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.

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

Majorana zero modes are usually attributed to topological superconductors. We study a class of twodimensional 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.Majorana zero modes (MZMs) or Majorana bound states are exotic excitations predicted to exist in the vortex cores [1, 2] of two-dimensional (2D) topological superconductors [3][4][5][6][7] and at the ends of 1D topological superconductors [8]. Spatially separated MZMs give rise to degenerate ground states, which encode qubits immune to local dechoerence [8, 9]. Furthermore, unitary transformations among the ground states can be implemented by braiding [10][11][12][13][14] or measurements [15, 16] of these modes, indicating that such qubits may become building blocks in topological quantum computation and information [17][18][19][20][21][22]. Therefore, MZMs have been vigorously pursued in condensed matter physics [23][24][25][26][27][28][29].There have been a great variety of proposals for topological superconductors, including 2D semiconductor heterostructures [30,31], topological insulatorsuperconductor proximity [32][33][34][35][36], 1D spin-orbit-coupled quantum wires [37][38][39][40][41][42][43][44][45], spiral magnetic chains on superconductors [46][47][48][49][50], Shockley mechanism [51, 52], and cold atom systems in 2D [53][54][55][56] and 1D [57,58] It is often implicitly assumed that topological superconductivity is a prerequisite for MZMs, accordingly, the chiral edge states go hand in hand with the vortex zero modes in 2D superconductors. In this Letter we show that certain topological defects [76][77][78][79][80][81] in 2D topologically trivial superconductors can support robust MZMs. Somewhat surprisingly, single-band superconductors suffice this purpose. The model Hamiltonian is related to the Hopf maps, which originally refer to nontrivial mappings from a 3D sphere S 3 to a 2D sphere S 2 , characterized by the integer Hopf invariant [82,83]. Mappings from a 3D torus T 3 to S 2 inherit the nontrivial topology from the mappings S 3 → S 2 . The Hopf invariant has found interesting applications in nonlinear σ models and spin systems [82,84], Hopf insulators [85][86][87][88][89][90], liquid crystals [91], and quench dynamics of Chern insulators [92,93].Our model describes topologically trivial superconductors with zero Chern number and no chiral edge state. Nevertheless, a topological point defect is characterized by a Hopf invariant defined in the (k x , k y , θ) space, where k x , k y are crystal momenta and θ is the polar angle [94] (Fig.1a). The parity (even/odd) of Hopf invariant determines the presence (absence) of robust MZMs, though the superconductor for ev- ery fixed θ is topologically trivial. Stimulated by this mechanis...

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.

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 , θ, 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.