Universal higher-order bulk-boundary correspondence of triple nodal points

Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. For spinless particles, the stability of a triple nodal point requires two ingredients: rotational symmetry of order three, four, or six; combined with mirror or space-time-inversion symmetry. However, despite ample studies of their classification, robust boundary signatures of triple nodal points have until now remained elusive. In this work, we first show that pairs of triple nodal points in semimetals and metals can be characterized by Stiefel-Whitney and Euler monopole invariants, of which the first one is known to facilitate higher-order topology. Motivated by this observation, we then combine symmetry indicators for corner charges and for the Stiefel-Whitney invariant in two dimensions with the classification of triple nodal points for spinless systems in three dimensions. The result is a complete higher-order bulk-boundary correspondence, where pairs of triple nodal points are characterized by fractional jumps of the hinge charge. We present minimal models of the various species of triple nodal points carrying higher-order topology, and illustrate the derived correspondence on ${\mathrm{Sc}}_3\mathrm{AlC}$ which becomes a higher-order triple-point metal in applied strain. The generalization to spinful systems, in particular to the WC-type triple-point material class, is briefly outlined.

Figure 1

Features of the model given in Eq. (1). (a) Real-space unit cell spanned by the lattice vectors ${\mathbf{a}}_{1,2,3}$ with the four orbitals: $(p_x,p_y)$ at Wyckoff position $1a$ and $d_{xy}$, $d_{x^2-y^2}$ at $1c$. (b) Single layer (for fixed $z$) of the three-dimensional lattice. The blue shaded regions indicate the projection of the unit cells with the orbitals indicated in the same colors as in (a). Black lines (where further symmetry-related lines are dropped to maintain clarity) indicate the in-plane hopping processes included in the model. (c) Band structure along the high-symmetry lines. (d) Brillouin zone spanned by the reciprocal lattice vectors ${\mathbf{G}}_{1,2,3}$ with high-symmetry points $\{\mathrm{\Gamma },Z,X,R,M,A\}$ shown. Nodal lines in the first, second, and third band gaps are displayed in orange, red, and blue color, respectively. The four triple points of the model are indicated by green arrows in (c) and (d).

Figure 2

(a) In-plane termination for the nanowire geometry considered in the main text. The blue shadowed regions denote unit cells with ionic charge $2\left|e\right|$ placed at the center (black circle). The electronic orbitals are illustrated as in Fig. 1. To establish terminology, we say that the depicted geometry consists of $3.5\times 3.5$ unit cells (cf. Sec. 5a). (b) Quantized hinge charges (blue solid line) identified using exact diagonalization (for details, see Sec. 5a), and second Stiefel-Whitney (2SW) class (yellow dashed line) obtained from the Wilson-loop winding; both plotted as a function of $k_z$ in the wire geometry. Cuts at fixed $k_z$ can each be interpreted as 2D flakes as shown in (a) that are characterized by a corner charge $Q_{⌞}\left(k_z\right)$. The edge charge of the model vanishes. For reference, the bulk band structure along $k_z$ for $k_x=k_y=0$ is displayed in gray with the 1D (2D) degenerate bands plotted by a thin (thick) line. (c), (d) Wilson-loop spectrum of the two occupied bands in the 2D BZ for fixed $k_z=0$ and $\pi$, respectively. For $k_z=0$ (c) we observe a winding with odd parity, indicating a nontrivial 2SW class.

Figure 3

(a) Two pairs of type-$\mathsf{A}$ triple points (TPs, yellow dots) formed by consecutive triplets of bands. All TPs are protected by $C_4$ or $C_6$ symmetry with respect to the $z$ axis. In $k_z$ direction the full extent of the Brillouin zone is shown. Nodal lines in the first, second, and third gaps of a four-band model are shown in orange, red, and blue, respectively. (b), (c) Closeup of the boxed region in (a) with the ellipsoid, on which the second Stiefel-Whitney (2SW) class is computed with respect to the principal (red) band gap, shown in purple. (c) Multiband nodal links that form after breaking the rotational symmetry. The red nodal ring carries a nontrivial value of the 2SW class computed on the purple ellipsoid. By continuity, the red nodal segment in (b) also carries a nontrivial 2SW class.

