# Nodal lines in momentum space: topological invariants and recent
realizations in photonic and other systems

Haedong Park  School of Physics and Astronomy, Cardiff University, Cardiff
CF24 3AA, United Kingdom  Wenlong Gao  Department of Physics, Paderborn
University, Warburger Straße 100, 33098 Paderborn, Germany  Xiao Zhang  School
of Physics, Sun Yat-sen University, Guangzhou 510275, China  Sang Soon Oh
<EMAIL_ADDRESS>School of Physics and Astronomy, Cardiff University,
Cardiff CF24 3AA, United Kingdom

###### Abstract

Topological insulators constitute one of the most intriguing phenomena in
modern condensed matter theory. The unique and exotic properties of
topological states of matter allow for unidirectional gapless electron
transport and extremely accurate measurements of the Hall conductivity.
Recently, new topological effects occurring at Dirac/Weyl points have been
better understood and demonstrated using artificial materials such as photonic
and phononic crystals, metamaterials and electrical circuits. In comparison,
the topological properties of nodal lines, which are one-dimensional
degeneracies in momentum space, remain less explored. Here, we explain the
theoretical concept of topological nodal lines and review recent and ongoing
progress using artificial materials. The review includes recent demonstrations
of non-Abelian topological charges of nodal lines in momentum space and
examples of nodal lines realized in photonic and other systems. Finally, we
will address the challenges involved in both experimental demonstration and
theoretical understanding of topological nodal lines.

Topology, nodal lines, non-Abelian, metamaterials, photonic crystals

††preprint: APS/123-QED

##  I Introduction

Degeneracies in energy-momentum relations, so-called band structures, play an
important role in many fields of physics from classical mechanics to condensed
matter physics and optics. Recently, the study on degenerate points in band
structures opened a new path to observe many exotic topological behaviors. For
instance, Dirac points, point degeneracies with a linear dispersion relation
in two-dimensional momentum space [1, 2], have been used to show chiral one-
way edge/surface states in a band gap [3, 4, 5, 6]. Weyl points, which are
also point degeneracies with a linear dispersion but in three-dimensional
momentum space [7, 8, 9], have been used to generate Fermi arc-like surface
states [10, 11, 12, 13, 14]. In particular, Weyl points are very interesting
and important in topological classification because they are stable meaning
that the Weyl points are robust to perturbations when only one of the
inversion ($\mathcal{P}$) symmetry and time-reversal ($\mathcal{T}$) symmetry
is broken in three-dimensional space [7].

By recovering the broken symmetry in a structure with Dirac/Weyl points, i.e.,
making it $\mathcal{P}$ and $\mathcal{T}$ symmetric, a nodal line [15, 16, 17,
18], a one-dimensional degeneracy, can be created. Nodal lines have drawn
attention because they can feature two-dimensional surface states bounded by
the projected nodal lines, called drumhead surface states [19, 20, 21], and
exhibit non-Abelian band topology [17]. Moreover, as nodal lines have higher
dimensions than Dirac/Weyl points, they show various shapes, for instance, a
simple nodal line, a nodal ring [22, 23, 24], nodal knots [25, 26, 27] for a
single nodal line or a Hopf link [28, 27, 29, 30, 31, 18], a nodal chain [32,
33, 34, 18] for multiple nodal lines.

To study the topological characteristics of the nodal lines, the starting
point is to define topological invariants which are the conserved quantities
when any topological phase transitions do not occur, i.e., the topological
phase do not change under certain perturbations [35]. Topological invariants
of Dirac/Weyl points and nodal lines are often called _topological charges_
similarly to electric/magnetic charges of electric/magnetic monopoles. While
the topological charge of a simple nodal line, that is the degeneracy between
two bands, can be described by a Berry phase [36], the definition of
topological charge for multiple nodal lines becomes more complex because three
or more bands are involved. Recently, it was shown that the topological
charges of the nodal lines in a three-band system can be described by
quaternion numbers, forming a non-Abelian group [17] and the experimental
observation of the quaternion charges has been reported [37].

Along with the progress in understanding band topology, recent years saw the
rapid development of topological photonics which studies and demonstrates the
topological states of photons in metamaterials (including metallic photonic
crystals) [38, 39, 40, 41, 22, 33, 24] and dielectric photonic crystals [7,
42, 43, 44, 18]. In contrast to electronic systems these photonic materials
have great advantages because one can design a structure and manipulate the
propagation properties of photons with more freedom and in a wide range of
frequency spectra. For this reason, many exciting advances have been made, for
example, Weyl fermions have been demonstrated using a dielectric photonic
crystal called double gyroid [7, 43, 44] and unidirectional edge modes have
been demonstrated using a magnetic photonic crystal [45, 10]. On the other
hand, however, the progress on nodal lines in photonics has been rather slow
compared to the study on Dirac/Weyl photonic crystals. The possible reasons
can be summarized as follows. First, investigating nodal lines in three-
dimensional momentum space requires more amount of computations than Dirac or
Weyl points. Second, it was very recent that nodal chains [33] or nodal links
[31, 46, 18] started to be realized although nodal ring by a dielectric
photonic crystal was already reported in Ref. [7]. Lastly, discussions on new
topological invariants of nodal lines such as non-Abelian topological charges
have been made very recently [17]. Therefore, reviewing the recent work on
nodal lines is very timely and summarising important concepts is essential for
more exciting outcomes that will be generated in the field of topological
photonics in the near future.

In this review, we will aim to cover the basic theory of topological physics
with a focus on topological nodal lines and introduce important examples of
topological nodal lines demonstrated in various artificial material systems.
In Section II, we describe the degeneracies in band structures including Weyl
points and nodal lines. In Section III, we explain Abelian and non-Abelian
topological invariants using two examples, the Berry phase and the Wilczek-Zee
connection. In Section IV, we highlight examples of topological nodal lines in
metamaterials, photonic crystals and photonic systems with synthetic
dimensions. In Section V, we discuss examples in electronic crystals, phononic
crystals and electrical circuits.

Figure 1:  (a),(b) Schematics of a Weyl point and a nodal line, respectively.

##  II Band degeneracies

In an electronic band structure, two adjacent bands may touch each other in
one or more $\mathbf{k}$-point(s) meaning that the two bands have the same
energy but different eigenstates. This is ‘degeneracy’. The concept of
degeneracy can also be applied to photonic/phononic band structures which are
the frequency-wavevector relations for waves in a periodic array of
photonic/phononic atoms. In three-dimensional momentum space, the degeneracies
can be classified as zero-, one-, and two-dimensional depending on their
dimensionality. In this section, we will explain zero- and one-dimensional
degeneracies which can be Weyl points and nodal lines, respectively, and
introduce diverse categories of nodal lines that are classified by their
shapes.

###  II.1 Zero-dimensional degeneracies: Weyl points

A representative example of a zero-dimensional degeneracy is a Weyl point [7,
8, 9]. The Weyl point was named thus because the dispersion around the
degenerate point is governed by the Weyl Hamiltonian
$H(\mathbf{k})=v_{1}k_{1}\sigma_{1}+v_{2}k_{2}\sigma_{2}+v_{3}k_{3}\sigma_{3}$
where $\sigma_{i}$ are the Pauli matrices. In three-dimensional momentum
space, the Weyl point acts as a monopole that emits or soaks the Berry flux
similar to a magnetic monopole where the magnetic flux departs or terminates.
The mathematical definition of the Berry flux will be introduced in Section
III.2. In the Weyl Hamiltonian, the $\sigma_{2}$ term can exist only when one
of the inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries
is broken [7]. This is a necessary condition for the existence of Weyl points.
If we set $v_{1}=v_{2}=v_{3}=1$, the eigenenergies of the Weyl Hamiltonian are
expressed as $E=\pm\left|\mathbf{k}\right|$. Thus, the band structure shows a
point degeneracy at $\mathbf{k}=\mathbf{0}$ and a linear dispersion around the
degeneracy, as shown in Figure 1a. The eigenstates of the Weyl Hamiltonian at
$\mathbf{k}\neq\mathbf{0}$ can be expressed as
$\psi_{1}=\left[\cos\left({\theta/2}\right),e^{i\phi}\sin\left({\theta/2}\right)\right]$
and
$\psi_{2}=\left[e^{-i\phi}\sin\left({\theta/2}\right),-\cos\left({\theta/2}\right)\right]$
using the spherical coordinate system $\left(r,\theta,\phi\right)$. Then, the
Berry curvature is expressed as $\pm 1/\left(2k^{2}\right)\hat{\mathbf{r}}$
where $\hat{\mathbf{r}}$ is the unit vector in the radial direction in
momentum space [47, 8] implying that the Weyl point becomes a sink or source
of the Berry flux.

In the early 2010s, significant efforts have been made to realize Weyl points
[48, 49, 50, 51, 52, 53, 54], and a double gyroid structure was theoretically
proposed as a Weyl photonic crystal in 2013 [7]. Two years after this
theoretical work, Weyl points were experimentally observed in the microwave
frequency range [8]. Following this first experimental demonstration, the
realization of Weyl points has been achieved using photonic crystals [10, 55,
56, 57, 9, 58, 59, 43, 60, 44], phononic crystals [61, 3, 14, 62, 63, 64, 65,
66], metals [67, 68, 69, 70] and semimetals [71, 72, 73, 74, 75, 76, 77].

Figure 2:  Classification of nodal lines.

###  II.2 One-dimensional degeneracies: nodal lines

One-dimensional degeneracies (co-dimension $N-1$ in a $N$-dimensional space),
called ‘nodal lines’, can be found when two bands touch each other on a line
in momentum space [78, 15, 16, 17, 18], as illustrated in Figure 1b. The nodal
lines can be classified by their shapes and the connectivities between other
nodal lines, as shown in Figure 2.

