(a) We first consider the conventional crystalline symmetries and the
transpose symmetries with the generators all indicated by $g$. Since no global
symmetries are required to protect EPs in 2D non-Hermitian systems, unlike
Hermitian Hamiltonian preserving chiral symmetry, there is no algebra between
crystalline symmetry operators and other global symmetry operators. By using
the symmetry equation (55), the Hamiltonians at these two points possess the
identical characteristic polynomial

$\displaystyle\det\big{[}E-\mathcal{H}_{\rm{nH}}(\bm{k})\big{]}$
$\displaystyle=\det\big{[}E-U_{g}({\bm{k}})\mathcal{H}_{\rm{nH}}^{1/t}({\bm{k}})U_{g}^{-1}({\bm{k}})\big{]}$
$\displaystyle=\det\big{[}E-\mathcal{H}_{\rm{nH}}(g{\bm{k}})\big{]}.$ (56)

Hence, the discriminants at these two points are identical

$\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}]({\bm{k}})=\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}](g{\bm{k}}).$
(57)

As symmetry $g$ connects two EPs ($g\bm{K}_{\rm{EP}}$ and $\bm{K}_{\rm{EP}}$),
the relation of the discriminant numbers at these two points is given by

$\displaystyle\nu_{\rm{disc}}(g\bm{K}_{\rm{EP}})$
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{\rm{EP}}\right)}\det(g)d\Big{(}\ln\big{(}\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}](g{\bm{k}})\big{)}\Big{)}$
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{\rm{EP}}\right)}\det(g)d\Big{(}\ln\big{(}\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}]({\bm{k}})\big{)}\Big{)}$
$\displaystyle=\det(g)\nu_{\rm{disc}}({\bm{K}}_{\rm{EP}})$ (58)

The explicit relations between the two EPs for the two different types of the
crystalline symmetries are given by

$\nu_{\rm{disc}}(\hat{C}_{n}\bm{K}_{\rm{EP}})=\nu_{\rm{disc}}(\bm{K}_{\rm{EP}}),\nu_{\rm{disc}}(\hat{M}\bm{K}_{\rm{EP}})=-\nu_{\rm{disc}}(\bm{K}_{\rm{EP}}),$
(59)

which are identical to the relation of the generators $C_{n}^{+}$ and $M^{+}$
in Table 2 for the Hermitian chiral-symmetric systems without any time-
reversal symmetry. Thus, as shown in Table 6, the list of the EP no-go theorem
for the 17 WGs formed by $U_{g}$ and $U_{g}\mathcal{K}^{t}$ can inherit the
cases of $C_{n}^{+}$ and $M^{+}$ in class AIII in Table 3, except for EPs
located at rotation centers.

To understanding an EP located at a rotation center, we consider $n$-fold
rotation symmetry $C_{n}$. Being different from the Hermitian chiral symmetric
systems, the charge of the non-Hermitian EP at a rotation center, say
$\bm{K}_{\rm{EP}}^{r}$, is limited by an additional restriction. In order to
show the restriction, in the expression (54) of the discriminant number, the
integral path $\Gamma$ enclosing $\bm{K}_{\rm{FP}}^{r}$ is divided into $n$
integral paths related by the $C_{n}$ rotation symmetry

$\Gamma\left(\bm{K}_{\rm{FP}}^{r}\right)=\cup_{l=0}^{n-1}\hat{C}_{n}^{l}\Gamma_{1/n}\left(\bm{K}_{\rm{FP}}^{r}\right),$
(60)

where $\Gamma_{1/n}\left(\bm{K}_{\rm{FP}}^{r}\right)$ denotes the one n-th of
the closed integral path $\Gamma\left(\bm{K}_{\rm{FP}}^{r}\right)$ with the
two end points ${\bm{k}}_{o}$ and $\hat{C}_{n}{\bm{k}}_{o}$. After decomposing
the integral of the winding number to $n$ separate integral paths, we can
reorganize the discriminant number equation