Figure 4

Relation between a nodal-line segment carrying a nontrivial second Stiefel-Whitney monopole charge, and a pair of two-dimensional insulators characterized by the ${\mathbb{Z}}_2$-valued 2SW class. The black frame represents the complete momentum-space extent of the Brillouin zone in the two horizontal directions (solid black lines), but not in the vertical direction (dashed black lines). (a) The monopole charge of the red nodal-line segment is calculated on a surface enclosing it (innermost purple ellipsoid), which can be continuously inflated (second purple ellipsoid) until it fills the whole BZ in $k_x$ and $k_y$ directions (purple cuboid). (b) The purple cuboid in (a) is equivalent to two horizontal planes (actually tori) $T_1$ and $T_2$ at the appropriate $k_z$ values.

Figure 5

(a) Type-$\mathsf{A}$ triple points (yellow dots). The red (blue) line indicates nodal lines formed by the lower (upper) two bands of a three-band model. In $k_z$ direction the full extent of the Brillouin zone is shown. (b), (c) Multiband nodal link formed from (a) by breaking the rotational symmetry protecting the triple points. The flux (gray $\pm 1$'s) depends on the orientation of the two blue nodal lines at the intersection points (gray) with the disk bounded by the red nodal ring [71]. The total flux vanishes in (b) giving rise to $\chi =0$, but is finite in (c) such that $\left|\chi \right|=2$. In the presence of mirror $m_z:k_z\mapsto -k_z$ symmetry, the difference between the situations (b) and (c) is reflected in the mirror eigenvalues of the three bands at $k_z=0$ vs $k_z=\pi$ (cf. the discussion in Sec. 6b).

Figure 6

(a) Nodal-line (NL) configuration (here colored red vs blue for NLs formed by the lower vs upper two bands of a three-band model) in the first Brillouin zone (BZ, black frame) and part of the second BZ. The two blue displayed NLs are displaced by a reciprocal lattice vector $\mathbf{b}$ (green arrow). The blue NL is intersected by a 2D plane (pale brown) at two inequivalent points (gray dots). The shown orientation of the blue NLs is compatible with ${\varphi }_2+{\varphi }_3=\pi$ (cf. Sec. 3b). (b) Band nodes (blue points) on the 2D plane in (a), with winding numbers $\pm 1$ inherited from the orientation of the NLs in 3D. The two upper point nodes have opposite orientation, because they pairwise annihilate (blue arrows pointing to the gray disk) when the vertical plane is shifted away from the NL composition. The point nodes displaced by $\mathbf{b}$ have winding numbers differing by a factor $s=e^{i({\varphi }_2+{\varphi }_3)}=\pm 1$ [19].

Figure 7

(a) The $C_n$-symmetric cross-section geometries for which the hinge charges are defined, and for which the values listed in Table 2 apply. The corners (gray regions) in the cross sections correspond to the hinges in the 3D system. (b) Deformation of the $C_6$-symmetric sample (inner hexagon) to a $C_3$-symmetric sample (outer triangle). Two corners (blue and green) of the hexagon with corner charges $Q_{⌞}^a$ and $Q_{⌞}^b$ merge into a single corner (red) of the triangle with corner charge $Q_{⌞}^{ab}=Q_{⌞}^a+Q_{⌞}^b$.

Figure 8

(a) Definition of the closed contours $\mathrm{\Gamma }\left(\varphi \right)=\gamma {\left(\varphi \right)}^{-1}\circ \gamma \left(0\right)$ (green) on which the Wilson-loop operator $\mathcal{W}\left(\varphi \right)$ is computed. The spherical surface (purple) is covered by these contours as the argument is increased in the range $\varphi \in [0,2\pi )$. If the surface is not crossed by any other nodal lines (NLs) than the orange and blue ones, the Euler monopole charge $\chi$ can be defined and is determined by the winding of the Wilson-loop spectrum. This is the case, for example, if there are no red nexus points enclosed in the surface. (b) In other cases, e.g., if there is exactly one red nexus point enclosed, red NLs crossing the surface are always present. This is indicated by a $\pi$ Berry phase on the path $\mathrm{\Gamma }(2\pi /n)$, where $n$ is the order of the rotational symmetry. The nontrivial Berry phase implies that $\mathrm{\Gamma }(2\pi /n)$ encircles an odd number of NLs in the principal gap (red NLs), as illustrated.