As shown in Figure 2a, a nodal line can have a loop shape to form a nodal ring
[7, 79, 80, 81, 82, 83, 84, 85, 86, 87, 22, 88, 89, 23, 29, 24, 90] or knot,
such as a trefoil knot [25, 26, 27], a double trefoil knot [26, 91], a
cinquefoil knot [26, 25], or a figure 8-knot [27]. Such knots cannot be
transformed into a nodal ring without cutting or intersecting them.

If two rings intersect each other, they form a nodal chain [92, 93, 94, 95,
96, 97, 98]. If two rings are tied without touching so that they cannot be
separated by cutting or passing them, they form a nodal link [99, 100, 101,
25, 102, 103, 92, 98, 91, 104, 105, 95, 28, 27, 29, 30] or Solomon’s knots
[91, 26] (Figure 2b). Each ring of the nodal chain/link can be formed by the
same or a different pair of bands. Especially, if the two rings of the nodal
chain originate from different set of bands in a three-band system, the three
bands meet at a single point where the nodal rings touch [17, 106, 107, 108].
This is called a triple point.

Multiple nodal rings may form an infinite nodal chain [32, 109, 33, 94, 34,
18] or a link [28, 106, 18] due to the periodicity of momentum space (Figure
2c). Although Figure 2c shows only a one-dimensional infinite chain and a
link, they can also form a two- or three-dimensional infinite chain [33] or a
link [28].

In some cases, multiple nodes of different types can appear mixed. First, the
mixed shape of the nodal rings appear as earring nodal links [17, 29],
multiple Hopf links [25, 91, 95, 110], mixed nodal links [29], and the linked
nodal ring and a chain (Figure 2d) [31, 111]. Second, the nodal lines and a
nodal ring/chain can be linked to show the non-touching between nodal lines
and rings [29, 31, 107]. Inversely, the nodal lines and nodal ring/link can be
chained to show the touching between nodal lines and ring [32, 94, 112, 111]
or the touching between nodal lines and link [17, 111], as shown in Figure 2e.
Finally, if the linked nodal ring and chain [46] and the nodal lines touch,
the linked nodal ring, chain, and lines [31] are generated (Figure 2f).

For the above classification, we make one assumption for brevity. As we will
look at a periodic system, the momentum space is periodic. When a nodal line
crosses a Brillouin zone boundary, it intersects at the same point at the
opposite Brillouin zone boundary. Therefore, a nodal line and a double helix
(Figure 2b) [34, 15, 113, 114, 115, 116, 117, 118] can be considered as a
nodal ring and a nodal link, respectively. In the above, however, we assumed
that the first Brillouin zone is distinct from the neighboring Brillouin
zones, which means that a nodal line that crosses the boundary extends from
negative infinity to positive infinity.

##  III Topological invariants

The physical behaviors of bands in momentum space are interpreted using
several kinds of topological invariants. The topological invariant is a
quantized number that characterizes the topological status of a given system,
and the Chern number or Berry phase are examples of the topological
invariants. The topological invariants are related to the various phenomena
(e.g., surface states) of the topological insulators, which act as an
insulators in their bulk and permit the electronic/photonic/phononic waves on
their boundaries. For example, in $\mathcal{T}$-symmetry broken topological
insulators, one-way surface states are formed at the interface of two band-
gapped materials due to the difference in Chern numbers between two bands
[119, 45, 120, 35]. In pseudo-$\mathcal{T}$-symmetry broken or
$\mathcal{P}$-symmetry broken topological insulators, surface states exist due
to the difference in the spin-Chern numbers or valley-Chern numbers [121, 122,
123]. In addition to the topological invariants of these gapped phases,
several efforts also have been carried out to understand relations between
surface states and the topological invariants of gapless phases (band
degeneracies), such as Dirac points [124], Weyl points [7, 125] or nodal lines
[17]. Thus, describing the topological invariants of the band degeneracy is an
important step to understanding the degeneracy and finding appropriate
applications.

In the following, we review Abelian and non-Abelian topological invariants. We
introduce the Berry phase [36] as an example of the Abelian topological
invariants. We also explain the Wilczek-Zee phase [126], which is the starting
point toward the non-Abelian topological quaternion charges [17]. Simple
examples of the quaternion charges are shown, and the use of correlation
vectors in full-vector field systems is explained. Finally, the patch Euler
patch class is explained with an example of non-Abelian topology. For more
information, refer to Ref. [36] for the Berry phase, Ref. [126] for the
Wilczek-Zee connection, Ref. [17] for the non-Abelian quaternion charges, and
Ref. [127] for the Euler class derived using the Wilson loop.

###  III.1 Abelian and non-Abelian topological invariants

As noted in Section II, a Weyl point and a nodal line are zero- and one-
dimensional degeneracies, respectively. In fact, a nodal line in three-
dimensional space has the similar features to a Dirac point [1, 2, 128, 42,
129, 130, 131, 132, 133, 4, 134, 135, 5] which is a point degeneracy in two-
dimensional momentum space. The nodal line and the Dirac point have the same
co-dimension $N-1$ and commonly correspond to
$H(\mathbf{k})=v_{1}k_{1}\sigma_{1}+v_{3}k_{3}\sigma_{3}$ that does not have
the $\sigma_{2}$ term compared to the Weyl Hamiltonian mentioned in Section
II.1. In addition, calculating the topological invariant of a nodal line and a
Dirac point starts from considering a closed loop around the degeneracies,
whereas calculating the topological charge of a Weyl point is associated with
the Berry flux on the surface enclosing the Weyl point.

In case of the Abelian charge of nodal lines, to describe the topological
nature of multiple degeneracies between the same pair of bands, e.g., in a
two-band system, the topological invariants are obtained by simply summing up
the invariants of all the degeneracies. However, such an invariant cannot
express the full topological nature of the multi-band systems. For example,
when the Abelian charges are used, the nodal lines between the first and
second bands and between the second and third bands commonly exhibit a
topological charge of $\pm\pi$. Thus, this invariant cannot distinguish which
bands make the degeneracy, and the relation of the charges between a different
pair of bands cannot be described.

The non-Abelian band topology gives a solution for the multi-band systems
[17]. Degeneracies by a different pair of bands have different topological
charges, the quaternion numbers. The mutual interaction between the different
pairs of bands can be written clearly, and the topological charges satisfy the
anticommutative relation.

###  III.2 Berry phase

Berry phase is a geometrical phase that is obtained by a system when it moves
along a closed path in a parameter space [36]. The Berry phase is path-
dependent and is useful in studying the topology of the parameter space by
providing a way to calculate topological invariants. The mathematical
description of the Berry phase starts with a Hamiltonian that depends on time-
varying parameters $\mathbf{k}=\left[k_{1},k_{2},\cdots\right]$, i.e.,
$H=H\left(\mathbf{k}\left(t\right)\right)$. We denote the orthonormal bases of
$H\left(\mathbf{k}\left(t\right)\right)$ as
$\left|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle$:

$H\left(\mathbf{k}\left(t\right)\right)\left|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle=E_{n}\left(\mathbf{k}\left(t\right)\right)\left|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle,$
(1) $\left\langle
u_{m}\left(\mathbf{k}\left(t\right)\right)\middle|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle=\delta_{mn}.$
(2)

Here, we assume that the eigenstates in Eqs. (1)-(2) are given by not only
$\left|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle$ but also
$e^{i\gamma_{n}\left(t\right)}\left|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle$.
A state $\left|\psi\left(t\right)\right\rangle$ satisfying the time-dependent
Schrödinger equation is considered. If the parameter $\mathbf{k}$ changes
adiabatically (that is, $\mathbf{k}$ varies slowly with time), the state
$\left|\psi\left(t\right)\right\rangle$ that was in $n$-th state at $t=0$
remains in the same state at $t=T$ where $T$ is the period of the cycle [136,
137]. The state $\left|\psi\left(t\right)\right\rangle$ is given by [36]

$\left|\psi\left(t\right)\right\rangle=e^{-\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}E_{n}\left(\mathbf{k}\left(t^{\prime}\right)\right)}e^{i\gamma_{n}\left(t\right)}\left|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle$
(3)

where the first exponential term is the dynamical phase factor. By
substituting $\left|\psi\left(t\right)\right\rangle$ into the time-dependent
Schrödinger equation, the geometric phase $\gamma_{n}$ can be obtained as an
integral form. If a closed loop $\Gamma$ in $\mathbf{k}$-space is considered
such that $\mathbf{k}\left(0\right)=\mathbf{k}\left(T\right)$, we have

$\gamma_{n}=\oint_{\Gamma}{\mathbf{A}}_{n}\cdot d{\mathbf{k}}$ (4)

where

${\mathbf{A}}_{n}\left(\mathbf{k}\right)=i\left\langle
u_{n}\left(\mathbf{k}\right)\middle|\nabla_{\mathbf{k}}u_{n}\left(\mathbf{k}\right)\right\rangle.$
(5)

As we consider a closed loop, the phase difference
$\zeta\left(\mathbf{k}\left(T\right)\right)-\zeta\left(\mathbf{k}\left(0\right)\right)$
for the gauge transformation
$\left|u_{n}\left(\mathbf{k}\right)\right\rangle\rightarrow
e^{i\zeta\left(\mathbf{k}\right)}\left|u_{n}\left(\mathbf{k}\right)\right\rangle$
should be an integer multiple of $2\pi$, so that $\gamma_{n}$ in Eq. (4)
becomes gauge-invariant. Here, $\gamma_{n}$ and
$\mathbf{A}_{n}\left(\mathbf{k}\right)$ are called Berry phase and Berry
connection, respectively [36, 138, 47]. The Berry flux (mentioned in Section
II.1) is given by $\iint{\mathbf{F}}_{n}\cdot d^{2}\mathbf{k}$ where
${\mathbf{F}}_{n}\left(\mathbf{k}\right)=\nabla_{\mathbf{k}}\times{\mathbf{A}}_{n}\left(\mathbf{k}\right)$
is called the Berry curvature [35]. Regarding the closed loop $\Gamma$, a more
practical form of Eqs. (4) and (5) is

