# Phase transitions of non-Abelian charged nodal links in a spring-mass system

Haedong Park  School of Physics and Astronomy, Cardiff University, Cardiff
CF24 3AA, United Kingdom  Stephan Wong  School of Physics and Astronomy,
Cardiff University, Cardiff CF24 3AA, United Kingdom  Adrien Bouhon  TCM
Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE,
United Kingdom  Robert-Jan Slager  TCM Group, Cavendish Laboratory, University
of Cambridge, Cambridge CB3 0HE, United Kingdom  Sang Soon Oh
<EMAIL_ADDRESS>School of Physics and Astronomy, Cardiff University,
Cardiff CF24 3AA, United Kingdom

###### Abstract

Although a large class of topological materials have uniformly been identified
using symmetry properties of wave functions, the past two years have seen the
rise of multi-gap 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 multi-gap
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 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 multi-gap topology.

††preprint: APS/123-QED

_Introduction_.—The uncovering of topological insulators has opened an active
field of physics [1, 2] and the past decades have seen rapid progress on these
effective topological phases. Topological classifications have in particular
profited from the interplay with space group symmetries [3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Beyond these works on single gap
topologies, recent studies on the multi-gap conditions are however steadily
rising [21, 22]. The crux of these developments is the insight that
degeneracies in a multi-gap system may carry non-Abelian charges [23, 24, 25],
and that braiding a degeneracy around another in a different gap can hence
render a system with similarly-valued charges between two bands [23, 24, 26,
27, 28]. Such braiding and stability/instability of band degeneracies can then
be described using Euler class [29, 25, 30, 24, 27, 28], an integer-valued
multi-gap topological invariant, that is calculated over the Brillouin zone or
a patch thereof in momentum space. That is, a non-zero integer-valued Euler
class indicates that the band touchings in the patch are topologically
obstructed to annihilate. Multi-gap topologies have been predicted to
culminate in new effects such as specific monopole-antimonopole generation
[30] that was recently observed in trapped ion experiments [31] as well as
signatures in structural phase transitions [32, 33], strained and magnetic
electronic systems [24, 34], and phononic systems [27, 28, 35, 36, 37].

Given these new features of non-Abelian charged degeneracies and multi-gap
topological physics, a simple, intuitive, and easily tuneable, but
experimentally viable system is of interest. A highlight candidate in this
regard are classical spring-mass systems [38, 39, 40, 41, 42, 43, 44, 45].
Especially because, if a spring-mass system is constituted in three-
dimensional real space, it may exhibit nodal line degeneracies that can have
an intricate interplay with non-Abelian charges [26, 22]. A nodal line is a
set of degeneracies in one-dimensional curved shape formed in three-
dimensional momentum space [46]. Various types of the nodal lines, e.g., nodal
rings [47, 48], nodal chains [49, 50, 51, 52, 53, 54], nodal links [51, 52,
53, 54, 55, 56, 57, 58, 59, 60], and nodal knots [58, 59, 61], have already
been reported using metals [23, 49, 57], semimetals [51, 52, 56, 59, 61, 62],
electrical circuits [58], photonic media [63, 48, 50, 54, 55, 64, 65], and
phononic crystals [47, 60, 66, 67, 36]. In addition, the non-Abelian multi-gap
charges, or frame rotation charges [23, 24, 27, 28], in three-band systems are
especially insightful and amount to quaternion numbers. Consequently, if a
simple phononic crystal having three bands exhibits nodal lines it acts as a
good platform to observe quaternion charges.

Here, we predict phase transitions of multi-gap nodal lines quantified by
Euler class using a class of simple and easily realizable spring-mass systems.
The non-Abelian charged nodal lines are realized using phonon waves in a
classical spring-mass system [68, 69, 70, 71, 39, 38, 40, 41, 42, 43, 44, 45,
72, 73, 74]. A unit cell of our system consists of only one mass and several
springs, and the system’s mechanical behavior is thus described by classical
mechanical equations of motion. Some springs concerning on-site potentials
[42, 43, 44, 45, 72, 73, 74] eliminate the degeneracy at the $\Gamma$-point
and enable a three-band nodal link. This heuristically amounts to removing the
Nambu-Goldstone zero modes [37] by having different on-site potential. By
braiding one nodal line around the other, we propose the principle of
generating a nodal link from a trivial state featuring nodal rings. The phase
transition is realized using the braiding process carried out by tuning the
spring constants. Subsequently, we apply Euler class to characterize whether
the nodal lines are topologically stable or can be annihilated pairwise. We
finally also discuss the possible states and transitions of the nodal lines
from the Euler class viewpoint.

_Spring-mass system_.—We consider the spring-mass system in a simple
orthorhombic lattice, where the unit cell consists of a single mass and eleven
springs, as shown in Fig. 1. We assume that all masses of the springs are
ignored, and only the sphere’s mass $m$ is considered. Spring constants
concerning inter-site energies are denoted by $C_{ij}$, where the subscripts
$i$ and $j$ mean the connection directions. A spring whose spring constant is
denoted by $C_{ii}$ connects the first nearest neighbor masses along
${\mathbf{a}}_{i}$-direction where ${\mathbf{a}}_{i}$ ($i=1,2,3$) is the
lattice vector. Springs with $C_{ij}$ ($i\neq j$) connect the second nearest
neighbor masses along ${\mathbf{a}}_{i}\pm{\mathbf{a}}_{j}$ direction. We also
consider two springs for the on-site potentials, and their spring constants
are denoted by $K_{1}$ and $K_{2}$ [see Fig. 1(b)]. While phononic crystals
generally exhibit the triple point degeneracy at the $\Gamma$-point, these
anisotropic on-site springs lift the degeneracy.

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 inter-site 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 inter-site 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).

A unit cell in the three-dimensional array is denoted by integer-valued
$\left[h_{1},h_{2},h_{3}\right]$ where all subscripts correspond to
${\mathbf{a}}_{1}$, ${\mathbf{a}}_{2}$, and ${\mathbf{a}}_{3}$-directions,
respectively. The equation of motion for the displacement
${\mathbf{u}}_{h_{1},h_{2},h_{3}}$ of a mass in the unit cell
$\left[h_{1},h_{2},h_{3}\right]$ is given by
$m{\ddot{\mathbf{u}}}_{h_{1},h_{2},h_{3}}=\sum_{n}{\mathbf{f}}_{n}$, where the
right-hand term is the summation of the spring force ${\mathbf{f}}_{n}$
exerted on the mass. The displacement vector can be expressed as
${\mathbf{u}}_{h_{1},h_{2},h_{3}}\left({\mathbf{x}},t\right)={\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}={\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot\left(h_{1}{\mathbf{a}}_{1}+h_{2}{\mathbf{a}}_{2}+h_{3}{\mathbf{a}}_{3}\right)}e^{-i\omega
t}$. Substituting this into the equation of motion leads to the following
eigenvalue problem:

$\displaystyle\frac{2}{m}\left[\sum_{i=1}^{3}H_{ii}+\sum_{i,j}H_{ij}+\frac{1}{2}\sum_{l}{K_{l}\left({\hat{\mathbf{d}}}_{l}\otimes{\hat{\mathbf{d}}}_{l}\right)}\right]{\mathbf{u}}_{\mathbf{k}}$
$\displaystyle=\left(\omega_{\mathbf{k}}\right)^{2}{\mathbf{u}}_{\mathbf{k}},$
(1)

where
$H_{ii}=C_{ii}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{i}\right)}\right\\}{\hat{\mathbf{a}}}_{i}\otimes{\hat{\mathbf{a}}}_{i}$
and
$H_{ij}=C_{ij}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{i}+{\mathbf{k}}\cdot{\mathbf{a}}_{j}\right)}\right\\}{\hat{\mathbf{a}}}_{ij}\otimes{\hat{\mathbf{a}}}_{ij}$
are the Hamiltonians for the inter-site interactions along ${\mathbf{a}}_{i}$
and ${\mathbf{a}}_{i}\pm{\mathbf{a}}_{j}$-directions, respectively.
${\hat{\mathbf{a}}}_{i}$, ${\hat{\mathbf{a}}}_{ij}$, and
${\hat{\mathbf{a}}}_{i\bar{j}}$ are the unit vectors along ${\mathbf{a}}_{i}$,
${\mathbf{a}}_{i}+{\mathbf{a}}_{j}$, and ${\mathbf{a}}_{i}-{\mathbf{a}}_{j}$,
respectively. In $H_{ij}$ ($ij=12,23,31,1\bar{2},2\bar{3},3\bar{1}$), if the
subscript $j$ is $\bar{1}$, $\bar{2}$, or $\bar{3}$, then
${\mathbf{k}}\cdot{\mathbf{a}}_{j}$ is replaced with
$-{\mathbf{k}}\cdot{\mathbf{a}}_{j}$. The last term entails the on-site
potential, and
${\hat{\mathbf{d}}}_{l}={\mathbf{d}}_{l}/\left|{\mathbf{d}}_{l}\right|$ is the
unit vector along the on-site spring connection. The detailed derivations are
in Supplemental Material, Sec. A [75]. Although Fig. 1 shows an orthorhombic
unit cell and the inter-site interactions in Eq. (1) are only along the first
and second nearest neighbors, we can also consider unit cells in the other
lattice systems and the interactions between the third nearest neighbors.
Supplemental Material, Sec. A and B also cover these general cases [75].

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 $\pm\bm{k}$ and $\pm\bm{i}$,
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
$\left[{\mathbf{u}}_{\mathbf{k}}^{1},{\mathbf{u}}_{\mathbf{k}}^{2}\right]$
(b-c) and
$\left[{\mathbf{u}}_{\mathbf{k}}^{2},{\mathbf{u}}_{\mathbf{k}}^{3}\right]$ (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.

_Braiding of non-Abelian charged nodal lines_.—The real-valued 3×3 Hamiltonian
in Eq. (1) is symmetric and positive definite for any wavevector $\mathbf{k}$.
Thus, the three eigenstates ${\mathbf{u}}_{\mathbf{k}}^{1}$,
${\mathbf{u}}_{\mathbf{k}}^{2}$, and ${\mathbf{u}}_{\mathbf{k}}^{3}$ form an
$\operatorname{SO}(3)$ orthonormal frame. Unless the degeneracies form an
accidental triple point [27, 76], as enforced e.g. by the Goldstone modes [36,
37], the generically stable band crossings in $\mathcal{PT}$-symmetric systems
are nodal lines [77] formed by only two adjacent bands. The non-Abelian frame
charge for a nodal line is determined by which bands are connected by the
nodal line and which band is gapped [23, 24, 26, 27, 28, 37]. A closed loop
that encloses a nodal line is considered. Along this loop, one obtains the
resulting frame charge by keeping track of the rotation of the orthonormal
frame $\left\\{{\mathbf{u}}_{\mathbf{k}}^{n}\right\\}_{n=1}^{3}$ via parallel
transport. The nodal lines of a three-band system turn out to be characterized
by the elements of the quaternion group
${\mathbb{Q}}=\left\\{\pm\bm{i},\pm\bm{j},\pm\bm{k},-1,+1\right\\}$ [23, 24].
The frame charge contains the information about which bands form a nodal line
in a multi-gap system. For example, the frame charge of the nodal line formed
by the first and second (by the second and third) bands can be denoted as
$\pm\bm{k}$ ($\pm\bm{i}$), as shown in Fig. 2(a). The detailed explanations
are in Supplemental Material, Sec. C [75]. The closed loop can also be set to
encircle two or more nodal lines. Its frame charge is expressed as the
multiplication of the frame charges of each nodal line, and it satisfies the
anti-commutation relations of the quaternion charges.