Figure 9

Convention for defining the jump of the fractional hinge charge ($\mathrm{\Delta }{Q}_{⌞}$) and of the second Stiefel-Whitney monopole ($\mathrm{\Delta }{w}_2$) associated with a pair of triple points (TP, yellow dots) in a four-band configuration. The black lines indicate the band structure along a high-symmetry line, with the 2D irreducible corepresentation (ICR) ${\rho }^{\text{2D}}$ passing from the bottom (orange) to the central (red) to the upper (blue) energy gap with increasing momentum $k_z$ due to crossing two 1D ICRs ${\rho }_a^{\text{1D}}$ and ${\rho }_b^{\text{1D}}$. The green dashed line ($\mu$) indicates half-filling at various $k_z$. The quantities $\mathrm{\Delta }{Q}_{⌞}$ and $\mathrm{\Delta }{w}_2$ associated with the TP pair are defined as the characteristics of the 2D cut in the region where ${\rho }^{\text{2D}}$ is unoccupied (i.e., above $\mu )$ from which we subtract the characteristics of the 2D cut in which ${\rho }^{\text{2D}}$ is filled (i.e., below $\mu$). In practice, the physical chemical potential has a value independent of momentum, and will therefore locally deviate from the $\mu$ defined above. Nevertheless, generically, the system will still be at half-filling for extended parts of the orange and blue regions which lie far enough from the TP pair. The differences $\mathrm{\Delta }{Q}_{⌞}$ and $\mathrm{\Delta }{w}_2$ are therefore still well defined and unchanged compared to the situation with variable $\mu \left(k_z\right)$.

Figure 10

Illustration of possible nodal-line (NL) compositions for the three considered triple-point pair configurations. (a) Two type-$\mathsf{A}$ triple points (TPs) have no attached NL arcs. (b) Two type-$\mathsf{B}$ TPs both exhibit attached NL arcs in the principal gap (red), but these NLs can compactly tie the two TPs together, leaving the 2D bulk gapped in both the orange and in the blue $k_z$ range. (c) If one of the TPs is type $\mathsf{A}$ and the other type $\mathsf{B}$, the NL arcs in the principal gap necessarily cross into either the orange or into the blue $k_z$ range, making the corresponding 2D cuts gapless.

Figure 11

Illustration of certain more intricate nodal-line (NL) compositions which can also arise near triple-point pairs (TPPs). (a) Type-$(\mathsf{A},\mathsf{A})$ TPP with two red nexus points whose NLs are tied together inside the red $k_z$ range, leaving the 2D bulk gapped in both the orange and in the blue $k_z$ range despite the nexus points. Here, both the monopole charges and the hinge-charge jump can be defined. (b) Analogous situation as in the previous panel; however, here the NLs extend into the orange and blue $k_z$ ranges rendering the corresponding 2D cuts gapless. In this case, neither the monopole charges nor the hinge-charge jump can be defined. (c) Type-$(\mathsf{B},\mathsf{B})$ TPP with NL arcs extending outside the red $k_z$ range, making the relevant 2D cuts gapless. The monopole charges and the hinge-charge jump are undefined. This scenario should be contrasted to Fig. 10, which shows a similar situation with the NL arcs tied together inside the red $k_z$ range.

Figure 12

Verification of the second Stiefel-Whitney monopole charge $w_2=1$ carried by the red NL segment in the model defined in Eq. (1). (a) Nodal configuration near the high-symmetry line $\mathrm{\Gamma }Z$ and the ellipsoid (purple) on which the monopole charge is computed. (b) Wilson-loop spectrum ${\mathcal{W}}_{12}$ of the lower two bands computed on that ellipsoid, parametrized by the latitude $\theta \in [-\pi /2,\pi /2]$. The spectrum shows a single winding (odd parity) and thus implies $\left|\chi \right|=1$ and $w_2=1$.

