# Knot Topology of Exceptional Point and Non-Hermitian No-Go Theorem

Haiping Hu<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  Shikang Sun  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Shu Chen  Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Yangtze River Delta Physics Research Center,
Liyang, Jiangsu 213300, China

###### Abstract

Exceptional points (EPs) are peculiar band singularities and play a vital role
in a rich array of unusual optical phenomena and non-Hermitian band theory. In
this paper, we provide a topological classification of isolated EPs based on
homotopy theory. In particular, the classification indicates that an $n$-th
order EP in two dimensions is fully characterized by the braid group Bn, with
its eigenenergies tied up into a geometric knot along a closed path enclosing
the EP. The quantized discriminant invariant of the EP is the writhe of the
knot. The knot crossing number gives the number of bulk Fermi arcs emanating
from each EP. Furthermore, we put forward a non-Hermitian no-go theorem, which
governs the possible configurations of EPs and their splitting rules on a two-
dimensional lattice and goes beyond the previous fermion doubling theorem. We
present a simple algorithm generating the non-Hermitian Hamiltonian with a
prescribed knot. Our framework constitutes a systematic topological
classification of the EPs and paves the way towards exploring the intriguing
phenomena related to the enigmatic non-Hermitian band degeneracy.

Introduction. Non-Hermitian (NH) systems are ubiquitous in physics [1, 2, 3,
4, 5, 6, 7, 8, 9], as epitomized by various photonic platforms with gain and
loss [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 4, 25, 26,
27]. One of the most remarkable features of NH systems is that they exhibit
level degeneracy of their complex eigenenergies, called exceptional points
(EPs) [28, 29]. At an EP, two or more eigenvalues, and their corresponding
eigenvectors, simultaneously coalesce (i.e., the Hamiltonian is defective),
giving rise to phenomena without any analogs in the Hermitian realm. The
intriguing properties of EPs have been widely exploited such as in
unconventional transmission or reflection [30], enhancing sensing [31, 32,
33], single-mode lasing [34, 35] and non-reciprocal phase transitions [36].

EPs can be categorized into different types [37]. An EP is of $n$-th order or
$n$-fold (EPn) if $n$ eigenstates simultaneously coalescence, i.e., the
Hamiltonian is diagonalized into an $n\times n$ Jordan normal form. Similar to
the well-known Dirac point or Weyl point in Hermitian systems, EP can be
characterized by assigning an integer invariant like the vorticity of
eigenvalues [38] or discriminant number [39]. Despite the explosive
theoretical and experimental researches of exceptional degeneracies during the
past few years, a comprehensive understanding of their exotic features and
topological classification has been achieved yet. The characterization using
discriminant number is far from satisfactory and incomplete. First, EPs of
different types (especially of higher-order) that cannot smoothly evolve from
one to another may have the same integer invariant introduced before. Second,
NH Hamiltonians generally have complex eigenenergies compared to Hermitian
systems, bringing critical differences in their topological characterizations
[40, 41, 42, 43, 44]. Near an EP, the energy levels may braid and tangle
together and a full description of EPs requires the information of their
nearby braiding patterns and branch cuts, hence they cannot be captured solely
by simple integer invariants. Moreover, the richness of the band braiding near
an EP may induce novel types of EPs previously unidentified.

In this paper, we formulate a homotopic classification of EPs in arbitrary
dimensional NH systems (or generic parameter space), which enables the
calculation of topological invariants using the machinery well-developed in
algebraic topology. Particularly in 2D, our main findings are: i) An isolated
$n$-th order EP is fully characterized by braid group Bn, which reduces to
integer group $\mathbb{Z}$ for EP2. Our classification reveals the existence
of infinitely many kinds of EPs which one-to-one correspond to the geometry
knots. ii) We demonstrate that the quantized vorticity (or discriminant
number) and the number of bulk Fermi arcs emanating from the EP is the writhe
and crossing number of the knot, respectively. iii) Based on the
classification, we put forward a NH no-go theorem that goes beyond the
previous fermion doubling theorem and dictates the possible configurations of
EPs and their splitting rules on a 2D lattice. iv) To facilitate experimental
realizations, we show how to generate a NH Hamiltonian hosting an EP of a
prescribed knot pattern. Our framework bridging the NH physics, algebraic
topology and knot theory, manifests the beauty and diversity of EPs and opens
a broad avenue for investigating the exotic features of NH band degeneracies.

Figure 1: Exceptional points (EPs) with energy-level braiding and knot
topology. (a) Braid diagram marked by Artin’s braid word notation: $\tau_{i}$
($\tau_{i}^{-1}$) represents the $i$-th strand crosses over/under the $i+1$-th
strand when traveling upwards. Closure of the braid by identifying the up and
bottom ends forms a knot (in this case, it is the figure-8 knot). (b)-(e) Band
structures (real part) and their associated knots along a closed path
enclosing the EP (lower panels) for some representative EPs. (b) EP2 of
unknot, braid word: $\tau_{1}$, the characteristic polynomial (ChP) is
$\lambda^{2}-z$, with $z=k_{x}+ik_{y}$; (c) EP2 of trefoil knot, braid word:
$\tau_{1}^{3}$, the ChP is $\lambda^{2}-z^{3}$; (d) EP3 of figure-8 knot,
braid word $(\tau_{1}^{-1}\tau_{2})^{2}$. The ChP for figure-8 knot is given
by $f_{F8}=1/64[64\lambda^{3}-12\lambda
a^{2}[3(z\bar{z})^{2}+2z^{2}(z\bar{z})-2\bar{z}^{2}(z\bar{z})]-14a^{3}(z^{2}+\bar{z}^{2})(z\bar{z})^{2}-a^{3}(z^{4}-\bar{z}^{4})(z\bar{z})]$
with $a=1/4$; (e) EP4 of $L6a1$ link, braid word:
$(\tau_{1}^{-1}\tau_{3}^{-1}\tau_{2})^{2}$.

Homotopy classification. Let us consider an $n$-band NH Hamiltonian
$H(\bm{k})=H(k_{1},k_{2},...,k_{d})$ of $d$D system (or parameter space) and
suppose an isolated $n$-fold EP located at $\bm{k}=0$. At the EP, the
Hamiltonian can be brought to the Jordan normal form (For simplicity, the EP
is set at zero energy unless otherwise noted). Deviating from the EP, the
energy levels are separated from each other. We denote the eigenvalues and
corresponding eigenvectors of $H(\bm{k})$ as
$X_{n}=(z_{1},z_{2},...,z_{n};\psi_{1},\psi_{2},...,\psi_{n})$. As an isolated
EP in $d$D can be enclosed by a $(d-1)$D sphere $S^{d-1}$. From the homotopy
point of view, classifying the band structures near the EP is to find all the
nonequivalent classes of the non-based maps from $S^{d-1}$ to $X_{n}$, denoted
as $[S^{d-1},X_{n}]$. While our main focus here is for isolated EPs, the
classification scheme can be directly extended to some other kinds of
exceptional degeneracies [45, 46, 47, 48, 49] like exceptional lines in 3D.
For generic exceptional band degeneracy of dimension $d_{E}$ ($d_{E}=0,1,2$
corresponds to exceptional point/line/surface, respectively), the topology can
be revealed by taking a nearby surface of dimension $D_{s}=d-d_{E}-1$, with
the topological invariants carried by the homotopy invariants
$[S^{D_{s}},X_{n}]$.

The eigenvalue part is the configuration space of ordered $n$-tuples
$\textrm{Conf}_{n}(\mathbb{C})$. As each eigenvector is unique up to
multiplying a nonzero phase factor, the eigenvector space (denoted as $\Psi$)
is $\Psi\cong\textrm{U}(n)/\textrm{U}^{n}(1)$ [49]. By further removing the
redundancy of permutation of eigenvalues (together with their associated
eigenvectors), we have the classifying space
$X_{n}\cong(\textrm{Conf}_{n}\times\Psi)/S_{n}$, with $S_{n}$ the symmetric
group of degree $n$. Using the relations
$\pi_{1}(\textrm{Conf}_{n})=\textrm{PB}_{n}$ (pure braiding group) [50, 51]
and $\pi_{m}(\textrm{Conf}_{n})=0$ ($m\geq 2$) [52], and long sequence of
homotopy relations [53, 40, 41], we have:

$\displaystyle\pi_{1}(X_{n})=\textrm{B}_{n},~{}~{}~{}d=2;$ (1)
$\displaystyle\pi_{2}(X_{n})=\mathbb{Z}^{n-1},~{}~{}~{}d=3;$ (2)
$\displaystyle\pi_{d-1}(X_{n})=\pi_{d-1}(\textrm{U}(n)),~{}~{}~{}d\geq 4.$ (3)

For $d=2,n=2$, the homotopy group $\textrm{B}_{2}\cong\mathbb{Z}$, consistent
with the previous characterization of EP2 through vorticity [38] or
discriminant number [39]. The above homotopy groups are all Abelian except for
$d=2,n\geq 3$, which is the braid group $\textrm{B}_{n}$. If we travel along a
circle around a 2D EP, the energy levels braid together. Eq. (1) indicates
that the topology of a 2D EPn is entirely captured by such braiding pattern.
The ordinary non-defective point degeneracy has trivial band braidings. Eq.
(2) means a 3D EPn is characterized by assigning a Chern number to each of the
separable $n$ bands, with the constraints that they sum to zero.

The non-based map relates to the homotopy group through the action of the
fundamental group [53]:

$\displaystyle[S^{d-1},X_{n}]\cong\pi_{d-1}(X_{n})/\pi_{1}(X_{n}).$ (4)

The right-hand side is the orbit set. $[S^{d-1},X_{n}]$ is not necessary a
group and usually decomposed into several distinct sectors, induced by the
fundamental group. While the non-based homotopy can be worked out in a case-
by-case manner, in 2D, $[S^{1},X_{n}]$ is the conjugacy class of the braid
group Bn [40, 41, 42]. In fact, choosing a different starting point on the
encircling path may end up with another braid, which however, conjugate to the
original one, hence they are in the same conjugacy class. In 3D,
$[S^{2},X_{n}]$ is a collection of $n$ integer Chern numbers. Due to the un-
sortability of the complex energy levels, the two integer sets
$[s_{1},s_{2},...,s_{n}]$ and $[s_{b_{1}},s_{b_{2}},...,s_{b_{n}}]$ (the set
induced by the band permutation of any $b\in\pi_{1}(X_{n})$) are identified as
the same [40, 41].