Topological phase transitions can be achieved through braiding of the frame
charges, which flips their signs due to their non-Abelian nature [23, 24, 26,
27, 28]. As mentioned in the above, a closed loop can encircle two nodal
lines. If the two nodal lines are equivalently (oppositely) charged, the frame
charge is $-1$ ($+1$) meaning that they carry a non-trivial (trivial) charge
[23, 24]. The resultant frame charge can vary depending on the choice of the
path [24]. Let us consider a closed loop $C$ that encircles both nodal lines
$L_{1}$ and $L_{2}$ and does not pass nodal rings $R_{1}$ and $R_{2}$ in Fig.
2(b). Here, we assume that the frame charge calculated along this loop is
$-1$, and thereby the frame charges of $L_{1}$ and $L_{2}$ are the same. We
braid $P_{1}$ and $P_{2}$ in Fig. 2(b) around $Q_{1}$ and $Q_{2}$,
respectively; they move under $L_{1}$ [see Fig. 2(c)], then over $L_{1}$ [see
Fig. 2(d)]. In Fig. 2(d), if the closed loop $C$ passes through $R_{1}$ (or
$R_{2}$), the frame charge becomes $+1$. The same explanation can be applied
to the nodal rings $R_{1}$ and $R_{2}$ in Fig. 2(b)-(d). In other words, after
completing the braiding process of $P_{1}$ and $P_{2}$ around the $Q_{1}$ and
$Q_{2}$, respectively, all their signs are flipped [see Fig. 2(d)]. Because
the frame charges of $Q_{1}$ and $Q_{3}$ (and $Q_{2}$ and $Q_{4}$) are trivial
in Fig. 2(d), they can be annihilated pairwise, as shown in Fig. 2(e).
Finally, they reach the nodal link in Fig. 2(f).

The stability of the nodal lines in the nodal link can be characterized by the
Euler class calculated on the closed patch which nodal lines pass through [24,
22]. If the Euler class is zero (non-zero), the nodal lines in the patch are
unstable (stable) [24]. The surfaces ${\mathcal{D}}_{12}$ and
${\mathcal{D}}_{23}$ in Fig. 2(f) enclose $R_{3}$ and $R_{2}$, respectively.
In other words, through ${\mathcal{D}}_{12}$, only nodal lines of $R_{1}$ and
$R_{2}$ exit, and no other lines pierce ${\mathcal{D}}_{12}$. Likewise,
through ${\mathcal{D}}_{23}$, only nodal lines of $R_{3}$ and $R_{4}$ pass. If
we observe the outward orientations at all the four nodes where $R_{1}$ or
$R_{2}$ touch ${\mathcal{D}}_{12}$, as marked in Fig. 2(f), we infer that the
nodal lines of $R_{1}$ and $R_{2}$ are stable due to the non-zero Euler class.
On the other hand, the patch ${\mathcal{D}}_{23}$ has both inward and outward
nodal lines. Thus, we expect the zero-valued Euler class and unstable $R_{3}$
and $R_{4}$. This means that $R_{3}$ and $R_{4}$ can be reversely-transformed
toward Fig. 2(e) and (d).

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 $\Delta C_{12}=42$ and $\Delta C_{31}=15$. (b) Braided state after
tuning $\Delta C_{12}$ to $20$. (c) Phase transition by $\Delta C_{31}$ to
$10.06$. Degeneracies by $\omega_{\mathbf{k}}^{3}-\omega_{\mathbf{k}}^{2}$
temporarily exhibit a nodal chain. (d) Nodal link by $\Delta C_{31}$ to $5$.
The parameters for the nodal link in (d) is listed in Table S1 of Supplemental
Material, Sec. B [75].

_Phase transitions in spring-mass system_.—The phase transition from trivial
nodal lines to a nodal link can be realized in our spring-mass system. Eq. (1)
receives the lattice vectors and spring constants as input parameters, and
outputs the eigenfrequencies $\omega_{\mathbf{k}}^{n}$ and eigenstates
${\mathbf{u}}_{\mathbf{k}}^{n}$, where the superscript $n$ is the band number
($n=1,2,3$). We use the values listed in Table S1 of Supplemental Material,
Sec. B [75], except for $C_{12}$, $C_{1\bar{2}}$, $C_{31}$, and
$C_{3\bar{1}}$. If we write $C_{12}=C_{12}^{0}+\Delta C_{12}$,
$C_{1\bar{2}}=C_{12}^{0}-\Delta C_{12}$, $C_{31}=C_{31}^{0}+\Delta C_{31}$,
and $C_{3\bar{1}}=C_{31}^{0}-\Delta C_{31}$ (where $C_{12}^{0}=75$ and
$C_{31}^{0}=35$), the four spring constants can be controlled by tuning only
$\Delta C_{12}$ and $\Delta C_{31}$. The degeneracies by
$\omega_{\mathbf{k}}^{2}-\omega_{\mathbf{k}}^{1}$ and
$\omega_{\mathbf{k}}^{3}-\omega_{\mathbf{k}}^{2}$ are plotted as red and blue
nodal lines, respectively, in Fig. 3. Note that all the nodal lines are
inversion symmetric about the $\Gamma$-point because Eq. (1) is
$\mathcal{T}$-symmetric and $\mathcal{P}$-symmetric. We start with the trivial
nodal lines [see Fig. 3(a)] by selecting $\Delta C_{12}=42$ and $\Delta
C_{31}=15$. The nodal lines $L_{1}$ and $L_{2}$ in Fig. 3(a) have the same
orientations directing the same boundary if the closed loop is placed like
Fig. 2(b). We change $\Delta C_{12}$ to $20$, then the rings $R_{1}$ and
$R_{2}$ become bigger, as shown in Fig. 3(b). The frame charges of $L_{1}$ and
$L_{2}$ are now opposite if the closed loop’s path is set like Fig. 2(d). This
configuration enables the pairwise annihilation of the $L_{1}$ and $L_{2}$
through the nodal rings $R_{1}$ and $R_{2}$, as shown in Fig. 3(c), by
decreasing $\Delta C_{31}$ to $10.06$. Further decreasing $\Delta C_{31}$ to
$5$ makes the final nodal link in Fig. 3(d). The frame charge analyses for the
situations in Fig. 3 are in Supplemental Material, Sec. C [75].

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 ${\mathcal{D}}_{23}$.

_Stability and instability of nodal lines_.—The Euler class [25, 24, 27, 28,
30] quantifies the pairwise annihilation and stability of the nodal lines’
transition in Fig. 3. We focus the discussions on the nodal link in Fig. 3(d).
Let us imagine a cube that encloses $R_{3}$ and denote its surface as
${\mathcal{D}}_{12}$ [refer Fig. 2(f)]. This ${\mathcal{D}}_{12}$ surface is
pierced only by $R_{1}$ and $R_{2}$, the degeneracies by
$\omega_{\mathbf{k}}^{1}-\omega_{\mathbf{k}}^{2}$. The Euler class $\chi$
calculated from ${\mathbf{u}}_{\mathbf{k}}^{1}$ and
${\mathbf{u}}_{\mathbf{k}}^{2}$ over ${\mathcal{D}}_{12}$, denoted by
$\chi_{12}\left({\mathcal{D}}_{12}\right)$, is $+2$. This indicates that the
four nodal lines of $R_{1}$ and $R_{2}$ piercing through ${\mathcal{D}}_{12}$
are stable, and their orientations are commonly outward of the surface. Any
two of these four are not annihilated on ${\mathcal{D}}_{12}$ until $R_{3}$
and $R_{4}$ are annihilated like Fig. 3(a)-(c).

Let us consider another cube that wraps $R_{2}$ and name its surface
${\mathcal{D}}_{23}$ [refer Fig. 2(f)]. $R_{3}$ and $R_{4}$ pass through the
left and right patches of ${\mathcal{D}}_{23}$, respectively, while $R_{1}$
and $R_{2}$ do not touch ${\mathcal{D}}_{23}$. The total Euler class
$\chi_{23}\left({\mathcal{D}}_{23}\right)$ is zero, and this means that
$R_{3}$ and $R_{4}$ can merge together. In detail, the zero-valued total Euler
class is due to the cancellation of the Euler classes on the left and right
patches of ${\mathcal{D}}_{23}$. The two nodal lines’ orientations on the one
boundary are inward, and the other two are oriented outward on the opposite
boundary. Thus, the annihilation of $R_{3}$ and $R_{4}$ can occur only by
merging two nodal lines respectively from $R_{3}$ and $R_{4}$ because their
orientations are opposite for ${\mathcal{D}}_{23}$ surface. This corresponds
to the inverse process from Fig. 3(d) to (b). The detailed process of
calculating the Euler classes is in Supplemental Material, Sec. D [75].

Although the transition of nodal lines has been discussed in Ref. [59], we
consider this from the Euler class point of view. Due to the non-zero Euler
classes on the left and right faces of ${\mathcal{D}}_{23}$ in Fig. 2(f), the
nodal ring $R_{3}$ cannot be separated into two rings in the region between
$R_{1}$ and $R_{2}$. Instead, two nodal lines in $R_{3}$ can touch each other
while keeping their orientations. The same explanation is also applied to
$R_{4}$. Let us recall Fig. 2(e), and focus on the touching in $R_{1}$. The
two nodal lines are heading towards the junction, while the others are
departing from the junction. Then, we infer a transition that two nodal lines
heading in the same direction change their connectivity through nodal chain,
while keeping the same Euler class, as illustrated in Fig. 4. In this case,
the nodal chain where the nodal lines intersect each other can be considered
as the critical state. The detailed discussions and results are in
Supplemental Material, Sec. E [75].

Our spring-mass system is $\mathcal{T}$-symmetric thereby the band structure
is inversion symmetric about the $\Gamma$-point throughout the momentum space.
Thus, we note that the $L_{1}$ and $L_{2}$ in Fig. 4 cannot be transformed
into a double helix [59, 78, 79, 80, 81, 82], and $R_{1}$ and $R_{2}$ in Fig.
3 cannot form a simple Hopf-link [52, 51, 83, 58, 26] or an $\infty$-shaped
structure twisted from a single loop, because these structures do not have an
inversion symmetry.

