Nodal-line semimetals and their variance

TL;DR

This work surveys nodal-line semimetals (NLSMs) as a distinct topological phase where band touchings form lines in 3D momentum space, protected by mirror, PT, or nonsymmorphic symmetries. It develops a comprehensive taxonomy by nodal structures, dispersions, and textures, including linked, chained, and nexus networks, with explicit two- and multi-band models illustrating invariant structures and their electromagnetic fingerprints. The review connects nodal topology to observable phenomena such as drumhead surface states, Landau level spectra, anisotropic optical responses, and hyperbolic polaritons, and highlights how electron correlations give rise to magnetism, superconductivity, and Kondo phenomena in NLSMs. It also discusses practical directions, like heterostructures and THz optoelectronics, where the unique geometry and topology of nodal lines can be harnessed for novel devices and quantum technologies.

Abstract

Topological nodal-line semimetals (NLSMs) are a new family of topological materials characterized by electronic band crossings that form lines in the Brillouin zone. These NLSMs host exotic nodal-line structures and exhibit distinct features such as drumhead surface states and unique electromagnetic responses. This review classifies various NLSM types based on their nodal structures and protecting symmetries, highlighting that these nodal-line structures can form links, knots, and chains. We discuss their characteristic electromagnetic responses, including Landau level spectroscopy, optical conductivity, and permittivity. Furthermore, the strong correlation effects in these NLSMs modify their semimetallic phases and lead to novel quantum phases where magnetism and superconductivity intertwine.

Figures (4)

• Figure 1: We summarize various types of NLSM, highlighting the nodal-line structures, symmetry constraints, and materials candidates. The theoretical computations are highlighted in $*$ and the experimental observations are marked by $\dagger$, respectively.
• Figure 2: NLSMs protected by non-symmorphic symmetries (glide reflection or screw rotation): (a) Glide Mirror Protections: Any red path on the glide plane connecting two time-reversal invariant points $\Gamma_1$ and $\Gamma_2$ will host a degenerate point. Connecting all such degenerate points forms a nodal line (blue circle). (b) The glide eigenvalues $\pm i e^{-i k_x/2}$ at two different time-reversal invariant points $\Gamma_1$ and $\Gamma_2$ are $\pm i$ and $\pm 1$ are switched. I,e,, $\pm i$ at $\Gamma_1$ becomes $+1$ or $-1$ at $\Gamma_2$. The energy spectrum exhibits an hourglass structure and giving rise to a degenerate point. Screw rotation Protections: If is an additional inversion symmetry is present, its combination with the screw rotation effectively forms a glide mirror operator. Consequently, a nodal ring can be protected by the combined symmetry, similar to the case of a glide-mirror-protected NLSM.
• Figure 3: Classification of energy dispersions near the nodal line: (a) Type I, (b) Type II, and (c) Type III.
• Figure 4: (a) The phase diagram of magnetic orders as a function of the Hubbard interaction $U$. Below a threshold $U_c$, the surface develops ferromagnetic order. Above the threshold $U_c$, the bulk develops antiferromagnetic order. The torus-like Fermi surface of the NLSMs, which can be parameterized by two angles $(\theta,\phi)$.