# Identifying topologically critical band from pinch-point singularities in
spectroscopy

Han Yan (闫寒<EMAIL_ADDRESS>Theory of Quantum Matter Unit, Okinawa Institute
of Science and Technology Graduate University, Onna-son, Okinawa 904-0412,
Japan  Department of Physics & Astronomy, Rice University, Houston, TX 77005,
USA  Smalley-Curl Institute, Rice University, Houston, TX 77005, USA

###### Abstract

In this paper, we investigate the relationship between pinch point
singularities observed in energy- and momentum-resolved spectroscopy and
topologically non-trivial gapless points. We show that these singularities are
a universal signature, and that the Berry flux encoded must be $n\pi$ for an
$n-$fold pinch point under suitable symmetry protection. Our results apply to
most systems and are independent of their microscopic details. Hence they
provide a new way to identify topological phases without requiring detailed
knowledge of the microscopic model. Our work can be readily applied in
spectroscopy experiments on various platforms.

Introduction. —  Topologically critical gapless points in the electron and
magnon band structures are significant and frequently-occurring features of
quantum matter. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Usually, to
unambiguously determine if a gapless point is accidental or topological, one
needs to reconstruct the Hamiltonian by reading the band dispersion relations
from the energy and momentum-resolved spectroscopy, while also acquiring a
significant amount of microscopic information including lattice structure,
symmetries etc. Such spectroscopy techniques include angle-resolved
photoemission spectroscopy (ARPES) [14] and scanning tunneling microscopy
(STM) for electron bands [15, 16], and inelastic neutron scattering (INS) for
magnon bands [17], and polariton photoluminescence (PP) for photonic lattices
[18].

In this work, we discuss a different way to unambiguously identify
topologically non-trivial gapless points from spectroscopy alone, without
knowing much else about the system. This approach utilizes the often-ignored
information: the spectroscopic intensity distribution on the bands. Due to the
winding of the wave-function around a topologically non-trivial gapless point,
the spectroscopy intensity on the two bands can show a universal,
characteristic singular pattern which we call an $n-$fold pinch point [Fig. 1]
[19]. Further more, if the system admits a suitable symmetry, the pinch point
is guaranteed to be topologically critical and encodes a Berry curvature of
$n\pi$.

Table 1:  A survey of known experiments exhibiting the pinch points [Figs. 1, 3]. For items labeled with a star ($\star$), the pinch points can be seen directly from or inferred from the references. For items without the star, there are no direct observation of the pinch points due to technological limits. But we predict that the pinch points can be observed in principle. The gapped pinch point case means the two bands have a small gap opening as shown in Fig. 3(c). “2-fold splits” means the 2-fold pinch point splits into two copies of 1-fold pinch point, as shown in Fig. 3(b). See first paragraph of main text for abbreviations for experimental methods.  | Pinch point | Material | Experiment | Ref.
---|---|---|---|---
$\star$ | 1-fold | graphene | ARPES | [20, 21, 22, 23]
$\star$ | 1-fold |  | graphene
---
(quasicrystal)
ARPES | [24]
$\star$ | 1-fold | CoTiO3 | INS | [25, 26]
$\star$ | 1-fold, gapped | YMn6Sn6 | INS | [27]
$\star$ | 1-fold, gapped | CrI3 | INS | [28, 29]
$\star$ | 1-fold, gapped | CrBr3 | INS | [30]
$\star$ | 1-fold, gapped | CrSeTe3 | INS | [31]
$\star$ | 1-fold, gapped | Fe3Sn2 | ARPES | [32]
$\star$ | 2-fold |  | graphene
---
(bilayer)
STM | [33]
$\star$ | 2-fold | Nd2Zr2O7 | INS | [34, 35, 36, 37]
| 2-fold | Ca10Cr7O28 | INS | [38, 39, 40]
$\star$ | 2-fold, gapped | FeSe | ARPES | [41, 42]
$\star$ | 2-fold, gapped | CoSn | ARPES, STM | [43, 44]
$\star$ | 2-fold, gapped | Lu2V2O7 | INS | [45]
| 2-fold, gapped | Cu(1,3-bdc) | INS | [46]
$\star$ |  | 2-fold splits
---
or opens gap
| photonic orbital
---
graphene
PP | [18]
Figure 1:  Schematic illustration of $n-$fold pinch points. (a) Band touching
scenarios discussed in this paper, highlighted in red. (b-1) 1-fold pinch
points imprinted on the two bands. (b-2) the spectroscopy density distribution
on the lower band. (b-3) The corresponding configuration of the wavefunction,
which winds $\pi$ around the pinch point. It encodes a Berry curvature of
$\pm\pi$ at the gapless point. (c,d,e) 2,3,4-fold pinch points illustrated
same way as (b).