_Conclusions_.—We show that recently predicted multi-gap topologies and the
relation with the stability of nodal lines become of importance in an easily
implementable class of spring-mass systems. The on-site springs in our
classical system make it possible to generate a nodal link in three-
dimensional momentum space. Tuning the inter-site spring constants drives the
braiding and phase transition of the nodal lines. Finally, the non-zero (zero)
Euler class calculated over a patch indicates that the nodal lines in the
patch are stable (unstable). Our system thus provides a platform for these
novel pursuits. Indeed, we note that a spring-mass system with two masses in a
unit cell, which has six bands, can be a good platform for four-band models
[21, 22] allowing for more diverse braiding processes and multi-gap
topologies. Furthermore, adjusting spring constants is easier than selecting
materials with appropriate elastic moduli in a continuum scale problem because
a spring constant is simply determined by material properties and geometrical
factors. Therefore, we hope that our system can give an impetus to further
explore experimental realization of multi-gap topologies.

###### Acknowledgements.

The work by H. P, S. W and S. S. O has been funded by the European Regional
Development Fund through the Welsh Government (80762-CU145 (East)). A. B. is
funded by a Marie-Sklodowska-Curie fellowship, grant no. 101025315. R. J. S
acknowledges funding from a New Investigator Award, EPSRC grant EP/W00187X/1,
as well as Trinity college, Cambridge.

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Supplemental Material for
“Phase transitions of non-Abelian charged nodal links in a spring-mass system”

Haedong Park,1 Stephan Wong,1 Adrien Bouhon,2 Robert-Jan Slager,2 Sang Soon
Oh1,*

1School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United
Kingdom

2TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE,
United Kingdom

*<EMAIL_ADDRESS>
###### Contents

  1. A 3$\times$3 eigenvalue problem for spring-mass system
    1. A.1 Spring forces for spring-mass system
    2. A.2 Derivation of the Hamiltonian for spring-mass system
  2. B Nodal links in seven lattice systems
  3. C Frame charges
    1. C.1 Charges $\pm\bm{k}$ and $\pm\bm{i}$ of nodal lines
    2. C.2 Stability/instability of nodal lines, and their braiding
    3. C.3 Consistency between frame charges and Euler classes of the nodal link
  4. D Euler class
    1. D.1 Euler form and Euler class
    2. D.2 Euler class for $R_{1}$ and $R_{2}$ in the nodal link
    3. D.3 Euler class for $R_{3}$ and $R_{4}$ in the nodal link
    4. D.4 Numerical calculation of Euler class
  5. E Transition from nodal link

##  A 3$\times$3 eigenvalue problem for spring-mass system

###  A.1 Spring forces for spring-mass system

In this section, the general form of the Hamiltonian for the three-dimensional
spring-mass system is derived. The unit cell of our spring-mass system for
phononic modes consists of one mass $m$ and several massless springs, as
displayed in Fig. S1(a). The type of the lattice system, i.e., whether it is
triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, or
cubic, is determined by the lattice vectors, ${\mathbf{a}}_{1}$,
${\mathbf{a}}_{2}$, and ${\mathbf{a}}_{3}$. The springs are classified as two
groups: inter-site and on-site. The inter-site springs connect the masses in
adjacent unit cells. Their spring constants are denoted as $C_{ii}$, $C_{ij}$,
and $C_{ijk}$, as illustrated in Fig. S1(b)-(e). The subscript $ii$ means that
the spring is placed along ${\mathbf{a}}_{i}$ so that it connects the first
nearest neighbor masses. The subscript $ij$
($ij=12,23,31,1\bar{2},2\bar{3},3\bar{1}$) indicates the second nearest
neighbor connection along ${\mathbf{a}}_{i}+{\mathbf{a}}_{j}$ (if $j>0$) or
${\mathbf{a}}_{i}-{\mathbf{a}}_{j}$ (if $j<0$) [see Fig. S1(b)-(d)].
$C_{123}$, $C_{\bar{1}23}$, $C_{1\bar{2}3}$, and $C_{12\bar{3}}$ indicate the
spring constants of the springs that connect the third nearest neighbor masses
along the directions ${\mathbf{a}}_{1}+{\mathbf{a}}_{2}+{\mathbf{a}}_{3}$,
$-{\mathbf{a}}_{1}+{\mathbf{a}}_{2}+{\mathbf{a}}_{3}$,
${\mathbf{a}}_{1}-{\mathbf{a}}_{2}+{\mathbf{a}}_{3}$, and
${\mathbf{a}}_{1}+{\mathbf{a}}_{2}-{\mathbf{a}}_{3}$, respectively [see Fig.
S1(e)]. The on-site springs connect the mass and a non-vibrating fixed point.
Their spring constants are denoted by $K_{l}$ ($l=1,2,\ldots,N$), and the
number of the on-site springs has no limit in our model. We use $N=2$
throughout this study.

To derive the equation of motion for three-dimensional infinite array, we
denote a particular unit cell by an integer-valued vector
$\left[h_{1},h_{2},h_{3}\right]$, where each component is the index along
${\mathbf{a}}_{1}$, ${\mathbf{a}}_{2}$, and ${\mathbf{a}}_{3}$-directions,
respectively. Then, for instance, a unit cell
$\left[h_{1},h_{2},h_{3}+1\right]$ means the next cell to the cell
$\left[h_{1},h_{2},h_{3}\right]$ along ${\mathbf{a}}_{3}$-direction. For the
mass’s displacement ${\mathbf{u}}_{h_{1},h_{2},h_{3}}$ in the current unit
cell, the equation of motion is

$m{\ddot{\mathbf{u}}}_{h_{1},h_{2},h_{3}}=\sum_{i=1}^{3}{\mathbf{f}}_{i}^{1st}+\sum_{i,j}{\mathbf{f}}_{ij}^{2nd}+\sum_{i,j,k}{\mathbf{f}}_{ijk}^{3rd}+\sum_{l}{\mathbf{f}}_{l}^{0},$
(S1)

where ${\mathbf{f}}_{i}^{1st}$, ${\mathbf{f}}_{ij}^{2nd}$, and
${\mathbf{f}}_{ijk}^{3rd}$ are the spring forces between the first, second,
and third nearest neighbors, respectively, and ${\mathbf{f}}_{l}^{0}$ is the
spring force for the on-site energy. We assume that all spring forces are
proportional to the distance change between two masses where each spring
connects. The direction of each spring force is assumed to be the same as the
initial direction of the spring.

Next, we consider three cells $\left[h_{1}-1,h_{2},h_{3}\right]$,
$\left[h_{1},h_{2},h_{3}\right]$, and $\left[h_{1}+1,h_{2},h_{3}\right]$. The
distance change between the last two and first two cells that are respectively
expressed as
$\left({\mathbf{u}}_{h_{1}+1,h_{2},h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1}$
and
$\left({\mathbf{u}}_{h_{1}-1,h_{2},h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1}$,
where ${\hat{\mathbf{a}}}_{i}$ is the unit vector in ${\mathbf{a}}_{i}$
direction. The net spring force between the first nearest neighbors in
${\hat{\mathbf{a}}}_{1}$-direction is

${\mathbf{f}}_{1}^{1st}=\left\\{C_{11}\left({\mathbf{u}}_{h_{1}+1,h_{2},h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1}\right\\}{\hat{\mathbf{a}}}_{1}+\left\\{C_{11}\left({\mathbf{u}}_{h_{1}-1,h_{2},h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1}\right\\}{\hat{\mathbf{a}}}_{1}.$
(S2a) Likewise, the net spring forces between the first nearest neighbors
along ${\hat{\mathbf{a}}}_{2}$\- and ${\hat{\mathbf{a}}}_{3}$-directions are
respectively expressed as $\displaystyle{\mathbf{f}}_{2}^{1st}$
$\displaystyle=$
$\displaystyle\left\\{C_{22}\left({\mathbf{u}}_{h_{1},h_{2}+1,h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{2}\right\\}{\hat{\mathbf{a}}}_{2}+\left\\{C_{22}\left({\mathbf{u}}_{h_{1},h_{2}-1,h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{2}\right\\}{\hat{\mathbf{a}}}_{2}$
(S2b) $\displaystyle{\mathbf{f}}_{3}^{1st}$ $\displaystyle=$
$\displaystyle\left\\{C_{33}\left({\mathbf{u}}_{h_{1},h_{2},h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{3}\right\\}{\hat{\mathbf{a}}}_{3}+\left\\{C_{33}\left({\mathbf{u}}_{h_{1},h_{2},h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{3}\right\\}{\hat{\mathbf{a}}}_{3}.$

Figure S1:  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.

For the second nearest neighbors, we should consider six directions,
${\hat{\mathbf{a}}}_{12}$, ${\hat{\mathbf{a}}}_{1\bar{2}}$,
${\hat{\mathbf{a}}}_{23}$, ${\hat{\mathbf{a}}}_{2\bar{3}}$,
${\hat{\mathbf{a}}}_{31}$, and ${\hat{\mathbf{a}}}_{3\bar{1}}$, where
${\hat{\mathbf{a}}}_{ij}$ and ${\hat{\mathbf{a}}}_{i\bar{j}}$ are the unit
vectors in ${\mathbf{a}}_{i}+{\mathbf{a}}_{j}$ and
${\mathbf{a}}_{i}-{\mathbf{a}}_{j}$ directions, respectively. Then, the spring
forces between the second nearest neighbors ${\mathbf{f}}_{ij}^{2nd}$ are
given by

