From triple-point materials to multiband nodal links

We study a class of topological materials which in their momentum-space band structure exhibit threefold degeneracies known as triple points. Focusing specifically on $\mathcal{P}\mathcal{T}$-symmetric crystalline solids with negligible spin-orbit coupling, we find that such triple points can be stabilized by little groups containing a three-, four-, or sixfold rotation axis, and we develop a classification of all possible triple points as type A vs type B according to the absence vs presence of attached nodal-line arcs. Furthermore, by employing the recently discovered non-Abelian band topology, we argue that a rotation-symmetry-breaking strain transforms type-A triple points into multiband nodal links. Although multiband nodal-line compositions were previously theoretically conceived and related to topological monopole charges, a practical condensed-matter platform for their manipulation and inspection has hitherto been missing. By reviewing the known triple-point materials with weak spin-orbit coupling and by performing first-principles calculations to predict new ones, we identify suitable candidates for the realization of multiband nodal links in applied strain. In particular, we report that an ideal compound to study this phenomenon is ${\mathrm{Li}}_2\mathrm{Na}\mathrm{N}$, in which the conversion of triple points to multiband nodal links facilitates a largely tunable density of states and optical conductivity with doping and strain, respectively.

Figure 1

Three species of triple points in spinless $\mathcal{P}\mathcal{T}$-symmetric systems. They are characterized by the number $N_a$ of attached NL arcs, which depends on symmetry as listed in Table 1. We adopt terminology of Ref. [28], and call TPs with $N_a=0$ ($\ne 0$) type A (type B).

Figure 2

Relation of type-A triple points (TPs) to multiband nodal links. Nodal lines (NLs) in the first (second) band gap are displayed in red (blue), and TPs are in yellow. (a) and (b) Band structure along a high-symmetry line and (c) and (d) the NL composition near the TPs before and after applying rotation-symmetry-breaking perturbation, respectively. The green path ${\gamma }_0$ carries quaternion charge $-1$. The orientations of the NLs (with reversals indicated by triangles of the corresponding color) follow the convention of Ref. [54].

Figure 3

${\mathrm{Li}}_2\mathrm{Na}\mathrm{N}$. (a) Crystal structure with Na (blue), Li (yellow), and N (green) atoms. The red plane indicates the surface termination of (g) and (h), and the purple arrow is its normal. (b) Band structure along high-symmetry lines: first-principles data (orange points) and fitted four-band tight-binding model (blue lines). The black arrow marks the triple point. (c) and (d) NLs inside the Brillouin zone without and with strain, respectively, and (e,f) close-ups for $0<k_z<0.15G_3$, colored orange, red, and blue for the first, second, and third energy gaps, respectively. Green arrows in (c) labeled ${\mathbf{G}}_{1,2,3}$ indicate primitive reciprocal vectors. (g) Projection of the NLs in (d) onto the surface Brillouin zone and the quaternion charge in each region (see the text). Coordinates $k_{2,3}$ and their units correspond to projections of ${\mathbf{G}}_{2,3}$. (h) Surface spectral function for $k_2=0$. (i) and (j) Interband optical conductivity $\text{Re}{\sigma }_{xx}$ (in $\mathrm{S}/\mathrm{cm}$) and density of states (DOS; in ${\mathrm{eV}}^{-1}$) computed from the fitted tight-binding model for the indicated values of strain $s$ [69]. In (d) and (f)–(h) we set $s=-0.01\mathrm{eV}$, which corresponds to tensile strain in the $y$ direction of $\approx 1\%$.