Non-Abelian topology of nodal-line rings in $\mathcal{PT}$-symmetric systems

Nodal lines inside the momentum space of three-dimensional crystalline solids are topologically stabilized by a $\pi$ flux of Berry phase. Nodal-line rings in $\mathcal{P}\mathcal{T}$-symmetric systems with negligible spin-orbit coupling (here described as “nodal class AI”) can carry an additional “monopole charge,” which further enhances their stability. Here, we relate two recent theoretical advancements in the study of band topology in nodal class AI. On the one hand, cohomology classes of real vector bundles were used to relate the monopole charge of nodal-line rings to their linking with nodal lines formed among the occupied and among the unoccupied bands. On the other hand, homotopy studies revealed that the generalization of the Berry-phase quantization to the case of multiple band gaps defines a non-Abelian topological charge, which governs the possible deformations of the nodal lines. The present work has three aims. First, we present how to efficiently wield the recently discovered non-Abelian topological charge. Second, we apply these methods to present an independent proof of the relation between the monopole charge and the linking structure, including all the fragile-topology exceptions. Finally, we show that the monopole charge flips sign when braided along a path with a nontrivial Berry phase. This facilitates a new type non-Abelian “braiding” of nodal-line rings inside the momentum space, that has not been previously reported. The work begins with a brief review of $\mathcal{P}\mathcal{T}$-symmetric band topology, and the geometric arguments employed in our theoretical analysis are supplemented in the appendices with formal mathematical derivations.

Figure 1

The black frame indicates a region of $\mathit{k}$-space, and the wavy lines ${\mathrm{L}}_G$ with $G\in \{-2,-1,0,+1,+2\}$ indicate NLs formed inside band gap $G$. In the online version, we use color scheme “ORBIT” (acronym for orange, red, blue, indigo, and tan) to plot NLs occurring inside the five consecutive band gaps. Closed paths $\mathrm{\Gamma }$ on which we evaluate the non-Abelian topological charge are plotted in green.

Figure 2

A point node ${\mathrm{L}}_G$ inside a plane can be enclosed by paths ${\mathrm{\Gamma }}_G^a$ and ${\mathrm{\Gamma }}_G^b$, both based at point $\mathrm{P}$. The two based paths cannot be continuously deformed into each other due to the presence of an additional point node ${\mathrm{L}}_{G^{\prime }}$. However, the two paths can be related by a conjugation with path ${\mathrm{\Gamma }}_{G^{\prime }}$ enclosing ${\mathrm{L}}_{G^{\prime }}$, leading to a conjugation relation between the topological charge on ${\mathrm{\Gamma }}_G^a$ and on ${\mathrm{\Gamma }}_G^b$, cf. Eqs. (11).

Figure 3

(a) Our convention to assign each NL inside a 3D $\mathit{k}$-space a unique orientation. We first fix a base point $\mathrm{P}$ from which we observe the NL composition. To define an orientation of a chosen NL segment, we consider closed paths (green ${\mathrm{\Gamma }}_i)$ composed of (1) a straight ray going from $\mathrm{P}$ to the vicinity of the NL segment, (2) a tight loop around the NL segment, and (3) the same ray in reverse. As explained in Sec. 3c, the NL orientation defined this way is reversed each time we see it passing under a NL formed inside the neighboring band gap. In the illustration, the orientation of a blue NL (${\mathrm{L}}_0$) is reversed when we see it passing under a red NL (${\mathrm{L}}_{-1}$). The reversal of the NL orientation is indicated by a triangular arrowhead with the corresponding color (“$⊳$”). (b) Two-dimensional projection of the situation displayed in the left panel. The colored dots along the projection of the blue (respectively, red) nodal line indicate the tightly enclosed points of the nodal lines. This projection makes it clear that paths ${\mathrm{\Gamma }}_0^a$ and ${\mathrm{\Gamma }}_0^b$ are related by conjugation with path ${\mathrm{\Gamma }}_{-1}$, in the same way as paths ${\mathrm{\Gamma }}_G^a$, ${\mathrm{\Gamma }}_G^b$, and ${\mathrm{\Gamma }}_{G^{\prime }}$ in Fig. 2. The centers of the tight-loop component of paths ${\mathrm{\Gamma }}_0^a$ and ${\mathrm{\Gamma }}_0^b$ here should be identified with the same point $L_G$ in Fig. 2.