Figure 13

Charge distribution of the model defined in Eq. (1) and for the geometry illustrated for fewer unit cells in Fig. 2. Assuming the compensating ionic charge to be located at Wyckoff position $1a$, we plot the total charge distribution at $k_z=0$ (a), (c) and at $k_z=\pi$ (b), (d). The total charge at each lattice site is indicated by a disk with its value encoded both in the area of the disk and the color scale (see legend on the right of each row). Note that the disks are scaled separately for each panel: compared to (a) the disks are enlarged by factors of (b) 2, (c) 3, and (d) 6 to increase visibility. To remove sub-unit-cell oscillations of the charge distribution observable in (a) and (b), we perform coarse graining of the data as explained in Appendix pp4-s1 and Fig. 23. The results of the coarse graining are shown in (c) and (d), respectively.

Figure 14

Results for the $C_6$-symmetric model with two type-$\mathsf{A}$ triple points (TPs) defined in Eq. (C3) and discussed in Sec. 5b. (a) Band structure with the two TPs indicated by black arrows. (b) Nodal-line structure in the full Brillouin zone with the high-symmetry points indicated. Nodal lines in the first, second, and third band gaps of the four-band model are shown in orange, red, and blue, respectively. (c) The second Stiefel-Whitney class can be determined from the Wilson-loop spectrum ${\mathcal{W}}_{12}$ of the lower two bands on horizontal cuts through the Brillouin zone (parametrized by the projection $k_1$ of the momentum onto the first reciprocal lattice vector). Blue/orange lines show the spectrum for $k_z=0$ (single winding) and green/red lines for $k_z=\pi$ (no winding). (d) Cross section through a hexagonal nanowire three shells wide; light blue hexagons indicate the model's unit cells. The orbitals are represented in color and all placed on top of each other in the center of the unit cell. The ionic charge is concentrated at the same position, indicated by the black circles. (e) Charge distribution for a cut through a 16-shell-wide nanowire at $k_z=0$. The charge at each site is indicated by a disk with its value encoded both in the area of the disk and the color scale (see legend on the right). The left half shows the charge distribution on the original lattice, and the right half shows the charge after two iterations of coarse graining [cf. Appendix pp4-s1 and Fig. 23]. A hinge charge of $Q_{⌞}=\frac{e}{6}$ with a strong localization is observed. (f) Hinge Brillouin zone with hinge charge $Q_{⌞}$ (blue solid line, left axis), second Stiefel-Whitney class $w_2$ (orange dashed line, right axis), and the projection of the bulk dispersion along $k_x=k_y=0$ (gray). The doubly degenerate band is displayed with a thicker line.

Figure 15

Results for the $C_6$-symmetric model with two type-$\mathsf{B}$ triple points (TPs) defined in Eq. (C5) and discussed in Sec. 5c. The organization of the panels is in one-to-one correspondence with Fig. 14. The key differences to observe are as follows: (b) Due to the TPs being type $\mathsf{B}$, there are nodal-line arcs attached to them, visible in the nodal-line structure. (c) The Wilson-loop spectra for both $k_z=0$ and $\pi$ show double and zero winding, respectively, implying $\left|\chi \right|=2$ and $w_2=0$. (e), (f) The hinge charge in the region around $k_z=0$ is $Q_{⌞}=-\frac{e}{3}$; the localization length is larger than for the model shown in Fig. 14, such that finite-size effects produce a visible deviation from the ideal quantized value of the corner charge in (f).

Figure 16

Nodal lines and Wilson-loop spectra for the $C_3$-symmetric models obtained by breaking the $C_6$ symmetry in the models discussed in Secs. 5b and 5c using the trigonal perturbation given in Eq. (C6) with ${\delta }_{C_3}=8\times {10}^{-4}$. (a), (d) Nodal lines in the full Brillouin zone (black frame). Nodal lines in the first, second, and third band gaps are shown in orange, blue, and red, respectively. (b), (e) Closeups of the red nodal-line segment. In both cases the triple points are clearly type $\mathsf{B}$ with three attached nodal-line arcs per energy gap. (c), (f) Wilson-loop spectra on the purple ellipsoids shown in (b) and (e), respectively. The single winding in (c) implies $\left|\chi \right|=1$ and $w_2=1$ and the double winding in (f) implies $\left|\chi \right|=2$ and $w_2=0$.