Experimentalists have actually observed this universal pattern in a various
materials [Table. 1], although the hidden connection has not been discussed
much. The simplest case, $1-$fold pinch point, is actually a Dirac cone, and
has been shown in ARPES experiments [20, 21, 22, 24] on several graphene-based
materials [6, 47, 48]. The $2-$fold pinch points appear in bilayer graphene
[33], FeSe [41, 42], and various frustrated lattice materials [34, 35, 36, 37,
38, 39, 40, 49, 50, 51, 52, 46, 43] . The $n-$fold pinch point’s implication
of the underlying Gauss’s law has been a focus in classical and quantum spin
liquids [53, 54, 55, 56, 57, 49, 58, 59, 60, 61, 62], but the connection to
Berry curvature has been rarely mentioned.

The universal patterns of pinch points has a high application value. Its
advantage relies on the fact that it does not require one to know much about
the microscopic details of the matter or to reconstruct the full Hamiltonian.
Also, this is particularly useful for two touching-bands with quadratic or
higher-order gap-opening dispersion, where the dispersion alone is not a very
distinguishing factor like the Dirac cones. Another application is to twisted
bilayer systems [63, 64, 65], where the full Hamiltonian for all bands is
practically impossible to reconstruct.

Dirac cone as 1-fold pinch point. —  We start by introducing the “pinch
point”. In this workwe work on 2D systems, but its generalization to 3D is
fairly straightforward. The simplest case — $1-$fold pinch point — is
imprinted on the most common ingredient of topological band systems: the
Weyl/Dirac cone [66, 26, 3].

Consider a local region in the reciprocal momentum space, where two bands have
a degenerate point set at $\bm{q}=\bm{0}$ [cf. Fig. 1(a)]. In the neighborhood
of $\bm{q}=\bm{0}$, they are also gapped from other bands, so we can focus on
the two-band subsystem only.

Spectroscopy measures the band structure as well as the intensity distribution
of certain correlation function on each band. We denote the upper and lower
band’s dispersion relations as $\omega_{-}(\bm{q})$ and $\omega_{+}(\bm{q})$.
The energy and momentum-resolved spectroscopy of the two bands are (assuming
infinitely fine resolution)

$\begin{split}\mathcal{S}_{+}(\omega,\bm{q})=\delta(\omega-\omega_{+}(\bm{q}))S_{+}(\bm{q}),\\\
\mathcal{S}_{-}(\omega,\bm{q})=\delta(\omega-\omega_{-}(\bm{q}))S_{-}(\bm{q}).\end{split}$
(1)

Here, we separated the dispersion $\delta(\omega-\omega_{\pm}(\bm{q}))$ and
the intensity distribution $S_{\pm}(\bm{q})$ for future convenience. The
intensity $S_{\pm}$ usually measures the amplitude of the wavefunction in a
particular basis. For example, in case of graphene, it measures
$\braket{(c_{A}^{\dagger}+c_{B}^{\dagger})(c_{A}+c_{B})}$ of the band
wavefunction, where $A,B$ are the two sublattice indices, so the basis
measured is $c_{A}+c_{B}$. The wavefunction $(1/\sqrt{2},1/\sqrt{2})^{T}$ has
maximal intensity, while $(1/\sqrt{2},-1/\sqrt{2})^{T}$ has zero.