$\gamma_{n}=\oint_{\Gamma}i\left\langle
u_{n}\left(\mathbf{k}\right)\middle|\frac{\partial}{\partial
k}u_{n}\left(\mathbf{k}\right)\right\rangle dk.$ (6)

From the orthonormality relation $\left\langle
u_{m}\middle|u_{n}\right\rangle=\delta_{mn}$, we have $\left\langle\partial
u_{m}/\partial k\middle|u_{n}\right\rangle+\left\langle u_{m}\middle|\partial
u_{n}/\partial k\right\rangle=2\operatorname{Re}\left(\left\langle
u_{m}\middle|\partial u_{n}/\partial k\right\rangle\right)=0$. Thus, the
integrand in Eq. (6) is purely imaginary, and $\gamma_{n}$ is real. If
$\left|u_{n}\right\rangle$ consists of only real numbers, $\gamma_{n}$ becomes
zero. In other words, to get a non-zero Berry phase,
$\left|u_{n}\right\rangle$ should consist of complex numbers having one or
more non-zero imaginary components. One representative example for this case
is the Weyl Hamiltonian with $v_{1}=v_{2}=v_{3}=1$, mentioned in Section II.1.
The Weyl Hamiltonian has $\sigma_{2}$ so that its eigenstates consist of
complex numbers. Possessing this $\sigma_{2}$ term corresponds to the
$\mathcal{P}\mathcal{T}$-symmetry breaking in three-dimensional space and is a
necessary condition for the existence of Weyl points [7].

###  III.3 Wilczek-Zee connection

Nodal lines are generated when both $\mathcal{P}$ and $\mathcal{T}$ symmetries
are conserved. Such a situation corresponds to the lack of $\sigma_{2}$ in the
Weyl Hamiltonian. In this case, the Berry phase in Eq. (6) becomes zero
because one can choose a gauge that keeps the Hamiltonian and its eigenvectors
real. Therefore, it is useful to define a non-vanishing topological invariant,
for example, the Wilczek-Zee connection [126] which will be explained in the
following.

A state $\left|\eta_{m}\left(t\right)\right\rangle$ of the $m$-th band
satisfying the Schrödinger equation can be expressed as a linear combination
of the basis $\left|u_{n}\left(\mathbf{k}\left(t\right)\right)\right\rangle$
in Eqs. (1) (2):

$\left|\eta_{m}\left(t\right)\right\rangle=\sum_{n}W_{mn}\left|u_{n}\right\rangle$
(7)

If we assume that $\left|\eta_{m}\left(t\right)\right\rangle$ remain
normalized, we have

$0=\left\langle\eta_{l}\middle|\frac{\partial\eta_{m}}{\partial
t}\right\rangle=\left\langle\eta_{l}\right|\sum_{n}\left[\frac{\partial
W_{mn}}{\partial
t}\left|u_{n}\right\rangle+W_{mn}\frac{\partial\left|u_{n}\right\rangle}{\partial
t}\right].$ (8)

This equation can be arranged as follows:

$\frac{\partial\mathbf{W}}{\partial
t}\left|\mathbf{u}\right\rangle=-\mathbf{W}\frac{\partial\left|\mathbf{u}\right\rangle}{\partial
t}$ (9)

or

$\left[{\mathbf{W}}^{-1}\frac{\partial\mathbf{W}}{\partial
t}\right]\left|\mathbf{u}\right\rangle=-\frac{\partial\left|\mathbf{u}\right\rangle}{\partial
t}$ (10)

where
$\left|\mathbf{u}\right\rangle=\left[\left|u_{1}\right\rangle,\left|u_{1}\right\rangle,\cdots\right]^{T}$
and $\left[\mathbf{W}\right]_{mn}=W_{mn}$. Using the orthonormality condition
in Eq. (2), we have the following skew-symmetric matrix:

$\left[{\mathbf{W}}^{-1}\frac{\partial\mathbf{W}}{\partial
t}\right]_{mn}=-\left\langle u_{n}\middle|\frac{\partial u_{m}}{\partial
t}\right\rangle=\left\langle u_{m}\middle|\frac{\partial u_{n}}{\partial
t}\right\rangle=A_{mn}.$ (11)

Then the path-ordered integrals of this can be expressed as

$\mathbf{W}=\operatorname{exp}\int_{0}^{T}\mathbf{A}\left(t^{\prime}\right)dt^{\prime},$
(12)

or if we consider a closed loop $\Gamma$, and if we recall that $\partial
u_{n}/\partial
t=\left(\nabla_{\mathbf{k}}u_{n}\right)\cdot\left(\partial{\mathbf{k}}/\partial
t\right)$, we can get the Wilson loop

$\mathbf{W}=\operatorname{exp}\left\\{\oint_{\Gamma}\mathbf{A}\left(\mathbf{k}\right)\cdot
d{\mathbf{k}}\right\\}=\operatorname{exp}\left\\{\oint_{\Gamma}\mathbf{A}\left(k\right)dk\right\\}.$
(13)

Here, the component of $\mathbf{A}\left(\mathbf{k}\right)$ is given by

$A_{mn}=\left\langle
u_{m}\middle|\nabla_{\mathbf{k}}u_{n}\right\rangle=\left\langle
u_{m}\middle|\frac{\partial u_{n}}{\partial k}\right\rangle$ (14)

and is called the Wilczek-Zee connection [126].

Figure 3:  Schematics of the non-Abelian quaternion charges. (a) A nodal line
and a closed loop encircling the nodal line. (b) An example of the eigenstates
along the closed loop. (c) Quaternion multiplication table. (d),(e),(f)
Eigenstates gathered at the origin indicating the topological charges
${\bm{k}}$, ${\bm{i}}$, and $+{\bm{j}}$, respectively.

###  III.4 Three-band system

In this section, the expressions derived in the previous section are applied
to a three-band system. Following formalism will be the basis of the rigorous
description of quaternion charges [17].

To calculate the topological charges of an arbitrary nodal line, first, a
closed loop $\Gamma\left(\alpha\right)$ ($\alpha\in\left(0,2\pi\right]$)
around the nodal line is considered (see Figure 3a). From the eigenstates of
the $m$\- and $n$-th bands ($m,n=1,2,3$), the Wilczek-Zee connection (Eq.
(14)) is rewritten as follows:

$\mathbf{A}\left(k\right)=\left[\begin{array}[]{ccc}A_{11}&A_{12}&A_{13}\\\
A_{21}&A_{22}&A_{23}\\\ A_{31}&A_{32}&A_{33}\\\ \end{array}\right].$ (15)

Eq. (11) leads to the skew-symmetric $\mathbf{A}\left(\mathbf{k}\right)$:

$\mathbf{A}\left(k\right)=\left[\begin{array}[]{ccc}0&A_{12}&-A_{31}\\\
-A_{12}&0&A_{23}\\\ A_{31}&-A_{23}&0\\\
\end{array}\right]={\bm{\beta}}\left(k\right)\cdot{\mathbf{L}}$ (16)

where ${\bm{\beta}}=\left[-A_{23},-A_{31},-A_{12}\right]$ and
$\left(L_{i}\right)_{jk}=-\epsilon_{ijk}$ [17, 31].

If the closed loop encircles the nodal line formed by the bands $m=1$ and
$n=2$, $\mathbf{A}\left(\mathbf{k}\right)$ is rewritten as

$\displaystyle\mathbf{A}\left(k\right)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}0&A_{12}&0\\\ -A_{12}&0&0\\\ 0&0&0\\\
\end{array}\right]$ (20) $\displaystyle=$
$\displaystyle{\bm{\beta}}_{12}\left(k\right)\cdot{\mathbf{L}}=-A_{12}L_{3},$
(21)

where ${\bm{\beta}}_{12}\left(k\right)=\left[0,0,-A_{12}\right]$. Substituting
Eq. (21) into Eq. (13) gives

$\displaystyle W$ $\displaystyle=$
$\displaystyle\operatorname{exp}\left\\{\oint_{\Gamma(\alpha)}{\bm{\beta}}_{12}\left(k\right)\cdot{\mathbf{L}}dk\right\\}$
(22) $\displaystyle=$
$\displaystyle\operatorname{exp}\left\\{\left(\oint_{\Gamma(\alpha)}-A_{12}dk\right)L_{3}\right\\}.$

The spin Wilczek-Zee connection which is a
$\mathfrak{spin}\left(N\right)$-valued 1-form is then written as [17]

$\bar{A}\left(k\right)={\bm{\beta}}_{12}\left(k\right)\cdot{\mathbf{t}}={\bm{\beta}}_{12}\left(k\right)\cdot\left(-\frac{i}{2}{\bm{\sigma}}\right),$
(23)

where the components of
${\bm{\sigma}}=\left[\sigma_{1},\sigma_{2},\sigma_{3}\right]$ are the Pauli
matrices and ${\mathbf{t}}=\left(-i/2\right){\bm{\sigma}}$.

