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Majorana zero modes are usually attributed to topological superconductors. We study a class of two-dimensional topologically trivial superconductors without chiral edge modes, which nevertheless host robust Majorana zero modes in topological defects. The construction of this minimal single-band model is facilitated by the Hopf map and the Hopf invariant. This work will stimulate investigations of Majorana zero modes in superconductors in the topologically trivial regime., 15 pages, 6 figures, including supplemental material. The published version is slightly shortened

In topological semimetals, the valence band and conduction band meet at zero-dimensional nodal points or one-dimensional nodal rings, which are protected by band topology and symmetries. In this Rapid Communication, we introduce "nodal-link semimetals", which host linked nodal rings in the Brillouin zone. We put forward a general recipe based on the Hopf map for constructing models of nodal-link semimetal. The consequences of nodal ring linking in the Landau levels and Floquet properties are investigated., Comment: 12 pages, 5 figures, including supplemental material. Published version

Topological nodal-line semimetals are characterized by one-dimensional lines of band crossing in the Brillouin zone. In contrast to nodal points, nodal lines can be in topologically nontrivial configurations. In this paper, we study the simplest topologically nontrivial forms of nodal line, namely, a single nodal line taking the shape of a knot in the Brillouin zone. We introduce a generic construction for various "nodal-knot semimetals", which yields the simplest trefoil nodal knot and other more complicated nodal knots in the Brillouin zone. The knotted-unknotted transitions by nodal-line reconnections are also studied. Our work brings the knot theory to the subject of topological semimetals., Comment: 7 pages, 5 figures. Figure quality improved.

Backscattering-immune chiral modes arise along certain line defects in three-dimensional materials. In this paper, we study Floquet chiral modes along Floquet topological defects, namely, the defects come entirely from spatial modulations of periodic driving. We define a precise topological invariant that counts the number of Floquet chiral modes, which is expressed as an integral on a five-dimensional torus parameterized by $(k_x,k_y,k_z,\theta,t)$. This work demonstrates the possibility of creating chiral modes in three-dimensional bulk materials by modulated driving. We hope that it will stimulate further studies of Floquet topological defects., Comment: 11 pages, 7 figures, including supplemental material. Published version

Unidirectional transports have been observed in two-dimensional systems, however, so far they have not been experimentally observed in three-dimensional bulk materials. In this theoretical work we show that the recently discovered Weyl materials provide a platform for unidirectional transports inside bulk materials. With high experimental feasibility, a complex Dirac mass can be generated and manipulated in the photonic Weyl crystals, creating unidirectionally propagating modes observable in transmission experiments. Possible realization in (electronic) Weyl semimetals is also studied. We show in a lattice model that, with a short-range interaction, the desired form of the Dirac mass can be spontaneously generated in a first-order transition., 9 pages, with supplemental material