The $1-$fold pinch point refers to the spectroscopic intensity
$S_{\pm}(\bm{q})$ distribution on the Dirac cone as illustrated in Fig. 1(b2).
The intensity distribution only depends on the angle around the gapless point.
On one band, it reaches zero on one side, and maximum on the other. The
intensity varies smoothly except at the gapless point, where it becomes
singular (i.e. not continuous). The other band has a similar pattern of
intensity distribution, but the strong and weak regions switch sides. This
pattern has in fact been observed in various experiments, as we summarized in
Table. 1.

$n-$fold pinch point. —  The pattern of $1-$fold pinch point can be
generalized to $n-$fold pinch point. The cases of $n=1,2,3,4$ are illustrated
in Fig. 1. The upper row panels show the entire energy and momentum resolved
spectroscopy, and the mid row panels show the intensity distribution
$S_{\pm}(\bm{q})$ without the dispersion.

The crucial ingredient of the $n-$fold pinch point is the singularity in the
intensity distribution $S_{\pm}(\bm{q})$. Near $\bm{q}=\bm{0}$, the intensity
only depends on the angle $\theta$ around $\bm{0}$. An $n-$fold pinch point
has $n$ dark wings where the intensity is low, and $n$ bright wings where the
intensity is high. The most symmetric form of the intensity distribution, by
choosing a suitable angle as $\theta=0$, is

$\begin{split}S_{+}(\bm{q})=A\cos^{2}(n\theta/2),\\\
S_{-}(\bm{q})=A\sin^{2}(n\theta/2).\end{split}$ (2)

Here, $A$ is just a scalar signifying the overall intensity. $S_{\pm}(\bm{q})$
at $\bm{q}=\bm{0}$ is singular, since one obtains different values of
$S_{\pm}(\bm{0})$ by approaching it from different directions.

For a specific lattice model, the intensity distribution can be mildly
distorted up to an isomorphic mapping. The more general form is

$\begin{split}S_{+}(\bm{q})=A\cos^{2}(n\Theta(\theta)/2),\\\
S_{-}(\bm{q})=A\sin^{2}(n\Theta(\theta)/2),\end{split}$ (3)

where $\Theta(\theta)$ is a smooth, monotonically increasing function as a
bijection from $[0,2\pi)$ to itself.

$\Theta(0)=0,\;\Theta(2\pi)=2\pi,\Theta^{\prime}(\theta)>0.$ (4)

Figure 2:  Representative continuous winding configurations of $\bm{\psi}_{-}$
consistent with a 2-fold pinch point. (a) $\bm{\psi}_{-}$ configurations with
winding number $\pm 1$, which are also smoothly varying. (b) $\bm{\psi}_{-}$
configurations with winding number $0$, which are not smoothly varying. (c)
Plot of $\psi_{-}^{1}$ as a function of angle $\theta$, showing non-smoothness
at $\theta=\pi/2,\ 3\pi/2$.

The main message of this paper is that such gapless $n-$fold pinch point is
guaranteed to be topologically nontrivial. By “non-trivial” we mean the
degenerate point is not accidental, and generally carries a non-zero Berry
curvature, except for some extremely fine-tuned cases. A much stronger result
is that if the system additionally admits a suitable symmetry (for example,
time reversal symmetry for spinless electron systems), then the Berry flux
encoded must be $n\pi$.

A remark is in order before we proceed to the proof. In this work we assume
that there is no “extrinsic” factors from the coupling between the system and
the probing particles (photons or neutrons), which may also exhibit pinch
point patterns. For ARPES, if the electron is in an anisotropic orbit, the
photon-electron coupling will pick up an angle-dependent projector depending
on the polarization of photons, which may yield pinch point patterns. A very
detailed discussion can be found in Ref. [67]. One needs to choose the photon
polarization (or use unpolarized photons) properly to avoid such effects. The
same principle applies to INS – for example, polarized neutrons couple to
spin-1/2 dimers in a fixed direction can also have similar projectors.
However, these extrinsic, “fake” pinch points are often not present in actual
experiment, or at least avoidable in principle. In all examples given in
Table. 1, one does not need to worry about them.