From Eqs. (22) and (23), the topological charge [17, 31] is then expressed as

$\displaystyle n_{\Gamma}$ $\displaystyle=$
$\displaystyle\operatorname{exp}\left\\{\oint_{\Gamma(\alpha)}\bar{A}\left(k\right)dk\right\\}$
(24) $\displaystyle=$
$\displaystyle\operatorname{exp}\left\\{\oint_{\Gamma(\alpha)}{\bm{\beta}}_{12}\left(k\right)\cdot\left(-\frac{i}{2}{\bm{\sigma}}\right)dk\right\\}$
$\displaystyle=$
$\displaystyle\operatorname{exp}\left\\{-\frac{i}{2}\sigma_{3}\oint_{\Gamma(\alpha)}-A_{12}dk.\right\\}.$

If the integral $\oint_{\Gamma(\alpha)}-A_{12}dk$ is $\pm\pi$, the charge
$n_{\Gamma}$ becomes $\mp i\sigma_{3}$.

Now, if the closed loop encircles the nodal line formed by the bands $m=2$ and
$n=3$, $\mathbf{A}\left(\mathbf{k}\right)$ in Eq. (16) is rewritten as
$\mathbf{A}\left(k\right)={\bm{\beta}}_{23}\left(k\right)\cdot{\mathbf{L}}=-A_{23}L_{1}$
where ${\bm{\beta}}_{23}\left(k\right)=\left[-A_{23},0,0\right]$. Substituting
this into Eq. (13) gives

$\displaystyle W$ $\displaystyle=$
$\displaystyle\operatorname{exp}\left\\{\left(\oint_{\Gamma(\alpha)}-A_{23}dk\right)L_{1}\right\\}.$
(25)

The spin Wilczek-Zee connection is
$\bar{A}\left(k\right)={\bm{\beta}}_{23}\left(k\right)\cdot{\mathbf{t}}={\bm{\beta}}_{23}\left(k\right)\cdot\left(-\frac{i}{2}{\bm{\sigma}}\right)$,
and the charge in Eq. (24) is then rewritten as

$\displaystyle n_{\Gamma}$ $\displaystyle=$
$\displaystyle\operatorname{exp}\left\\{-\frac{i}{2}\sigma_{1}\oint_{\Gamma(\alpha)}-\left[A\left(k\right)\right]_{23}dk.\right\\}.$
(26)

In the same way as the first case, we obtain $n_{\Gamma}=\mp i\sigma_{1}$ when
$\oint_{\Gamma(\alpha)}-A_{23}dk=\pm\pi$.

###  III.5 Non-Abelian quaternion charges

To describe the non-Abelian band topology, Q. Wu et al. [17] employed the
quaternion numbers, ${\mathbb{Q}}=\left\\{\pm\bm{i},\pm\bm{j},\pm\bm{k},\pm
1\right\\}$ (first written by the Irish mathematician, William Rowan Hamilton
in 1843). The basis elements ${\bm{i}}$, ${\bm{j}}$, and ${\bm{k}}$ are
defined such that ${\bm{i}}^{2}={\bm{j}}^{2}={\bm{k}}^{2}=-1$. Their
multiplication relations are ${\bm{i}}{\bm{j}}={\bm{k}}$,
${\bm{j}}{\bm{k}}={\bm{i}}$, and ${\bm{k}}{\bm{i}}={\bm{j}}$. They all
anticommute, that is, ${\bm{i}}{\bm{j}}=-{\bm{j}}{\bm{i}}$,
${\bm{j}}{\bm{k}}=-{\bm{k}}{\bm{j}}$, and
${\bm{k}}{\bm{i}}=-{\bm{i}}{\bm{k}}$. All these are summarized in Figure 3c.
Interestingly, the Pauli matrices $\left\\{\mp i\sigma_{1},\mp i\sigma_{2},\mp
i\sigma_{3},\pm I\right\\}$ exhibit the same properties as the quaternions and
are isomorphic so that we can map ${\bm{i}}\mapsto-i\sigma_{1}$,
${\bm{j}}\mapsto-i\sigma_{2}$, ${\bm{k}}\mapsto-i\sigma_{3}$, and $1\mapsto
I$. Due to the anticommutative properties of ${\bm{i}}$, ${\bm{j}}$,
${\bm{k}}$, or $-i\sigma_{1}$, $-i\sigma_{2}$, $-i\sigma_{3}$, their relations
are regarded as non-Abelian.

Thus, the topological charges calculated in Eqs. (24) and (26) can be regarded
as the quaternions $\pm{\bm{k}}$ and $\pm{\bm{i}}$, respectively.

###  III.6 Behavior of eigenstates of a 3$\times$3 Hamiltonian

We apply the above formalism to a system expressed by a $3\times 3$
Hamiltonian. First, the rightmost side of Eq. (22) is rewritten in rotation
matrix form:

$W=\left[\begin{array}[]{ccc}\operatorname{cos}\left(\phi_{12}\right)&-\operatorname{sin}\left(\phi_{12}\right)&0\\\
\operatorname{sin}\left(\phi_{12}\right)&\operatorname{cos}\left(\phi_{12}\right)&0\\\
0&0&1\end{array}\right],$ (27)

where

$\phi_{12}=\oint_{\Gamma(\alpha)}-A_{12}dk.$ (28)

Let us suppose that $\left|u_{\mathbf{k}}^{n}\right\rangle$ ($n=1,2,3$) are
the eigenstates of the Hamiltonian of the given system. To satisfy Eqs. (21)
and (27) we fix $\left|u_{\mathbf{k}}^{3}\right\rangle$ as
$\left[0,0,1\right]$. We also assume
$\left|u_{{\mathbf{k}}\left(\alpha\right)}^{1}\right\rangle=\left[\operatorname{cos}\left(g\left(\alpha\right)\right),\operatorname{sin}\left(g\left(\alpha\right)\right),0\right]$
and
$\left|u_{{\mathbf{k}}\left(\alpha\right)}^{2}\right\rangle=\left[-\operatorname{sin}\left(g\left(\alpha\right)\right),\operatorname{cos}\left(g\left(\alpha\right)\right),0\right]$
where $g\left(\alpha\right)$ is a real-valued arbitrary function that depends
on the position of the closed-loop $\Gamma\left(\alpha\right)$ parametrized by
$\alpha$. For convenience, we set $g\left(0\right)=0$. The integral in Eq.
(28) is written as

$\phi=\int_{\alpha=0}^{\alpha=2\pi}-\left\langle
u_{{\mathbf{k}}\left(\alpha\right)}^{1}\middle|\frac{\partial
u_{{\mathbf{k}}\left(\alpha\right)}^{2}}{\partial\alpha}\right\rangle
d\alpha=g\left(2\pi\right)$ (29)

In the above, we mentioned that the quaternion $\mp i\sigma_{3}$ is obtained
if this integral is $\pm\pi$. Thus, we can deduce $g\left(2\pi\right)=\pm\pi$,
and one can define $g\left(\alpha\right)=\pm\alpha/2$ [17, 31]. This means
that we can plot the eigenstates $\left|u_{\mathbf{k}}^{1}\right\rangle$ and
$\left|u_{\mathbf{k}}^{2}\right\rangle$ rotating around the fixed
$\left|u_{\mathbf{k}}^{3}\right\rangle$ by $\pm\pi$.

In the same manner, the rightmost side of Eq. (25) becomes

$W=\left[\begin{array}[]{ccc}1&0&0\\\
0&\operatorname{cos}\left(\phi_{23}\right)&-\operatorname{sin}\left(\phi_{23}\right)\\\
0&\operatorname{sin}\left(\phi_{23}\right)&\operatorname{cos}\left(\phi_{23}\right)\\\
\end{array}\right],$ (30)

where

$\phi_{23}=\oint_{\Gamma(\alpha)}-A_{23}dk.$ (31)

We assume $\left|u_{\mathbf{k}}^{1}\right\rangle=\left[1,0,0\right]$,
$\left|u_{{\mathbf{k}}\left(\alpha\right)}^{2}\right\rangle=\left[0,\operatorname{cos}\left(g\left(\alpha\right)\right),\operatorname{sin}\left(g\left(\alpha\right)\right)\right]$
, and
$\left|u_{{\mathbf{k}}\left(\alpha\right)}^{3}\right\rangle=\left[0,-\operatorname{sin}\left(g\left(\alpha\right)\right),\operatorname{cos}\left(g\left(\alpha\right)\right)\right]$
with the same $g\left(\alpha\right)$. The integral in Eq. (31) becomes

$\phi=\int_{\alpha=0}^{\alpha=2\pi}-\left\langle
u_{{\mathbf{k}}\left(\alpha\right)}^{2}\middle|\frac{\partial
u_{{\mathbf{k}}\left(\alpha\right)}^{3}}{\partial\alpha}\right\rangle
d\alpha=g\left(2\pi\right)$ (32)

and we get similar results; the eigenstates
$\left|u_{\mathbf{k}}^{2}\right\rangle$ and
$\left|u_{\mathbf{k}}^{3}\right\rangle$ rotate by $\pm\pi$ around
$\left|u_{\mathbf{k}}^{1}\right\rangle$ that corresponds to the quaternion
$\mp i\sigma_{1}$.

Thus, if the nodal line system is described by a 3$\times$3 Hamiltonian, the
eigenstates ${\mathbf{u}}_{\mathbf{k}}^{1}$, ${\mathbf{u}}_{\mathbf{k}}^{2}$,
${\mathbf{u}}_{\mathbf{k}}^{3}$ along the closed loop can be calculated and
plotted along an arbitrary coordinate system (see Figure 3b). After collecting
the eigenstates at the origin, the rotation behavior of the eigenstates
indicates the corresponding topological charge. For example, in Figure 3d,
${\mathbf{u}}_{\mathbf{k}}^{3}$ is fixed while ${\mathbf{u}}_{\mathbf{k}}^{1}$
and ${\mathbf{u}}_{\mathbf{k}}^{2}$ show the $+\pi$-rotation. Then, this is
considered as the quaternion charge ${\bm{k}}$. Figure 3e and f also show
similar behaviors, thereby their charges are ${\bm{i}}$ and ${\bm{j}}$,
respectively.