$\displaystyle{\mathbf{f}}_{12}^{2nd}$ $\displaystyle=$
$\displaystyle\left\\{C_{12}\left({\mathbf{u}}_{h_{1}+1,h_{2}+1,h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{12}\right\\}{\hat{\mathbf{a}}}_{12}+\left\\{C_{12}\left({\mathbf{u}}_{h_{1}-1,h_{2}-1,h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{12}\right\\}{\hat{\mathbf{a}}}_{12}$
(S2c) $\displaystyle{\mathbf{f}}_{1\bar{2}}^{2nd}$ $\displaystyle=$
$\displaystyle\left\\{C_{1\bar{2}}\left({\mathbf{u}}_{h_{1}+1,h_{2}-1,h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1\bar{2}}\right\\}{\hat{\mathbf{a}}}_{1\bar{2}}+\left\\{C_{1\bar{2}}\left({\mathbf{u}}_{h_{1}-1,h_{2}+1,h_{3}}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1\bar{2}}\right\\}{\hat{\mathbf{a}}}_{1\bar{2}}$
(S2d) $\displaystyle{\mathbf{f}}_{23}^{2nd}$ $\displaystyle=$
$\displaystyle\left\\{C_{23}\left({\mathbf{u}}_{h_{1},h_{2}+1,h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{23}\right\\}{\hat{\mathbf{a}}}_{23}+\left\\{C_{23}\left({\mathbf{u}}_{h_{1},h_{2}-1,h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{23}\right\\}{\hat{\mathbf{a}}}_{23}$
(S2e) $\displaystyle{\mathbf{f}}_{2\bar{3}}^{2nd}$ $\displaystyle=$
$\displaystyle\left\\{C_{2\bar{3}}\left({\mathbf{u}}_{h_{1},h_{2}+1,h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{2\bar{3}}\right\\}{\hat{\mathbf{a}}}_{2\bar{3}}+\left\\{C_{2\bar{3}}\left({\mathbf{u}}_{h_{1},h_{2}-1,h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{2\bar{3}}\right\\}{\hat{\mathbf{a}}}_{2\bar{3}}$
(S2f) $\displaystyle{\mathbf{f}}_{31}^{2nd}$ $\displaystyle=$
$\displaystyle\left\\{C_{31}\left({\mathbf{u}}_{h_{1}+1,h_{2},h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{31}\right\\}{\hat{\mathbf{a}}}_{31}+\left\\{C_{31}\left({\mathbf{u}}_{h_{1}-1,h_{2},h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{31}\right\\}{\hat{\mathbf{a}}}_{31}$
(S2g) $\displaystyle{\mathbf{f}}_{3\bar{1}}^{2nd}$ $\displaystyle=$
$\displaystyle\left\\{C_{3\bar{1}}\left({\mathbf{u}}_{h_{1}-1,h_{2},h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{3\bar{1}}\right\\}{\hat{\mathbf{a}}}_{3\bar{1}}+\left\\{C_{3\bar{1}}\left({\mathbf{u}}_{h_{1}+1,h_{2},h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{3\bar{1}}\right\\}{\hat{\mathbf{a}}}_{3\bar{1}}.$
(S2h)

There are four directions connecting the third nearest neighbors:
${\hat{\mathbf{a}}}_{123}$, ${\hat{\mathbf{a}}}_{\bar{1}23}$,
${\hat{\mathbf{a}}}_{1\bar{2}3}$, and ${\hat{\mathbf{a}}}_{12\bar{3}}$ that
are the unit vectors along the directions
${\mathbf{a}}_{1}+{\mathbf{a}}_{2}+{\mathbf{a}}_{3}$,
$-{\mathbf{a}}_{1}+{\mathbf{a}}_{2}+{\mathbf{a}}_{3}$,
${\mathbf{a}}_{1}-{\mathbf{a}}_{2}+{\mathbf{a}}_{3}$, and
${\mathbf{a}}_{1}+{\mathbf{a}}_{2}-{\mathbf{a}}_{3}$, respectively. Then, the
spring forces between the third nearest neighbors ${\mathbf{f}}_{ijk}^{3rd}$
are written as

$\displaystyle{\mathbf{f}}_{123}^{3rd}$ $\displaystyle=$
$\displaystyle\left\\{C_{123}\left({\mathbf{u}}_{h_{1}+1,h_{2}+1,h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{123}\right\\}{\hat{\mathbf{a}}}_{123}+\left\\{C_{123}\left({\mathbf{u}}_{h_{1}-1,h_{2}-1,h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{123}\right\\}{\hat{\mathbf{a}}}_{123}$
(S2i) $\displaystyle{\mathbf{f}}_{\bar{1}23}^{3rd}$ $\displaystyle=$
$\displaystyle\left\\{C_{\bar{1}23}\left({\mathbf{u}}_{h_{1}-1,h_{2}+1,h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{\bar{1}23}\right\\}{\hat{\mathbf{a}}}_{\bar{1}23}+\left\\{C_{\bar{1}23}\left({\mathbf{u}}_{h_{1}+1,h_{2}-1,h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{\bar{1}23}\right\\}{\hat{\mathbf{a}}}_{\bar{1}23}$
(S2j) $\displaystyle{\mathbf{f}}_{1\bar{2}3}^{3rd}$ $\displaystyle=$
$\displaystyle\left\\{C_{1\bar{2}3}\left({\mathbf{u}}_{h_{1}+1,h_{2}-1,h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1\bar{2}3}\right\\}{\hat{\mathbf{a}}}_{1\bar{2}3}+\left\\{C_{1\bar{2}3}\left({\mathbf{u}}_{h_{1}-1,h_{2}+1,h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1\bar{2}3}\right\\}{\hat{\mathbf{a}}}_{1\bar{2}3}$
(S2k) $\displaystyle{\mathbf{f}}_{12\bar{3}}^{3rd}$ $\displaystyle=$
$\displaystyle\left\\{C_{12\bar{3}}\left({\mathbf{u}}_{h_{1}+1,h_{2}+1,h_{3}-1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{12\bar{3}}\right\\}{\hat{\mathbf{a}}}_{12\bar{3}}+\left\\{C_{12\bar{3}}\left({\mathbf{u}}_{h_{1}-1,h_{2}-1,h_{3}+1}-{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{12\bar{3}}\right\\}{\hat{\mathbf{a}}}_{12\bar{3}}.$
(S2l)

We also describe the spring force corresponding to the on-site energy. By
considering the current cell $\left[h_{1},h_{2},h_{3}\right]$ only, the spring
force ${\mathbf{f}}_{l}^{0}$ in Eq. (S1) is written as

${\mathbf{f}}_{l}^{0}=-K_{l}\left({\hat{\mathbf{d}}}_{l}\otimes{\hat{\mathbf{d}}}_{l}\right){\mathbf{u}}_{h_{1},h_{2},h_{3}},$
(S2m)

where ${\hat{\mathbf{d}}}_{l}={\mathbf{d}}_{l}/\left|{\mathbf{d}}_{l}\right|$
is the unit vector along the spring connection, and $\otimes$ is the outer
product.

Eqs. (A.1) are summed to give the summations in Eq. (S1):

$\displaystyle\sum_{i=1}^{3}{\mathbf{f}}_{i}^{1st}$ $\displaystyle=$
$\displaystyle{\mathbf{f}}_{1}^{1st}+{\mathbf{f}}_{2}^{1st}+{\mathbf{f}}_{3}^{1st}$
(S3a) $\displaystyle\sum_{i,j}{\mathbf{f}}_{ij}^{2nd}$ $\displaystyle=$
$\displaystyle{\mathbf{f}}_{12}^{2nd}+{\mathbf{f}}_{1\bar{2}}^{2nd}+{\mathbf{f}}_{23}^{2nd}+{\mathbf{f}}_{2\bar{3}}^{2nd}+{\mathbf{f}}_{31}^{2nd}+{\mathbf{f}}_{3\bar{1}}^{2nd}$
(S3b) $\displaystyle\sum_{i,j,k}{\mathbf{f}}_{ijk}^{3rd}$ $\displaystyle=$
$\displaystyle{\mathbf{f}}_{123}^{3rd}+{\mathbf{f}}_{\bar{1}23}^{3rd}+{\mathbf{f}}_{1\bar{2}3}^{3rd}+{\mathbf{f}}_{12\bar{3}}^{3rd}$
(S3c) $\displaystyle\sum_{l}{\mathbf{f}}_{l}^{0}$ $\displaystyle=$
$\displaystyle{\mathbf{f}}_{1}^{0}+{\mathbf{f}}_{2}^{0}+\ldots.$ (S3d)

###  A.2 Derivation of the Hamiltonian for spring-mass system

Owing to translation symmetry, the displacement vector can be expressed as
${\mathbf{u}}_{h_{1},h_{2},h_{3}}\left({\mathbf{x}},t\right)={\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}={\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot\left(h_{1}{\mathbf{a}}_{1}+h_{2}{\mathbf{a}}_{2}+h_{3}{\mathbf{a}}_{3}\right)}e^{-i\omega
t}$. We substitute this into the following equation, which is just oppositely
signed of Eq. (S1):

$-m{\ddot{\mathbf{u}}}_{h_{1},h_{2},h_{3}}=-\sum_{i=1}^{3}{\mathbf{f}}_{i}^{1st}-\sum_{i,j}{\mathbf{f}}_{ij}^{2nd}-\sum_{i,j,k}{\mathbf{f}}_{ijk}^{3rd}-\sum_{l}{\mathbf{f}}_{l}^{0}.$
(S4)

The left-hand side becomes

$-m{\ddot{\mathbf{u}}}_{h_{1},h_{2},h_{3}}=m\omega^{2}{\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}.$ (S5)

To get the results for the first three terms in the right-hand side, we input
the above ${\mathbf{u}}_{h_{1},h_{2},h_{3}}\left({\mathbf{x}},t\right)$ into
Eqs. (A.1). For example, $-{\mathbf{f}}_{1}^{1st}$ becomes

$\displaystyle-{\mathbf{f}}_{1}^{1st}$ $\displaystyle=$
$\displaystyle-\left\\{C_{11}\left({\mathbf{u}}_{h_{1}+1,h_{2},h_{3}}+{\mathbf{u}}_{h_{1}-1,h_{2},h_{3}}-2{\mathbf{u}}_{h_{1},h_{2},h_{3}}\right)\cdot{\hat{\mathbf{a}}}_{1}\right\\}{\hat{\mathbf{a}}}_{1}$
(S6) $\displaystyle=$
$\displaystyle-\left\\{C_{11}\left(e^{i{\mathbf{k}}\cdot{\mathbf{a}}_{1}}+e^{-i{\mathbf{k}}\cdot{\mathbf{a}}_{1}}-2\right){\mathbf{u}}_{h_{1},h_{2},h_{3}}\cdot{\hat{\mathbf{a}}}_{1}\right\\}{\hat{\mathbf{a}}}_{1}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ $\displaystyle=$ $\displaystyle
2C_{11}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}\right)\right\\}\left({\mathbf{u}}_{\mathbf{k}}\cdot{\hat{\mathbf{a}}}_{1}\right){\hat{\mathbf{a}}}_{1}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}.$

Using
$\left({\mathbf{u}}_{\mathbf{k}}\cdot{\hat{\mathbf{a}}}_{1}\right){\hat{\mathbf{a}}}_{1}=\left({\hat{\mathbf{a}}}_{1}\otimes{\hat{\mathbf{a}}}_{1}\right){\mathbf{u}}_{\mathbf{k}}$
(which results from the vector identity
$\left({\mathbf{v}}\cdot{\mathbf{w}}\right){\mathbf{w}}=v_{i}w_{i}w_{j}=w_{j}w_{i}v_{i}=\left({\mathbf{w}}\otimes{\mathbf{w}}\right){\mathbf{v}}$),
the above equation becomes

$-{\mathbf{f}}_{1}^{1st}=2C_{11}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}\right)\right\\}\left({\hat{\mathbf{a}}}_{1}\otimes{\hat{\mathbf{a}}}_{1}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}.$ (S7a) We also convert the other spring forces in Eqs. (A.1) as follows:
$\displaystyle-{\mathbf{f}}_{2}^{1st}=2C_{22}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{2}\right)\right\\}\left({\hat{\mathbf{a}}}_{2}\otimes{\hat{\mathbf{a}}}_{2}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7b)
$\displaystyle-{\mathbf{f}}_{3}^{1st}=2C_{33}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)\right\\}\left({\hat{\mathbf{a}}}_{3}\otimes{\hat{\mathbf{a}}}_{3}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7c)

