak as a function of the magnetic field and study them analytically. We argue that many of the experimentally observed features hint at an actual spin-orbit interaction larger than the one typically assumed. However, even taking into account all the known ingredients of the experiments and exploring many parameter regimes for MFs, we are not able to reach full agreement with the reported data. Thus, a different physical origin for the observed zero-bias peak cannot be excluded.Comment: 7 pages, 7 figures; Published versio