Knot around EP. In the following, we focus on 2D EPs. It turns out there is a
one-to-one correspondence between the conjugacy class of braid group
$\textrm{B}_{n}$ and geometric knot in the solid torus [54], culminating in a
knot classification of the EPs. Different types of EPs are represented by
topologically distinct knots, thus characterized by knot invariants [51, 42],
e.g., Jones polynomials [55]. Intuitively, let us take a closed path
$\Gamma(\theta)~{}(\theta\in[0,2\pi])$ surrounding the EP. Along $\Gamma$, the
$\theta$-dependent energy levels $(z_{1},z_{2},...,z_{n})$ trace $n$ strands
which tangle together in the 3D space spanned by (ReE, ImE, $\theta$), as
depicted in Fig. 1(a). The trajectory returns to itself to form a closed knot
during the $\theta$-evolution from $0$ to $2\pi$ ($\theta=0$ and $\theta=2\pi$
is identical). The knot topology of a given EP is best represented using braid
word. It can be determined by projecting the above energy-level strings onto
the $\textrm{Im}E=0$ plane. As $\theta$ evolves, the strings undergo a
sequence of crossings. In Artin’s notation, we label a crossing as $\tau_{i}$
($\tau_{i}^{-1}$) if the $i$-th string crosses over (under) the $(i+1)$-th
string from left. $\tau_{i}$ satisfy the braid relation:
$\tau_{i}\tau_{j}=\tau_{j}\tau_{i}$ for $|j-i|\geq 2$, and
$\tau_{i}\tau_{i+1}\tau_{i}=\tau_{i+1}\tau_{i}\tau_{i+1}$. The entire level
set is specified by a product of braid operators. For example, $\tau_{1}^{n}$
corresponds to twisting two level strands $n$ times; $n=1,2,3$ represents
unknot, Hopf link, and Trefoil knot, respectively. We note the difference from
the 1D knotted separable bands [42], here the knots of EPs are attributed to
the nontrivial nearby braiding around the band singularities.

The knot near an EP can be regarded as the roots (or nodal set) of the
characteristic polynomial (ChP). Denote $z=k_{1}+ik_{2}$ and
$H(z)=H(k_{1},k_{2})$, the ChP reads

$\displaystyle
f(\lambda,z)=\det[\lambda-H(z)]=\textstyle{\prod}_{j=1}^{n}(\lambda-z_{j}).$
(5)

The well-known complex polynomial for the $(p,q)$-torus knot is
$f(\lambda,z)=\lambda^{p}-z^{q}$, with $p$ roots
$z_{j}=z^{q/p}e^{i2\pi(j-1)/p},~{}(j=1,2,...,p)$. By tracing a closed path
around the EP, the energy levels wind $q$ times around a circle in the
interior of the torus, and $p$ times around its axis of rotational symmetry.
The simplest Hamiltonian realizing the ($p$,$q$)-torus knot can thus be chosen
as:

$\displaystyle
H_{T_{p,q}}=\left(\begin{array}[]{cccc}0&0&0&(k_{1}+ik_{2})^{q}\\\ 1&0&0&0\\\
0&\ddots&\ddots&0\\\ 0&0&1&0\end{array}\right)_{p\times p}.$ (10)

Figs. 1(b)(c) plot the band structures of EP2 associated with $T_{2,1}$
(unknot) and $T_{2,3}$ (trefoil knot), respectively. The unknot case is the
most studied EP2 in the literature.

To find the Hamiltonian for an EPn with a prescribed knot (denoted as
$\mathcal{K}$), we need to construct the proper ChP $f(\lambda,z)$. As
$\lambda,z\in\mathbb{C}$, $f(\lambda,z)$ defines a mapping from
$\mathbb{C}^{2}$ to $\mathbb{C}$. The energy levels $(z_{1},z_{2},...,z_{n})$
are the nodal sets of the ChP and the EPn is the isolated singularity located
at $(\lambda,z)=(0,0)$. Mathematically, the closure of the band braiding near
the EPn is the algebraic knot around the singular point. Akbulut and King [56]
showed (albeit in a nonconstructive way) that any knot can arise as the knot
around such a singular point of a polynomial in the context of complex
hypersurfaces. As elaborated in Ref. [49], the desired Hamiltonian with the
desired knotted EP pattern can be intuitively constructed from the Fourier
parameterization of the associated braid diagram. Fig. 1(d) depicts an EP3
with its nearby energy levels forming a figure-8 knot, described by braid word
$(\tau_{1}^{-1}\tau_{2})^{2}$. Fig. 1(e) depicts an EP4 associated with $L6a1$
link of braid word $(\tau_{1}^{-1}\tau_{3}^{-1}\tau_{2})^{2}$. By further
replacing $k_{1,2}$ with $\sin k_{x,y}$ in $H_{\mathcal{K}}(z)$, we get a 2D
lattice Hamiltonian hosting the desired EPn with $\mathcal{K}$ knot topology.

Bulk Fermi arc & discriminant number. A direct physical consequence of the EP
topology is the appearance of bulk Fermi arcs [57]. For example, a pair of EPs
split from a Dirac point is connected by an open-ended Fermi arc, as
identified as the isofrequency contour in photonic experiments [57]. Here we
extend this notion to generic higher-order EPs and define the bulk Fermi arcs
as the loci in the ($k_{1}$, $k_{2}$) space when any two levels have the same
real energy, i.e., Re[$z_{i}$]=Re[$z_{j}$] for $i\neq j$. As the EP satisfies
this condition, a bulk Fermi arc must start from an EP and end at another; The
EPs are the endpoints of Fermi arcs. The constraint imposed by the condition
Re[$z_{i}$]=Re[$z_{j}$] yields some 1D loci in the $(k_{1},k_{2})$ space. The
bulk Fermi arcs trace open-ended curves emanating from the EPs. From the braid
representation, the bulk Fermi arc is formed whenever there is a crossing,
either over or under, in the braid diagram. Hence we have:

$\displaystyle n_{\textrm{arc}}=n_{+}+n_{-}=c(\mathcal{K}).$ (11)

Here $n_{\pm}$ denotes the number of over/under crossings, respectively.
$c(\mathcal{K})$ is the knot crossing number. The number of Fermi arcs
$n_{\textrm{arc}}$ emanating from each EP equals to $c(\mathcal{K})$
associated with the EP.

Moreover, the braid crossing dictates the discriminant invariant $\nu$ [39].
In its neat form, $\nu$ is the sum of band vorticity [38] for any pair of
band:

$\displaystyle\nu=-\frac{1}{2\pi}\sum_{i\neq
j}\oint_{\Gamma}\nabla_{\bm{k}}\arg(z_{i}-z_{j})\cdot d\bm{k}.$ (12)

Along the closed path $\Gamma$, the EP knot contributes to the vorticity
through braid crossings. An over/under crossing contributes a $\pm 1$,
yielding

$\displaystyle\nu=n_{+}-n_{-}=W(\mathcal{K}),$ (13)

where $W(\mathcal{K})\equiv n_{+}-n_{-}$ is the writhe of the knot
$\mathcal{K}$. It is obvious two different EPs of different braids may have
the same discriminant number, e.g., figure-8 and Borromean ring ($L6a4$). We
stress that, the full information of an EP is encoded in its knot structure;
either the bulk Fermi arc or discriminant number is determined by the braiding
and has limited discriminant power to specify an EP.

No-go theorem. The knot classification assigns a specific braiding pattern to
a given EP. On a 2D lattice, there may exist multiple different EPs featuring
distinct braidings. So what kind of EP configurations is allowed? It turns out
the possible EPs on a 2D lattice are governed by the following no-go theorem:

$\displaystyle b_{1}b_{2}...b_{J}\in[\textrm{B}_{n},\textrm{B}_{n}],$ (14)

where $J$ is the total number of isolated EPs in the 2D BZ [see Fig. 2(a)],
$b_{j}$ is the braid for the $j$-th EP along a small surrounding path.
$[\textrm{B}_{n},\textrm{B}_{n}]=\\{uvu^{-1}v^{-1}|(u,v)\in\textrm{B}_{n}\\}$
is the commutator subgroup. It is easy to check taking a conjugate element of
any $b_{i}$ or switching its order in Eq. (14) is irrelevant [58]. We can
continuously expand and deform the enclosing path to the Brillouin zone (BZ)
boundary as sketched in Fig. 2(a), without passing through any singularity.
The resulting overall band braiding $b_{1}b_{2}...b_{J}$ equals to the band
braiding $cdc^{-1}d^{-1}$ along the four edges of the BZ as the upper (left)
and lower (right) edge are identical. Specifically, if the two neighboring
braidings $c,d$ (or two braidings along any orthogonal line cut of the BZ)
commute, the composite braiding must be trivial
$b_{1}b_{2}...b_{J}=\textrm{I}$.

The no-go theorem indicates that the braiding of the EP inside the BZ is
determined by that of the boundary. It dictates the possible braiding
configurations of the isolated EPs and imposes strong constraints on their
splittings: the no-go theorem holds for all the isolated split EPs, e.g., a
Dirac point splits into two EPs of opposite charges. Note that the commutator
subgroup is the set of braids with total exponent sum zero in the braid
generator $\tau_{i}$. According to Eq. (13), the doubling theorem [39]
naturally arises: the sum of the discriminant number of all EPs vanishes.
Hence a single EP of non-zero discriminant invariant must be paired with
another one of opposite discriminant number.

Figure 2: Non-Hermitian no-go theorem on a 2D lattice. (a) Sketch of the proof
of the theorem. The $j$-th EP ($j=1,2,...,J$) is labeled by level braiding
$b_{j}$ along the nearby path (cyan). The paths can be continuously deformed
to the orange curves and finally to the BZ boundary. (b) Band structure of
model (17). $m=0.3$. (c) EPs and their associated Fermi arcs. The EP3 of
figure-8 knot (red dot) at the origin connects with the four neighboring
2-fold EPs (green dots) through bulk Fermi arcs (green lines). (d) Knots
formed along the red path $\Gamma_{1}$ (left panel) and purple path
$\Gamma_{2}$ (right panel) in (c).

We illustrate the no-go theorem through an explicit model:

$\displaystyle H=H_{F8}+m\left(\begin{array}[]{cc}c_{x,y}&0\\\
1-c_{x,y}&-c_{x,y}\end{array}\right)\oplus 0_{1\times 1},$ (17)

with $c_{x,y}=2-\cos k_{x}-\cos k_{y}$. Here $H_{F8}$ is the $3\times 3$
lattice Hamiltonian [49] of the figure-8 EP. Without the second term, $H_{F8}$
hosts four EP3 located at $(0,0)$, $(0,\pi)$, $(\pi,0)$, and $(\pi,\pi)$,
respectively. For a finite $m$, only the EP3 at $(0,0)$ survives. A typical
band structure is depicted in Figs. 2(b)(c) for $m=0.3$. Besides the EP3 of
figure-8 knot with braid word $(\tau_{1}^{-1}\tau_{2})^{2}$ at the origin,
four isolated 2-fold EPs, with braid word $\tau_{1}$, $\tau_{2}^{-1}$,
$\tau_{1}$, and $\tau_{2}^{-1}$, respectively, emerge nearby. Each EP2
connects to the central EP3 through a bulk Fermi arc. The number of Fermi arc
emanating from each EP coincides with the knot crossing number [see Eq. (11)].
The knot along a closed path is shown in Fig. 2(d). If the loop is small,
enclosing only the central EP3, the energy levels close to a figure-8 knot. In
contrast, if the loop encloses all the EPs, the knot is composed of three
unlinked components (i.e., trivial braiding) since
$\tau_{2}^{-1}\tau_{1}(\tau_{1}^{-1}\tau_{2})^{2}\tau_{2}^{-1}\tau_{1}=\textrm{I}$,
consistent with the no-go theorem.