Figure 4

(a) A red NL ${\mathrm{L}}_{-1}$ passing below a blue NL ${\mathrm{L}}_0$ inside some region (the cube) of $\mathit{k}$ space. The orientation of ${\mathrm{L}}_0$ is constant, while the orientation of ${\mathrm{L}}_{-1}$ is reversed at the overlay (indicated by “$⊳$”). (b) A red NL ${\mathrm{L}}_{-1}$ passing over a blue NL ${\mathrm{L}}_0$. In this case, the orientation of ${\mathrm{L}}_{-1}$ is the same everywhere, while the orientation of ${\mathrm{L}}_0$ is flipped at the overlay (indicated by “$⊳$”). The situation (a) cannot be continuously evolved into the situation (b) because the orientations of the four outgoing NL segments do not match. (c) Analogous situation with two NLs of the same color (here both blue). Since their topological charges commute, cf. Eq. (10c) for $G=G^{\prime }$, there is no orientation reversal at the overlay. This allows the two NLs ${\mathrm{L}}_0^a$ and ${\mathrm{L}}_0^b$ to freely move across each other. Furthermore, (d) near the critical points there is also a third phase, in which the two NLs of the same color reconnect into ${\mathrm{L}}_0^{+}$ and ${\mathrm{L}}_0^{-}$. The three phases near the critical point are analogous to Fig. 1 of Ref. [85].

Figure 5

Four examples of NL compositions consistent with the criteria formulated in Sec. 3e. The indexing (in the online version also the coloring) of the displayed NLs follow the scheme shown in Fig. 1. The orientation reversals of the NLs are indicated by triangular arrowheads (“$⊳$”) of the corresponding color. (a) A blue NL ring ${\mathrm{L}}_0$ is located above a red NL ring ${\mathrm{L}}_{-1}$. Since ${\mathrm{L}}_0$ and ${\mathrm{L}}_{-1}$ are formed inside consecutive band gaps, ${\mathrm{L}}_{-1}$ reverses orientation each time it goes under ${\mathrm{L}}_0$. Since both NL rings reverse orientation an even number of times (0 vs 2), the NL composition is admissible. (b) A composition of two red NL rings ${\mathrm{L}}_{-1}^{a,b}$ and two blue NL rings ${\mathrm{L}}_0^{a,b}$. Each NL ring is threaded by two NLs of the other color, leading to two orientation reversal along each NL ring, as required by the consistency criteria. (c) A composition of one red NL ring ${\mathrm{L}}_{-1}$, two blue NL rings ${\mathrm{L}}_0^{a,b}$, and one indigo NL ring ${\mathrm{L}}_{+1}$. NL rings ${\mathrm{L}}_{-1}$ (in band gap $G=-1$) and ${\mathrm{L}}_{+1}$ (in band gap $G=+1$) are each threaded by two blue NLs ${\mathrm{L}}_0^{a,b}$ (in a neighboring band gap $G=0$), leading to two orientation reversals. On the other hand, ${\mathrm{L}}_0^a$ as well as ${\mathrm{L}}_0^b$ are each threaded by ${\mathrm{L}}_{-1}$ and by ${\mathrm{L}}_{+1}$, both inside neighboring band gaps. This implies an even number of orientation reversal of each ${\mathrm{L}}_0^{a,b}$. We find that all NL rings in this composition reverse orientation an even number of times, implying its consistency. (d) A Hopf link formed a red NL ring ${\mathrm{L}}_{-1}$ and an indigo NL ring ${\mathrm{L}}_{+1}$. Since NLs with $G=\pm 1$ are not formed inside consecutive band gaps, they ignore the presence of each other, and no orientation reversals take place.

Figure 6

Manifestation of Eq. (10d), i.e., why four NLs of the same type and orientation are topologically trivial. (a–e) show a process that locally annihilates segments of four parallel red NLs. The process involves the creation of a pair of blue NL rings located inside a neighboring band gap. For a detailed discussion of the process, see Sec. 3f. The arrowheads (“$⊳$”) indicate orientation reversals of NLs of the corresponding color, as explained in Sec. 3c.

