Topological concepts in partially ordered vector spaces

In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations. In particular, we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin, Wolk and Vulikh. We prove that the considered topologies disagree for all infinite dimensional Archimedean vector lattices that contain order units. For reflexive Banach spaces equipped with ice cream cones we show that the order topology, the order bound topology and the norm topology agree and that order convergence is equivalent to norm convergence.

Intrinsic First and Higher-Order Topological Superconductivity in a Doped Topological Insulator

We explore higher-order topological superconductivity in an artificial Dirac material with intrinsic spin-orbit coupling. A mechanism for superconductivity due to repulsive interactions – pseudospin pairing – has recently been shown to result in higher-order topology in Dirac systems past a minimum chemical potential [1]. Here we apply this theory through microscopic modelling of a superlattice potential imposed on an inversion symmetric hole-doped semiconductor heterostructure, and extend previous work to include the effects of spin-orbit coupling. We find spin-orbit coupling enhances interaction effects, providing an experimental handle to increase the efficiency of the superconducting mechanism. We find that the phase diagram, as a function of chemical potential and interaction strength, contains three superconducting states – a first-order topological p + ip state, a second-order topological spatially modulated p + iτp state, and a second-order topological extended s-wave state, sτ. We calculate the symmetry-based indicators for the p + iτp and sτ states, which prove these states possess second-order topology. Exact diagonalisation results are presented which illustrate the interplay between the boundary physics and spin orbit interaction. We argue that this class of systems offer an experimental platform to engineer and explore first and higher-order topological superconducting states.

Structural buckling induced higher-order topology

ABSTRACT The higher-order topological insulator (HOTI) states, such as two-dimension (2D) HOTI featured with topologically protected corner modes at the intersection of two gapped crystalline boundaries, have attracted much recent interest. However, physical mechanism underlying the formation of HOTI states is not fully understood, which has hindered our fundamental understanding and discovery of HOTI materials. Here we propose a mechanistic approach to induce higher-order topological phases via structural buckling of 2D topological crystalline insulators (TCIs). While in-plane mirror symmetry is broken by structural buckling, which destroys the TCI state, the combination of mirror and rotation symmetry preserves in the buckled system, which gives rise to the HOTI state. We demonstrate that this approach is generally applicable to various 2D lattices with different symmetries and buckling patterns, opening a horizon of possible materials to realize 2D HOTIs. The HOTIs so generated are also shown to be robust against buckling height fluctuation and in-plane displacement. A concrete example is given for the buckled $\beta $-Sb monolayer from first-principles calculations. Our finding not only enriches our fundamental understanding of higher-order topology, but also opens a new route to discovering HOTI materials.