$\displaystyle\nu_{\rm{disc}}(\bm{K}_{\rm{EP}}^{r})$
$\displaystyle=\frac{i}{2\pi}\int_{\Gamma_{1/n}\left(\bm{K}_{\rm{EP}}^{r}\right)}\sum_{l=1}^{n-1}d\Big{(}\ln\big{(}\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}](\hat{C}_{n}^{l}\bm{k})\big{)}\Big{)}$
$\displaystyle=n\times\frac{i}{2\pi}\oint_{\Gamma_{1/n}\left(\bm{K}_{\rm{EP}}^{r}\right)}d\Big{(}\ln\big{(}\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}](\bm{k})\big{)}\Big{)}$
$\displaystyle=nj.$ (61)

where $j$ is an integer. The second line stems from that the symmetry relation
(55) between the two points (${\bm{k}},\hat{C}_{n}{\bm{k}}$) leads to
$\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}]({\bm{k}})=\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}](\hat{C}_{n}{\bm{k}})$.
Since the integrands at ${\bm{k}}_{o}$ and $\hat{C}_{n}{\bm{k}}_{o}$ are
identical, the integral path of
$\Gamma_{1/n}\left(\bm{K}_{\rm{EP}}^{r}\right)$ is effectively closed so that
the integral is quantized. Hence, as an EP is located at a $n-$fold rotation
symmetry center, its discriminant number is a multiple of $n$. This constraint
for the value of the discriminant number does not allow the EPs to inherit
some of the minimal configurations from the chiral symmetric systems. For
example, for WG$\\#10$ with $C_{4}^{+}$, one of the chiral symmetric minimal
configurations is that four Dirac points with $\pm 1$ charges quantized can be
separately located at $\Gamma,X,Y,M$, which are rotation centers. However, the
$C_{4}$ rotation symmetry from WG$\\#10$ forbids any non-Hermitian EP with
$\pm 1$ charge to appear at any rotation center. Hence, this configuration is
excluded in Table 6. On the other hand, we note that the non-Hermitian cases
in Table 1 only show the no-go theorem for the 17 conventional WGs
($C_{n},M_{x}$).

For the transpose symmetries with $U_{g}U_{g}^{*}=-1$, we have to take a
special care of degeneracy points. Back to the ten-fold classification of the
Hermitian systems, the different values of the symmetry operator squares
($T^{2},C^{2}=\pm 1$) correspond to distinct AZ symmetry classes. In
particular, class AII ($T^{2}=-1$) leads to the Kramers degeneracy at TRI
points. On the contrary, the Kramers degeneracy does not hold in the time-
reversal symmetric non-Hermitian systems. Instead, only when transpose
symmetry with $U_{g}U_{g}^{*}=-1$ leads to $\mathcal{H}_{\rm
nH}(g\bm{k})=U_{g}\mathcal{H}_{\rm nH}^{t}(\bm{k})U_{g}^{-1}$, where
$g=C_{n},M_{i}$, these $g$-symmetry-invariant points must exhibit at least
two-fold non-defective degeneracy Kawabata  _et al._ (2019b). In other words,
the invariant points cannot be a two-fold defective degeneracy EP. Although
without any symmetry constraint a non-defective degeneracy point can be easily
deformed to an EP Yang  _et al._ (2021a), under this transpose symmetry with
$U_{g}U_{g}^{*}=-1$ the degeneracy point in the $g$-symmetry-invariant
location is non-defective.

(b) Consider the conjugation crystalline symmetries and the conjugate-
transpose ones; their generators are all indicated by $\bar{g}$. By using the
symmetry equation (55), we can directly connect the determinant at ${\bm{k}}$
and $\bar{g}{\bm{k}}$

$\displaystyle\det\big{[}E-\mathcal{H}_{\rm{nH}}(\bar{g}\bm{k})\big{]}$
$\displaystyle=\det\big{[}E-U_{\bar{g}}({\bm{k}})\mathcal{H}_{\rm{nH}}^{*/{\dagger}}({\bm{k}})U_{\bar{g}}^{-1}({\bm{k}})\big{]}$
$\displaystyle=\det\big{[}E-\mathcal{H}_{\rm{nH}}^{*}({\bm{k}})\big{]}.$ (62)

The conjugation adds an extra minus sign to the relation equation (58) between
the two symmetry-connected discriminant numbers

$\displaystyle\nu_{\rm{PF}}(\bar{g}\bm{K}_{\rm{EP}})=-\det(\bar{g})\nu({\bm{K}}_{\rm{EP}})$
(63)

Hence, in Table 6 the EP no-go theorem for the conjugate 17 WGs
($\bar{C}_{n},\bar{M}$) corresponds to the Hermitian no-go theorem for
$C_{n}^{-}$ and $M^{-}$ in class AIII as listed in Table 3.