$\displaystyle-{\mathbf{f}}_{12}^{2nd}$ $\displaystyle=$ $\displaystyle
2C_{12}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}+{\mathbf{k}}\cdot{\mathbf{a}}_{2}\right)\right\\}\left({\hat{\mathbf{a}}}_{12}\otimes{\hat{\mathbf{a}}}_{12}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7d) $\displaystyle-{\mathbf{f}}_{1\bar{2}}^{2nd}$ $\displaystyle=$
$\displaystyle
2C_{1\bar{2}}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}-{\mathbf{k}}\cdot{\mathbf{a}}_{2}\right)\right\\}\left({\hat{\mathbf{a}}}_{1\bar{2}}\otimes{\hat{\mathbf{a}}}_{1\bar{2}}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7e) $\displaystyle-{\mathbf{f}}_{23}^{2nd}$ $\displaystyle=$
$\displaystyle
2C_{23}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{2}+{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)\right\\}\left({\hat{\mathbf{a}}}_{23}\otimes{\hat{\mathbf{a}}}_{23}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7f) $\displaystyle-{\mathbf{f}}_{2\bar{3}}^{2nd}$ $\displaystyle=$
$\displaystyle
2C_{2\bar{3}}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{2}-{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)\right\\}\left({\hat{\mathbf{a}}}_{2\bar{3}}\otimes{\hat{\mathbf{a}}}_{2\bar{3}}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7g) $\displaystyle-{\mathbf{f}}_{31}^{2nd}$ $\displaystyle=$
$\displaystyle
2C_{31}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{3}+{\mathbf{k}}\cdot{\mathbf{a}}_{1}\right)\right\\}\left({\hat{\mathbf{a}}}_{31}\otimes{\hat{\mathbf{a}}}_{31}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7h) $\displaystyle-{\mathbf{f}}_{3\bar{1}}^{2nd}$ $\displaystyle=$
$\displaystyle
2C_{3\bar{1}}\left\\{1-\cos\left({\mathbf{k}}\cdot{\mathbf{a}}_{3}-{\mathbf{k}}\cdot{\mathbf{a}}_{1}\right)\right\\}\left({\hat{\mathbf{a}}}_{3\bar{1}}\otimes{\hat{\mathbf{a}}}_{3\bar{1}}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}$ (S7i)

$-{\mathbf{f}}_{l}^{0}=-K_{l}\left({\hat{\mathbf{d}}}_{l}\otimes{\hat{\mathbf{d}}}_{l}\right){\mathbf{u}}_{\mathbf{k}}e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega
t}.$ (S7j)

By replacing all terms in Eq. (S4) as Eqs. (S7j) and dropping
$e^{i{\mathbf{k}}\cdot{\mathbf{x}}}e^{-i\omega t}$, we finally get

$H{\mathbf{u}}_{\mathbf{k}}=\omega^{2}{\mathbf{u}}_{\mathbf{k}}.$ (S8a) Here,
the real-valued $3\times 3$ Hamiltonian $H$ is given by $\displaystyle H$
$\displaystyle=$
$\displaystyle\frac{2}{m}\left[C_{11}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}\right)}\right\\}{\hat{\mathbf{a}}}_{1}\otimes{\hat{\mathbf{a}}}_{1}+C_{22}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{2}\right)}\right\\}{\hat{\mathbf{a}}}_{2}\otimes{\hat{\mathbf{a}}}_{2}+C_{33}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)}\right\\}{\hat{\mathbf{a}}}_{3}\otimes{\hat{\mathbf{a}}}_{3}\right.$
$\displaystyle+$
$\displaystyle\left.C_{12}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}+{\mathbf{k}}\cdot{\mathbf{a}}_{2}\right)}\right\\}{\hat{\mathbf{a}}}_{12}\otimes{\hat{\mathbf{a}}}_{12}+C_{1\bar{2}}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}-{\mathbf{k}}\cdot{\mathbf{a}}_{2}\right)}\right\\}{\hat{\mathbf{a}}}_{1\bar{2}}\otimes{\hat{\mathbf{a}}}_{1\bar{2}}\right.$
$\displaystyle+$
$\displaystyle\left.C_{23}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{2}+{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)}\right\\}{\hat{\mathbf{a}}}_{23}\otimes{\hat{\mathbf{a}}}_{23}+C_{2\bar{3}}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{2}-{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)}\right\\}{\hat{\mathbf{a}}}_{2\bar{3}}\otimes{\hat{\mathbf{a}}}_{2\bar{3}}\right.$
$\displaystyle+$
$\displaystyle\left.C_{31}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{3}+{\mathbf{k}}\cdot{\mathbf{a}}_{1}\right)}\right\\}{\hat{\mathbf{a}}}_{31}\otimes{\hat{\mathbf{a}}}_{31}+C_{3\bar{1}}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{3}-{\mathbf{k}}\cdot{\mathbf{a}}_{1}\right)}\right\\}{\hat{\mathbf{a}}}_{3\bar{1}}\otimes{\hat{\mathbf{a}}}_{3\bar{1}}\right.$
$\displaystyle+$
$\displaystyle\left.C_{123}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}+{\mathbf{k}}\cdot{\mathbf{a}}_{2}+{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)}\right\\}{\hat{\mathbf{a}}}_{123}\otimes{\hat{\mathbf{a}}}_{123}+C_{\bar{1}23}\left\\{1-\cos{\left(-{\mathbf{k}}\cdot{\mathbf{a}}_{1}+{\mathbf{k}}\cdot{\mathbf{a}}_{2}+{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)}\right\\}{\hat{\mathbf{a}}}_{\bar{1}23}\otimes{\hat{\mathbf{a}}}_{\bar{1}23}\right.$
$\displaystyle+$
$\displaystyle\left.C_{1\bar{2}3}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}-{\mathbf{k}}\cdot{\mathbf{a}}_{2}+{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)}\right\\}{\hat{\mathbf{a}}}_{1\bar{2}3}\otimes{\hat{\mathbf{a}}}_{1\bar{2}3}+C_{12\bar{3}}\left\\{1-\cos{\left({\mathbf{k}}\cdot{\mathbf{a}}_{1}+{\mathbf{k}}\cdot{\mathbf{a}}_{2}-{\mathbf{k}}\cdot{\mathbf{a}}_{3}\right)}\right\\}{\hat{\mathbf{a}}}_{12\bar{3}}\otimes{\hat{\mathbf{a}}}_{12\bar{3}}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{2}\sum_{l}{K_{l}\left({\hat{\mathbf{d}}}_{l}\otimes{\hat{\mathbf{d}}}_{l}\right)}\right].$

All terms in the right-hand side of the above equation are written as
${\mathbf{n}}\otimes{\mathbf{n}}$ form. Thus, Hamiltonian $H$ is symmetric and
positive definite for any wave vector $\mathbf{k}$. Its eigenstates
${\mathbf{u}}_{\mathbf{k}}^{1}$, ${\mathbf{u}}_{\mathbf{k}}^{2}$, and
${\mathbf{u}}_{\mathbf{k}}^{3}$ form an $\operatorname{SO}(3)$ orthonormal
frame.

The orthorhombic system in the main text does not have $H_{ijk}$ because all
the spring constants $C_{123}$, $C_{\bar{1}23}$, $C_{1\bar{2}3}$, and
$C_{12\bar{3}}$ are zero, as shown in Table S1.