Figure 7

Summary of our notation. We consider a NL ring ${\mathrm{L}}_0^1$ formed inside the $G=0$ band gap, which is not linked with other NLs ${\mathrm{L}}_0^{2,3,...}$ formed inside the same band gap. We find a two-dimensional disk ${\mathcal{D}}^2$ that is bounded by ${\mathrm{L}}_0^1$ and which is not intersected by ${\mathrm{L}}_0^{2,3,...}$. Nodal lines ${\mathrm{L}}_{G^{\prime }\ne 0}$ inside other band gaps $G^{\prime }\ne 0$ can be linked with ${\mathrm{L}}_0^1$, leading to intersection points ${\mathrm{L}}_{G^{\prime }}\cap {\mathcal{D}}^2={\left\{{\mathrm{P}}_{G^{\prime }}^a\right\}}_{a\in \mathcal{I}}$. Depending on the orientation of the NL just above the disk, we assign to each intersection point with $G^{\prime }=\pm 1$ a flux $w_{G^{\prime }}^a=\pm 1$ (for the illustrated example $w_{-1}^1=+1$, $w_{-1}^2=-1$, and we do not define $w_{+2}^1$). To assign ${\mathrm{L}}_0^1$ a monopole charge, we inflate ${\mathcal{D}}^2$ into a thin “pancake” containing ${\mathrm{L}}_0^1$. The surface of the pancake is topologically a sphere ${\mathcal{S}}^2$ that preserves the $G=0$ band gap. We relate the monopole charge on ${\mathcal{S}}^2$ to the linking structure of ${\mathrm{L}}_0^1$ with NLs inside the neighboring band gaps. The linking structures are encoded in the fluxes $w_{\pm 1}^a$, which we use to calculate the linking numbers defined in Eqs. (15). For better clarity, we do not draw the pancake surface ${\mathcal{S}}^2$ in the following figures. The triangular arrowheads (“$⊳$”) indicate orientation reversals of NLs of the corresponding color.

Figure 8

The monopole charge assigned to NL ring ${\mathrm{L}}_0^1$ may depend on the choice of the inscribed disk ${\mathcal{D}}^2$. The flat gray disk ${\mathcal{D}}_a^2$ (only a section of the disk is shown) exhibits four intersection points with ${\mathrm{L}}_{-1}$, all having the same orientation, leading to ${\eta }_{-}=+4$. In contrast, the cap-shaped green disk ${\mathcal{D}}_b^2$ is not intersected by ${\mathrm{L}}_{-1}$, leading to a different value ${\eta }_{-}=0$. The ambiguity arises because an additional NL ring ${\mathrm{L}}_0^2$ makes it impossible to continuously deform ${\mathcal{D}}_a^2$ into ${\mathcal{D}}_b^2$ while preserving the $G=0$ spectral gap. In Sec. 5, we relate this ambiguity to nontrivial braiding of the monopole charge around NLs of the same color.

Figure 9

The possible strategies to change the linking number ${\eta }_{-}$ of a blue ($G=0$) NL ring ${\mathrm{L}}_0^1$ with red ($G=-1$) NLs ${\mathrm{L}}_{-1}$. None of these strategies is succesful. [(a) and (b)] Moving a segment of ${\mathrm{L}}_{-1}$ across the disk produces two intersection points ${\mathrm{P}}_{-1}^{1,2}$ with opposite flux. [(c) and (d)] Local reconnection of two antiparallel segments of ${\mathrm{L}}_{-1}$ annihilates two intersection points ${\mathrm{P}}_{-1}^{1,2}$ with opposite flux. [(e) and (f)] Local annihilation of four parallel segments of ${\mathrm{L}}_{-1}$ is possible, following the strategy of Fig. 6, but produces two new NL rings ${\mathrm{L}}_0^{2,3}$ inside the $G=0$ band gap. Their presence makes it impossible to bulge the disk ${\mathcal{D}}_2$ into the region where the four segments of ${\mathrm{L}}_{-1}$ have annihilated, because ${\mathrm{L}}_0^2$ would temporarily close the $G=0$ band gap on the sphere ${\mathcal{S}}^2$ [the sphere is shown explicitly in (e)] obtained by inflating the disk. In (b) and (c), the triangular arrowheads (“$⊳$”) indicate the orientation reversal of NLs of the corresponding color. For better clarity, such reversals are not explicitly shown in (e) and (f).