Figure 17

Bulk properties of the compound ${\mathrm{Sc}}_3\mathrm{AlC}$. The band structure (a) without any strain and (b) with $6.6\%$ uniaxial compressive strain in $z$ direction, which generates a triple-point pair on the $\mathrm{\Gamma }Z$ line. Triple points are indicated by yellow arrows. (c) Closeup of the band structure near the triple-point pair with the relevant bands labeled by their irreducible corepresentations. (d) The Brillouin zone of the tetragonal lattice of strained ${\mathrm{Sc}}_3\mathrm{AlC}$. The high-symmetry points and mirror planes are indicated. (e) Nodal lines in the relevant three gaps (orange, blue, red according to increasing energy) determined from the Wannier tight-binding model for the mirror planes shown in (d). The triple point is clearly type $\mathsf{A}$. The cross sections of the ellipsoid on which we compute the second Stiefel-Whitney monopole charge with the mirror planes are indicated by black ellipses. (f) Wilson-loop spectrum of the relevant bands computed on the ellipsoid indicated in (e); the single winding implies $w_2=1$.

Figure 18

Hinge-charge jump observed in the nanowire geometry of ${\mathrm{Sc}}_3\mathrm{AlC}$. (a) Cross section of the nanowire geometry (extended along the $z$ direction) with $9.5\times 9.5$ unit cells. The unit cells are shown as blue shaded squares, and the boundary of the nanowire indicated with a black frame. Two-dimensional Wyckoff positions (WPs) are shown as green, red, and orange points with their labels in the inset that shows an enlarged single unit cell. The scandium atoms project onto WPs $1b$ and $2c$, the aluminum atoms onto $1a$, and the carbon atoms onto $1b$. Note the $C_4$ symmetry of the cross section with the center at WP $1b$. The black circle in the enlarged unit cell denotes the location of the ionic charge of $8\left|e\right|$ per unit cell. (b) Band structure of the nanowire (colored lines) with cross section shown in (a) as a function of momentum ($k_z$) in the hinge Brillouin zone. The projected bulk bands (transparent gray) are overlayed on the nanowire spectrum. The coloring of the bands (see legend) encodes the inverse participation ratio, with a larger value indicating localization on fewer sites. Consequently, hinge-localized states are colored red, surface-localized states green, and bulk states blue. The two energy gaps to the left and to the right of the studied triple-point pair are marked by a yellow square ($k_z=0.08$) and by a purple triangle ($k_z=0.34$), respectively. (c) Integrated corner charge (corresponding to the hinge at the selected values of $k_z$) for the two $k_z$ cuts with filling as indicated in (b) as a function of the side length $a$ of the square integration region [cf. gray squares in (d) and (e)]. The corner charge for $k_z=0.08$ converges to $\frac{e}{4}$ and for $k_z=0.34$ to 0 (yellow vs purple solid lines). (d), (e) Charge distribution coarse grained over one unit cell for $k_z=0.08$ and 0.34, respectively. The magnitude of the charge (in units of $e$) is shown by both the area of the circles as well as the color (see legend on the right). (f), (g) Charge distribution for the same values of $k_z$ after removing the contributions from the edge based on slab calculations (cf. Appendix pp4-s2). In (f) the localization of a nonzero charge on the corners is clearly visible.

Figure 19

Verification of the Euler monopole charge $\left|\chi \right|=2$ ($\chi =0$) on the red (blue) nodal-line segment in ${\mathrm{Li}}_2\mathrm{NaN}$, based on the four-band tight-binding model of Ref. [60]. (a)–(d) Show illustrations and data for the red nodal-line segment (centered at $A$) and (e)–(h) for the blue segment (centered at $\mathrm{\Gamma }$). (a), (e) Nodal line and triple-point configuration in the first one-and-a-half Brillouin zones $A\mathrm{\Gamma }A\mathrm{\Gamma }$. Nodal lines in the second and third band gaps are shown in red and blue, respectively; triple points are shown in yellow. The purple ellipsoid defines the surface on which we compute the Euler class. (b), (f) Linking structure of the corresponding nodal chain configuration after applying approximately $5\%$ tensile strain in $y$ direction. The green line shows the closed contour on which the Berry phases are computed. (c), (g) Spectrum of the Wilson-loop operators ${\mathcal{W}}_{ij}\left(\theta \right)$ of bands $i$ and $j$ as indicated in the axis labels of constant latitude contours on the purple ellipsoid in (a) and (e), respectively, parametrized by the latitude angle $\theta$. (d), (h) Euler curvature $F(\theta ,\phi )$ on the purple ellipsoid in (a) and (e), respectively, calculated using the algorithm from Ref. [73]. The curvature integrates to $\left|\chi \right|=2$ in (d) and to $\chi =0$ in (h).