Pinch points are topologically critical. —  We now prove that under suitable
symmetries, the $2-$fold pinch-point singularity is associated with a gapless
point with Berry flux $\pm 2\pi$ encoded . Here we take the symmetry be time
reversal ($\mathcal{T}$) for the spinless electrons, which forces the
Hamiltonian to be real. The proof can be easily generalized to $n-$fold pinch-
points and other proper symmetries (see end of section).

The physical picture is the following. The $n-$fold pinch point pattern (Fig.
1(b1-e1)) indicates that the spectroscopy intensity follows a squared-sine
function. This requires that the wavefunction of the corresponding band, which
is a two component real vector, rotates in a sine/cosine manner on a loop
around the gapless point, and accumulate a quantized total rotation angle
$n\pi$ (Fig. 1(b3-e3)) at the end. This is exactly the origin of the Berry
flux.

Now, onto the formal proof, we first set up the model with a few
simplifications without affecting its topological features. In the vicinity of
the pinch-point, we consider the relevant subsystem with two degrees of
freedom $\psi^{1},\ \psi^{2}$, and their corresponding two-level Hamiltonian.
The upper and lower bands correspond to two wavefunctions written as two
orthonormal, unit vectors of complex entries

$\bm{\psi}_{+}(\bm{q})=\begin{pmatrix}\psi_{+}^{1}\\\
\psi_{+}^{2}\end{pmatrix},\qquad\bm{\psi}_{-}(\bm{q})=\begin{pmatrix}\psi_{-}^{1}\\\
\psi_{-}^{2}\end{pmatrix}.$ (5)

The system is then described by a Hermitian Hamiltonian

$\mathcal{H}(\bm{q})=\bm{S}\begin{pmatrix}\omega_{+}(\bm{q})&0\\\
0&\omega_{-}(\bm{q})\end{pmatrix}\bm{S}^{\dagger}.$ (6)

where $\bm{S}=\left(\bm{\psi}_{+},\bm{\psi}_{-}\right)$ is the eigenvector
matrix, and $\omega_{\pm}(\bm{q})$ are the two bands’ dispersion relations.

Under the $\mathcal{T}$ symmetry, the eigenvectors and Hamiltonian are real.
Because the spectroscopy intensity follows a squared-sine function, it
requires

$|\psi_{-}^{1}(\bm{q})|\propto|q^{x}|/q=|\sin\theta|.$ (7)

This puts strong constraint on the possible eigenvector configurations. Some
of them are listed in Fig. 2(a,b).

We also require the Hamiltonian to be smooth, i.e., continuous and
differentiable to any order. This is generically true for systems with short-
rang interactions, and not so only for systems with certain fine-tuned long-
range interactions. Hence the requirement applies to most physically realistic
systems [68, 69]. It plays a key role in eliminating the physically
unrealistic cases shown in Fig. 2(b), because in those cases, the eigenvector
has to go though points where unsmooth change of its component(s) is bounded
to happen, even though there is no gap closing at those points. One of such
unsmooth components is illustrated in Fig. 2(c). A more detailed, technical
analysis is presented in the appendix.

Therefore, for $\psi_{-}^{1}$ to vary smoothly, it has to take the form
$\psi_{-}^{1}=\sin\theta$ over the entire circle, up to an overall minus sign.
After a similar analysis on $\psi_{-}^{2}$ and $\bm{\psi}_{+}$, we can
conclude that the viable eigenvectors in $\bm{q}-$space are those in Fig.
2(a), which are all topologically equivalent in eyes of Berry flux at the
gapless point. We may pick the eigenvectors to be

$\begin{split}&\bm{\psi}_{+}(\bm{q})=(-\cos\theta,\sin\theta)=\frac{1}{q}(-q^{y},q^{x}),\\\
&\bm{\psi}_{-}(\bm{q})=(\sin\theta,\cos\theta)=\frac{1}{q}(q^{x},q^{y}).\\\
\end{split}$ (8)

The eigenvectors and eigenvalues completely determines the Hamiltonian (cf. Eq
(6)). Written as an effective magnetic field coupled to Pauli matrices, it is

$\begin{split}\mathcal{H}(\bm{q})=&\frac{\omega_{+}(\bm{q})+\omega_{-}(\bm{q})}{2}\mathbb{I}\\\
&+\Delta(\bm{q})\frac{(q^{x})^{2}-(q^{y})^{2}}{2q^{2}}\sigma_{z}-\Delta(\bm{q})\frac{q^{x}q^{y}}{q^{2}}\sigma_{x}\\\\[10.00002pt]
\equiv&\frac{\omega_{+}(\bm{q})+\omega_{-}(\bm{q})}{2}\mathbb{I}+\bm{\sigma}\cdot\bm{B}(\bm{q}),\end{split}$
(9)

where $\Delta(\bm{q})=\omega_{+}(\bm{q})-\omega_{-}(\bm{q})$ is the energy
gap.