Figure 10

Changing the linking number ${\eta }_{-}$ of the blue ($G=0$) NL ring ${\mathrm{L}}_0^1$ in a $(3+1)$-band model by $\pm 2$. The change is achieved by moving an orange ($G=-2$) NL ${\mathrm{L}}_{-2}$ to the inside of the NL ring ${\mathrm{L}}_0^1$. The anticommutation of the charge $g_{-2}$ of orange NLs ${\mathrm{L}}_{-2}$ with the charge $g_{-1}$ of red ($G=-1$) NLs ${\mathrm{L}}_{-1}$ allows us to revert the flux of some of the intersection points ${\mathrm{P}}_{-1}^a$, thus changing the value of ${\eta }_{-}$ by multiples of 2. As a consequence, ${\eta }_{-}$ can be made zero, and the blue NL ring ${\mathrm{L}}_0^1$ can be completely unlinked. Triangular arrowheads (“$⊳$”) indicate the orientation reversals of NLs of the corresponding color. For a more detailed discussion of the process illustrated in (a)–(d), see Sec. 4e.

Figure 11

(a) A pair of blue ($G=0$) NL rings ${\mathrm{L}}_0^{1,2}$ which exhibit opposite values of the linking number ${\eta }_{-}$ with red ($G=-1$) NLs ${\mathrm{L}}_{-1}^{1,2}$, namely ${\eta }_{-}=\pm 2$. According to the arguments presented in Sec. 4, the linking numbers reveal that ${\mathrm{L}}_0^{1,2}$ carry opposite values of the monopole charge. (b) We merge the two NL rings ${\mathrm{L}}_0^{1,2}$ along a trajectory that does not pass under another blue NL. We observe that the combined NL ring ${\mathrm{L}}_0^{1+2}$ exhibits net linking number ${\eta }_{-}^{\mathrm{tot}}.=0$, meaning that its monopole charge is trivial. (c) Alternatively, we bring the blue NL rings ${\mathrm{L}}_0^{1,2}$ together along a trajectory that encloses another blue NL ${\mathrm{L}}_0^3$. In this process, the blue NL ${\mathrm{L}}_0^3$ moves in front of all the intersection points inside the NL ring ${\mathrm{L}}_0^2$ (namely, ${\mathrm{P}}_{-1}^3$ and ${\mathrm{P}}_{-1}^4$), inducing a reversal of their fluxes (i.e., the numbers $w_{-1}^3$ and $w_{-1}^4$). (d) When moved together along such an alternative path, the blue NL rings ${\mathrm{L}}_0^{1,2}$ exhibit the same linking number ${\eta }_{-}=+2$, such that the total linking after they are merged into ${\mathrm{L}}_0^{1+2}$ is ${\eta }_{-}^{\mathrm{tot}}.=4$, implying a nonvanishing monopole charge. The path-dependent ability of the two monopole charges to cancel resp. to add up indicates a nontrivial action induced by the Berry phase (carried by the blue NL ${\mathrm{L}}_0^3$) on the monopole charge (carried by the blue NL ring ${\mathrm{L}}_0^2$). For better clarity, in this figure we do not explicitly indicate orientation reversals of the NLs.