Figure 20

Demonstration of the topological obstruction of the red nodal-line segment in ${\mathrm{Li}}_2\mathrm{NaN}$ due to the nontrivial Euler monopole charge. The nodal lines (obtained from the tight-binding model in Ref. [60]) near the rotation axis $A\mathrm{\Gamma }A\mathrm{\Gamma }$ spanning over one-and-a-half Brillouin zones are shown, with nodes in the second and third band gaps shown in red and blue, respectively. The two triple points (yellow) can be forced to collide as described in the text by tuning a parameter $\delta$ of the tight-binding model. The triple points collide at $\delta =-8.6$, at which point the red NL segment has been reduced to only a single touching point at $A$. Increasing $\delta$ even further, a horizontal red nodal ring forms in the $ALH$ plane.

Figure 21

Three hexagonal Brillouin zones around a $K$ point of a $C_6$-symmetric model. Assuming $\left[M_1^{\left(2\right)}\right]=1mod2$ the total Berry phase ${\varphi }_{\mathrm{\Gamma }M\mathrm{\Gamma }}$ of the occupied bands along the closed contour $\mathrm{\Gamma }M\mathrm{\Gamma }$ equals $\pi$ (cf. Appendix pp1-s2b). Therefore, the Berry phase along the triangular contour (green) is also ${\varphi }_{▵}=3\pi =\pi mod2\pi$. The result implies that the triangle encloses an odd number of band nodes in the principal gap (red disks), i.e., the system is not an insulator.

Figure 22

Illustration of the closed loops considered in Appendix pp2-s5 for the particular case $n=4$. (a) The contour ${\mathrm{\Lambda }}_{\epsilon }(n,\varphi )$ (green) consists of two rotation-related semicircular arcs inside rotation-related mirror planes (here $m_x$ and $m_y$). If mirror-protected nodal lines occur along the contour, we avoid them with infinitesimal half-circles perpendicular to the mirror plane containing the avoided nodal lines. The infinitesimal circular contour ${\eta }_{\epsilon }^{\left(1\right)}\left(\varphi \right)$ enclosing the nodal line is illustrated as a dark orange ring; it is symmetric under $m_y$. (b) Composition of ${\mathrm{\Lambda }}_{\epsilon }(n,\varphi )$ with ${\eta }_{\epsilon }^{\left(1\right)}\left(\varphi \right)$ results in the new contour ${\mathrm{\Lambda }}_{\epsilon }^{\prime }(n,\varphi )$, which is symmetric under the diagonal mirror symmetry $m_{\overline{x}y}$.

Figure 23

Discrete coarse-graining method. (a) The charge on the original lattice (defined by the unit cells shaded in blue and sublattice sites shown as green disks) is redistributed to the target lattice (defined by the red diamond sites). Here the lattice corresponding to the projection of ${\mathrm{Sc}}_3\mathrm{AlC}$ (see Sec. 6a) is shown. Each site in the target lattice is assigned a coarse-graining cell (unit cell of that lattice; shaded in orange). The charge on each green site is then redistributed (black arrows) with equal weights (weight 1 shown by solid, $\frac{1}{2}$ by dashed, and $\frac{1}{4}$ by dotted lines) to all those red sites whose coarse-graining cell is adjacent to the green site. Only a few arrows are shown to maintain clarity of the illustration. (b), (c) Definitions of the original (green) and target (red) lattice and the coarse-graining cell (orange) for the $C_4$- and $C_6$-symmetric models discussed in Sec. 5, respectively. Note that in (c) the coarse-graining cells overlap; as a consequence, the original and the target lattice coincide except for a missing layer on the boundary.