From the Hamiltonian we can read off the effective magnetic field
$\bm{B}(\bm{q})$ to be

$\bm{B}(\bm{q})=\Delta(\bm{q})\left(-\frac{q^{x}q^{y}}{q^{2}},0,\frac{(q^{x})^{2}-(q^{y})^{2}}{2q^{2}}\right).$
(10)

The crucial property is that, on a loop around the pinch point, the two
components $(B^{x},B^{z})$ as a 2D vector field form a vortex of winding
number 2. In Fig. 3(a), the normalized $(B^{x},B^{z},B^{y})/B(\bm{q})$ is
plotted. Here we swapped $B^{y}$ and $B^{z}$ for better visualization. This is
known to encode a Berry curvature of $\pm 2\pi$ [13]. Another way to see this
is to directly compute the Berry flux enclosed by a loop
$\bm{q}=q(\cos\theta,\sin\theta)$ around the gapless point, defined as

$C_{\pm}=\int_{0}^{2\pi}\text{d}\theta\
i\bm{\psi}_{\pm}^{\dagger}\cdot\partial_{\theta}\bm{\psi}_{\pm}.$ (11)

We conclude our proof here.

This proof can be intuitively generalized to a general $n-$fold pinch point.
In Fig. 1, we plot the smoothly varying $\bm{\psi}_{-}$ for different cases.
Note that for odd $n$, $\bm{\psi}_{-}$ needs anti-periodic boundary condition
instead.

Without the symmetry protection, the different components of an eigenvectors
can have different complex phases, so the proof above does not apply any
longer, and the Berry-flux at the gapless point is generally not quantized.
One example of such scenarios is to add different phases to $\psi^{1}$ and
$\psi^{2}$ in Eq. (8), which can yield a finite Berry flux contribution when
plugged into Eq. (11), or even render to total Berry zero. In this case,
$\bm{B}(\bm{q})/B$ still travels back and forth twice from the north pole to
south pole on the unit sphere for a path of $\bm{q}$ around the pinch point.
It can be, however, not two great circles, but some general curve. The solid
angle enclosed by the path, which is the Berry flux, is then not quantized.

Finally, the picture above also shows that other symmetry protections can also
work, if the normalized effective magnetic field is restricted to move on a
fixed great circle between the two poles.

Figure 3: Transformation of 2-fold pinch point and corresponding effective
magnetic field configuration. For better visibility we set the upper band to
be transparent and focus on the lower band. (a) The 2-fold pinch point and
winding number-2 magnetic field in the effective Hamiltonian. (b) Upon
introducing $\delta B^{z}$, the gapless point splits into two Dirac points.
The 2-fold pinch point splits into two 1-fold pinch point. (c) Upon
introducing $\delta B^{y}$, the two bands become gapped and pick up a Chern
number. The singularity disappears.

Splitting and gapping the pinch point. —  How the topologically critical
gapless points transform under different perturbations is a well-studied
topic. In this section we revisit some of these transformations, with a focus
on the corresponding pinch point phenomenology.

Fig. 3(b) shows a 2-fold pinch point spliting into two 1-fold pinch points
(Dirac cones). Correspondingly, the Berry curvature of $2\pi$ is also split
into the two 1-fold pinch points each carrying Barry flux $\pi$, assuming the
proper symmetry protection. In this process the overall Berry flux is
conserved.