###  III.7 Correlations for full-vector field problems

Let us think more about the rotation behaviors of the eigenstates
$\left|u_{\mathbf{k}}^{1}\right\rangle$,
$\left|u_{\mathbf{k}}^{2}\right\rangle$, and
$\left|u_{\mathbf{k}}^{3}\right\rangle$. If we denote the starting point of
the closed loop as ${\mathbf{k}}_{0}$ and choose the orthonormal coordinates
placed along $\left|u_{{\mathbf{k}}_{0}}^{1}\right\rangle$,
$\left|u_{{\mathbf{k}}_{0}}^{2}\right\rangle$, and
$\left|u_{{\mathbf{k}}_{0}}^{3}\right\rangle$, these eigenstates
$\left|u_{{\mathbf{k}}_{0}}^{n}\right\rangle$ are the same as the unit vectors
of the coordinate system. For an arbitrary orthonormal coordinate system
$\left|e^{n}\right\rangle$, the eigenstates
$\left|u_{\mathbf{k}}^{n}\right\rangle$ can be mapped to
$\left|{\left(u^{\prime}\right)}_{\mathbf{k}}^{n}\right\rangle$ by the
following rotation matrix

${\mathbf{R}}=\sum_{n=1}^{3}\left|e^{n}\right\rangle\left\langle
u_{{\mathbf{k}}_{0}}^{n}\right|.$ (33)

The resulting new eigenstates
$\left|{\left(u^{\prime}\right)}_{\mathbf{k}}^{n}\right\rangle$ are explicitly
written as

$\left|{\left(u^{\prime}\right)}_{\mathbf{k}}^{n}\right\rangle=\left[\left\langle
u_{{\mathbf{k}}_{0}}^{1}|u_{\mathbf{k}}^{n}\right\rangle,\left\langle
u_{{\mathbf{k}}_{0}}^{2}|u_{\mathbf{k}}^{n}\right\rangle,\left\langle
u_{{\mathbf{k}}_{0}}^{3}|u_{\mathbf{k}}^{n}\right\rangle\right]^{T}.$ (34)

Now, we want to render a plot similar to Figure 3d-f when the system is
described not only by a 3$\times$3 matrix but also a geometry-dependent
Hamiltonian, e.g., as in photonic crystals. In this case, an eigenstate
$\left|\psi_{\mathbf{k}}^{n}\right\rangle$ is a function of the three-
dimensional position vector [139]. Similar to Eq. (34) the following
correlations can be defined [18]:

${\mathbf{C}}_{\mathbf{k}}^{n}=\left[\left\langle\psi_{{\mathbf{k}}_{0}}^{1}|\psi_{\mathbf{k}}^{n}\right\rangle,\left\langle\psi_{{\mathbf{k}}_{0}}^{2}|\psi_{\mathbf{k}}^{n}\right\rangle,\left\langle\psi_{{\mathbf{k}}_{0}}^{3}|\psi_{\mathbf{k}}^{n}\right\rangle\right].$
(35)

In the same manner, if the correlations ${\mathbf{C}}_{\mathbf{k}}^{1}$ and
${\mathbf{C}}_{\mathbf{k}}^{2}$ rotate by $\pm\pi$ around
${\mathbf{C}}_{\mathbf{k}}^{3}$, it corresponds to the quaternion $\mp
i\sigma_{3}$. And if the correlations ${\mathbf{C}}_{\mathbf{k}}^{2}$ and
${\mathbf{C}}_{\mathbf{k}}^{3}$ rotate by $\pm\pi$ around
${\mathbf{C}}_{\mathbf{k}}^{1}$, it corresponds to the quaternion $\mp
i\sigma_{1}$.

Figure 4:  Schematic of a three-band system exhibiting the non-Abelian band
nodes in two-dimensional momentum space. (a) Band structure with oppositely
charges nodes. The blue triangles and red circles indicate the nodes whose
charges are $\pm\bm{i}$ and $\pm\bm{j}$, respectively. The open and filled
symbols correspond to the plus and minus signs, respectively. (b) Braiding of
the open triangle node around the open circle node. (c),(d) Band structure
with the nodes whose charges changed by a factor of $-1$ after the tuning of
(b). The patch Euler class is now $1$. Reproduced with permission from Ref.
[124]. Copyright 2021, Nature Portfolio (a-d).

###  III.8 Evolution of degeneracies and the patch Euler class

In the previous sections, the non-Abelian topological charges in three-band
systems were introduced. On top of discovering the nodal line systems with
non-Abelian charges, it has been studied how the non-Abelian charged
degeneracies evolve with the tuning of the Hamiltonian. Such evolutions
include the annihilation or creation of the degeneracies [140, 141, 93, 142,
143]. To understand whether the degeneracies are annihilated or not, the patch
Euler class was introduced in Ref. [127, 142].

In a recent experimental work [124], B. Jiang et al. demonstrated the
evolution of degeneracies with non-Abelian charges and patch Euler class. The
experimentally observed evolution of degeneracies clearly shows the creation
and annihilation of degeneracies although they considered Dirac nodes in two-
dimensional momentum space instead of the nodal lines in three-dimensional
momentum space. For the three-band system in two-dimensional momentum space
illustrated in Figure 4a, the nodes between the first and second (second and
third) bands have the charges $\pm\bm{i}$ ($\pm\bm{j}$). Here, the two charges
$\pm\bm{i}$ ($\pm\bm{j}$) have opposite signs. Thus, the oppositely charged
$\pm\bm{i}$ (or $\pm\bm{j}$) can be annihilated pairwise and the patch Euler
class is zero. However, braiding the open triangle around the open circle (see
Figure 4b) makes these two flip their charges to positive as shown in Figure
4(c). As a result, the patch Euler class becomes one (see Figure 4d). A patch
Euler class of $\pm 1$ means that when the two nodes merge together, they form
a stable quadratic node with frame charge $q=-1$ rather than undergoing
pairwise annihilation.

##  IV Topological nodal lines in photonic systems

Figure 5:  Examples of nodal line topological metamaterials. (a) Schematic of
the metallic mesh structure for generating a nodal chain. (b) The illustration
of the nodal chain consists of two types of nodal lines (blue and red) in the
cubic Brillouin zone. (c) Equifrequency contour and band structure of the non-
Abelian nodal line metamaterial reminiscent of a biaxial crystal at low
frequencies (d) Schematic of the unit cell and the nodal lines in the
Brillouin zone. (e) The transitions between orthogonal nodal chain, in-plane
nodal chain, and separated nodal lines are observed in the illustrated
bianisotropic metamaterial. Reproduced with permission from Ref. [33].
Copyright 2018, Nature Portfolio (a-b); Reproduced with permission from Ref.
[31]. Copyright 2020, American Physical Society (c-d); Reproduced with
permission from Ref. [46]. Copyright 2021, Nature Portfolio (e).

As we described in Section III.5, nodal line degeneracies can be described by
the Weyl Hamiltonian without $\sigma_{2}$ term. Therefore, the goal of
designing a structure with nodal lines is to find geometrical or coupling
parameters for a structure that has such a dispersion. Although there is no
general recipe that can be applied to different systems, the spatial symmetry
consideration can be a good guide to find nodal lines because a structure with
nodal lines respects $\mathcal{P}$ and $\mathcal{T}$ symmetry as mentioned
earlier. For instance, if we know a structure with Weyl degeneracy, that
respects $\mathcal{P}$ or $\mathcal{T}$ symmetry only, we can start from the
structure to recover both symmetries. Alternatively, if we start with a
structure with excessive symmetries including $\mathcal{P}$ and $\mathcal{T}$
symmetries, which is the case for most of the Bravais lattices, we need to
introduce perturbations in the direction of reducing the number of symmetries.
Numerical simulations are often used to see how the dimension of degeneracies
changes during the symmetry reduction process.

In this section, we will introduce examples of nodal lines in photonic systems
using metamaterials, metallic photonic crystals and dielectric photonic
crystals.

###  IV.1 Metamaterials and metallic photonic crystals

Photonic metamaterials emerge as a prominent light matter interaction platform
and have attracted enormous research interest within the past two decades
[144, 145, 146, 147]. Utilizing the quasi-homogeneity and by manipulating the
constituent deep-subwavelength units, so-called meta-atoms, they can create
collective responses to photons far beyond the scope of natural materials, for
instance, negative refraction [148, 149], strong anisotropy [150, 151],
hyperbolicity [152, 153], strong optical activity [154, 155], etc.
Metamaterials have provided a reliable and convenient guideline towards
artificial photonic materials, assisted by the rich design experience
accumulated in the past two decades. Providing new degrees of freedom in
photonic material design, researchers have found rich fundamentally new
physics and applications with metamaterials.

Topological photonic metamaterials have emerged in recent years as a salient
topic within the grand regime of topological photonics. For instance,
bianisotropic metamaterials were used to realize a photonic topological
insulator [38]. In fact, topological phenomena in continuous photonic medium
have a long-standing history including the renowned Pancharatnam-Berry phase
in polarization space [156], and the conical diffraction in biaxial crystals
that is a direct consequence of the quantized Berry phase of the Dirac point
[157].

The first topological metamaterial was designed in 2015 [39], in which a
composite response from hyperbolicity and chirality introduces Fermi surfaces
with distinct Chern numbers that uni-directional surface state connects.
Various topological semimetals have also been discovered in metamaterials,
including Weyl nodes [40] and Dirac nodes [41]. Naturally, metamaterials, or
equivalently effective medium methods, play an important role in the
construction of topological nodal lines.

