Orthonormal Compactly Supported Wavelets with Optimal Sobolev Regularity

Numerical optimization is used to construct new orthonormal compactly supported wavelets with Sobolev regularity exponent as high as possible among those mother wavelets with a fixed support length and a fixed number of vanishing moments. The increased regularity is obtained by optimizing the locations of the roots the scaling filter has on the interval (pi/2,\pi). The results improve those obtained by I. Daubechies [Comm. Pure Appl. Math. 41 (1988), 909-996], H. Volkmer [SIAM J. Math. Anal. 26 (1995), 1075-1087], and P. G. Lemarie-Rieusset and E. Zahrouni [Appl. Comput. Harmon. Anal. 5 (1998), 92-105].Comment: 18 pages, 8 figure

Engineering nodal topological phases in Ising superconductors by magnetic superstructures

Recently it was discovered that superconductivity in transition metal dichalcogenides (TMDs) is strongly affected by an out-of-plane spin-orbit coupling (SOC). In addition, new techniques of fabricating 2d ferromagnets on van der Waals materials are rapidly emerging. Combining these breakthroughs, we propose a realization of nodal topological superconductivity in TMDs by fabricating nanostructured ferromagnets with an in-plane magnetization on the top surface. The proposed procedure does not require application of external magnetic fields and applies to monolayer and multilayer (bulk) systems. The signatures of the topological phase include Majorana flat bands that can be directly observed by Scanning Tunneling Microscopy (STM) techniques. We concentrate on NbSe$_2$ and argue that the proposed structures demonstrating the nodal topological phase can be realized withing existing technology.Comment: 8 pages, 7 figure

Chern mosaic - topology of chiral superconductivity on ferromagnetic adatom lattices

In this work we will explore the properties of superconducting surfaces decorated by two-dimensional ferromagnetic adatom lattices. As discovered recently [R\"ontynen and Ojanen, Phys. Rev. Lett. \textbf{114}, 236803 (2015)], in the presence of a Rashba spin-orbit coupling these systems may support topological superconductivity with complex phase diagrams and high Chern numbers. We show how the long-range hopping nature of the effective low-energy theory generically gives rise to a phase diagram covered by a \emph{Chern mosaic} -- a rich pattern of topological phases with large Chern numbers. We study different lattice geometries and the dependence of energy gaps and abundance of different phases as a function of system parameters. Our findings establish the studied system as one of the richest platforms for topological matter known to date.Comment: 9 pages, 8 figure