Table S1:  Parameter sets to realize the nodal links shown in Fig. S2. The mass $m$ in Eq. (S8) is set as one. Lattice system |  Parameter sets
---|---
Triclinic |  ${\mathbf{a}}_{1}=\left[1,0,0\right]$, ${\mathbf{a}}_{2}=\left[-0.5,1.0825,0\right]$, ${\mathbf{a}}_{3}=\left[-0.3,-0.2,1\right]$
|  $C_{11}=30$, $C_{22}=110$, $C_{33}=280$
|  $C_{12}=95$, $C_{1\bar{2}}=55$, $C_{23}=62$, $C_{2\bar{3}}=42$,
$C_{31}=40$, $C_{3\bar{1}}=30$
|  $C_{123}=50$, $C_{\bar{1}23}=0$, $C_{1\bar{2}3}=30$, $C_{12\bar{3}}=0$
|  $K_{1}=40$, ${\mathbf{d}}_{1}={\mathbf{a}}_{1}$
|  $K_{2}=180$,
${\mathbf{d}}_{2}=0.3924{\mathbf{a}}_{1}+0.1848{\mathbf{a}}_{2}+1.0{\mathbf{a}}_{3}$
Monoclinic |  ${\mathbf{a}}_{1}=\left[1,0,0\right]$, ${\mathbf{a}}_{2}=\left[0,1.2,0\right]$, ${\mathbf{a}}_{3}=\left[0,-0.2,1\right]$
|  $C_{11}=30$, $C_{22}=110$, $C_{33}=280$
|  $C_{12}=95$, $C_{1\bar{2}}=55$, $C_{23}=62$, $C_{2\bar{3}}=42$,
$C_{31}=40$, $C_{3\bar{1}}=30$
|  $C_{123}=0$, $C_{\bar{1}23}=0$, $C_{1\bar{2}3}=0$, $C_{12\bar{3}}=20$
|  $K_{1}=40$, ${\mathbf{d}}_{1}={\mathbf{a}}_{1}$
|  $K_{2}=180$, ${\mathbf{d}}_{2}=0.1667{\mathbf{a}}_{2}+1.0{\mathbf{a}}_{3}$
Orthorhombic |  ${\mathbf{a}}_{1}=\left[1,0,0\right]$, ${\mathbf{a}}_{2}=\left[0,1.1,0\right]$, ${\mathbf{a}}_{3}=\left[0,0,1.2\right]$
|  $C_{11}=30$, $C_{22}=110$, $C_{33}=280$
|  $C_{12}=95$, $C_{1\bar{2}}=55$, $C_{23}=26$, $C_{2\bar{3}}=26$,
$C_{31}=40$, $C_{3\bar{1}}=30$
|  $C_{123}=0$, $C_{\bar{1}23}=0$, $C_{1\bar{2}3}=0$, $C_{12\bar{3}}=0$
|  $K_{1}=40$, ${\mathbf{d}}_{1}={\mathbf{a}}_{2}$
|  $K_{2}=160$, ${\mathbf{d}}_{2}={\mathbf{a}}_{3}$
Tetragonal |  ${\mathbf{a}}_{1}=\left[1,0,0\right]$, ${\mathbf{a}}_{2}=\left[0,1,0\right]$, ${\mathbf{a}}_{3}=\left[0,0,1.2\right]$
|  $C_{11}=30$, $C_{22}=110$, $C_{33}=280$
|  $C_{12}=95$, $C_{1\bar{2}}=55$, $C_{23}=52$, $C_{2\bar{3}}=52$,
$C_{31}=40$, $C_{3\bar{1}}=30$
|  $C_{123}=0$, $C_{\bar{1}23}=0$, $C_{1\bar{2}3}=0$, $C_{12\bar{3}}=0$
|  $K_{1}=40$, ${\mathbf{d}}_{1}={\mathbf{a}}_{2}$
|  $K_{2}=160$, ${\mathbf{d}}_{2}={\mathbf{a}}_{3}$
Rhombohedral |  ${\mathbf{a}}_{1}=\left[1,0,0.438\right]$, ${\mathbf{a}}_{2}=\left[-1/2,\sqrt{3}/2,0.438\right]$, ${\mathbf{a}}_{3}=\left[-1/2,-\sqrt{3}/2,0.438\right]$
|  $C_{11}=30$, $C_{22}=110$, $C_{33}=280$
|  $C_{12}=95$, $C_{1\bar{2}}=55$, $C_{23}=85$, $C_{2\bar{3}}=35$,
$C_{31}=40$, $C_{3\bar{1}}=30$
|  $C_{123}=70$, $C_{\bar{1}23}=0$, $C_{1\bar{2}3}=25$, $C_{12\bar{3}}=0$
|  $K_{1}=40$, ${\mathbf{d}}_{1}={\mathbf{a}}_{2}-{\mathbf{a}}_{3}$
|  $K_{2}=160$,
${\mathbf{d}}_{2}={\mathbf{a}}_{1}+{\mathbf{a}}_{2}+{\mathbf{a}}_{3}$
Hexagonal |  ${\mathbf{a}}_{1}=\left[1/2,-\sqrt{3}/2,0\right]$, ${\mathbf{a}}_{2}=\left[1/2,\sqrt{3}/2,0\right]$, ${\mathbf{a}}_{3}=\left[0,0,1.5\right]$
|  $C_{11}=30$, $C_{22}=110$, $C_{33}=280$
|  $C_{12}=95$, $C_{1\bar{2}}=55$, $C_{23}=26$, $C_{2\bar{3}}=26$,
$C_{31}=40$, $C_{3\bar{1}}=30$
|  $C_{123}=35$, $C_{\bar{1}23}=10$, $C_{1\bar{2}3}=0$, $C_{12\bar{3}}=65$
|  $K_{1}=260$, ${\mathbf{d}}_{1}={\mathbf{a}}_{1}+{\mathbf{a}}_{2}$
|  $K_{2}=160$, ${\mathbf{d}}_{2}={\mathbf{a}}_{3}$
Cubic |  ${\mathbf{a}}_{1}=\left[1,0,0\right]$, ${\mathbf{a}}_{2}=\left[0,1,0\right]$, ${\mathbf{a}}_{3}=\left[0,0,1\right]$
|  $C_{11}=30$, $C_{22}=110$, $C_{33}=280$
|  $C_{12}=80$, $C_{1\bar{2}}=80$, $C_{23}=26$, $C_{2\bar{3}}=26$,
$C_{31}=40$, $C_{3\bar{1}}=30$
|  $C_{123}=35$, $C_{\bar{1}23}=10$, $C_{1\bar{2}3}=0$, $C_{12\bar{3}}=65$
|  $K_{1}=60$, ${\mathbf{d}}_{1}={\mathbf{a}}_{2}$
|  $K_{2}=240$, ${\mathbf{d}}_{2}={\mathbf{a}}_{3}$

##  B Nodal links in seven lattice systems

In this section, the seven lattice systems containing a single mass in a unit
cell are considered. The input parameters for Eq. (S8) are lattice vectors
[see Fig. S1(a)], thirteen inter-site spring constants [see Fig. S1(b)-(e)],
and information of on-site springs including the spring constants $K_{l}$ and
the directions ${\hat{\mathbf{d}}}_{l}$ [see Fig. S1(f)]. For each lattice
system defined by ${\mathbf{a}}_{1}$, ${\mathbf{a}}_{2}$, and
${\mathbf{a}}_{3}$, the parameter sets to realize the nodal links are listed
in Table S1. The real space systems by those parameters and the resulting
nodal links in momentum space are shown in the left and right figures in each
panel of Fig. S2. The red and blue nodal lines are the degeneracies by
$\omega_{\mathbf{k}}^{2}-\omega_{\mathbf{k}}^{1}$ and
$\omega_{\mathbf{k}}^{3}-\omega_{\mathbf{k}}^{2}$, respectively. In all cases,
the blue-colored nodal rings encircle the $\Gamma$-point, the other blue-
colored nodal rings span the first Brillouin boundaries, two red-colored nodal
rings exist around the $\Gamma$-point, and these red and blue nodal rings are
infinitely connected to reveal the nodal links.

All the seven cases have two on-site springs. In each case, the two constants
$K_{1}$ and $K_{2}$ of these springs are different. These make the
eigenfrequencies of the $3\times 3$ Hamiltonian $H$ [Eq. (S8)] different at
the $\Gamma$-point. Two nodal rings $R_{1}$ and $R_{2}$ do not intersect each
other at this point, and the triple point degeneracy is also not observed
here.

The triclinic, monoclinic, rhombohedral, hexagonal, and cubic cases have at
least one spring for the third-nearest neighbor springs, and their values in
each case are different (see Table S1). Furthermore, in all cases except the
cubic lattice, $C_{12}$ and $C_{1\bar{2}}$ are different, and $C_{31}$ and
$C_{3\bar{1}}$ also have different values. These spring constants make the
lattice have no rotation- or mirror-symmetries. They preserve
$\mathcal{P}$-symmetry and $\mathcal{T}$-symmetry only.

Figure S2:  Seven lattice systems and their nodal links. The left figures in
each panel are the real space plots according to Table S1. 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(d) in the main text.

##  C Frame charges

###  C.1 Charges $\pm\bm{k}$ and $\pm\bm{i}$ of nodal lines

Figure S3:  Frame charges $\bm{k}$ and $\bm{i}$ 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 systems. (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.

In the main text, the frame charge of the nodal line by the first and second
(by the second and third) bands were denoted as $\pm\bm{k}$ ($\pm\bm{i}$)
among ${\mathbb{Q}}=\left\\{\pm\bm{i},\pm\bm{j},\pm\bm{k},-1,+1\right\\}$. In
this section, we see how they can be achieved, and briefly review the frame
charge [23, 22, 24, 21, 26]. We select the orthorhombic system shown in the
main text [or in Fig. S2(c)] as an example. First, we investigate the frame
charge of the nodal ring $R_{2}$. We conider a closed loop parametrized by
$\alpha\in\left[0,2\pi\right]$ encircling $R_{2}$, as shown in the right inset
of Fig. S3(a). Along the loop, the eigenstates
${\mathbf{u}}_{\mathbf{k}}^{1}$, ${\mathbf{u}}_{\mathbf{k}}^{2}$, and
${\mathbf{u}}_{\mathbf{k}}^{3}$ are calculated [see Fig. S3(b)] where the
superscripts are the band numbers. The tails of all the eigenstates are
collected at the origin of an arbitrary orthogonal coordinate system, for
example,
${\mathbf{u}}_{{\mathbf{k}}_{0}}^{1}$-${\mathbf{u}}_{{\mathbf{k}}_{0}}^{2}$-${\mathbf{u}}_{{\mathbf{k}}_{0}}^{3}$
coordinate system [see Fig. S3(c)]. The eigenstates reveal that all the
${\mathbf{u}}_{\mathbf{k}}^{3}$s for $\alpha\in\left[0,2\pi\right]$ are fixed
along one direction while ${\mathbf{u}}_{\mathbf{k}}^{1}$ and
${\mathbf{u}}_{\mathbf{k}}^{2}$ exhibit $+\pi$-rotations. The rotation matrix
for these behaviors is
$R_{12}\left(\alpha\right)=e^{\left(\alpha\textfractionsolidus 2\right)L_{3}}$
where $\left(L_{i}\right)_{jk}=-\varepsilon_{ijk}$. Rewriting
$R_{12}\left(\alpha\right)$ by its lift in the double cover
$\operatorname{Spin}\left(3\right)$ gives
$\bar{R}_{12}\left(\alpha\right)=e^{-i\left(\alpha/2\right)\left(\sigma_{3}/2\right)}$
with $\alpha\in\left[0,4\pi\right]$ [23, 55]. Therefore,
$\bar{R}_{12}\left(\alpha=2\pi\right)=-i\sigma_{3}$, and the frame charge of
Fig. S3(b-c) is the quaternion number $\bm{k}$ [23, 55, 54]. The same analysis
can be used to the loop that encloses $R_{3}$ in Fig. S3(a). After calculating
the eigenstates along the loop [see Fig. S3(d)], the rotation matrix for the
results in Fig. S3(e) is
$R_{23}\left(\alpha\right)=e^{\left(\alpha\textfractionsolidus
2\right)L_{1}}$, and lifting in the double cover Spin(3) gives
$\bar{R}_{23}\left(\alpha\right)=e^{-i\left(\alpha/2\right)\left(\sigma_{1}/2\right)}$.
Therefore, we get $\bar{R}_{23}\left(\alpha=2\pi\right)=-i\sigma_{1}$
indicating the frame charge $\bm{i}$ [23, 55, 54].

The results on the frame charges $\pm\bm{k}$ and $\pm\bm{i}$ [shown in Fig.
S3(c),(e)] show that the eigenstates of the adjacent two bands concerning the
nodal line commonly show the $\pi$-disclination, that is, the signs of the
eigenstates at $\mathbf{k}\left(\alpha=0\right)$ and
$\mathbf{k}\left(\alpha=2\pi\right)$ are opposite. The eigenstates of the
remaining bands do not exhibit $\pi$-disclination. Then, we observe the
rotation of the former two sets of the eigenstates by $\pi$ around the latter
sets of the eigenstates. As already mentioned, the frame charges are
determined by which bands show the $\pi$-disclinations and which band is
fixed.

###  C.2 Stability/instability of nodal lines,
and their braiding

Figure S4:  Nodal lines and closed loops for frame charge characterizations.
Nodal lines in (a), (b)-(d), and (e)-(f) correspond to Fig. 3(a), (b), and
(d), respectively. Frame charge $n\left(C_{i}\right)$ 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.

In this section, we analyze the behavior of nodal lines in Fig. 3 in the main
text using the frame charges. The following discussions provide further
insight in understanding Fig. 2 and 3 in the main text.

The frame charges $-1$ and $+1$ provide useful information on the
stability/instability of nodal lines, as Euler class does [23, 26]. To
characterize nodal lines’ stability/instability, a closed loop that encircles
even number of nodal lines is placed. We suppose the right-handed rule; if one
nodal line’s frame charge is positive, its orientation is the thumb’s
direction when we grab the line along the closed loop’s direction. The same
(opposite) frame charges indicate that the total frame charge is $-1$ ($+1$),
their orientations are the same (opposite), and they are stable (can be pair-
annihilated).