Discussions. To conclude, we have established a homotopy classification of
isolated EPs and demonstrated that the knots tied up by energy levels fully
characterize the 2D EPs. Based on this new classification scheme, we have
demonstrated how to construct NH Hamiltonians corresponding to a given knot.
We further proposed a no-go theorem for the EPs on a 2D lattice. Our scheme
elegantly relates the bulk Fermi arc and discriminant number to the crossing
number and writhe of the knot, respectively. It is worth to mention that the
knot topology we presented refers to the knotted structures intrinsic to non-
Hermitian EPs from the homotopy perspective, which should not be confused with
the usual topological phase in the presence of a well-defined chemical
potential. Transitions between different kinds of knotted EPs must be through
band touching and rearranging.

The various EP knots and their associated NH Hamiltonians could in principle
be realized in platforms such as photonic lattice [25, 59, 60] or electric
circuits [61, 62, 63, 64, 65]. For the former, the asymmetric coupling between
the ring resonators can be implemented via auxiliary microring cavities. The
consequent bulk Fermi arcs should be extracted through resonances in
frequencies [57]. For the latter, the NH Hamiltonian is simulated by the
circuit Laplacian, with its band structures given by the admittance spectra.
We note that the model Hamiltonian in Eq. (17) no-go theorem is merely chosen
for illustrative purpose of the no-go theorem and the figure-8 EP therein
requires the fine tuning of parameters. However, the stability of such knotted
EPs can in fact, be ensured by e.g., PT, CP, pseudo-Hermiticity, pseudo-
chirality symmetries [66, 67] due to the symmetry reduction of constraints of
EPs. Under a symmetry-breaking perturbation, the knotted EP may locally split
into other types of band singularities; however, the geometry knot formed by
encircling the overall band singularities after splitting remains unchanged,
as dictated by the no-go theorem. The various knotted EPs discovered here
should naturally emerge as the critical point of the nonreciprocal phase
transition [36]. Beyond the isolated EPs considered here, more intricate line-
or surface-degeneracies may exist. For example, in 2D, encircling the EP line
yields the so-called singular knots. It would be interesting to extend the
analysis to such singular knots and reveal their physical consequences.

###### Acknowledgements.

This work is supported by the NSFC under Grants No.11974413 and the Strategic
Priority Research Program of Chinese Academy of Sciences under Grant No.
XDB33000000.

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##  Appendix A Supplementary Materials

In this Supplementary Material, we provide details on (I) the classification
scheme of exceptional degeneracy from homotopy perspective; (II) the homotopy
equivalence between space $U(n)/U^{n}(1)$ and $GL(n,\mathbb{C})/C^{*n}$; and
(III) the construction of the characteristic polynomial and its associated
Hamiltonian corresponding to a prescribed knotted exceptional point (EP).

###  A.1 (I) Classification scheme of exceptional degeneracy from homotopy
perspective

The classification scheme for isolated EPs can be extended to more generic
exceptional degeneracies like exceptional lines and surfaces. Intuitively, the
various types of exceptional degeneracies can be regarded as topological
“defects” in the parameter space. The exceptional point/line/surface is taken
as point/line/surface defect, respectively. In a similar vein as the
characterization of defects in topological insulators [68], the strategy here
is to take a small nearby surface enclosing the considered exceptional
degeneracy and to seek the topological invariants on the enclosing surface.
Let us suppose a $d_{E}$-dimensional exceptional degeneracy ($d_{E}=0,1,2$
corresponds to exceptional point/line/surface) in $d$ dimensional parameter
space. The enclosing surface is then chosen as $D_{s}=d-d_{E}-1$, and the
exceptional degeneracy is governed by the homotopy invariant

$\displaystyle[S^{D_{s}},X_{n}],$ (18)

i.e., the nonequivalent class of the non-based maps from $S^{D_{s}}$ to
$X_{n}$. Here $S^{D_{s}}$ is the $D_{s}$-sphere, $X_{n}$ is the space of
$n$-band non-Hermitian Hamiltonians as defined in the main text. It is clear
that an isolated EP ($d_{E}=0$) is characterized by the homotopy invariant
$[S^{d-1},X_{n}]$.

Figure 3: Sketch of some typical exceptional degeneracies and their homotopy
characterizations. (a) Exceptional point in 2D, $(d,d_{E})=(2,0)$. (b)
Exceptional point in 3D, $(d,d_{E})=(3,0)$. (c) Exceptional line in 3D,
$(d,d_{E})=(3,1)$. The exceptional degeneracies are marked in red and the
enclosing surface is marked in green.

Several typical exceptional degeneracies in 2D and 3D are sketched in Fig. 3.
Fig. 3(a) is for an isolated EP in 2D. It is characterized by the homotopy
invariant $[S^{1},X_{n}]$. In the main text, $[S^{1},X_{n}]$ is identified as
the conjugacy class of braid group $B_{n}$ and further one to one corresponds
to the geometric knot formed by the eigenenergies nearby the EP. Fig. 3(c) is
for an exceptional line (red line) in 3D. It topology is carried by the
encircling loop (green) around the. In this case $d=3,d_{E}=1$ and $D_{s}=1$.
It is also characterized $[S^{1},X_{n}]$, or conjugacy class of braid group.
The same conclusion can be obtained for an exceptional surface in 4D parameter
space with $d=4,d_{E}=2$. Fig. 3(b) is for an EP in 3D with $d=3,d_{E}=0$
(similarly for an exceptional line in 4D). It is characterized by
$[S^{2},X_{n}]$, which is a collection of Chern numbers as discussed in the
main text. In the presence of certain symmetries, more intriguing defects
like, exceptional line in 2D ($d_{E}=1,d=2$) and exceptional surface
($d_{E}=2,d=3$) in 3D may appear. For these cases, $D_{s}=0$. The zero-th
homotopy group $\pi_{0}(X_{n})$, the set of path components of the space
$X_{n}$, does not reveal too much information. The stability of such band
singularities should be analyzed in a case-by-case manner, e.g., using the
resultant vector as in [66].

###  A.2 (II) The homotopy equivalence between space $U(n)/U^{n}(1)$ and
$GL(n,\mathbb{C})/\mathbb{C}^{*n}$

For non-Hermitian Hamiltonian with $n$ separable bands (i.e., $E_{m}\neq
E_{n}$ for any $m\neq n$ with some fixed parameters), the space of the
eigenvectors is $GL(n,\mathbb{C})/\mathbb{C}^{*}$ [41]. Here
$GL(n,\mathbb{C})$ comes from $n$ linearly independent complex vectors,
$\mathbb{C}^{*}=\mathbb{C}-\\{0\\}$ comes from the redundancy of biorthogonal
rescaling by some nonzero complex numbers [5]. In the following, we showcase
that the eigenvector space $GL(n,\mathbb{C})/\mathbb{C}^{*n}$ is homotopy
equivalent to $U(n)/U^{n}(1)$ in the main text, with $U(n)$ the $n\times n$
unitary matrices. Compared to the general linear group, the unitary group is
more commonly used in the context of algebraic topology. The equivalence is
due to the two facts: (1) $U(n)$ is a deformation retraction [53] of
$GL(n,\mathbb{C})$; (2) The punctured complex plane $\mathbb{C}^{*}\cong
S^{1}\cong U(1)$, ($\cong$ means homeomorphic, it preserves all the
topological properties of two spaces). While the first statement can be
explicitly visualized through the Gram-Schmidt process, which takes $n$
linearly independent vectors into $n$ orthogonal vectors in a continuous way,
here we give an exact proof: up to homeomorphism, $GL(n,\mathbb{C})\cong
U(n)\times\mathbb{C}^{\frac{n(n+1)}{2}}$.

That being said is a direct consequence of the polar decomposition and
Cholesky decomposition in linear algebra.

Polar decomposition

For any real (complex) $n\times n$ invertible matrix $A$, there is a unique
orthogonal (unitary) matrix $R$ and positive-definite symmetric matrix $S$ so
that $A=RS$.

Cholesky decomposition

Any symmetric positive definite matrix $S$ can be written uniquely as
$S=LL^{\dagger}$ where $L$ is a lower-triangular matrix with strictly positive
diagonal entries.

Further note that there are $\frac{n(n+1)}{2}$ nonzero elements in an $n\times
n$ lower-triangular matrix and $(0,\infty)\cong(-\infty,\infty)$ up to
homeomorphism. Hence statement (1) is proved. Since $\mathbb{C}$ is
contractible to a point (with trivial homotopy groups), it can be neglected
from the homotopy perspective. It follows that the homotopy groups of the
above two spaces $U(n)/U^{n}(1)$ and $GL(n,\mathbb{C})/\mathbb{C}^{*n}$ are
the same.

###  A.3 (III) Construction of the characteristic polynomial and Hamiltonian
associated with a given knotted exceptional point

In the main text, we have demonstrated that any isolated EPn can be assigned a
knot structure $\mathcal{K}$ due to the exact correspondence between the
geometric knot and the conjugacy class of braid group. Here we showcase how to
construct the characteristic polynomial and Hamiltonian associated with an EPn
with desired braiding pattern (or knot structure $\mathcal{K}$).
Mathematically, the closure of the band braiding near the EPn is the algebraic
knot around the singular point.

The characteristic polynomial $f(\lambda,z)=\det(\lambda-H(k_{1},k_{2}))$ with
$\lambda,z=k_{1}+ik_{2}\in\mathbb{C}$, defines a mapping from $\mathbb{C}^{2}$
to $\mathbb{C}$. The energy levels $(z_{1},z_{2},...,z_{n})$ are the nodal
sets of $f(\lambda,z)=0$ and the EPn is the isolated singularity located at
$(\lambda,z)=(0,0)$. $f(\lambda,z)=0$ is semiholomorphic and can be written as
a complex polynomial in $\lambda$, $z$, and $\bar{z}$. To proceed, we need an
auxiliary polynomial $f_{\theta}:\mathbb{C}\times S^{1}\rightarrow\mathbb{C}$,
with the knot $\mathcal{K}$ as its nodal set. This process has been elaborated
in [42, 69]. The key lies in the Fourier parameterization of the braid diagram
associated with the knot. Having obtained $f_{\theta}$, we can replace $e^{\pm
i\theta}$ with $z$ and $\bar{z}$, respectively:
$f(\lambda,z)=f_{\theta}(e^{i\theta}\rightarrow
z;e^{-i\theta}\rightarrow\bar{z})$. In some cases, however, such an analytical
continuation does not necessarily guarantee the EPn is the isolated singular
point of $f(\lambda,z)$. To overcome this issue, we slightly modify the
characteristic polynomial to take the following form ($g>0$ is some integer)
[70]:

$\displaystyle
f(\lambda,z)=(z\bar{z})^{ng}f(\frac{\lambda}{(z\bar{z})^{g}},\frac{z}{\sqrt{z\bar{z}}}).$
(19)

Through the above procedures, we get the characteristic polynomial of e.g.,
the figure-8 knot: $f_{F8}=1/64[64\lambda^{3}-12\lambda
a^{2}[3(z\bar{z})^{2}+2z^{2}(z\bar{z})-2\bar{z}^{2}(z\bar{z})]-14a^{3}(z^{2}+\bar{z}^{2})(z\bar{z})^{2}-a^{3}(z^{4}-\bar{z}^{4})(z\bar{z})]$
with $a=1/4$. Similarly , we can obtain the characteristic polynomial for more
complicated knots, e.g., the $L6a1$ link of braid word
$(\tau_{1}^{-1}\tau_{3}^{-1}\tau_{2})^{2}$.