An ideal photonic nodal line was discovered in a type-I hyperbolic
metamaterial [22]. Here ‘ideal’ refers to that the nodal line is free from
coexisting trivial modes in the bulk. The band crossing happens between
effective longitudinal and transverse modes in which interactions are
eliminated due to their mismatched field polarizations. Despite being an
accidental degeneracy, the band crossing is imposed by the engineered nonlocal
response in the metamaterial exerted by the glide reflection symmetry. In
terms of crystallographic symmetry, these nodal lines are protected by mirror
symmetry and by introducing mirror symmetry breaking terms, for instance
bianisotropy, these nodal lines are instantly gapped and give rise to vortex-
like distributed Berry curvatures [24].

An ideal type-II nodal line has been discovered recently in Bragg reflection
mirror type layered photonic crystals as the phase transition point between
trivial and non-trivial Zak phase regimes [158]. Definition of the ‘type-II’
follows the classification of Weyl points [71], meaning the highly tilted
contact between bands. It exhibits a ring-like contact between electron and
hole pocket, distinguished from the donut-like Fermi surface in type-I nodal
line semimetals [22]. The nodal chain was experimentally introduced using a
three-dimensional metallic-mesh structure (Figure 5a) in microwave scale [33],
which was the original design of a metallic metamaterial with extremely low
plasma frequency [159], though the nodal chain was found far above the plasma
frequency and cannot be explained by effective medium theory. The nodal chain
in Figure 5b, although it consists of two colored nodal rings, is formed by
the same adjacent two bands. This study also examined the drumhead surface
states, a sheet of surface dispersion enclosed by the projected nodal line
bulk states on the surface Brillouin zone.

Another example of crystallographic symmetry mediated optical response of
meta-atoms was demonstrated in the discovery of hourglass nodal lines in a
photonic metamaterial [23]. Although it may seem that band topologies can be
solely determined by global crystallographic symmetries, the interplay between
them and local optical responses is surprisingly rich in new physics, for
instance, the hidden symmetries that are unforeseen by crystallographic group
theory [160]. It is, however, worth noting that without exquisite design, the
touching point between equi-frequency contours in natural biaxial crystal
forms a 3-dimensional nodal chain if the bands at higher momentum are
considered cut off and flattened by the Brillouin zone boundary (Figure 5c).
Utilizing this property and the extreme anisotropy provided by metamaterials,
researchers have constructed and measured nodal-link metamaterials in the
microwave regime [31]. The nodal link in this metamaterial is formed by the
lowest three bands (see Figure 5c, d). The surface bound states in the
continuum are another achievement of this study. Moreover, in a metamaterial
with explicitly broken inversion symmetry through bianisotropic optical
activities (Figure 5e), transition between different types of nodal chains is
observed by engineering the optical resonances of meta-atoms [46]. One
remarkable aspect of these studies is that the topological nature of the nodal
line was well described by the non-Abelian band topology [17], and can be
conveniently derived from the effective Hamiltonian model stemming from the
effective medium theory without referring to microscopic electromagnetic
fields within the structures.

Figure 6:  Examples of nodal lines in dielectric photonic crystals. (a)
Inversion symmetric double gyroid structure that has air spheres. (b) Band
structure of (a) that exhibits a nodal ring in the momentum space. (c)
Inversion symmetric and anisotropic double diamond structure. (d) Nodal link,
nodal chain, and nodal lines in momentum space of the structure in (c).
Reproduced with permission from Ref. [7]. Copyright 2013, Nature Portfolio
(b); Reproduced with permission from Ref. [18]. Copyright 2021, American
Chemical Society (c-d).

###  IV.2 Dielectric photonic crystals

In this section, we explain how to realize nodal lines in dielectric materials
with two examples: double gyroid and double diamond structures. Although
fabricating dielectric photonic crystals is challenging, these crystals have
advantages such as scalability and convenience in theoretical descriptions.

The nodal ring was theoretically realized using an $\mathcal{P}$-symmetric
double gyroid [7]. The well-known single gyroid in $O$-symmetry is defined by
a set $\mathbf{x}=\left[x_{1},x_{2},x_{3}\right]$ such that

$\displaystyle g\left(\mathbf{x}\right)$ $\displaystyle=$
$\displaystyle\operatorname{sin}\left(X_{1}\right)\operatorname{cos}\left(X_{2}\right)+\operatorname{sin}\left(X_{2}\right)\operatorname{cos}\left(X_{3}\right)$
(36) $\displaystyle+$
$\displaystyle\operatorname{sin}\left(X_{3}\right)\operatorname{cos}\left(X_{1}\right)>g_{c}>0,$

and its space group is $I4_{1}32$ (No. 214) [59, 43]. Here,
$X_{i}=\left(2\pi/a\right)x_{i}$ is a local coordinate where $a$ is a lattice
constant. In the study in Ref. [7], the reduced-symmetric single gyroid is
created by introducing an air sphere of radius $0.13a$ located at
$\left[1/4,-1/8,1/2\right]a$ in the single gyroid. Then, a double gyroid was
created by combining this single gyroid and its counterpart while satisfying
inversion symmetry, as shown in Figure 6a. The photonic band structure reveals
that the set of degeneracies between the forth and fifth bands form the nodal
ring, as shown in Figure 6b.

Very recently, a double diamond photonic crystal exhibiting nodal link, nodal
chain, and nodal lines all at once was reported [18]. When the lattice vectors
are given by ${\mathbf{a}}_{1}=a/2\left[0,1,1\right]$,
${\mathbf{a}}_{2}=a/2\left[1,0,1\right]$, and
${\mathbf{a}}_{1}=a/2\left[1,1,0\right]$ with the lattice constant $a$, the
double diamond is defined by a set $\mathbf{x}$ satisfying
$f\left(\pm\mathbf{x}\right)>f_{c}>0$. Each inequality with plus or minus sign
corresponds to each single diamond, so that the two single diamonds are
inversion symmetric. Here, the function $f\left(\mathbf{x}\right)$ is given by

$\displaystyle f\left(\mathbf{x}\right)$ $\displaystyle=$ $\displaystyle
A_{0}\operatorname{sin}\left(X_{1}+X_{2}+X_{3}\right)$ (37) $\displaystyle+$
$\displaystyle\sum_{i=1}^{3}A_{i}\operatorname{sin}\left(X_{1}+X_{2}+X_{3}-2X_{i}\right)$

where
$\mathbf{X}=\left[X_{1},X_{2},X_{3}\right]=\left(2\pi/a\right)\left(\mathbf{x}-\gamma/2\sum_{i=1}^{3}{\mathbf{a}}_{i}\right)$
is a local coordinate that expresses the translation of the single diamonds
along the $\pm\left[1,1,1\right]$-directions, adjusted by the coefficient
$\gamma$. Selecting $A_{0}=A_{1}=A_{2}=A_{3}$ and $\gamma=0$ generates the
conventional diamond structure [161, 162, 163, 164, 165]. However, this study
selected different coefficients $A_{i}$ and non-zero $\gamma$ (see Figure 6c)
to destroy as many symmetries as possible. The resulting structure is
anisotropic so that the double diamond has only inversion and translational
symmetries.

In momentum space the degeneracies between the first and second bands form the
nodal chain, the degeneracies between the third and fourth bands form the
simple nodal lines, and the degeneracies between the third, fourth, and fifth
bands form the nodal link (see Figure 6d). The nodal chain is centered at the
$\Gamma$-point. Two pink nodal lines (between the first and second bands)
depart the $\Gamma$-point and arrive on a single boundary. Another two pink
nodal lines are centrosymmetric to the first two pink nodal lines. The two
boundaries that these nodal lines touch are parallel, and their normal vectors
are commonly along the ${\mathbf{b}}_{2}$-direction. Thus, the nodal chain is
infinitely connected along the ${\mathbf{b}}_{2}$-direction.

Meanwhile, the nodal link consists of the non-touching orange (between the
third and fourth bands) and cyan rings (between the fourth and fifth bands),
as shown in Figure 6d. The cyan ring is centered at the $\Gamma$-point, and
the center of the orange ring is located on a boundary. Due to the periodicity
of the first Brillouin zone, this link is infinitely connected.

One more significance of this research [18] is that the correlation vectors in
Eq. (35) were first introduced. By employing the correlation vectors, the non-
Abelian topological nature of the nodal link could be directly calculated from
the full-vector field eigenstates, instead of using the $3\times 3$ effective
Hamiltonian, thus the topological charges
${\mathbb{Q}}=\left\\{\pm\bm{i},\pm\bm{j},\pm\bm{k},\pm 1\right\\}$ were
completely deduced from the numerically calculated nodal link.

A simple post-processing calibration was also proposed in this photonic study
[18]. To get the topological charge $\pm\bm{j}$, a closed loop that encloses
two nodal lines exhibiting the topological charges $\pm\bm{k}$ and $\pm\bm{i}$
should be considered. Although we illustrate the quaternion charge $\pm\bm{j}$
in Figure 3f, the eigenstates or correlations calculated by Figure 3a and b do
not generally show results as in Figure 3f. In many cases, they exhibit the
$\pi$-disclinations of ${\mathbf{u}}_{\mathbf{k}}^{3}$ and
${\mathbf{u}}_{\mathbf{k}}^{1}$ at ${\mathbf{k}}_{0}$ of the loop, implying
the topological charge $\pm\bm{j}$. Here, the $\pi$-disclinations mean that,
before and after winding along the loop, the directions of
${\mathbf{u}}_{\mathbf{k}}^{3}$ and ${\mathbf{u}}_{\mathbf{k}}^{1}$ are
rotated by $\pi$. To observe the rotation behaviors of the eigenstates or
correlations more clearly, this study [18] introduces a post-processing
calibration method. First, at a point $\mathbf{k}$, a rotation matrix
$\mathbf{R}\left(\mathbf{k}\right)$ is defined so that it maps
${\mathbf{u}}_{\mathbf{k}}^{2}$ to ${\mathbf{u}}_{{\mathbf{k}}_{0}}^{2}$
around
$\mathbf{r}\left(\mathbf{k}\right)={\mathbf{u}}_{\mathbf{k}}^{2}\times{\mathbf{u}}_{{\mathbf{k}}_{0}}^{2}$.
Then, ${\mathbf{u}}_{\mathbf{k}}^{1}$ and ${\mathbf{u}}_{\mathbf{k}}^{3}$ are
also rotated by $\mathbf{R}\left(\mathbf{k}\right)$. This process is done for
all $\mathbf{k}$ on the closed loop, to get a result like Figure 3f.