(c) For a WG containing two generators, there are four groups based on the
combinations —
$(C_{n},M),(C_{n},\bar{M}),(\bar{C}_{n},M),(\bar{C}_{n},\bar{M})$. The last
two can be easily extended by the recipe above. Due to the relations between
the symmetry-connected EPs (58,63), the non-Hermitian no-go theorem for
$C_{n},\bar{M}$/$\bar{C}_{n},M$ inherits the Hermitian no-go theorem for
$C_{n}^{+},M^{-}$/$C_{n}^{-},M^{+}$ in class AIII. However, we note that the
constraint (61) for EP at rotation centers also excludes some minimal
configurations from the inheritance of $C_{n}^{+}$ in Table 3. For example,
WG$\\#14$ with $C_{3}$ and $M_{y}$, WG$\\#16$ with $\bar{C}_{6}$, and
WG$\\#17$ with $\bar{C}_{6}$ and $\bar{M}_{x}$ can have an EP at $K$ with $+3$
charge and another EP at $K^{\prime}$ with $-3$ charge and the the minimal
configuration of the two EPs connected by symmetries labeled by
$\tilde{6}(K,K^{\prime})$.

Table 6: The minimal absolute topological number $\nu_{\rm{abs}}$ for non-Hermitian “Exceptional points” and “Fermi Points”. In the column of rotation centers, $\bigtimes$ indicates the absence of the topological points with non-zero charges in the rotation centers, while the labels of the rotation centers with $n\mathbb{Z}$ indicates that topological points can be located at the listed centers and the topological numbers must be multiples of integer $n$. Label $\sim$ above some absolute numbers $\nu_{\rm{abs}}$ indicates all topological points connected by symmetries in the minimal configuration. “MLs” is an abbreviation for mirror lines and $G$ indicates topological points at general points. The non-zero $\nu_{\rm{abs}}$ without location specification denotes that all topological points in the minimal configuration are located at general points.  WG | Generators | Rotation centers | $\nu_{\rm{abs}}$
---|---|---|---
$\\#$1 |  | No centers | $2$
$\\#$2 | $C_{2}$ | $(\Gamma,X,Y,M):2\mathbb{Z}$ | 4 ($G$, $\Gamma,\ X,\ Y,$ or $M)$
| $\bar{C}_{2}$ | $\bigtimes$ | $\tilde{2}$
$\\#$3,4,5 | $M_{x}$ | $\bigtimes$ | $\tilde{2}$
| $\bar{M}_{x}$ | No centers | 2 (MLs, $\Gamma,\ X,\ Y,$ or $M)$ or $4$
$\\#$6,7,8,9 |  $C_{2}$ and $M_{x}$ | $\bigtimes$ | $\tilde{4}$
|  $C_{2}$ and $\bar{M}_{x}$ | $(\Gamma,X,Y,M):2\mathbb{Z}$ | 4 (MLs, $\Gamma,\ X,\ Y,$ or $M)$ or $8$
|  $\bar{C}_{2}$ and $M_{x}/\bar{M}_{x}$ | $\bigtimes$ |  $\tilde{2}$ MLs or $\tilde{4}$
$\\#$10 | $C_{4}$ |  $(\Gamma,M):4\mathbb{Z}$ & $(X,Y):2\mathbb{Z}$ | 8 ($G,\ \Gamma,\ X,\ Y,$ or $M)$
