Topological phase transitions of non-Abelian charged nodal lines in spring-mass systems

Although a large class of topological materials has uniformly been identified using symmetry properties of wave functions, the past two years have seen the rise of multigap topologies beyond this paradigm. Given recent reports of unexplored features of such phases, platforms that are readily implementable to realize them are therefore desirable. Here we demonstrate that multigap topological phase transitions of non-Abelian charged nodal lines arise in classical phonon waves. By adopting a simple spring-mass system, we construct nodal lines of a three-band system. The braiding process of the nodal lines is readily performed by adjusting the spring constants. The generation and annihilation of the nodal lines are then analyzed using the Euler class. Finally, we retrieve topological transitions from trivial nodal lines to a nodal link. Our work provides a simple platform that can offer diverse insights to not only theoretical but also experimental studies on multigap topology.

Figure 1

A spring-mass phononic crystal. (a) Illustration of the spring-mass system in an orthorhombic lattice system. The yellow spheres have the mass $m$. The red and green are intersite springs that connect the first and second nearest neighbors, respectively. Each blue spring for the on-site potential connects a sphere (mass) and the fixed support, the gray structure. The red and blue springs are set to be parallel to each other along ${\mathbf{a}}_2$ and ${\mathbf{a}}_3$ directions. The intersite springs pass through the holes of the support, while the on-site springs are attached to the support. (b) The unit cell of (a) considered in Eq. (1).

Figure 2

Schematics of braiding nodal lines. (a) Frame charges of the nodal lines in a three-band system. The open and filled symbols indicate opposite charges, and the circles and rectangles mean $\pm \mathit{k}$ and $\pm \mathit{i}$, respectively. (b)–(d) Braiding of $P_1$ and $P_2$ around $Q_1$ and $Q_2$, respectively. After braiding, all these four points change their signs. The dotted curves are the Dirac strings where $[{\mathbf{u}}_{\mathbf{k}}^1,{\mathbf{u}}_{\mathbf{k}}^2]$ (b)–(c) and $[{\mathbf{u}}_{\mathbf{k}}^2,{\mathbf{u}}_{\mathbf{k}}^3]$ (d) flip their signs. (e) Phase transition between (d) and (f). $Q_1$ and $Q_3$ that were in opposite frame charges in (d) are now annihilated. $Q_2$ and $Q_4$ do the same. (f) Final formation of the nodal link. The closed surfaces enclosing the rings $R_3$ and $R_2$ are denoted as ${\mathcal{D}}_{12}$ and ${\mathcal{D}}_{23}$, respectively. In (b)–(f), red and blue arrows on the nodal lines indicate their orientations deduced from their frame charges.

Figure 3

Evolution of nodal lines. The red and blue nodal lines are degeneracies by ${\omega }_{\mathbf{k}}^2-{\omega }_{\mathbf{k}}^1$ and ${\omega }_{\mathbf{k}}^3-{\omega }_{\mathbf{k}}^2$, respectively. (a) Trivial state with $\mathrm{\Delta }{C}_{12}=42$ and $\mathrm{\Delta }{C}_{31}=15$. (b) Braided state after tuning $\mathrm{\Delta }{C}_{12}$ to 20. (c) Phase transition by $\mathrm{\Delta }{C}_{31}$ to 10.06. Degeneracies by ${\omega }_{\mathbf{k}}^3-{\omega }_{\mathbf{k}}^2$ temporarily exhibit a nodal chain. (d) Nodal link by $\mathrm{\Delta }{C}_{31}$ to 5. The parameters for the nodal link in (d) is listed in Table 1 of Appendix pp2.

Figure 4

Allowed transition of nodal lines $L_1$ and $L_2$ according to the Euler class. $L_1$ and $L_2$ carry the same frame charge which gives Euler class of $+1$ for ${\mathcal{D}}_{23}$.

Figure 5

Generalized spring-mass system. (a) Unit cell containing a mass and all springs dealt in the generalized $3\times 3$ matrix. (b)–(d) Mass and springs on the red, green, and blue planes in (a), respectively. (e) Mass and springs along the diagonal directions of the unit cell. (f) Schematic figure of the spring concerning on-site potential.

Figure 6

Seven lattice systems and their nodal links. The left figures in each panel are the real space plots according to Table 1. The spring whose spring constant is zero is not plotted. The right figures in each panel are the resulting nodal links in the first Brillouin zones. The red and blue nodal lines are by ${\omega }_{\mathbf{k}}^2-{\omega }_{\mathbf{k}}^1$ and ${\omega }_{\mathbf{k}}^3-{\omega }_{\mathbf{k}}^2$, respectively. The light blue nodal lines around boundaries mean that they are outside the current Brillouin zone. (c) is equivalent to Fig. 3 in the main text.