Figure 24

Procedure to remove the edge signal from the charge distribution of a $C_4$-symmetric system. It is assumed that the total edge charge per unit cell (of the ribbon geometry) vanishes. The solid lines in both panels represent the boundary of the lattice. (a) Ribbon geometry extended in the horizontal direction with $2N-1=7$ unit cells in the vertical direction (corresponding to $N=4$ inequivalent layers). Unit cells of the lattice are indicated by dots, colored according to their distance to the boundary. We denote by $q_i$ the charge per unit cell on the $i\text{th}$ layer (which is constant within the layer). (b) Flake geometry for the same system. Here, shells of unit-cell sites (dots) correspond to the layers from the ribbon geometry (indicated by the corresponding color). The charge on the whole flake is redistributed according to the arrows indicated in the bottom-left part of the system. The amount of charge that is transferred from a site that belongs to the $j\text{th}$ shell to the next one is given by ${\sum }_{i=1}^j{q}_j$, where $q_i$ are the charges determined in the ribbon geometry [cf. (a)]. Note that the charge redistribution flows perpendicular to the boundary; therefore, no charge is transferred between the corners.

Figure 25

Definition of nodal-line (NL) orientations via the non-Abelian invariant, illustrated on the example of multiband nodal link [60] of a three-band model; red (blue) line indicates a NL formed by the lower (upper) two bands. The non-Abelian invariant is computed on paths $\gamma$ (light green) and $\tilde{\gamma }$ (dark green), where $P^{\prime }$ is shifted relative to $P$ by a reciprocal lattice vector. In the three-band example shown here, $\mathfrak{q}\left(\gamma \right)=g_2$ (corresponding to a nodal line in the second gap, shown in blue) and bands 2 and 3 have different Berry phases ${\varphi }_2\ne {\varphi }_3$. Therefore, the orientation of the upper blue nodal ring is opposite to the one of the lower ring, $\mathfrak{q}\left(\gamma \right)=-g_2$.

Figure 26

Computation of Euler monopole charges of a pair of single triple points in ${\mathrm{Na}}_2\mathrm{LiN}$ with the surface centered at $\mathrm{\Gamma }$. (a) Shows the Brillouin zone, the high-symmetry points, and the planes in which the nodal lines are shown in (b). (b) Size of the four relevant gaps (orange, red, blue, and purple, according to increasing energy; at the triple point both the red and blue gaps are closed) in the two mirror planes shown in (a) encoded by the intensity of the color (and with a cutoff at a gap of $0.02\mathrm{eV}$). The triple point (yellow) and the central nodal line are emphasized by appropriately colored overlays. (c) Wilson-loop spectrum computed for the relevant bands on the ellipsoid indicated in (b). We observe that the Wilson-loop spectrum winds twice, which implies $\left|\chi \right|=2$.

Figure 27

Computation of Euler monopole charges of a pair of single triple points in ${\mathrm{Na}}_2\mathrm{LiN}$ with the surface centered at $Z$. The organization of the panels is in one-to-one correspondence with Fig. 26 (the same cutoff of $0.02\mathrm{eV}$ is used). In (c) we observe that the Wilson-loop spectrum winds twice, which implies $\left|\chi \right|=2$.

Figure 28

Computation of Euler monopole charges of a pair of single triple points in ${\mathrm{Li}}_2\mathrm{NaN}$ with the surface centered at $\mathrm{\Gamma }$. The organization of the panels is in one-to-one correspondence with Fig. 26 (cutoff $0.02\mathrm{eV}$). In (c) we observe that the Wilson-loop spectrum winds twice, which implies $\chi =0$.

Figure 29

Computation of Euler monopole charges of a pair of single triple points in ${\mathrm{Li}}_2\mathrm{NaN}$ with the surface centered at $A$. The organization of the panels is in one-to-one correspondence with Fig. 26 (cutoff $0.02\mathrm{eV}$). In (c) we observe that the Wilson-loop spectrum winds twice, which implies $\left|\chi \right|=2$.

Figure 30

Computation of Euler monopole charges of a pair of single triple points in ${\mathrm{TiB}}_2$ with the surface centered at $\mathrm{\Gamma }$. The organization of the panels is in one-to-one correspondence with Fig. 26 (cutoff $0.05\mathrm{eV}$). In (c) we observe that the Wilson-loop spectrum winds twice, which implies $\left|\chi \right|=2$.