To obtain the associated non-Hermitian Hamiltonian, we expand the
characteristic polynomial in the series of $\lambda$, which takes the form:
$f(\lambda,z)=\lambda^{n}-\sum_{j=0}^{n-1}\zeta_{j}(z)\lambda^{j}$. It is
clear at the location of EPn, $z=0$, $\zeta_{j}(z)=0$. A natural choice of the
knotted Hamiltonian is

$\displaystyle
H_{\mathcal{K}}(z)=\left(\begin{array}[]{cccc}\zeta_{n-1}(z)&~{}~{}~{}...{}{}{}&\zeta_{1}(z)&\zeta_{0}(z)\\\
1&0&0&0\\\ 0&\ddots&\ddots&0\\\ 0&0&1&0\\\ \end{array}\right)_{n\times n}.$
(24)

Obviously $\det[\lambda-H_{\mathcal{K}}(z)]=f(\lambda,z)$. Fig. 1 of the main
text showcases several typical non-Hermitian bands with the knotted EPs
located at the origin through the above procedures. On a 2D lattice, we simply
replace $k_{1,2}$ with $\sin k_{x,y}$ in $H_{\mathcal{K}}(z)$ and get a 2D
lattice Hamiltonian hosting the desired EPn of the $\mathcal{K}$ knot.


# Non-Hermitian topological invariant of photonic band structures undergoing
inversion

Paul Bouteyre1<EMAIL_ADDRESS>Dung Xuan Nguyen2,3 Guillaume
Gachon1 Taha Benyattou1 Xavier Letartre1 Pierre Viktorovitch1 Ségolène
Callard1 Lydie Ferrier1<EMAIL_ADDRESS>Hai Son Nguyen1,4 hai-
<EMAIL_ADDRESS>1Univ Lyon, Ecole Centrale de Lyon, CNRS, INSA Lyon,
Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL, UMR5270, 69130 Ecully,
France  2Brown Theoretical Physics Center and Department of Physics, Brown
University, 182 Hope Street, Providence, Rhode Island 02912, USA  3Center for
Theoretical Physics of Complex Systems, Institute for Basic Science(IBS),
Daejeon, Korea, 34126  4IUF, Université de France

(August 28, 2024)

###### Abstract

The interplay between symmetry and topology led to the discovery of symmetry-
protected topological phases in Hermitian systems, including topological
insulators and topological superconductors. However, the intrinsic symmetry-
protected topological characteristics of non-Hermitian systems still await
exploration. Here, we investigate experimentally the topological transition
associated with the inversion of non-Hermitian band structures in an optical
lattice. Intriguingly, we demonstrate that the winding number associated with
the symmetry-protected bound state in the continuum is not a conserved
quantity after band inversion. To define a topological invariant, we propose
the skyrmion number given by spawning in momentum space a pseudo-spin with the
polarisation vortex as the in-plane component and the band-index as the
pseudo-spin direction at the origin. This leads to a topological transition
from an antimeron to an meron-like texture through band inversion, while
always conserving the half-charge skyrmion number. We foresee the use of
skyrmion number to explore exotic singularities in various non-Hermitian
physical system.

In 1929, Von Neumann and Wigner suggested, as an anecdote, the existence of
peculiar solutions of Schrödinger equation which are perfect bound states
although lying in the continuum spectrum of extended states [1]. Six decades
later, the origin of Bound States in the Continuum (BICs) was elegantly
unraveled in 1985 by Friedrich and Wintgen, who explained the suppression of
the leaking to the continuum as the result of destructive interference between
two resonances when coupled to the same radiation channel [2]. Nowadays, BICs
have been evidenced in many physical platform, such as atomic and molecular
system [3, 4, 5], water waves [6], solid-state and mesoscopic physics [7, 8,
9], and photonic structures [10, 11]. In particular, engineering BICs with
custom-cut design in nanophotonics has led to tremendous success in various
applications such as lasing[11, 12], ultra-fast optical switch[13] and
sensing[14, 15].

The seminal work of Zhen et al.[16] has shown that each BIC in photonic
lattices corresponds to a polarization singularity associated with a
polarization vortex in momentum space. These vectorial vortices have now been
experimentally demonstrated by different groups[17, 18]. Ultimately, the
winding number of such a polarization vortex is suggested to be a conserved
topological charge of non-Hermitian photonics[16, 19]. This interpretation is
now widely accepted since the winding number has been theoretically and
experimentally demonstrated as a conserved quantity in different scenarios
such as BIC merging, splitting, annihilation, and creation[16, 20, 21, 22, 23,
24]. Most of these works have focused on mechanisms either involving
accidental BICs[16, 20, 23], or breaking lattice symmetry[21, 22].

However, symmetries play a crucial role in classifying topological phases of
matters [25, 26] as well as in interpreting topological photonics systems
[27]. On the other hand, the topological order can be altered with the band
inversion mechanism while preserving the symmetries [28, 29]. Therefore
investigating the topological nature of photonic symmetry-protected BICs under
band inversion mechanism is fundamentally important.

In this work, we experimentally investigate the topological nature of
symmetry-protected BICs undergoing band inversion transformation in leaky
photonic lattice. Using a home-built band dispersion tomography setup, we map
the polarization texture in momentum space together with the dispersion
surface of each photonic band. Our results show that the polarization vortex
associated with the symmetry-protected BIC changes sign after the band
inversion. The results suggest that the winding number is not a good
topological number as soon as the band inversion mechanism is involved. We
then propose the definition of skyrmion number [30, 31] for non-Hermitian
topological charges, which combines the winding number of symmetry-protected
BICs and their band index. The latter can be interpreted as the direction of
the pseudo-spin at the origin, leading to a meron-like texture in the momentum
space.

Figure 1: (a) Sketch of the sample: photonic lattices consisting of gold
gratings of period $a=600\,$nm and $50\,$nm thickness, on top of a dielectric
mirror (4 pairs of $SiO_{2}/Si$ Bragg mirror). The width $w$ of gold stripes
is in the range of $0.25a$ to $0.7a$. (b) Top view of a Scanning Electronic
Microscope (SEM) image of the sample. (c) Field distribution of lowest energy
eigenmodes at normal incidence (i.e. $k_{x}=k_{y}=0$). (d) Energy (in the unit
of $a/\lambda$) of the eigenmodes as a function of the aspect ratio $w/a$. The
open diamond symbols are experimental data and the dash lines are results
obtained from RCWA simulations. (e,g) Band structure of the photonic lattice
in the case of (e) $U<0$ and (g) $U>0$ experimentally measured by angle-
resolved reflectivity contrast (left panels), and numerically simulated by
RCWA method (right panels). These band structures are depicted along $q_{x}$
(with $q_{y}=0$) and along $q_{y}$ (with $q_{x}\approx 0$). Here
$q_{x}=k_{x}.a/2\pi$ and $q_{y}=k_{y}.a/2\pi$, and the energy axis is the
normalized pulsation, given by $\hat{\omega}=(\omega-\omega_{0})a/2\pi c$
where $\omega_{0}$ is the mid-gap frequency. (f,h) Tomography of the
dispersion surface in the case of (f) $U<0$ and (h) $U>0$ reconstructed from
band structure measurements (left panels) and RCWA simulations (right panels).
Figure 2: (a to d) Mapping of the farfield polarization orientation $\phi$ in
momentum space $(q_{x},q_{y})$ from respectively (a) the upper and (b) lower
branch before the band inversion; (c) the upper and (d) lower branch after the
band inversion. The farfield polarization maps are reconstructed from (i)
experimental data, (ii) Finite Element Method (FEM) simulations, and effective
Hamiltonian ((iii) colormap , (iv) vectorial representation).

The photonic lattice that we examine, shown in Figs.1(a,b), consists of a thin
metal sub-wavelength grating on top of a dielectric mirror (the structure is
periodically modulated along $x$-axis and invariant by translation along
$y$-axis). Such designs have been considered in previous works as Tamm plasmon
photonic crystals[32], but here we refer to it simply as a case study of
photonic lattices. Indeed, most of the results in this work should be
reproduced in any other sub-wavelength photonic lattice exhibiting an in-plane
mirror symmetry along the corrugation direction $x\rightarrow-x$ and
translational invariance in the $y$ direction. In this work, we will focus on
Transverse Electric (TE) modes, keeping in mind that similar effects would be
obtained for Transverse Magnetic (TM) modes (the only condition is that the
splitting TE-TM is strong enough so that there is single polarization mode at
the operating frequency). At the $\Gamma$ point, the two lowest TE eigenmodes
exhibit opposite parity with respect to the in-plan mirror symmetry, as seen
in Fig.1(c). The antisymmetric mode is not allowed to couple to plane waves of
the radiative continuum, thus corresponding to a symmetry-protected BIC. On
the other hand, the symmetric mode can radiate to the free space. Due to the
presence of nonradiative losses via absorption in metal, both modes have a
finite quality factor. Since some papers only use the term BIC when the
quality factor is infinite (and quasi-BIC or ultrahigh-Q otherwise), to avoid
any misleading, for the rest of the paper, the symmetry-protected BIC will be
referred to as the dark mode, while the radiative eigenmode will be referred
to as the bright mode.

For an arbitrary point in momentum space in the vicinity of the $\Gamma$
point, the formation of these eigenmodes is captured in a simple $2\times 2$
effective non-Hermitian Hamiltonian with counter propagating guided modes as
the basis (see Supplemental Materials for detailed derivations):

$H=\left[\frac{q_{y}^{2}}{2n_{g}}+i\left(\hat{\gamma}+\hat{\gamma_{nr}}\right)\right]\mathbb{I}_{2}+\left(U-i\hat{\gamma}\right)\sigma_{1}+\frac{q_{x}}{n_{g}}\sigma_{3}$
(1)

where $\sigma_{1}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$,
$\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ are the Pauli matrices
and $q_{x,y}=k_{x,y}a/2\pi$ are the normalized momentum coordinates. As
parameters, $U$ is the diffractive coupling strength, $\hat{\gamma}$ is the
radiative losses of the folded guided modes to the radiative continuum and
also responsible for the radiative coupling between these guided modes via the
radiative continuum; $\hat{\gamma}_{nr}$ is the nonradiative losses; and
$n_{g}$ is the group index of the uncoupled guided modes. Note that we
normalized the diffractive coupling $U$, linewidths $\hat{\gamma}$, and
$\hat{\gamma}_{nr}$ by the energy $hc/a$ to make the system dimensionless and
as general as possible. The energy-momentum band structure is given by the
real part of the complex eigenvalues from (1), while the imaginary counterpart
gives the losses. In the following, the photonic band having the dark(bright)
mode at $\Gamma$ point will be called the dark(bright) branch. From (1), one
may show that the gap separating the two bands amounts to 2$U$ and the
curvature of the dark(bright) band has the same(opposite) sign as the one of
$U$ (See Supplemental Materials for details.). As a consequence, for each
value of $q_{y}$, we can define a band inversion corresponding to a “swapping”
between two band dispersion along $q_{x}$ when $U$ changes sign[33]: if $U>0$,
the upper branch corresponds to the dark branch and if $U<0$, the upper branch
is the bright one.