Figure 7

Frame charges $\mathit{k}$ and $\mathit{i}$ of the orthorhombic system. (a) Closed loops that encircle the nodal rings $R_2$ and $R_3$, respectively. (b), (d) Eigenstates ${\mathbf{u}}_{\mathbf{k}}^1, {\mathbf{u}}_{\mathbf{k}}^2$, and ${\mathbf{u}}_{\mathbf{k}}^3$ on the loops in (a), respectively, in an arbitrary coordinate system. (c), (e) Eigenstates whose tails are gathered at the origin. In (c), the dark-red and -green curved arrows mean the traces of the eigenstates ${\mathbf{u}}_{\mathbf{k}}^1$ and ${\mathbf{u}}_{\mathbf{k}}^2$ rotating around ${\mathbf{u}}_{\mathbf{k}}^3$, respectively. In (f) the dark-green and -blue curved arrows mean the traces of the eigenstates ${\mathbf{u}}_{\mathbf{k}}^2$ and ${\mathbf{u}}_{\mathbf{k}}^3$ rotating around ${\mathbf{u}}_{\mathbf{k}}^1$, respectively.

Figure 8

Nodal lines and closed loops for frame charge characterizations. Nodal lines in (a), (b)–(d), and (e)–(f) correspond to Figs. 3, 3 and 3, respectively. Frame charges $n\left(C_i\right)$ calculated on each loop $C_i$ are also denoted. Note that the open and solid symbols (circles and rectangles) do not mean the absolute frame charge of each nodal line, but they are shown to illustrate the same or opposite orientations between two nodal lines enclosed by a closed loop.

Figure 9

Calculation of Euler class ${\chi }_{12}\left({\mathcal{D}}_{12}\right)$. (a) Patch ${\mathcal{D}}_{12}$ pierced by $R_1$ and $R_2$. Local coordinates $k_a$ and $k_b$ for each face are also marked as red and blue arrows. $k_a\times k_b$ is always outward of the box. The integral of Euler connection is performed along the counterclockwise direction of each boundary, as displayed by the green dotted arrows. (b) Euler form numerically calculated over face 2 of ${\mathcal{D}}_{12}$. The red dots indicate the nodal lines, and the dotted lines are the Dirac strings. The yellowish region is a result of the region being near the curved section of $R_1$, as shown in (a). Face 5's Euler form is the same. On the other faces, the surface integral of Euler form and boundary integral of Euler connection are canceled out.

Figure 10

Calculation of Euler class ${\chi }_{23}\left({\mathcal{D}}_{23}\right).$ (a), (c) Patch ${\mathcal{D}}_{23}$ pierced by $R_3$ and $R_4$. (b), (d) Euler form numerically calculated over faces 2 and 5 of ${\mathcal{D}}_{23}$, respectively. The blue dots indicate the nodal lines, and the dotted line is the Dirac string. On the other faces of ${\mathcal{D}}_{23}$, the surface integral of Euler form and boundary integral of the Euler connection are canceled out.

Figure 11

Schematics of calculating Euler class. (a) Discretized domain to calculate the Euler class. This domain consists of $N^2$ rectangular meshes. Eigenfrequencies and eigenstates are calculated at red dots and meshes' vertices, respectively. Meshes that contain band degeneracies are marked as red rectangles. The thick dotted black path means the Dirac string. Each mesh is indexed by $(i,j)$. Eigenstates smoothing on the boundary and Euler connection integral are performed along the thick green arrow direction. (b) Enlarged cell $(i,j)$. $P_{12}, P_{23}, P_{34}$, and $P_{14}$ are midpoints of each edge.

Figure 12

Allowed transition of nodal lines and nodal rings between (a), (d), (g) and (c), (f), (i), through (b), (e), (h). Schematics (a)–(c) are about the transition of nodal lines $L_1$ and $L_2$. Schematics (d)–(f) display the transition of $R_1$ and $R_2$ on the zone boundary. If $R_2$ is in the current zone, $R_1$ is in the next zone. (g)–(i) Realization of (a)–(c) and (d)–(f) using the spring-mass system. The transition is performed by decreasing $\mathrm{\Delta }{C}_{31}$ from 2 (g) through 0 (h) to $-2$ (i), with setting $K_1=140$ and keeping the other variables in Table 1.