The frame charge calculated along the loop $C_{1}$ in Fig. S4(a) is $-1$,
meaning that $L_{1}$ and $L_{2}$ have the same orientations. $C_{1}$ is not
placed through $R_{1}$ and $R_{2}$. If $R_{1}$ and $R_{2}$ grow as shown in
Fig. 3(b) in the main text, and if $C_{1}$ still does not pass through these
two rings (or $C_{1}$ detours the rings), the frame charge remains $-1$.
However, if the closed loop $C_{1}$ is set as shown in Fig. S4(b) [as
illustrated in Fig. 2(d) in the main text], the frame charge becomes $+1$, the
trivial case, so that they can transform into the nodal chain and nodal link
in Fig. 3(c) and (d) in the main text.

Meanwhile, we investigate the frame charge of $R_{2}$. For the loop $C_{2}$ in
Fig. S4(c), the frame charge is $+1$, so that $R_{2}$ can be deformed to
shrink and disappear if it follows the path drawn by $C_{2}$, like Fig. S4(a).
However, if we consider a loop $C_{3}$ that detours both $L_{1}$ and $L_{2}$
[see Fig. S4(d)], the frame charge becomes $-1$: $R_{2}$ cannot be deformed to
disappear if it follows the path by $C_{3}$. After deforming $L_{1}$ and
$L_{2}$ into $R_{3}$ and $R_{4}$, the relation between $C_{3}$ and $R_{2}$
remains [see Fig. S4(e)]. Thus, the frame charge is $-1$, and $R_{2}$ becomes
stable. The frame charge of $R_{1}$ can be explained in the same manner.

###  C.3 Consistency between frame charges and Euler classes of the nodal link

This section discusses the stability and instability of the nodal rings
$R_{1}$-$R_{2}$ and $R_{3}$-$R_{4}$ using the frame charges. Results generated
in this section are consistent with the analyses using Euler classes that will
be mentioned in Section D.2 and D.3. First, we set a loop $C_{4}$ tying
$R_{1}$ and $R_{2}$, as shown in Fig. S4(f). The frame charge calculated along
$C_{4}$ is $-1$. This means that the orientations of both $R_{1}$ and $R_{2}$
are the right-handed outward directions of the $C_{4}$ surface. As already
mentioned regarding Fig. S4(e), two nodal lines of $R_{2}$ are outward of
$C_{3}$ surface. The situation of $R_{1}$ is the same. Therefore, all the four
nodal lines of $R_{1}$ and $R_{2}$ apart from the $\Gamma$-point are commonly
outward.

The situation of the nodal rings $R_{3}$ and $R_{4}$ is different. The charges
along both $C_{5}$ and $C_{6}$ in Fig. S4(g) are $-1$, thus, the same charges
are generated in each ring $R_{3}$ and $R_{4}$. However, on the surface of the
loop $C_{7}$ in Fig. S4(h), the nodal rings $R_{3}$ and $R_{4}$ are opposite
because the frame charge obtained along $C_{7}$ is $+1$. Therefore, $R_{3}$
and $R_{4}$ are not stable regarding $C_{7}$ connection.

##  D Euler class

###  D.1 Euler form and Euler class

For real eigenstates $\left|u_{\mathbf{k}}^{m}\right>$ and
$\left|u_{\mathbf{k}}^{n}\right>$ of any two adjacent bands $m$ and $n$, the
Euler form is given by

$\operatorname{Eu}^{mn}\left({\mathbf{k}}\right)=\left<\nabla_{\mathbf{k}}u_{\mathbf{k}}^{m}\mid\times\mid\nabla_{\mathbf{k}}u_{\mathbf{k}}^{n}\right>.$
(S9)

The Euler class, an integer topological invariant, is given by [29, 25, 30,
24, 27, 28]

$\chi_{mn}\left({\mathcal{D}}\right)=\frac{1}{2\pi}\left[\int_{\mathcal{D}}\operatorname{Eu}^{mn}{\textrm{d}}k_{a}{\textrm{d}}k_{b}-\oint_{\partial{\mathcal{D}}}{\mathbf{a}}\left({\mathbf{k}}\right)\cdot{\textrm{d}}{\mathbf{k}}\right].$
(S10)

where
${\mathbf{a}}\left({\mathbf{k}}\right)=\left<u_{\mathbf{k}}^{m}\mid\nabla_{\mathbf{k}}u_{\mathbf{k}}^{n}\right>$
is the Euler connection. The Euler class is the difference between the surface
integral of the Euler form over a patch $\mathcal{D}$ and the boundary
integral of the Euler connection. If there is no nodal line passing through
the patch, this surface can be filled with smooth
$\left|u_{\mathbf{k}}^{m}\right>$. By Stokes’ theorem, Eq. (S10) becomes zero.
If the patch is pierced by a nodal line, the intersecting point becomes a
singularity, and Eq. (S10) is not zero.

###  D.2 Euler class for $R_{1}$ and $R_{2}$ in the nodal link

Figure S5:  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 caused by that the region
is near from 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 cancelled out.

In the momentum space of our orthorhombic spring-mass system, two nodal rings
$R_{1}$ and $R_{2}$, degeneracies by
$\omega_{\mathbf{k}}^{2}-\omega_{\mathbf{k}}^{1}$, are generated, as shown in
Fig. 3 in the main text. To calculate Euler class, we set a cubic patch
centered on the $\Gamma$-point, denoted by ${\mathcal{D}}_{12}$ [see Fig.
S5(a)]. The patch completely encloses $R_{3}$, degeneracies by
$\omega_{\mathbf{k}}^{3}-\omega_{\mathbf{k}}^{2}$. In other words, there is no
nodal line by $\omega_{\mathbf{k}}^{3}-\omega_{\mathbf{k}}^{2}$ passing the
${\mathcal{D}}_{12}$ surface. Only $R_{1}$ and $R_{2}$ intersect this surface.

We divide ${\mathcal{D}}_{12}$ into six square faces. To calculate the surface
integral of Eq. (S10) for each face, $k_{a}$ and $k_{b}$ in Eq. (S10) are set
such that $k_{a}\times k_{b}$ is outward of the cube. Each face contains zero
or even number of nodal line piercings. If a face contains the piercings, (i)
we skip calculating Eq. (S10) around the nodal lines where the eigenstates are
not sufficiently continuous, and (ii) the eigenstates’ gauge are adjusted to
make them smooth across Dirac string. We also calculate the boundary integral
in Eq. (S10). The summation of the boundary integral for all the edges of the
cube should be zero because all the paths in Fig. S5(a) are cancelled out. By
using this, we adjust the signs of Euler classes over the six faces to make
the boundary integral summation zero.

The Euler form for Face 2 in Fig. S5(a) is shown in Fig. S5(b). Face 2 has two
piercings, and its Euler class is $+1$. Thus, the two nodal lines are stable,
and they have the same orientations. Face 5 exhibits the same results. For
Faces 1 and 3, Euler classes, the sum of the surface integral of the Euler
form and boundary integral of the Euler connection, are zero. Thus, the
overall result $\chi_{12}\left({\mathcal{D}}_{12}\right)$ becomes $+2$.

The above analysis only reveals the summation of Euler classes over
${\mathcal{D}}_{12}$’s six faces; it does not show a direct result of Euler
class by $R_{1}$ and $R_{2}$. Thus, we lower Face 1’s position along
$-{\mathbf{b}}_{3}$ direction to make it contain two nodal lines by $R_{1}$
and $R_{2}$, respectively. In this case, Euler class calculated over Face 1 is
$+1$, and we can ensure that all the nodal lines of $R_{1}$ and $R_{2}$ are
outward of ${\mathcal{D}}_{12}$. All the results in this section are
consistent with Fig. S4(e),(f) in Section C.2.

###  D.3 Euler class for $R_{3}$ and $R_{4}$ in the nodal link

Figure S6:  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
Face 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 Euler connection are cancelled out.

The nodal link mentioned in the previous section also reveals the nodal rings
$R_{3}$ and $R_{4}$, degeneracies by
$\omega_{\mathbf{k}}^{3}-\omega_{\mathbf{k}}^{2}$. We set a box
${\mathcal{D}}_{23}$ that contains $R_{2}$. Only $R_{3}$ and $R_{4}$ intersect
this patch through Faces 2 and 5 [see Fig. S6(a),(c)]. We calculate Euler
class in Eq. (S10) for all the six faces of the box by considering the same
things as in the previous section. Euler classes for Faces 2 and 5 are $+1$
and $-1$, respectively. Another four faces exhibit zero-valued Euler class.
Thus, the total result $\chi_{23}\left({\mathcal{D}}_{23}\right)$ is $0$, and
therefore the transition between Fig. 3(c) and (d) is possible.

Because both Euler classes calculated over Faces 2 and 5 are non-zero, the two
nodal lines of a single ring escaping the same face of the box cannot be
annihilated. Thus, the transition of $R_{3}$ and $R_{4}$ to nodal lines
$L_{1}$ and $L_{2}$ in Fig. 3(c) in the main text is done by annihilation of
each strand of $R_{3}$ and $R_{4}$. This can be more clearly seen if we pull
Face 6 in Fig. S6(c) (the face parallel to ${\mathbf{b}}_{1}$ and
${\mathbf{b}}_{3}$) along ${\mathbf{b}}_{2}$-direction so that it contains
totally two degeneracies that respectively belong to $R_{3}$ and $R_{4}$; its
Euler class is zero. All these discussions are consistent with Fig. S4(g),(h)
in Section C.2.

###  D.4 Numerical calculation of Euler class

Figure S7:  Schematics of calculating Euler class. (a) Discretized domain to
calculate 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 $\left(i,j\right)$. Eigenstates smoothing on the boundary and
Euler connection integral are performed along the thick green arrow direction.
(b) Enlarged cell $\left(i,j\right)$. $P_{12}$, $P_{23}$, $P_{34}$, and
$P_{14}$ are mid points of each edge.  Figure S8:  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 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 $\Delta C_{31}$ from $2$ (g)
through $0$ (h) to $-2$ (i), with setting $K_{1}=140$ and keeping the other
variables in Table S1.

Throughout this study, we use a two-dimensional surface as a domain to
calculate Euler class Eq. (S10). We assume that a domain contains zero or two
degeneracies, and we also suppose that we already know whether the domain has
degeneracies from a plot of nodal lines. Thus, the rough positions of
degeneracies are given input parameters. The degeneracies are by
$\omega_{\mathbf{k}}^{n}-\omega_{\mathbf{k}}^{m}$ ($n>m$), and there are not
any other degeneracies by $\omega_{\mathbf{k}}^{m}-\omega_{\mathbf{k}}^{m-1}$
or $\omega_{\mathbf{k}}^{n+1}-\omega_{\mathbf{k}}^{n}$. As Eq. (S10) consists
of a surface and boundary integrals, we also divide the numerical process into
two parts: numerical calculation over the surface and along the boundary.