Using our model for demonstration, this can be done by introducing a
perturbation of constant $B^{z}>0$ for the Hamiltonian in (9), and take
$\Delta(\bm{q})=cq^{2}$,

$\bm{B}_{\text{z-tuned}}(\bm{q})=\left(-cq^{x}q^{y},0,\frac{c}{2}((q^{x})^{2}-(q^{y})^{2})+\delta
B^{z}\right).$ (12)

The original gapless point at $\bm{q}=\bm{0}$ is then split into two linearly
dispersive gapless points at $\bm{q}=(0,\pm\sqrt{2\delta B^{z}/c})$. the
original winding number-$2$ vortex of $\bm{B}(\bm{q})$ splits into two
vortices, each with winding number $1$ [Fig. 3(b)], consistent with the Berry
flux conservation.

The critical gapless point can also be gapped, and induces well-defined,
opposite non-zero Berry curvature on the two bands locally. We can consider
perturbing the effective magnetic field in the following way,

$\bm{B}_{\text{y-tuned}}(\bm{q})=\left(-cq^{x}q^{y},\delta
B^{y},\frac{c}{2}((q^{x})^{2}-(q^{y})^{2})\right).$ (13)

As a consequence, the normalized $\bm{B}_{\text{y-tuned}}/B_{\text{y-tuned}}$
form half a skyrmion, or a meron, as illustrated in Fig. 3(c). Since the
skyrmion is of winding number $2$, the two bands get local Berry curvature
$\pm 2\pi$. The $\pm$ sign depends on the sign of $\delta B^{y}$, and cannot
be distinguished from the spectroscopy pattern.

The pinch point singularity disappears as the gap opens. The spectroscopic
intensity on two bands becomes smooth at the center, but gradually recovers
the pinch point pattern when zoomed out. An example of this will be discussed
in detail in a separate work studying the Kagome model [70]. Similar examples
can also be found in Refs. [71, 72].

Discussion. —  The main message of this work is that the $n-$fold pinch points
observed in energy-momentum resolved spectroscopy indicate that the gapless
point is topologically critical, and encodes Berry flux $n\pi$ if there is a
suitable symmetry protection. We have proven this for time reversal symmetry,
and also provided a survey of experiments (Table. 1) that observe the
universal phenomenon. This result does not rely on further microscopic details
of the system, hence has great potential in experimental application across
several platforms including ARPES, STM, and INS.

Another useful lesson is that the reversed conclusion is often but not always
true: a topologically critical gapless point may appear as a $n-$fold pinch
point. Some of these cases has been discussed in Refs. [26, 73, 66], although
the connection to pinch points were not mentioned. It is not always true,
because the winding of certain wavefunction may not be captured by the
correlation function measured by spectroscopy. The Honeycomb and Kagome
lattice magnon models studied in Ref. [74, 75, 58] are such examples, in which
some critical gapless points appear as 1FPP and 2FPP, but some are completely
dark in the structure factor due to extrinsic, accidental cancellation of form
factor.

The bigger picture we gain from this work is that the spectroscopy contains a
huge amount of information, of which a lot are still waiting to be exploited.
For example, instead of points, other topologically critical loci of band
degeneracy should also manifest universal, characteristic patterns. This idea
also applies to interacting or non-Hermitian Hamiltonians. Developing the
zoology of them will be an important and useful piece of phenomenological
study.

## Acknowledgment

We specially thank Nic Shannon for inspiring discussions at the beginning
stage of this work, and Owen Benton for enlightening discussions on Berry
flux. We also thank Andreas Thomasen for his helpful review of the paper. H.Y.
is supported by the Theory of Quantum Matter Unit at Okinawa Institute of
Science and Technology, the Japan Society for the Promotion of Science (JSPS)
Research Fellowships for Young Scientists, and the National Science Foundation
Division of Materials Research under the Award DMR-191751 at different stages
of this project.

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Appendix for “Identifying topologically critical band from pinch-point
singularities in spectroscopy”

In this section, we give a detailed proof of Eq. (8). That is, the squared-sin
pattern of the spectra distribution indicates that the corresponding
eigenvector has to be equivalent to Eq. (8), or those Fig. 2(a). The
configurations in Fig. 2(b) are forbidden.