Figure 3:  (a) Three-dimensional vector $\bm{M}$ which is composed of the
polarization vector $\bm{\epsilon}=(E_{x},E_{y})/E_{0}$, and the band index
$U$ as the vertical coordinate. The pseudo-spin $\bm{m}$ is then defined by
$\bm{m}=\bm{M}/||{\bm{M}||}$. (b,c) The topological transition from antimeron
texture (b) to (c) meron texture of the pseudo-spin $\bm{m}$ through the band
inversion. The upper panels of (b,c) show the texture of $\bm{m}$ in the
momentum space, while the lower panels show the texture of $\bm{m}$ along the
$q_{x}$ and $q_{y}$ axes.

We notice that the energy of the dark mode is very sensitive to the aspect
ratio $w/a$ while the one of the bright mode is not. In particular, Figure. 1
(d) shows that the dark mode can be tuned continuously across the bright mode
when scanning the aspect ratio, corresponding a tuning of $U$ from $-0.01$ to
$0.01$. To demonstrate the band inversion experimentally, we present the
following results of two configurations corresponding to $U=\pm 0.005$. Their
band structures along the two symmetry directions are experimentally obtained
by angle-resolved reflectivity contrast measurements, and numerically
calculated by Rigorous coupled-wave analysis (RCWA) simulations[34]. These
band structures are represented in Fig. 1(e,g), showing perfect agreement
between experimental and simulation data. For the dispersions along $q_{x}$
(with $q_{y}=0$), the dark branch is revealed by its signal vanishing at
$\Gamma$ point, while for the dispersion along $q_{y}$ (with $q_{x}\approx
0$), it corresponds to the branch exhibiting sharp linewidth and weaker
signal. These results clearly evidenced the band inversion when $U$ switches
sign. Finally, the dispersion surface $\hat{\omega}(q_{x},q_{y})$ is
tomographically reconstructed by putting together a mega-data of angle-
resolved reflectivity contrast measurement along different orientation sets
(see Methods and Supplemental Materials for the detailed experimental setup).
The experimental data and simulation results of the two-mode surfaces before
and after the band inversion, shown in Fig. 1(f,h), exhibit again a perfect
agreement.

We now focus on the polarization vortex pinned at the symmetry-protected BIC
undergoing band inversion. Such a vortex is characterized by the accumulation
of the orientation angle $\phi(\bm{q})$, the orientation of the farfield
polarization corresponding to the dark branch, when encircling the symmetry-
protected BIC. The pattern $\phi(\bm{q})$ can be analytically calculated from
the eigenvectors of the effective Hamiltonian (1) and one may show that the
vortex winding number, defined by[16]
$W=\frac{1}{2\pi}\oint_{\mathcal{C}}d\bm{q}\nabla_{\bm{q}}\phi$, is simply
(see Supplemental Materials for full derivation):

$W=sgn(U).$ (2)

The result suggests that the vortex winding number changes sign after the band
inversion and thus is not a conservative quantity. It is quite surprising
because most of the reported works in the literature suggest the conservation
of BIC winding number, which is a topological invariant. To verify
experimentally and numerically this intriguing feature, we perform angle-
resolved reflectivity contrast measurements and FEM simulations in different
polarizations to obtain the tomography reconstruction of the farfield
polarization pattern for both photonic branches. The results, presented in
Fig. 2, convincingly demonstrate the change of sign of winding number after
the band inversion. Indeed, while in the case of the bright mode, no
polarization vortices can be observed in the farfield (see Fig. 2 (a) and
(d)), the dark mode exhibits a polarization vortex whose winding direction
switches when $U$ changes sign, confirming that the winding number is not
conserved through the band inversion mechanism and therefore is not a
preserved topological number (see Fig. 2 (b) and (c)).

Since the symmetry-protected BIC switches band and the sign of its winding
number under band inversion, we propose a definition of a non-Hermitian
topological charge that combines the winding number of symmetry-protected BICs
and their band index $p=sgn(U)$ defined through the dimensionless diffracting
coupling strength. We first consider a three-dimensional vector $\bm{M}$ which
is composed of the normalized electric field
$\bm{\epsilon}=(E_{x},E_{y})/E_{0}$ and the dimensionless diffracting coupling
strength $U$ (see fig. 3 (a)). Here, $E_{x}(q_{x},q_{y})$ and
$E_{y}(q_{x},q_{y})$ are electric field amplitudes in the farfield of the
photonic lattice, and $E_{0}$ is the amplitude of the confined electric field
of TE-guided mode in non-corrugated structure (More information in the
theoretical section of the Supplemental Materials). This vector represents
both the polarization vortices in its in-plane component, and band-flip in its
out-of-plane component. Intuitively, the dynamic of our non-Hermitian band
structures is captured by the precession in momentum space of the pseudo-spin
that is defined by $\bm{m}=\bm{M}/||{\bm{M}||}$. Indeed, the lower panels in
Figs. 3 (b) and (c) present the pseudo-spin $\bm{m}$ along the $q_{x}$ and
$q_{y}$ axes before (b) and after (c) the band inversion. At $\Gamma$ point
$(q_{x}=q_{y}=0)$, because of the nature of the symmetry-protected BICs, the
polarization is not defined and $\bm{\epsilon}=(0,0)$. As a consequence, the
pseudo-spin only has a vertical component $m_{z}$ and is oriented in the
direction determined by the band index: $\bm{m}=(0,0,p)$. Away from the
$\Gamma$ point, for $q=\sqrt{q_{x}^{2}+q_{y}^{2}}\gg U$, the norm of the
polarization vector $\bm{\epsilon}$ is non-null and becomes much greater than
$U$. Consequently, the pseudo-spin $\bm{m}$ is oriented in the plane, and its
vertical component $m_{z}$ is close to zero, resulting in a half flip of the
pseudo-spin $\bm{m}$ along the $q_{x}$ and $q_{y}$ axes. The upper panels of
Figs. 3 (b) and (c) show the pseudo-spin $\bm{m}$ in momentum space before (b)
and after (c) the band inversion. As described previously, the pseudo-spin
$\bm{m}$ is half flipped in the vertical direction along the radial axes,
while its radial components depend on the polarization vector winding around
the BIC at $q_{x}=q_{y}=0$ as shown in Fig. 2. The combination of the half-
flip of the vertical component $m_{z}$ and the winding of the polarization
vector $\bm{\epsilon}$ results in a precession of the pseudo-spin $\bm{m}$
around the BIC, which can be described as a skyrmion in the momentum space:
the precession of a flipping magnetic spin, and we use the definition of the
skyrmion number [31]

$n=\frac{1}{4\pi}\int
dq_{x}dq_{y}\bm{m}\cdot\left(\frac{\partial\bm{m}}{\partial
q_{x}}\times\frac{\partial\bm{m}}{\partial q_{y}}\right).$ (3)

The skyrmion number represents the number of times the pseudo-spin winds
around the unit sphere; here is $n=pW/2=1/2$ (from the half-flip of the
pseudo-spin) corresponding to a meron. We note that the usual definition of
pseudo-magnetic moment for photonic skyrmion number is only the polarization
vector with the pseudo-spin given by the circular polarization degree [35,
36]. Here the polarization vector of our photonic mode is purely linear (see
Fig.2) and only contributes to the in-plane component of $\bm{m}$, and the
pseudo-spin is purely related to the band order. Interestingly, as shown in
Figs. 3 (b) and (c), the pseudo-spin texture switchs from anti-meron to meron
[37] through the band inversion but its skyrmion number of $1/2$ is conserved
and can be considered as the definition of non-Hermitian topological charges
of symmetry-protected BICs. The skyrmion number defined in this paper reminds
us of the definition of skyrmion number in the two-band model of Chern
insulator [28], in which the skyrmion number determines the Hall conductance.
However, the skyrmion number (and thus the Hall conductance) defined in the
model of Ref [28] can change after a band inversion. Therefore, the definition
of our skyrmion number is not directly related to the skyrmion number in Chern
insulators.

In conclusion, we have shown experimentally that the winding number of the
polarization around the symmetry-protected bound state in the continuum (BIC)
of leaky photonic modes in a 1D optical lattice is not a conserved quantity as
it changes sign during a band inversion. The experimental results were
perfectly matched with numerical simulations and are nicely explained with a
model of effective Hamiltonian. Our observations showed that the winding
number of polarization vortex is not a topological constant in the general
case. Consequently, we suggested combining the winding number and the band
index to redefine the topological number as a skyrmion number in the momentum
space. In our proposal, the band index can be interpreted as the direction of
the pseudo-spin at the origin, leading to a meron-like texture in the momentum
space. This picture associates the band inversion with a transformation
between anti-meron and meron topological spin texture while always conserving
the half-charge skyrmion number. Here our skyrmion number plays a similar role
as the total Chern number in topological insulators, where the total Chern
number is preserved before and after band inversion. In future work, we will
apply our formalism to photonic band structures with multiple BICs. We expect
the definition of skyrmion number in this paper can be applied to analyze the
singularities of other non-Hermitian topological systems [38].

Acknowledgement

The authors thank Shanhui Fan and Pierre Delplace for fruitful discussions.The
work is partly funded by the French National Research Agency (ANR) under the
project POPEYE (ANR-17-CE24-0020) and the IDEXLYON from Université de Lyon,
Scientific Breakthrough project TORE within the Programme Investissements
d’Avenir (ANR-19-IDEX-0005). DXN was supported partly by Brown Theoretical
Physics Center and by IBS-R024-D1.

Author contributions

H.S.N and L.F initiated the research project and supervised throughout the
project. All experimental measurements as well as their analysis, were
performed by P.B. The sample was designed and fabricated by L.F. The
experimental setup was built by H.S.N and P.B. Effective Hamiltonian model
were developed by D.X.N, X.L, P.V and H.S.N. Numerical simulations were
performed by P.B and T.B. All authors discussed the results and were involved
in writing the manuscript.

Supplementary Information is available for this paper.

Correspondence and requests for materials should be addressed to Paul
Bouteyre, Lydie Ferrier and Hai Son Nguyen.

Mehods

Sample fabrication: The bottom Bragg mirror is constituted by 4 $\lambda$/4n
pairs of amorphous silicon and silica, deposited by using PECVD (Plasma
Enhanced Chemical Vapor Deposition) on a silicon substrate. The thickness and
refractive index of each layers have been characterized by ellipsometry
measurements. The periodic metallic patterns on top of the Bragg mirror have
been defined by using electron beam lithography followed by a 50nm gold
evaporation. All patterns are realeased thanks to a lift-off with acetone at
the end of the process.

Experimental set-up: All experimental results are obtained from angle-resolved
reflectivity contrast measurements using a home-made Fourier setup. Details of
the setups, technical points on tomographic bands reconstruction and
polarization orientation measurements are provided in the Supplemental
Material.

