Higher-order Topological Superconductors Research Articles

Higher-order topological superconductors as generators of quantum codes

We show that interactions can drive a class of higher order topological superconductors (HOTSC) into symmetry enriched topologically ordered phases exemplified by topological quantum error correcting codes. In two dimensions, interacting HOTSC realize various topologically ordered surface and color codes. In three dimensions, interactions can drive HOTSC protected by subsystem symmetries into recently discovered fracton phases. We explicitly relate fermion parity operators underlying the gapless excitations of the HOTSC to the Wilson algebra of symmetry enriched quantum codes. Arrays of crossed Majorana wires provide an experimental platform for realizing fracton matter and for probing the quantum phase transition between HOTSC and topologically ordered phase.

Majorana Corner Modes with Solitons in an Attractive Hubbard-Hofstadter Model of Cold Atom Optical Lattices.

Higher-order topological superconductors hosting Majorana-Kramers pairs (MKPs) as corner modes have recently been proposed in a two-dimensional quantum spin Hall insulator proximity-coupled to unconventional cuprate or iron-based superconductors. Here, we show that such MKPs can be realized using a conventional s-wave superfluid with a soliton in cold atom systems governed by the Hubbard-Hofstadter model. The MKPs emerge in the presence of interaction at the "corners" defined by the intersections of line solitons and the one-dimensional edges of the system. Our scheme is based on the recently realized cold atom Hubbard-Hofstadter lattice and will pave the way for observing possible higher-order topological superfluidity with conventional s-wave superfluids or superconductors.

Three-dimensional superconductors with hybrid higher-order topology

We consider three dimensional superconductors in class DIII with a four-fold rotation axis and inversion symmetry. It is shown that such systems can exhibit higher order topology with helical Majorana hinge modes. In the case of even-parity superconductors we show that higher order topological superconductors can be obtained by adding a small pairing with the appropriate $C_4$ symmetry implementation to a topological insulator. We also show that a hybrid case is possible, where Majorana surface cones resulting from non-trivial strong topology coexist with helical hinge modes. We propose a bulk invariant detecting this hybrid scenario, and numerically analyse a tight binding model exhibiting both Majorana cones and hinge modes.

Higher-order topological insulators and superconductors

<sec>Very recently, higher-order topological insulators and superconductors have attracted wide attention and aroused the great interest of researchers. Owing to their unconventional bulk-boundary correspondence, higher-order topological insulators and superconductors possess novel boundary modes whose dimensions are always lower than the first-order (or say conventional) topological insulators and superconductors, provided that their bulk dimensions are the same. The essence of higher-order topological phase is the formation of Dirac-mass domain walls on the gapped one-dimensional lower boundary. Roughly speaking, the origins of the formation can be classified as " intrinsic” and " extrinsic” type. For the former case, the formation of domain walls is forced by symmetry, suggesting that the resulting higher-order topological phases can be taken as topological crystalline phases. For this case, the domain walls are quite robust if the corresponding symmetry is preserved. For the latter case, the domain walls are formed simply because the one-dimensional lower boundary modes are gapped in a nontrivial way, however, the nontrivial way is not forced by symmetry. For this case, the domain walls are also stable against perturbations as long as the separations between them are large enough. The domain walls can have various patterns, which indicates that the higher-order topological phases are very rich. </sec><sec>In this paper, we first reveal the connection between the higher-order topological phase and the first-order topological phase. Concretely, we show how to realize a higher-order topological phase by breaking some symmetries of a first-order topological phase, as well as stacking lower-dimensional first-order topological systems in an appropriate way. After these, we review the recent progress of theoretical and experimental study of higher-order topological insulators and superconductors. For the higher-order topological insulators, we find that the electronic materials are still laking though a lot of experimental realizations have been achieved. For higher-order topological superconductors, we find that their experimental realization and investigation are still in the very primary stage though quite a lot of relevant theoretical studies have been carried out. In order to comprehensively understand this newly-emerging field there are still many things to be done.</sec>