Figure 12

Schematic illustration of the considerations in Appendix pp2-s2. The projection $\mathsf{G}\stackrel{\pi }{\to }\mathsf{G}/\mathsf{H}$ corresponds to the fibration in Eq. (A1). The closed path $\mathrm{\Gamma }:I\to \mathsf{G}/\mathsf{H}$ shown in blue (right) is lifted to an open-ended path ${\mathrm{\Gamma }}_{\uparrow }:I\to \mathsf{G}$ (left). The path ${\mathrm{\Gamma }}_{\uparrow }\left(I\right)$ starts at the identity $e$ and ends at an element $h$ contained inside the connected component $h{\mathsf{H}}_0$ of coset ${\mathsf{H}}_e$. Similarly, the continuous based map $f:I^p\to \mathsf{G}/\mathsf{H}$ shown in green (right) is lifted into $f_{\uparrow }:I^p\to \mathsf{G}$ (left) in a way that the top face $J^{p-1}\subset \partial I^p$ of the cube is mapped to the identity $e$ [121, 144]. The gray sheets indicate connected components ${\mathsf{H}}_0$ and $h{\mathsf{H}}_0$ of coset ${\mathsf{H}}_e$. The image $\mathrm{im}{f}_{\uparrow }$ looks like a cap (green), while the image of the composition of ${\mathrm{\Gamma }}_{\uparrow }$ with $f_{\uparrow }{|}_{\partial I^p}\equiv \partial f_{\uparrow }$ looks like a tube (blue). The black rings along the tube indicate several images ${\mathrm{\Gamma }}_{\uparrow }\left(t\right)·\partial f_{\uparrow }$ with fixed $t\in I$, and they project onto the black dots shown along $\mathrm{\Gamma }\left(I\right)$ inside $\mathsf{G}/\mathsf{H}$. Since the green cap and the blue tube share a boundary, we can compose them. Finally, we shift this composition such that its boundary lies inside ${\mathsf{H}}_0$. This is achieved by a right multiplication with $h^{-1}$. We find that the composition of $f$ with $\mathrm{\Gamma }$ induces the automorphism ${▹}_h\in \mathrm{Aut}\left[{\pi }_p(\mathsf{G}/\mathsf{H})\right]$, called the action of ${\pi }_1(\mathsf{G}/\mathsf{H})$ on ${\pi }_p(\mathsf{G}/\mathsf{H})$, which we describe in the text. For $p=2$, the automorphism is given explicitly by Eq. (B3).

Figure 13

A map $f^{♭}:S^1\times I\to M$ such that the green portion, i.e., $S^1\times \left(\partial I\right)\cup \mathrm{P}\times I$, is mapped to a base point $\mathfrak{m}\in M$.

Figure 14

A map $f^{♭}$ as in Fig. 13 is isomorphic to a map $f\in \text{Map}{[S_{\mathrm{BZ}}^2,M]}_0$ via the above construction. The first horizontal arrow corresponds to mapping the cylinder $S^1\times I$ to $S^2$ such that the top and bottom circles getted mapped onto chosen points in $S^2$. The second arrow maps $S^2$ to itself such that the entire green portion gets mapped to a single point.

Figure 15

Maps $f^{♭}$ and $\mathrm{\Gamma }\in \text{Map}{[S_{\mathrm{BZ}}^1,M]}_0$ can be composed by stacking cylinders.

Figure 16

Maps $f_{1,2}^{♭}\in \text{Map}{[S_{\text{BZ}}^2,M]}_0$ can be composed by stacking cylinders such that all the marked (green) portions are mapped to the marked (green) point $\mathfrak{m}\in M$.

Figure 17

The action of ${\pi }_1(M,\mathfrak{m})$ on ${\pi }_2(M,\mathfrak{m})$ is understood as a concatenation of cylinders. The bottom cylinder and the top cylinder are foliations of circles, where each foliation is mapped to the same point in $M$. The bottom and the top circles are mapped to the green point in $M$. Similarly, the middle cylinder is essentially a map from $S^2$ to $M$ with all the red regions mapped to the (arbitrary) red point in $M$.

Figure 18

Abe homotopy group ${\kappa }_2(M,\mathfrak{m})$ can be defined as homotopy classes of based maps from a pinched torus to $M$. As a set, ${\kappa }_2(M,\mathfrak{m})$ can be identified with ${\pi }_1(M,\mathfrak{m})\times {\pi }_2(M,\mathfrak{m})$, however the product structure on ${\kappa }_2(M,\mathfrak{m})$ corresponds to a semidirect product.

Figure 19

There exists an injection ${\mathrm{\Psi }}_{*}:{\pi }_2(M,\mathfrak{m})\to {\kappa }_2(M,\mathfrak{m})$ such that ${\mathrm{\Psi }}_{*}\left(g\right)=g^{\mathfrak{a}}={\psi }_1\circ {\psi }_2\circ g$. The map ${\psi }_1$ shrinks the top and the bottom circles of the cylinder to the north and the south poles of the 2-sphere, respectively, while ${\psi }_2$ is a smooth map that sends the north pole to an arbitrary point on the 2-sphere while keeping the south pole fixed.

Figure 20

There exists an inclusion $\iota :{\pi }_1(M,\mathfrak{m})\hookrightarrow {\kappa }_2(M,\mathfrak{m})$ obtained by “thickening” a closed loop into a pinched torus.