Numerical simulation: Numerical simulations are performed with Rigorous
Coupled-Wave Analysis (RCWA) method and Finite Element Method (FEM). The RCWA
simulations have been performed with the S4 package provided by the Fan Group
at the Stanford Electrical Engineering Department.[34] The FEM simulations
have been performed with the Comsol software.

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— SUPPLEMENTAL MATERIAL —

##  I Theoretical model

In this section, we provide the detailed derivation of the effective
Hamiltonian representing leaky Bloch modes in a general photonic lattice of
period $a$ along x-axis and exhibiting a lateral mirror symmetry
$x\rightarrow-x$, while being invariant by translation along $y$-axis.

###  I.1 In-plane guides modes in the vicinity of the $\Gamma$ points

In a perturbative approach, the Bloch modes of the photonic lattice can be
described in the basis of propagating in-plan guided modes
$\lvert\bm{\beta}\rangle$ of an effective medium with $\bm{\beta}$ being the
propagation vector of the guided mode. In the vicinity of $\Gamma$ points, we
can decompose a given propagation vector $\bm{\beta}$ as:

$\bm{\beta_{m}}(\bm{k})=\bm{k}+\frac{2\pi m}{a}\bm{u_{x}}$ (S1)

where $\bm{k}=k_{x}\bm{u_{x}}+k_{y}\bm{u_{x}}$ with $k_{x},k_{y}\ll 2\pi/a$
and $m\in\mathbb{Z}$ is the index of the 1D Brillouin zone along x-axis. We
also used $\bm{u_{x}}$ and $\bm{u_{y}}$ as the unit vectors in $\hat{x}$ and
$\hat{y}$ directions correspondingly.

Here, we only focus on the band structure in the vicinity of the lowest
$\Gamma$ point (i.e. $\Gamma_{1}$ corresponding to $\lvert m\rvert=1$). As
sketched in Fig. S1a, each point $(k_{x},k_{y})$ in the first Brillouin zone
corresponds to two guided modes $\lvert\bm{\beta_{\pm 1}}\rangle$ with

$\bm{\beta_{\pm 1}}(\bm{k})=\bm{k}\pm\frac{2\pi}{a}\bm{u_{x}}.$ (S2)

Figure S1: (a) The momentum space in the vicinity of the first $\Gamma$ point
of a lattice with period $a$ along x-axis and invariant translation along
y-axis is divided into Brilloin zones with index $m\in\mathbb{Z}$.(b) The
points $(k_{x},k_{y})$ in the Brillouin zones $m=\pm 1$ correspond to two
momentum vector $\bm{\beta_{\pm 1}}=(k_{x}\pm 2\pi
m/a)\bm{u_{x}}+k_{y}\bm{u_{y}}$. $E_{\pm 1}$ indicate the corresponding
electric field for TE guided modes. (c) Sketch of coupling mechanisms between
$\lvert\bm{\beta_{-1}}\rangle$, $\lvert\bm{\beta_{+1}}\rangle$ and the
radiative continuum. This includes the diffractive coupling of strength $U$,
the radiative losses of rate $\gamma$ and the nonradiative losses of rate
$\gamma_{nr}$.

####  I.1.1 Dispersion characteristic of guided modes

Assuming that the dispersion characteristic in the vicinity of the
$\Gamma_{1}$ point is linear with the slope given by the effective group index
$n_{g}$, one may show that the dispersion characteristic $\omega_{\pm
1}(k_{x},k_{y})$ is given by:

$\displaystyle\begin{split}\omega_{\pm
1}(\bm{k})&=\omega_{\Gamma_{1}}+\frac{c}{n_{g}}\left(\lvert\bm{\beta_{\pm
1}}(\bm{k})\rvert-\frac{2\pi}{a}\right)\\\
&\approx\omega_{\Gamma_{1}}\pm\frac{c}{n_{g}}k_{x}+\frac{ca}{4\pi
n_{g}}k_{y}^{2}\end{split}$ (S3)

where $c$ is the speed of light and $\omega_{\Gamma_{1}}$ is the pulsation of
this guided mode at $\Gamma_{1}$ point.

It is preferable to work with dimensionless quantities. We thus redefine the
wavevectors, as well as the pulsations as:

$\bm{q}=\bm{k}a/2\pi$ (S4)

and

$\hat{\omega}_{\pm 1}(\bm{q})=\left(\omega_{\pm
1}(\bm{k})-\omega_{\Gamma_{1}}\right)a/2\pi c.$ (S5)

Using this quantity instead of $\bm{k}$ and $\omega_{\pm 1}$, the dispersion
characteristic (S3) can be rewritten as:

$\hat{\omega}_{\pm
1}(\bm{q})\approx\pm\frac{q_{x}}{n_{g}}+\frac{q_{y}^{2}}{2n_{g}}$ (S6)

where $q_{x}=k_{x}a/2\pi$ and $q_{y}=k_{y}a/2\pi$.

####  I.1.2 Polarization vector of the guided modes

Polarizations of guided modes are classified into TE and TM modes (i.e. the
electric field $\bm{E}$ and magnetic field $\bm{H}$ are in-plan and
perpendicular to the propagation vector respectively). Here we suppose that
there is only one type of guided modes in the spectral range of interest, thus
the setup is with single polarization. Moreover, for the sake of simplicity,
in this work, we only consider TE modes, knowing that a similar derivation can
be obtained for TM modes. For the propagation vector from (S2), the
corresponding polarization vector $\bm{u_{\pm 1}}$ is given by $\bm{u}_{\pm
1}\cdot\bm{\beta}_{\pm 1}=0$, leading to:

$\bm{u_{\pm 1}}(\bm{q})=\cos\theta_{\pm 1}\bm{u_{x}}+\sin\theta_{\pm
1}\bm{u_{y}},$ (S7)

with

$\left(\cos\theta_{\pm 1},\sin\theta_{\pm
1}\right)=\frac{\left(-q_{y},q_{x}\pm 1\right)}{\sqrt{(q_{x}\pm
1)^{2}+q_{y}^{2}}}$ (S8)

###  I.2 Construction and Analysis of the effective non-Hermitian Hamiltonian

####  I.2.1 Coupling mechanisms: diffractive coupling and radiative coupling

Now the implementation of a periodic corrugation will induce a coupling $U$
between $\lvert\bm{\beta_{-1}}\rangle$ and $\lvert\bm{\beta_{+1}}\rangle$.
Each of them also couples to the radiative continuum with the coupling
strength $\gamma$ (see Fig. S1b). In a general case with a presence of non-
radiative losses (if the structures include lossy materials such as metals or
absorptive dielectrics), the total losses would be increased by an amount
$\gamma_{nr}$. As consequence, in the basis
$\Psi=\left(\lvert\bm{\beta_{+1}}\rangle,\lvert\bm{\beta_{-1}}\rangle\right)$,
the system can be described by the following effective non-Hermitian
Hamiltonian:

$H(\bm{q})=\left(\begin{matrix}\hat{\omega}_{+1}&U\\\
U&\hat{\omega}_{-1}\end{matrix}\right)+i\left(\begin{matrix}\gamma+\gamma_{nr}&\gamma\cos\alpha\\\
\gamma\cos\alpha&\gamma+\gamma_{nr}\end{matrix}\right)$ (S9)

The first term of $H$ is Hermitian, describing the free oscillation of each
mode and the coupling between them via diffractive coupling. The second term
of this Hamiltonian is anti-Hermitian. Its diagonal components describe the
total losses including both radiative an non-radiative ones; while its anti-
diagonal components describes the coupling between
$\lvert\bm{\beta_{+1}}\rangle$ and $\lvert\bm{\beta_{-1}}\rangle$ via the
radiative continuum. The coefficient $\cos\alpha=\bm{u_{-1}}.\bm{u_{+1}}$ is
due to the fact that only the same electric field component can interfere. We
note that in the limit of $q_{x},q_{y}\ll 1$, this coefficient can be
effectively replaced by $q_{y}^{2}-1$.

####  I.2.2 Non-Hermitian Hamiltonian: Eigenvalues of Quasi-normal eigenmodes

Using the Pauli matrices $\sigma_{1}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$,
$\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$, the Hamiltonian (S9) can
be rewritten as:

$H=d_{0}\mathbb{I}_{2}+d_{1}\sigma_{1}+d_{3}\sigma_{3}$ (S10)

where

$\displaystyle d_{0}$
$\displaystyle=\frac{q_{y}^{2}}{2n_{g}}+i\left(\gamma+\gamma_{nr}\right),$
(S11) $\displaystyle d_{1}$ $\displaystyle=U+i\gamma\cos\alpha,$ (S12)
$\displaystyle d_{3}$ $\displaystyle=\frac{q_{x}}{n_{g}}.$ (S13)

.

The eigenvalues $\Omega_{\pm}$ and corresponding eigenstates
$\lvert\pm\rangle$ of (S10) is given by:

$\Omega_{\pm}=d_{0}\pm\sqrt{d_{1}^{2}+d_{3}^{2}},$ (S14)

and

$\lvert\pm\rangle=\left(\begin{matrix}1\\\ C_{\pm}\end{matrix}\right)$ (S15)

with

$C_{\pm}=-\frac{d_{3}}{d_{1}}\pm\sqrt{\left(\frac{d_{3}}{d_{1}}\right)^{2}+1}.$
(S16)

As eigenmodes of a non-Hermitian system, $\lvert\pm\rangle$ are quasinormal
modes[39] with many mathematical difficulties, for example the way of
normalization. Another unusual behaviour of these quasinormal modes is that in
general $\lvert+\rangle$ and $\lvert-\rangle$ are not orthogonal between each
other. There are two peculiar configurations:

  * •

$d_{3}=\pm id_{1}$ : Simultaneous coalescence of eigenvalues and eigenstates:
$\Omega_{+}=\Omega_{-}$ and $\lvert+\rangle=\lvert-\rangle$. This
configuration corresponds to Exceptional points[40, 41, 42].

  * •

$d_{3}=0$ : The eigenmodes become normal with $C_{\pm}=\pm 1$. In this
configuration, $\lvert+\rangle$ and $\lvert-\rangle$ are two orthogonal states
as the ones in Hermitian system.

###  I.3 Photonic band structures and Farfield polarizations

####  I.3.1 Complex eigenvalues of Bloch resonances and band inversion
mechanism

The explicit form of eigenvalues $\Omega_{\pm}$ are given by:

$\displaystyle\Omega_{\pm}(\bm{q})$
$\displaystyle=\frac{q_{y}^{2}}{2n_{g}}+i\left(\gamma+\gamma_{nr}\right)\pm\sqrt{\frac{q_{x}^{2}}{n_{g}^{2}}+\left(U+i\gamma\cos\alpha\right)^{2}}$
$\displaystyle=\hat{\omega}_{\pm}+i\gamma_{r\pm}$ (S17)

The real part $\hat{\omega}_{\pm}$ of $\Omega_{\pm}(\bm{q})$ corresponds to
the normalized eigen-frequency. The imaginary counter part $\gamma_{\pm}$
corresponds to the losses including both radiative and nonradiative losses.
Finally, the radiative losses $\gamma_{r\pm}$ corresponding to the leaking of
these eigenmodes to the radiative continuum) can be extracted by taking out
the nonradiative part from the total losses (i.e. subtracting $\gamma_{nr}$
from $\gamma_{\pm}$). One may show that:

$\displaystyle\hat{\omega}_{\pm}(\bm{q})$
$\displaystyle\approx\frac{q_{y}^{2}}{2n_{g}}\pm
U\left[1+\frac{q_{x}^{2}}{2n_{g}(U^{2}+\gamma^{2})}\right],$ (S18a)
$\displaystyle\gamma_{r\pm}(\bm{q})$ $\displaystyle\approx\gamma\left[1\mp
1\pm\frac{q_{x}^{2}}{2n_{g}(U^{2}+\gamma^{2})}\pm q_{y}^{2}\right]$ (S18b)

for $q_{x},q_{y}\ll 1$.

