Multifold Hopf semimetals

Abstract

Three-dimensional topological semimetals exhibit linear energy band crossing points that act as monopole sources of Berry curvature. Here, an alternative class of semimetals is introduced, featuring linear N -fold crossing points each of which acts as a source of a Berry dipole. Continuum and lattice models for such multifold Hopf semimetals (MHSMs) are proposed for N = 3, 4, 5. Weak field magnetotransport properties of MHSMs are revealed: a single Berry dipole generates not only an anomalous Hall effect, but also dissipative current contributions to linear order in B, similar to the axial anomaly, chiral magnetic, and magnetochiral effects familiar from a pair of coupled Weyl nodes. For strong B, the Landau level spectrum highlights the importance of quantum geometry: intraband corrections to the Onsager quantization condition crucially depend on the angle between B and the Berry dipole, and interband corrections reflect the strong coupling between degenerate orbits. Finally, it is shown that MHSMs mediate topological phase transitions between N -band Hopf insulators, suggesting them as an ideal playground to explore the physics of delicate topological semimetals and insulators.

Optical conductivity of the threefold Hopf semimetal

“…1 Recently, a novel type of topological nodal semimetal, the so-called Hopf semimetal, has been theoretically proposed and predicted to possess multifold point nodes at which three or more bands are attached. 5 Around the nodes, the dispersion of energy bands is flat or linear, and it is qualitatively the same as that of multifold point node semimetals 4,6 . However, the Berry curvature possesses a dipole structure different from the isotropic monopole in these multifold point-node semimetals 7,8 .…”

“…The anisotropic Berry curvature shows that the internal anisotropy can be found in electronic properties even if the energy dispersion is completely isotropic. In previous work 5 , it was proposed that the anisotropy of the internal electronic structure can be confirmed by the Hall conductivity. However, the band structure of a multifold nodal semimetal cannot be confirmed in the measurement simultaneously.…”

“…The theoretical analysis is performed by using a minimal model proposed in Ref. 5. Around the point node, the electronic states are described by…”

“…Here, ξ = ±1 indicates the direction of the dipole structure, d = (0, 0, ξ). In previous work 5 , two symmetries were proposed for the protection of the point node: axial rotation symmetry and chiral symmetry. The axial rotation operator L d + Σ d is defined with the angular momentum operator, L d = d • (i∇ q × q), projected on the d-vector and the effective spin projection operator is given by…”

Extrinsic and Intrinsic Nonlinear Hall Effects across Berry-Dipole Transitions

“…Furthermore, it is worth pointing out that Berry-dipole transitions are not restricted to two-band realizations. There also exist N -fold (N ≥ 3) Berry-dipole critical points with linear dispersion [49]. Compared to the two-band realizations of Berry-dipole transitions, an important advantage of the Nband (N ≥ 3) realizations is that their corresponding tightbinding Hamiltonians can only involve nearest-neighbor hoppings, and are thus more feasible in experiments.…”