To perform the surface integral, we prepare $N\times N$ rectangular meshes on
the domain, as shown in Fig. S7(a). On the domain, we can think of two sets of
points: one group of vertices of the meshes, and the other group of the
meshes’ central points marked as red dots in Fig. S7(a). The number of points
in the first and second groups are $\left(N+1\right)^{2}$ and $N^{2}$,
respectively.

Next, we calculate eigenfrequencies using Eq. (S8) at the red dots of Fig.
S7(a) to search which meshes have the band degeneracies. The aforementioned
input parameters, the degeneracies’ rough positions, are used here. We regard
that a degeneracy is in a mesh if the mesh is around one of the degeneracies’
rough position, and if
$\Delta\omega_{mn}=\omega_{\mathbf{k}}^{n}-\omega_{\mathbf{k}}^{m}$ at the
mesh is minimum compared to surrounding meshes. After determining two meshes
of degeneracies, we define a Dirac string that connects the two meshes [see
the dotted path in Fig. S7(a)]. The Dirac string can take various routes, but
we use the simple path that bends one time at most.

The eigenstates ${\mathbf{u}}_{\mathbf{k}}^{1}$,
${\mathbf{u}}_{\mathbf{k}}^{2}$, and ${\mathbf{u}}_{\mathbf{k}}^{3}$ of Eq.
(S8) are then calculated at the vertices of all the meshes. At each point, the
eigenstates should satisfy the right-handed rule
${\mathbf{u}}_{\mathbf{k}}^{1}\cdot\left({\mathbf{u}}_{\mathbf{k}}^{2}\times{\mathbf{u}}_{\mathbf{k}}^{3}\right)=+1$.
Practically, we use the condition
${\mathbf{u}}_{\mathbf{k}}^{1}\cdot\left({\mathbf{u}}_{\mathbf{k}}^{2}\times{\mathbf{u}}_{\mathbf{k}}^{3}\right)\geq
0$ considering the numerical errors.

All the eigenstates should be smoothened by adjusting their gauges
(determining their signs). First, we make them continuous along the boundary
in the counterclockwise directions, as marked with the green arrow in Fig.
S7(a). Then, the smoothing process is carried out from the boundary to inner
regions of the domain. This smoothing should not go across the Dirac string.
Thus, all the smoothing process stop in the vicinity of the Dirac string.

To calculate the Euler form in Eq. (S9), we mark four vertices of a mesh as
$P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$, as shown in Fig. S7(b). In the previous
steps, the eigenstates ${\mathbf{u}}_{\mathbf{k}}^{m}$ and
${\mathbf{u}}_{\mathbf{k}}^{n}$ at these points are already prepared. Let us
denote ${\mathbf{u}}_{\mathbf{k}}^{m}$ as ${\mathbf{U}}_{P1}$,
${\mathbf{U}}_{P2}$, ${\mathbf{U}}_{P3}$, and ${\mathbf{U}}_{P4}$. Likewise,
${\mathbf{u}}_{\mathbf{k}}^{n}$ is also rewritten as ${\mathbf{V}}_{P1}$,
${\mathbf{V}}_{P2}$, ${\mathbf{V}}_{P3}$, and ${\mathbf{V}}_{P4}$. If the
current mesh is on the Dirac string, the eigenstates at $P_{2}$, $P_{3}$,
$P_{4}$ should be smoothened using the eigenstates at $P_{1}$. On each edge of
the mesh, we set midpoints $P_{12}$, $P_{23}$, $P_{34}$, $P_{14}$, as denoted
in Fig. S7(b). Eigenstates ${\mathbf{U}}$ and ${\mathbf{V}}$ at each midpoint
are obtained from their averages at both ends of the edge. Euler form in Eq.
(S9) is now rewritten as

$\displaystyle\operatorname{Eu}$ $\displaystyle=$
$\displaystyle\frac{{\mathbf{U}}_{P23}-{\mathbf{U}}_{P14}}{{\textrm{d}}k_{a}}\cdot\frac{{\mathbf{V}}_{P34}-{\mathbf{V}}_{P12}}{{\textrm{d}}k_{b}}$
(S11) $\displaystyle-$
$\displaystyle\frac{{\mathbf{U}}_{P34}-{\mathbf{U}}_{P12}}{{\textrm{d}}k_{b}}\cdot\frac{{\mathbf{V}}_{P23}-{\mathbf{V}}_{P14}}{{\textrm{d}}k_{a}},$

where ${\textrm{d}}k_{a}=\left|P_{23}-P_{14}\right|$ and
${\textrm{d}}k_{b}=\left|P_{34}-P_{12}\right|$ are the edges’ lengths of the
mesh. With the mesh’s area
${\textrm{d}}A=\left|P_{2}-P_{1}\right|\left|P_{4}-P_{1}\right|$, we sum up
all the Euler form in Eq. (S11) as follows:

$\int_{\mathcal{D}}\operatorname{Eu}{\textrm{d}}k_{a}{\textrm{d}}k_{b}=\sum_{i,j}\operatorname{Eu}_{ij}{\textrm{d}}A_{ij},$
(S12)

where $i$ and $j$ are each mesh’s indices from the left to right and bottom to
top in Fig. S7(a), respectively. If the current mesh has a degeneracy or is
around a degeneracy where the eigenstates are ill-defined (that is, if at
least one pair among ${\mathbf{U}}_{P1}$ to ${\mathbf{U}}_{P4}$ has the inner
product below the tolerance 0.99), $\operatorname{Eu}_{ij}$ in this mesh is
not summed to the above equation by assigning zero to
$\operatorname{Eu}_{ij}$.

Meanwhile, we use the smoothened eigenstates on the boundary to calculate the
boundary integral of Eq. (S10). We rewrite the boundary integral as follows:

$\oint_{\partial{\mathcal{D}}}\left<u_{\mathbf{k}}^{m}\mid\nabla_{\mathbf{k}}u_{\mathbf{k}}^{n}\right>\cdot{\textrm{d}}{\mathbf{k}}=\oint_{\partial{\mathcal{D}}}{\mathbf{U}}\cdot\frac{\partial\mathbf{V}}{\partial
k}\partial k.$ (S13)

The boundary has four vertices. At any points except the ones on the vertices,
$\partial\mathbf{V}/\partial k=D\mathbf{V}$ is performed by the two- or four-
point central difference method, i.e,
$D{\mathbf{V}}_{i}=\left({\mathbf{V}}_{i+1}-{\mathbf{V}}_{i-1}\right)/h$ or
$D{\mathbf{V}}_{i}=\left(-{\mathbf{V}}_{i+2}+8{\mathbf{V}}_{i+1}-8{\mathbf{V}}_{i-1}+{\mathbf{V}}_{i-2}\right)/12h$,
respectively, where $h$ is the spacing between adjacent points. At the
beginning and finishing vertices, marked as 1 in Fig. S7(a), the three-point
forward and backward differences are used, respectively, i.e.,
$D{\mathbf{V}}_{i}=\left(-3{\mathbf{V}}_{i}+4{\mathbf{V}}_{i+1}-{\mathbf{V}}_{i+2}\right)/2h$
and
$D{\mathbf{V}}_{i}=\left({\mathbf{V}}_{i-2}-4{\mathbf{V}}_{i-1}+3{\mathbf{V}}_{i}\right)/2h$,
respectively. At vertices 2, 3, and 4, both three-point forward and backward
differences are calculated followed by averaging them. Then, the integral in
Eq. (S13) becomes

$\oint_{\partial{\mathcal{D}}}{\mathbf{U}}\cdot\frac{\partial\mathbf{V}}{\partial
k}\partial k=\sum_{i}{\mathbf{U}}_{i}\cdot D{\mathbf{V}}_{i}h.$ (S14)

Thus, the difference between Eqs. (S12) and (S14) generates the final result
of Euler class:

$\chi=\frac{1}{2\pi}\left[\sum_{i,j}\operatorname{Eu}_{ij}{\textrm{d}}A_{ij}-\sum_{i}{\mathbf{U}}_{i}\cdot
D{\mathbf{V}}_{i}h\right].$ (S15)

We set $N=600$ throughout this study, that is, a domain consists of $600\times
600$ meshes. The eigenstates are calculated at $601\times 601$ vertices of
meshes.

##  E Transition from nodal link

In the main text, we infer the allowed transition, that is, two nodal lines
heading the same direction [$L_{1}$ and $L_{2}$ in Fig. S8(a)] can change
their connectivity with keeping $+1$ Euler class. During this transition from
Fig. S8(a) to (c), a critical state is given by a nodal chain, as shown in
Fig. S8(b).

Similar predictions can be made for the nodal rings $R_{1}$ and $R_{2}$. At
the Brillouin zone boundary, $R_{2}$ in current zone and $R_{1}$ in the next
zone are getting closer, like Fig. S8(d). The Euler class
$\chi\left({\mathcal{D}}_{12}\right)$ over the marked patch in Fig. S8(d) is
$+1$. While this condition is satisfied, the nodal rings can be deformed to
Fig. S8(e) and (f). Here, the nodal chain in Fig. S8(e) is the critical state
between Fig. S8(d) and (f).

Based on the above prediction, we realize these transition by tuning the
spring constants in the spring-mass system. First, from the nodal link in Fig.
3(d) in the main text [or in Fig. S2(c) with Table S1], we increase the on-
site potential spring $K_{1}$ to $140$ and decrease $\Delta C_{31}$ to $2$
(where $\Delta C_{31}$ is for control of $C_{31}$ and $C_{3\bar{1}}$ by
$C_{31}=C_{31}^{0}+\Delta C_{31}$, $C_{3\bar{1}}=C_{31}^{0}-\Delta C_{31}$,
and $C_{31}^{0}=35$) to prepare Fig. S8(g). Now we decrease $\Delta C_{31}$ to
zero. Then, the spring constants become $C_{31}=C_{3\bar{1}}$, and we have
Fig. S8(h). This critical state exhibits two nodal chains formed by
$L_{1}$-$L_{2}$ around the $\Gamma$-point and $R_{1}$-$R_{2}$ on the zone
boundary. Because we have $C_{23}=C_{2\bar{3}}$ and the two on-site springs’
directions are respectively the same as ${\mathbf{a}}_{2}$ and
${\mathbf{a}}_{3}$, this spring-mass system is in $2/m$ ($C_{2h}$) symmetry.
After further decreasing $\Delta C_{31}$ to $-2$, the nodal chains by
$L_{1}$-$L_{2}$ and $R_{1}$-$R_{2}$ are separated into two nodal lines and
nodal rings, respectively [Fig. S8(i)]. Compared to Fig. S8(g), the
connectivity between $L_{1}$ and $L_{2}$ are exchanged. In addition, $R_{2}$
was placed at higher position along ${\mathbf{b}}_{3}$-direction than $R_{1}$
in Fig. S8(g), and this is now reversed.