###  IV.3 Optical frequency synthetic dimension

As described in the previous sections, diverse topologies of nodal lines can
be created using metamaterials and photonic crystals because one can design a
periodic structure and tune geometric and optical parameters. However, the
freedom in photonic designs is not simply limited to a spatially periodic
system but one can extend the design freedom into the frequency domain based
on a new concept called ‘synthetic dimension’. Indeed, this can provide
another possibility to create more complex topologies. Very recently, K. Wang
et al. [166] showed that topologies such as unknots, Hopf links and trefoils
can be created using two coupled ring resonators and a dielectric waveguide by
modulating the phase and amplitude in one of the ring resonators. In their
work, the frequency synthetic dimension is created by multiple resonance
frequencies of the unperturbed ring resonators which are periodic in frequency
space and the complex energy spectra are obtained by measuring the detuning of
the resonance frequencies. By doing so, the wavevector-energy space
$(k,\mathrm{Re}(E),\mathrm{Im}(E))$ becomes the parameter space where the
optical bands with different topologies can exist. It is worth noting that the
topology of two bands is considered instead of nodal lines originating from
two different bands.

##  V Topological nodal lines in other systems

In this section, we review recent achievement in finding nodal lines in
electronic crystals, phononic crystals and electrical circuits. Historically,
the study on nodal lines in electronic crystals started earlier than all other
systems. Several review papers have been published on nodal line semimetals
[167, 168]. Therefore, in our review, we highlight a few important
achievements in electronic crystals.

Figure 7:  Nodal lines and nodal chains in electronic crystals. (a) Drumhead
surface states (green) bounded by the nodal line in Ca3P2. (b) Surface states
in Mg3Bi3 measured by ARPES. (c) Nodal chains. (d) nodal lines in IrF4.
Reproduced with permission from Ref. [85]. Copyright 2021, American Physical
Society (a); Reproduced with permission from Ref. [169]. Copyright 2019,
WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (b); Reproduced with permission
from Ref. [32]. Copyright 2016, Nature Portfolio (c); Reproduced with
permission from Ref. [17]. Copyright 2019, American Association for the
Advancement of Science (d).  Figure 8:  Nodal lines in phononic crystals. (a)
Schematic of a granular metamaterial toward phononic nodal chain. Four types
of interactions between two beads in the nearest-neighbors are also
illustrated. (b). Simulation results showing the nodal chain in momentum
space. (c) An illustration of the tight-binding Hamiltonian in Eq. (38) with
different values of $t$ and $\delta t$. (d) Photograph (left) and schematics
(right) of a phononic crystal exhibiting nodal rings. (e-f) Simulation and
experimental results showing nodal rings, respectively. Reproduced with
permission from Ref. [97]. Copyright 2019, Nature Portfolio (c-f); Reproduced
with permission from Ref. [89]. Copyright 2019, Nature Portfolio (c-f).

###  V.1 Electronic crystals

The discovery of nodal lines starts from the prediction of the cubic
antiperovskite material Cu3NX, where X={Ni; Cu; Pd; Ag; Cd}, as
${\mathbb{Z}}_{2}$ protected topological semimetals when ignoring spin-orbit
interaction [19, 20]. This material holds one-dimensional Dirac line nodes and
two-dimensional nearly-flat surface states, protected by $\mathcal{P}$ and
$\mathcal{T}$ symmetries. In particular, the 2D surface states are bounded by
the projected Dirac line nodes and because of this, they were called drumhead
states (Figure 7a) in the field afterwards. In the same work by Y. Kim et al.
[20], they showed that nearly flat surface states exist in Cu3NX.
Additionally, Y. H. Chan et al. showed that the drumhead surface states of
Ca3P2 exist due to a quantized Berry phase and the ${\mathbb{Z}}_{2}$
topological invariants were defined similarly as in strong topological
insulators [85]. It is worth noting that, before the surge of the search for
the drumhead edge states, a topologically protected flat band has drawn
attention because it can promote surface superconductivity with an infinite
density of states [170]. Recently, the drumhead surface states were also shown
in phononic crystals [89]. Y. Wang et al. analyzed the flatness and
boundedness of photonic drumhead surface states using a simple cubic lattice
of metals [21].

Soon after, other nodal line materials in the absence of spin-orbital
interaction were predicted, such as alkaline-earth compounds AX2 (A = Ca, Sr,
Ba; X = Si, Ge, Sn) [171], the CaP3 family of materials [172, 173], some of
which are experimentally demonstrated, such as CaCdSn [174] and Mg3Bi3 [169]
(Figure 7b). Furthermore, topological semimetals can be classified into two
types according to the tilting degree of the fermion cone. Type-II nodal lines
in CaPd [175] and Mg3Bi3 [169] were proposed. Finally, topological spinful
nodal lines were found to exist in TlTaSe2 [176], ZrSiSe and ZrSiTe [177] by
including spin-orbit interaction as well.

More recently, diverse topologies have been reported in electronic crystals.
For example, nodal chains were predicted in IrF4 (Figure 7c) [140] and nodal
links were shown in Sc [17]. Interestingly, non-Abelian topological invariants
(see Subsection III.5) were shown to exist in those multiple nodal line
structures and this work has inspired much additional research on non-Abelian
topology including the one in photonic crystals [43]. In addition, it was
recently reported that braiding happens in electrons with strain and phase
transitions [178, 179].

###  V.2 Phononic crystals

Phononic crystals, also known as acoustic crystals, are also a good platform
to demonstrate the topological physics because the couplings between meta-
atoms can be easily controlled and the displacement field of the modes can be
easily measured giving the full profile of the modes and band structure.

A nodal chain by the phononic wave was theoretically proposed using a granular
metamaterial [97]. For the beads consisting a face-centered cubic arrangement,
one can assume that tension, shear, bending, and torsion exist on all contacts
between any two nearest-neighbors grains, denotes as $K_{N}$, $K_{S}$,
$G_{B}$, and $G_{T}$, respectively, in Figure 8a. By applying these
assumptions to the linear equations of motion for each bead, the result in
Figure 8b clearly shows the nodal chain in momentum space.

W. Deng et al. proposed a phononic crystal that exhibits nodal rings in
momentum space [89]. The effective Hamiltonian for this realization is written
as

$H=d_{x}\sigma_{1}+d_{y}\sigma_{2}$ (38)

where

$d_{x}=-2\operatorname{cos}k_{x}-2\operatorname{cos}k_{y}-2t\operatorname{cos}k_{z}$
(39)

and

$d_{y}=-2\delta t\operatorname{sin}k_{z}$ (40)

are the sublattice pseudospins. For $t>0$ and $\delta t>0$, the hopping
amplitude in the $z$-direction is $-t\pm\delta t$ as illustrated in Figure 8c
by thin and thick vertical rods. The hopping in the $x$\- and $y$-directions
are $-1$ as represented in Figure 8c. The eigenvalue of Eq. (38) is
$E=\pm\sqrt{d_{x}^{2}+d_{y}^{2}}$, and the degeneracies are formed when
$d_{x}=d_{y}=0$. This indicates that the pseudospins are arranged as that
corresponds to nodal rings formed on the $k_{z}=n\pi$ plane ($n$ is an
integer).

For experimental observation of the nodal rings, they prepared a layer-
stacked-phononic crystal made of plastic stereolithography material. In the
three-dimensional structures shown in Figure 8d, there are several types of
holes, the propagation paths of sound waves. Along the $z$-direction, two
types of holes exist. The smaller and larger holes correspond to the thinner
and thicker rods in Figure 8c. Along the horizontal directions, the sound
waves meet the same types of rectangular holes, and this corresponds to the
same size of rods in Figure 8c. With this phononic crystal design, the nodal
rings could be observed in momentum space (see Figure 8e and f), as predicted
by Eq. (38).

Figure 9:  Experimental realization of Euler class using a phononic crystal.
(a) Schematics of a kagome tight-binding model with tunning parameters $t$ and
$t^{\prime}$. (b) Phononic band structure of (a). (c) Experimental design of
(a). (d) Simulation (white plots) and experimental band structure of (c).
Reproduced with permission from Ref. [124]. Copyright 2021, Nature Portfolio
(a-d).

Very recently, B. Jiang et al. used a two-dimensional phononic crystal to
observe non-Abelian topological charges and topological phase transitions
[124]. Although this work is not directly related to nodal lines in three-
dimensional momentum space, this work has its significance in experimental
demonstration of the non-Abelian phononic nodes (degenerate point in two-
dimensional momentum space) and the patch Euler class mentioned in Section
III.8. They employed a tight-binding model of a kagome lattice as shown in
Figure 9a. The hopping between lattice points A, B, and C can be adjusted by
$t$ and $t^{\prime}$. Band structures by tuning these two variables exhibit
several types of non-Abelian charges and Euler classes, as shown in Figure 9b.
Then, the tight-binding model is realized using cylindrical acoustic
resonators as shown in Figure 9c to observe topological phase transitions and
the new topological invariant ($-1$ of the Euler class) (see Figure 9d) .