Eq. (S18a) describes band dispersion of the eigenmodes in the vicinity of the
two band-edges. It shows that the gap separating the two bands amounts to
2$U$. Moreover, the curvature of the dispersion of $\lvert+\rangle$ has the
same sign as the one of $U$. As consequence, for each value of $q_{y}$, we can
define a band inversion corresponding to a “swapping” between two band
dispersion along $q_{x}$ when $U\rightarrow-U$. If $U>0$, the upper branch
corresponds to $\lvert+\rangle$ and if $U<0$, the upper branch corresponds to
$\lvert-\rangle$.

Now we will show that the band inversion concept is applied not only for the
band dispersion but also for whole the complex eigenvalues with complex gap.
Indeed, from (I.3.1), for a given $q_{x}$ and for $\forall q_{y}\ll 1$, the
two complex eigenvalues are separated by:

$\kappa(q_{x})\approx
2\sqrt{\frac{q_{x}^{2}}{n_{g}^{2}}+\left(U-i\gamma\right)^{2}}$ (S19)

The gap $\Delta$ is defined by the minimum of $|\kappa(q_{x})|$. One may show
that this gap corresponds to the value $q_{x}=\pm
n_{g}\sqrt{max(\gamma^{2}-U^{2},0)}$, and is given by:

$\begin{cases}|U|>\gamma:\;\;\Delta=2(U-i\gamma)\\\
|U|\leq\gamma:\;\;\Delta=2\left[sgn(U)-i\right]\sqrt{|U|\gamma}.\end{cases}$
(S20)

As consequence, the system is always gapped except when $U=0$. When sweeping
$U$ from negative to positive, the complex gap is closed at $U=0$ and then
reopened again as soon as $U>0$. We note that $U=0$ corresponds to the
Exceptional Points configuration mentioned in the previous section.

Finally, from (S18b), we note that $\gamma_{r+}=0$ at $q_{x}=q_{y}=0$, and
increasing as $q_{x}^{2}$ and $q_{y}^{2}$. As consequence $\lvert+\rangle$ is
a BIC at $q_{x}=q_{y}=0$. Since this state can still exhibit losses via
nonradiative channel, we prefer to call it dark state to underline the
nonradiating nature of $\lvert+\rangle$. Logically, the state $\lvert-\rangle$
will be refereed to as bright state since the radiative loss is maximized at
$q_{x}=q_{y}=0$. As a consequence, the band inversion mechanism swaps the
bright and the dark states.

####  I.3.2 Eigenmodes of the effective Hamiltonian

The explicit form of eigenstates $\lvert\pm\rangle$ are given by:

$\lvert\pm\rangle=\lvert\bm{\beta_{+1}}\rangle+C_{\pm}\lvert\bm{\beta_{-1}}\rangle$
(S21)

with :

$C_{\pm}=-\frac{q_{x}}{n_{g}(U-i\gamma\cos\alpha)}\pm\sqrt{1+\left[\frac{q_{x}}{n_{g}(U-i\gamma\cos\alpha)}\right]^{2}}.$
(S22)

At $q_{x}=q_{y}=0$, Eq. (S22) gives us $C_{\pm}\rvert_{q_{x}=q_{y}=0}=\pm 1$,
and the eigenmodes is simplifed as:

$\lvert\pm_{q_{x}=q_{y}=0}\rangle=\lvert\bm{\beta_{+1}}\rangle\pm\lvert\bm{\beta_{-1}}\rangle$
(S23)

The $\pi$ phase between $\lvert\bm{\beta_{+1}}\rangle_{q_{x}=q_{y}=0}$ and
$\lvert\bm{\beta_{-1}}\rangle_{q_{x}=q_{y}=0}$ implies that
$\lvert+_{q_{x}=q_{y}=0}\rangle$ is an antisymmetric state while
$\lvert-_{q_{x}=q_{y}=0}\rangle$ is a symmetric one. As a consequence,
$\lvert+\rangle$ cannot couple to the radiative continuum at $q_{x}=q_{y}=0$
due to symmetry mismatch and is a symmetry protected BIC. This result is quite
inline with the one provided by the eigenvalue calculations.

####  I.3.3 Nearfield pattern

The nearfield of the guided modes $\lvert\bm{\beta_{\pm 1}}\rangle$ is
described by plane waves propagating in-plane (x,y) with vertical confinement
profile $f(z)$:

$\displaystyle\begin{split}\bm{\mathcal{E}_{\pm
1}}&=\mathcal{E}_{0}f(z)\exp{\left(i\bm{\beta_{\pm
1}}.\bm{r}_{||}\right)}\bm{u_{\pm 1}}\\\
&=\mathcal{E}_{0}f(z)\exp{\left(i\frac{2\pi\bm{q}.\bm{r}_{||}}{a}\right)}\exp{\left(\pm
i\frac{2\pi x}{a}\right)}\bm{u_{\pm 1}}\end{split}$ (S24)

where the propagation wavevector $\bm{\beta_{\pm 1}}$ is from (S2),
polarization vector $\bm{u_{\pm 1}}$ is from (S7), $\bm{r}_{||}=(x,y)$ is the
in-plane coordinates, $\bm{q}=(q_{x},q_{y})$ is the in-plane wavevector and
the constant $\mathcal{E}_{0}$ is a the field amplitude.

Since the eigenmodes $\lvert\pm\rangle$ are given by superpositions of
$\lvert\bm{\beta_{+1}}\rangle$ and $\lvert\bm{\beta_{-1}}\rangle$, the
nearfield patterns of $\lvert\pm\rangle$ are also dictated by the same
superposition rule:

$\displaystyle\begin{split}\bm{\mathcal{E}_{\pm}}&=\bm{\mathcal{E}_{+1}}+C_{\pm}\bm{\mathcal{E}_{-1}}\\\
&=\mathcal{E}_{0}f(z)\exp{\left(i\frac{2\pi\bm{q}.\bm{r}_{||}}{a}\right)}\left[\exp{\left(i\frac{2\pi
x}{a}\right)}\bm{u_{+1}}+C_{\pm}\exp{\left(-i\frac{2\pi
x}{a}\right)}\bm{u_{-1}}\right].\end{split}$ (S25)

At $\bm{q}=\bm{0}$, Eqs. (S7),(S8) and(S22) imply that $\bm{u_{\pm
1}}(\bm{q}=\bm{0})=\pm\bm{u_{y}}$ and $C_{\pm}(\bm{q}=\bm{0})=\pm 1$. As a
consequence, the nearfield is given by::

$\displaystyle\begin{cases}\bm{\mathcal{E}_{+}}(\bm{q}=\bm{0})&=2i\mathcal{E}_{0}f(z)\sin{\left(\frac{2\pi
x}{a}\right)}\bm{u_{y}}\;\;\rightarrow\textit{ odd parity with respect to the
lateral mirror symmetry}\\\
\bm{\mathcal{E}_{-}}(\bm{q}=\bm{0})&=2\mathcal{E}_{0}f(z)\cos{\left(\frac{2\pi
x}{a}\right)}\bm{u_{y}}\;\;\rightarrow\textit{ even parity with respect to the
lateral mirror symmetry}\end{cases}$ (S26)

As a consequence, $\lvert+\rangle(\bm{q}=\bm{0})$ cannot couple to the
radiative continuum due to symmetry mismatch with plane waves of even parity
in this continuum. Otherwise, the coupling between
$\lvert-\rangle(\bm{q}=\bm{0})$ and the radiative continuum is permitted.
Therefore $\lvert+\rangle(\bm{q}=\bm{0})$ corresponds to symmetry-protected
BIC. We note that the nearfield distribution given by (S26) is in perfect
aggrement with the numerical simulation results in Fig. 1(c) of the main text.

####  I.3.4 Polarization pattern of the farfield in momentum space

The Bragg scattering mechanism due to periodic corrugation fold the guided
modes $\lvert\bm{\beta_{1}}\rangle$ from $\bm{\beta_{\pm
1}}=\bm{k}\pm\frac{2\pi}{a}\bm{u_{x}}$ to the same wavevector $\bm{k}$. As a
consequence, these guided modes once folded radiate to the freespace as plane
wave with $\bm{k}$ as the in-plane component. In the vicinity of the normal
emission, these radiating waves are approximately described as plane waves of
the same wavevector $\frac{\omega_{\Gamma_{1}}}{c}\bm{u_{z}}$ while preserving
the polarization $\bm{u_{\pm 1}}$ of the unfolded modes:

$\bm{E_{\pm
1}}(\bm{q})=E_{0}\exp\left(i\omega_{\Gamma_{1}}z/c\right)\bm{u_{\pm
1}}(\bm{q})$ (S27)

where the amplitude $E_{0}$ of radiative waves is proportional to the
amplitude $mathcal{E_{0}}$ of the unfolded modes.

The farfield patterns of the hybrid modes $\lvert\pm\rangle$ are also dictated
by the same superposition rule of the egeinmodes and nearfield pattern:

$\displaystyle\begin{split}\bm{E_{\pm}}(\bm{q})&=\bm{E_{+1}}(\bm{q})+C_{\pm}\bm{E_{-1}}(\bm{q})\\\
&=E_{0}\exp\left(i\omega_{\Gamma_{1}}z/c\right)\left(\bm{u_{+1}}+C_{\pm}\bm{u_{-1}}\right).\end{split}$
(S28)

We remind that at $\bm{q}=\bm{0}$, $\bm{u_{\pm
1}}(\bm{q}=\bm{0})=\pm\bm{u_{y}}$ and $C_{\pm}(\bm{q}=\bm{0})=\pm 1$.
Therefore, the farfield at $\Gamma$ point is given by:

$\displaystyle\begin{cases}\bm{E_{+}}(\bm{q}=\bm{0})&=0\;\;\rightarrow\textit{
destructive interference}\\\
\bm{E_{-}}(\bm{q}=\bm{0})&=2E_{0}\exp\left(i\omega_{\Gamma_{1}}z/c\right)\bm{u_{y}}\,\rightarrow\textit{
constructive interference}\end{cases}$

This result shows that the farfield of $\lvert+\rangle$ has a singularity at
$q_{x}=q_{y}=0$, which is in good agreement with its dark nature demonstrated
previously.

