Nodal-link semimetals

Topological phases of matter have been among the most active research subjects in condensed matter physics. They can be broadly classified as two major classes. The first class of phases, including topological insulators and superconductors[1–6], and other symmetry protected topological phases[7], have gapped bulks with nontrivial topological structures characterized by topological invariants[8–15], dictating the existence of robust gapless modes on the boundary. More recently, the second major class of topological materials, known as topological semimetals, have attracted widespread attentions. In the noninteracting limit, they are characterized by topologically robust k-space band-touching manifolds, which can be zero-dimensional (0D) nodal points or one-dimensional (1D) nodal rings (or nodal lines). The bulk Dirac[16–25] and Weyl points[26–40] are responsible for novel phenomena related to chiral anomaly[41–51]. Moreover, bulk Weyl points entail surface Fermi arcs, while nodal rings[52–71] imply flat surface band (drumhead states) that may trigger interesting correlation effects[72]. Nodal rings have been predicted (e.g., Cu3PdN[60, 61],Hg3As2[73], Ca3P2[63, 74], 3D carbon networks[58], CaP3[75], Alkali Earth Metals[76, 77]) and experimentally studied in quite a few materials (e.g., PbTaSe2[62], ZrSiTe[78], ZrSiS[79–83]). Notably, nodal rings can be driven to Floquet Weyl points by circularly polarized light[84–88], accordingly, the drumhead surface states become Fermi arcs. Unlike nodal points, nodal rings allow richer topological structures. They can touch at special points[60, 61, 89, 90], enabling formations of nodal chains[91, 92]. In this Letter, we introduce new types of topological semimetals, dubbed nodal-link semimetals, which host nontrivially linked nodal rings[e.g., Fig.1(f)]. We put forward minimal two-band models for these phases. The difference between essential links and fake links is highlighted: unlike fake links, essential links cannot be smoothly unlinked. We investigate generic physical consequences of nontrivial linking; in particular, a global toroidal π Berry phase generates a half-integer shift of Landau level index when the magnetic field is perpendicular to the ring plane. In addition, a suitable periodic external field can drive nodal-link semimetal to a Floquet Hopf insulator. Model.–Nodal ring models should have two bands at least. A two-band Bloch Hamiltonian can generally be written as