Figure 10:  (a) Mathematical models construct a nodal knot/link from a braid.
A braid closure can be embedded onto the three-dimensional Brillouin zone
torus differently through different choices of $F\left(\mathbf{k}\right)$
functions. Depending on its topological charge density distribution, it
produces different numbers of nodal knots in the Brillouin zone, i.e. either a
single copy ($F_{1}$) or two copies ($F_{2}$) related by mirror symmetry. (b)
Experimental setup for impedance measurement of the Hopf-link circuit.
Reproduced with permission from Ref. [27]. Copyright 2020, Nature Portfolio
(a-b).  Figure 11:  Illustration of different constituents of the non-
Hermitian trefoil knot circuit. (a) On-site hopping and nearest-neighbor
hopping along the $x$ direction, (b) next-nearest-neighbor hopping along the
$x$ direction, (c) nearest-neighbor hopping along the $y$ direction, (d)
nearest-neighbor hopping along the $z$ direction, (e) next-nearest-neighbor
hopping along the $y$ direction, and (f) diagonal hopping within the $x-y$
plane. Reproduced with permission from Ref. [180]. Copyright 2021, Nature
Portfolio (a-f).

###  V.3 Electrical circuits

Knots, such as everyday-life ropes are intricate nodal lines. They are
difficult to construct because they require finely tuned long-ranged hoppings.
To realize models with those long-ranged hoppings, it is naturally suggested
to use artificial structures, which allow for unprecedented control over
individual couplings [27]. Most importantly, electrical circuits, whose
connections transcend locality and dimensionality constraints, put the
implementation of couplings between distant sites of a high-dimensional system
and nearest-neighbor connections on equally accessible footing. This advantage
is found to be crucial to the realization of nodal knots, which contains many
strong long-ranged hoppings.

Having explained the necessity to implement nodal knots using electrical
circuits, we now describe how they can be concretely implemented and detected.
An electrical $RLC$ circuit is an undirected network with nodes
$\alpha=1,\cdots,N$ connected by resistors, inductors, capacitors or
combinations of them. Its behavior can be fully characterized by Kirchhoff’s
law as $I_{\alpha}=J_{\alpha\beta}V_{\beta}$, where $I_{\alpha}$ is the
external current at junction $\alpha$ and $V_{\beta}$ is the voltage at
junction $\beta$. Each entry $J_{\alpha\beta}$ of the Laplacian $J$
contributes $r_{ab}$ to the Laplacian, where $r_{ab}=R,i\omega L,(i\omega
C)^{-1}$ for a single resistor, inductor and capacitor and whose combinations
follows. The strictly reciprocal (symmetric) form of $J_{\alpha\beta}$
constrains the possible Laplacian, which prevents nodal knot circuits from
being developed using mathematical models of nodal knots proposed before,
since those imply broken reciprocity. Thus in order to construct nodal knot
circuits, new models which preserve reciprocity need to be invented [27].

A very recent work discovered a method to overcome this obstacle, which goes
beyond existing approaches that requires broken reciprocity, by pairing nodal
knots with their mirror images to realize pairs of nodal knots in a fully
reciprocal setting [27]. The key insight is that pairs of nodal knots can
preserve reciprocity while a single knot cannot, such that they can be
realized in a circuit as shown in Figure 10a. A highlight of this work is the
experimental verification of surface drumhead states in a design of the nodal
Hopf link circuit shown in Figure 10b. This experimental setting is used to
extract the admittance band structure through linearly independent
measurements, whereas the number of measurements needed is the same as the
number of inequivalent nodes $N$. Each step consists of a local excitation in
this circuit and a global measurement of the voltage response from which one
can extract all components of the Laplacian in reciprocal space. By
diagonalizing the Laplacian $J(k)$, the admittance band structure can be
found, reflecting that the Laplacian plays the same role as the Hamiltonian in
an electronic material.

There are other advantages that we have not mentioned so far in using
electronic circuits to construct nodal knot as well as other nodal lines. Take
the non-Hermitian nodal knots as an example, the positive, negative and non-
reciprocal couplings needed for non-Hermitian nodal knots can be realized
through carefully chosen combinations of $RLC$ components and operation
amplifiers, which can introduce non-reciprocal feedback needed for the skin
effect [180] that $RLC$ components can not. By active elements in those
circuits we can realize non-Hermitian models straightforwardly in electrical
circuits, given the non-Hermitian trefoil circuit as an example in Figure 11,
which reflects the flexibility of topolectrical circuits. In those circuits,
topological zero modes manifest themselves through a divergent impedance which
we call topolectrical resonances. Finally, but not least, electronic circuits
provide possibilities to simulate nodal lines and drumhead states beyond three
dimensions, in an analogous way as simulating topological insulators in Class
AI with electronic circuits in four dimensions [181, 182].

##  VI Conclusions and outlook

To summarize, we have explained the concept of nodal lines in band structures
and showed that a wide range of types from a simple nodal line, a nodal link
and a chain to mixed nodal lines can exist. Indeed, the extension from zero-
dimensional degeneracy allows for a wider range of topology for the
degeneracies in band structures of particles/waves including electrons,
photons and phonons. Then, we have reviewed the theoretical description of the
topological invariants of nodal lines with the Berry phase and the Wilczek-Zee
connection based on the two seminal papers by M. V. Berry [36] and Wilczek and
Zee [99]. This provides us with an essential toolset to describe the topology
of nodal lines. Using the Wilczek-Zee connection, we explained how non-Abelian
topology can be considered in a three-band system, following the recent work
on non-Abelian topology by Q. Wu et al. [17].

Finally, we have reviewed recent advances in implementing nodal lines using
metamaterials and photonic crystals. These two photonic systems have
successfully demonstrated those exotic behaviors and they remain very
promising in finding new topological states related to nodal lines. However,
they are not the only systems and they also benefit from the earlier work in
other fields. To supplement, we have introduced the examples of nodal lines in
electronic crystals, phononic crystals, and electrical circuits. Moreover,
non-Hermitian systems, which have complex energy eigenvalues due to the
exchange of energy with the environment, can be used to extend the dimension
of parameters space. Indeed, new non-Hermitian systems with optical ring
resonators and RLC circuits are being introduced to implement more complex
topology of nodal lines such as knots and we expect many undiscovered
topological nodal lines will be implemented soon with these new approaches.

However, there are still challenges in understanding the physical consequence
of topological phase of nodal lines and also implementing the proposed ideas
in artificial materials. First, no bulk-edge/surface correspondence for the
non-Abelian charges has been mathematically proven [124] although one can
naturally think that there could be a relation similar to the bulk-
edge/surface correspondence in Chern insulators, which is Abelian. To answer
this question, more in-depth theoretical investigation on multiple nodal lines
systems is required. Second, the refractive index required to have nodal links
and other topology using dielectric photonic crystals is too high, normally
requiring a refractive index higher than 3.5 [43]. This limits the choices of
materials in the microwave range, adding another challenge in fabrication. For
example, L. Lu et al. used a ceramic material with a high refractive index to
observe Weyl points experimentally, and they drilled the material in different
directions to prepare a Weyl photonic crystal [8]. Third, the fabrication of
nanostructures for dielectric photonic crystals working at optical wavelengths
can be challenging. As the nodal lines are normally implemented in three-
dimensional momentum space that requires a three-dimensional array of high
index dielectric materials. A self-assembly method using block copolymers [44]
can be used but it requires the additional step of inserting high-refractive
index materials and it is hard to control the local geometrical perturbations
as we want. A direct-laser writing or other polymerization methods can be used
but they still require an additional step to add high-refractive index
material. However, these challenges could be overcome in a few years
considering the rapid progress in nanofabrication techniques.

There are also unexplored areas in relation to nodal line physics in photonic
systems. First, 3D nanoplasmonic systems can be used to implement and observe
nodal lines of surface plasmon polaritons. In 2D topological photonics, there
is already a theoretical study that shows unidirectional propagation and
corner states in 2D metallic arrays [183]. However, due to the difficulty in
fabricating 3D plasmonic structures, no nodal lines in 3D plasmonic band
structures have been observed yet. Second, quantum optic systems can be used
to implement the nodal line. An array of coupled quantum emitters has been
studied in 2D topological photonics structures [184]. If we extend the system
to 3D structures, 3D exciton polaritonic systems could be a good platform to
study the topology of nodal lines. Recently, also an array of dipolar
molecules in 3D optical lattices has been proposed to implement Hopf
insulators [185].

The remaining question is what applications could be enabled using nodal
lines. There are very few reports or proposals regarding practical
applications and it is hard to discuss a general way of applying nodal lines.
For electronic crystals with nodal lines, high-temperature surface
superconductivity using drumhead surface states [170] has been one major
motivation of study of nodal lines. Additionally, applications for surface
ferromagnetism [186] and high harmonic generation [187] have been
theoretically proposed. For photonic applications, high density of states of
surface states in nodal lines systems [21, 22] are expected to enhance
spontaneous emission, resonant scattering, nonlinearities and blackbody
radiation. The bound states in the continuum related to nodal lines [31] may
find applications in lasing [188] and sensing [189]. Although direct
applications seem limited for the moment, a better understanding of
topological phases of nodal lines would give us rich knowledge and interesting
physics of artificially designed materials system as well as electronic
materials. We believe this would open a new avenue to exciting applications as
well as expand our human knowledge.

###### Acknowledgements.

The authors thank Robert-Jan Slager, Andreas Pusch and Stephan Wong for
critical reading and comments. The work is part-funded by the European
Regional Development Fund through the Welsh Government (80762- CU145 (East)).
This research is also supported by the National Natural Science Foundation of
China (Grant No.11874431), the National Key R&D Program of China (Grant No.
2018YFA0306800), and the Guangdong Science and Technology Innovation Youth
Talent Program (Grant No.2016TQ03X688).

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