To start with, we have the freedom to choose the basis $(1,0)$ to be the one
on which the spectroscopy measures the amplitude of the wavefunction. This is
because the spectroscopy is measuring the expectation value of certain
operator of scalar degree of freedom. So we have

$\begin{split}S_{+}(\bm{q})\propto|(1,0)\cdot(\psi^{1}_{+},\psi^{2}_{+})^{\mathrm{T}}(\bm{q})|^{2}=|\psi_{+}^{1}(\bm{q})|^{2},\\\
S_{-}(\bm{q})\propto|(1,0)\cdot(\psi^{1}_{-},\psi^{2}_{-})^{\mathrm{T}}(\bm{q})|^{2}=|\psi_{-}^{1}(\bm{q})|^{2}.\end{split}$
(S1)

In the example of graphene, $\psi^{1}\sim c_{A}+c_{B}$ is measured by ARPES,
and $\psi^{2}\sim c_{A}-c_{B}$. We also assume that the pinch point pattern is
in its most symmetric form

$\begin{split}S_{+}(\bm{q})=A\cos^{2}(\theta)=A\frac{(q^{y})^{2}}{q^{2}},\\\
S_{-}(\bm{q})=A\sin^{2}(\theta)=A\frac{(q^{x})^{2}}{q^{2}}.\end{split}$ (S2)

A small distortion of the pinch point in form of Eq. (3) does not affect our
conclusions on its topological features.

Combining Eq. (S1) and Eq. (S2), we see that

$|\psi_{-}^{1}(\bm{q})|\propto|q^{x}|/q=|\sin\theta|.$ (S3)

Since $\bm{\psi}_{-}$ is a unit vector, we then also know

$|\psi_{-}^{2}(\bm{q})|\propto|q^{y}|/q=|\cos\theta|,$ (S4)

There are several ways to arrange continuously varying $\bm{\psi_{-}}$ on a
circle around the pinch point and satisfy conditions of Eq. (S3) and Eq. (S4).
Some representatives are illustrated in Fig. 2(a,b).

The configurations in the same row are physically identical, since they can be
related to each other by introducing minus signs to the basis $\psi^{1}$ or
$\psi^{2}$. The configurations in Fig. 2(a), compared to those in Fig. 2(b),
are physically different: the first row has winding number $\pm 1$, and the
second has winding number $0$.

A key observation is that $\psi_{-}^{1}$ configurations with zero winding
number are not smooth, hence do not yield a smooth Hamiltonian. Taking the
first case of Fig. 2(b) as an example, $\psi_{-}^{1}$ takes a sharp turn at
$\theta=\pm\pi/2$ as shown in the Fig. 2(c). However, $\psi_{-}^{2}=\pm 1$,
and varies smoothly following a sine curve at these two point. The Hamiltonian
is

$\mathcal{H}(\bm{q})=\bm{S}\begin{pmatrix}\omega_{+}(\bm{q})&0\\\
0&\omega_{-}(\bm{q})\end{pmatrix}\bm{S}^{\dagger}.$ (S5)

where $\bm{S}=\left(\bm{\psi}_{+},\bm{\psi}_{-}\right)$. So its off-diagonal
terms component, containing $\omega_{-}\psi_{-}^{1}\psi_{-}^{2}$, is then not
smooth.

One may want to amend this problem by “softening” the sharp turn of
$\psi_{-}^{1}$ at $\theta=\pi/2,\ 3\pi/2$, hoping that it will yield a
distorted pinch point described by some function $\Theta(\theta)$ in Eq. (3)
and has a smooth Hamiltonian. However, in this case we always have
$\psi_{-}^{1\prime}=\psi_{-}^{2\prime}=0$, so that
$\Theta^{\prime}(\theta_{0})=0$ at the points $\Theta(\theta_{0})=\pi/2,\
3\pi/2$, which is not the proper pinch point pattern defined in Eq. (4)
anymore. Such unauthentic pinch points can be distinguished in experiments
since $\Theta(\theta)$ can be measured.

Therefore, the eigenvector configurations in Fig. 2(b) are forbidden. It has
to be equivalent to Eq. (8), or those Fig. 2(a), which encodes a Berry flux at
the gapless point as shown in the main text.