For an arbitrary coordinate $(q_{x},q_{y})$, the electric field $\bm{E_{\pm}}$
is reconstructed in Cartesian basis as:

$\bm{E_{\pm}}=E_{x,\pm}\bm{u_{x}}+E_{y,\pm}\bm{u_{y}}$ (S29)

where the complex components $E_{x,\pm}$ and $E_{y,\pm}$ are given by:

$\displaystyle E_{x,\pm}$
$\displaystyle=E_{0}\exp\left(i\omega_{\Gamma_{1}}z/c\right)\left(\cos\theta_{+1}+C_{\pm}\cos\theta_{-1}\right),$
(S30a) $\displaystyle E_{y,\pm}$
$\displaystyle=E_{0}\exp\left(i\omega_{\Gamma_{1}}z/c\right)\left(\sin\theta_{+1}+C_{\pm}\sin\theta_{-1}\right).$
(S30b)

where $\sin\theta_{\pm 1}$, $\cos\theta_{\pm 1}$ and $C_{\pm}$ are from Eqs.
(S8) and(S22). The orientation (i.e. angle $\phi_{\pm}$) of
$\bm{E_{\pm}}(\bm{q})$ with respect to $\bm{u_{x}}$ (see Fig.S2a) is given
by:[43]

$\tan
2\phi_{\pm}=\frac{2\text{Re}(E_{x,\pm}^{*}.E_{y,\pm})}{|E_{x,\pm}|^{2}-|E_{y,\pm}|^{2}}$
(S31)

The ellipticity of the polarization, described by angle $\chi_{\pm}$ (see
Fig.S2a), is given by: [43]

$\sin
2\chi_{\pm}=\frac{2\text{Im}(E_{x,\pm}^{*}.E_{y,\pm})}{|E_{x,\pm}|^{2}+|E_{y,\pm}|^{2}}$
(S32)

Figure S2: (a) Elliptic polarization of light with orientation angle $\phi$
and ellipticity $\chi$. (b) Graphical presentation of the 1:1 correspondence
between $\varphi$ and $\psi$.

####  I.3.5 Winding number of the polarization vortex

From the orientation angle $\phi_{\pm}(\bm{q})$, the winding number around
$q_{x}=q_{y}=0$ of the vector field $\bm{E_{\pm}}(\bm{q})$ is defined as:

$W_{\pm}=\frac{1}{2\pi}\oint_{\mathcal{C}}d\bm{q}\nabla_{\bm{q}}\phi_{\pm}.$
(S33)

where $\mathcal{C}$ is a closed circulation encircling $q_{x}=q_{y}=0$. One
may uses directly Eq. (S31) and calculating $W_{\pm}$ for a arbitrary path
$\mathcal{C}$. However, knowing that the only possible singularity is located
at $q_{x}=q_{y}=0$, $W_{\pm}$ does not depend on the choice of $\mathcal{C}$,
we will choose an adequate $\mathcal{C}$ so that the integral can be easily
calculated. To do so, we define the complex number $z=\rho e^{i\varphi}$ with:

$\displaystyle\text{Re}(z)$
$\displaystyle=\rho\cos\varphi=\frac{q_{x}}{2n_{g}\sqrt{U^{2}+\gamma^{2}}},$
(S34a) $\displaystyle\text{Im}(z)$ $\displaystyle=\rho\sin\varphi=q_{y}.$
(S34b)

The winding number will be calculated by encircling in the counter-clockwise
direction around $q_{x}=q_{y}=0$ when fixing $\rho\ll 1$ constant and varying
$\varphi$. The condition $\rho\ll 1$ implies that $\lvert q_{x}\rvert,\lvert
q_{y}\rvert\ll 1$. Thus from (S8),(S22), we can approximate:

$\begin{cases}\sin\theta_{\pm 1}\approx\pm 1\\\ \cos\theta_{\pm
1}\approx-\rho\sin\varphi\\\ C_{\pm}\approx\pm
1-2\rho\cos\varphi\frac{U+i\gamma}{\sqrt{U^{2}+\gamma^{2}}}\end{cases}$ (S35)

Injecting to (S30a),(S30b), that leads to:

– For the $\lvert-\rangle$ state :

$\displaystyle\frac{E_{x,-}}{E_{0}}$ $\displaystyle\approx
2\rho^{2}\cos\varphi\sin\varphi\frac{U+i\gamma}{\sqrt{U^{2}+\gamma^{2}}}\approx
0$ $\displaystyle\frac{E_{y,-}}{E_{0}}$ $\displaystyle\approx
2+2\rho\cos\varphi\frac{U+i\gamma}{\sqrt{U^{2}+\gamma^{2}}}\approx 2$
$\displaystyle\rightarrow\text{always vertical polarization,i.e.
}\phi_{-}=\pi/2$ $\displaystyle\rightarrow\boxed{W_{-}=0}.$

– For the $\lvert+\rangle$ state :

$\displaystyle\frac{E_{x,+}}{E_{0}}$ $\displaystyle\approx-2\rho\sin\varphi$
$\displaystyle\frac{E_{y,+}}{E_{0}}$ $\displaystyle\approx
2\rho\cos\varphi\frac{U+i\gamma}{\sqrt{U^{2}+\gamma^{2}}}$

using (S31), we obtain then:

$\displaystyle\tan 2\phi_{+}$ $\displaystyle=\tan
2\varphi\frac{U}{\sqrt{U^{2}+\gamma^{2}}}$ $\displaystyle=sgn(U)\tan{2\psi}$

where angle $\psi$ is obtained from $\varphi$ by shrinking an ellipse of semi-
major axe $\sqrt{U^{2}+\gamma^{2}}$ and semi-mino axe $|U|$ into a circle of
radius $\sqrt{U^{2}+\gamma^{2}}$ (see Fig.S2b). Hence

$\phi_{+}=sgn(U)\psi$ (S36)

Moreover, we note that the pathway $\mathcal{C}$ of counter-clockwise
direction around $q_{x}=q_{y}=0$ corresponds to $\varphi$ varies from 0 to
2$\pi$, due to the definition (S34a) and (S34b). Moreover, as indicated in
Fig.S2b, $\psi$ will also sweep from 0 to 2$\pi$. Thus

$\displaystyle\begin{split}\oint_{\mathcal{C}}d\varphi=2\pi.sgn(U)\rightarrow\boxed{W_{+}=sgn(U)}.\end{split}$
(S37)

As a matter of fact, the winding number of the polarization vortex associated
with $\lvert+\rangle$ is given by the sign of $U$, thus changes sign after a
band-inversion (i.e. $U$ changes sign).

##  II Experimental methods

Figure S3: Experimental setup (a) Sketch of the home-made Fourier set-up for
the angle-resolved reflectivity measurements. Band inversion mechanism (b to
f) $k_{x}$-resolved contrast reflectivity maps of five different structures
with different aspect ratios $w/a$ (diffractive coupling $U$): (b) 0.34
(-0.0075) (c) 0.4 (-0.0045) (d) 0.425 (0) (e) 0.47 (0.0025) (f) 0.505 (0.005).
For the aspect ratios $w/a<0.44$ ($U<0$), the lower branch corresponds to the
dark mode (BIC), and the upper branch to the bright mode. When the aspect
ratio increases continuously, the dark and bright modes get closer to each
other until they form two exceptional points for $w/a=0.44$ ($U=0$). The two
modes are swapped when the aspect ratios becomes $w/a>0.44$ ($U>0$), and the
two modes move away from each other when the aspect ratio further increases.
This process is the band inversion mechanism. (b) and (f) corresponds to the
structures studied in the article.

####  II.0.1 Angle-resolved reflectivity measurements

All experimental results in this work are obtained from angle-resolved
reflectivity measurements using a home-made Fourier setup [see Fig.S3 (a)].
The Fourier plane of the structure located at the focal length of the
microscope objective is projected with the ’Fourier’ and ’Focus’ lenses to the
entrance of the spectrometer. The slit of the spectrometer only selects the
Fourier image in the $k_{x}$ direction for a given $k_{y}$, which can be
controlled by the y position of the ’focus’ lens. The signal is then dispersed
by the spectrometer diffraction grating and collected by a CCD camera
resulting in a $k_{x}$-resolved measurement of the bands.

As illustration, Figs. S3 (b) to (f) depicts angle-resolved measurements from
five different structures. These structures have been chosen to depict the
band inversion mechanism.

####  II.0.2 Tomographic bands reconstruction

To perform tomographic bands reconstruction, 50 $k_{x}$-resolved measurements
for $k_{y}$ varying from $-k_{max}$ to $k_{max}$ were performed for all the
structures. For each $k_{x}$-resolved measurements at given $k_{y}$, the
measurement is fitted with two Lorentzian functions giving the modes’
wavelength [see blue and red dashed line in Fig.  S4 (a)] and amplitude
resonances (i.e. intensities). From the modes’ wavelength, we can construct
the dispersion surface of the modes as illustrated in Fig. S4 (b).

Figure S4: Tomography band reconstruction (a) Sketch illustrating the
$k_{x}$-resolved measurements of the optical bands for different $k_{y}$ in
the case of the structure with the diffractive coupling strength $U<0$. Only
five measurements are shown here for simplicity. For each $k_{x}$-resolved
measurements at given $k_{y}$, each $k_{x}$ slices are fitted with two
Lorentzian functions giving the modes wavelength $\lambda_{m}(k_{x},k_{y})$
(see blue and red dashed line), as well as their resonant amplitude
$I_{m}(k_{x},k_{y})$. (b) Tomographic bands reconstruction of the structure
with the diffractive coupling strength $U<0$. The projections in the momentum
plane and in the ($k_{x},a/\lambda$) plane are taken from the $k_{y}$-scan
measurements presented in this figure. The projection in the
($k_{y},a/\lambda$) plane is taken from additional measurements in which the
sample was rotated by 90∘ to directly measure the $k_{y}$-resolved dispersion
at $k_{x}$=0.

####  II.0.3 Measurement of the polarization vortex

Different polarization elements (polarizers, half-wave plates, quater-wave
plates) may be introduced for polarization-resolved measurements. The
tomographic experiment is performed for different polarizations to retrieve
polarization-resolved intensities of each mode [see Fig. S5  (a,b)]. From
these results, we obtain the mapping of Stoke parameters in momentum space
[see Fig. S5  (c)]. Finally, from the Stoke parameters, the mapping of the
polarization orientation $\Phi$ is retrieved [see Fig.  S5  (d)].

Figure S5: Polarization orientation measurements (a) Tomographic bands
reconstruction of the structure with the diffractive coupling strength $U<0$
for four different linear polarizations: Horizontal, Diagonal, Vertical, and
Anti-Diagonal. The color-map texture indicates the intensity of the modes at
each position in momentum space. A drop in intensity can be observed on the
low-energy mode around the BIC at $q_{x}=q_{y}=0$ (b) Momentum-resolved
intensity map of the low-energy mode in (a). (c) Momentum-resolved Stokes
parameters $S_{1}=(I_{H}-I_{V})/(I_{H}+I_{V})$ and
$S_{2}=(I_{D}-I_{A})/(I_{D}+I_{A})$ of the low-energy mode extracted from the
intensity map in (b). (d) Farfield polarization orientation of the low-energy
mode with a diffractive coupling $U<0$ calculated using the stokes parameters
in (c). This method was used to measure the polarization orientation of all
the modes studied in the article.