| $\bar{C}_{4}$ | $\bigtimes$ | $\tilde{4}$
$\\#$11,12 |  $C_{4}$ and $M_{x}$ | $\bigtimes$ | $\tilde{8}$
|  $C_{4}$ and $\bar{M}_{x}$ |  $(\Gamma,M):4\mathbb{Z}$ & $(X,Y):2\mathbb{Z}$ | 8 (MLs, $\Gamma,\ X,\ Y,$ or $M)$ or 16
|  $\bar{C}_{4}$ and $M_{x}/\bar{M}_{x}$ | $\bigtimes$ |  $\tilde{4}$ MLs or $\tilde{8}$
$\\#$13 | $C_{3}$ | $(\Gamma,K,K^{\prime}):3\mathbb{Z}$ | 6 $(G,\ \Gamma,\ K,\ K^{\prime},$ or $M_{1/2/3})$
| $\bar{C}_{3}$ | $\bigtimes$ | $0$
$\\#$14 |  $C_{3}$ and $M_{y}$ | ($K,K^{\prime}):3\mathbb{Z}$ |  $\tilde{6}$ or $\tilde{6}(K,K^{\prime})$
|  $C_{3}$ and $\bar{M}_{y}$ | $(\Gamma,K,K^{\prime}):3\mathbb{Z}$ | 6 $($MLs$,\ \Gamma,$ or $M_{1/2/3})$ or 12 $(G,\ K,$ or $K^{\prime})$
|  $\bar{C}_{3}$ and $M_{y}/\bar{M}_{y}$ | $\bigtimes$ | $0$
$\\#$15 |  $C_{3}$ and $M_{x}$ | $\bigtimes$ | $\tilde{6}$
|  $C_{3}$ and $\bar{M}_{x}$ | $(\Gamma,K,K^{\prime}):3\mathbb{Z}$ | 6 $($MLs$,\ \Gamma,\ K,\ K^{\prime},$ or $M_{1/2/3})$ or $12$
|  $\bar{C}_{3}$ and $M_{x}/\bar{M}_{x}$ | $\bigtimes$ | $0$
$\\#$16 | $C_{6}$ | $(\Gamma):6\mathbb{Z}~{}\&~{}(K,K^{\prime}):3\mathbb{Z}~{}\&~{}(M_{1/2/3}):2\mathbb{Z}$ | 12 $(G,\ \Gamma,\ K,\ K^{\prime},$ or $M_{1/2/3})$
| $\bar{C}_{6}$ | $(K,K^{\prime}):3\mathbb{Z}$ |  $\tilde{6}$ or $\tilde{6}(K,K^{\prime})$
$\\#$17 |  $C_{6}$ and $M_{x}$ | $\bigtimes$ | $\tilde{12}$
|  $C_{6}$ and $\bar{M}_{x}$ | $(\Gamma):6\mathbb{Z}~{}\&~{}(K,K^{\prime}):3\mathbb{Z}~{}\&~{}(M_{1/2/3}):2\mathbb{Z}$ | 12 $($MLs$,\ \Gamma,\ K,\ K^{\prime},$ or $M_{1/2/3})$ or $24$
|  $\bar{C}_{6}$ and $M_{x}$ | $\bigtimes$ |  $\tilde{6}$ MLs or $\tilde{12}$
|  $\bar{C}_{6}$ and $\bar{M}_{x}$ | $(K,K^{\prime}):3\mathbb{Z}$ |  $\tilde{6}$ MLs or $\tilde{6}(K,K^{\prime})$ or $\tilde{12}$

####  V.1.2 Examples

To host EPs, the minimal dimension of the non-Hermitian Hamiltonian must be
$2\times 2$, since at least two states coalesce. First, we calculate the
explicit form of the discriminant for a generic $2\times 2$ non-Hermitian
Hamiltonian

$\mathcal{H}_{2\times
2}({\bm{k}})=h_{0}({\bm{k}})\sigma_{0}+h_{x}({\bm{k}})\sigma_{x}+h_{y}({\bm{k}})\sigma_{y}+h_{z}({\bm{k}})\sigma_{z}.$
(64)

The eigenenergies are given by
$E_{\pm}=h_{0}\pm\sqrt{h_{x}^{2}+h_{y}^{2}+h_{z}^{2}}$. Using Eq. 53, we have
a simple form of the discriminant

$\displaystyle\operatorname{Disc}_{E}[\mathcal{H}_{2\times
2}]({\bm{k}})=(E_{+}-E_{-})^{2}$ $\displaystyle=$ $\displaystyle
4(h_{x}^{2}+h_{y}^{2}+h_{z}^{2})=4(h_{0}^{2}-\det\mathcal{H}_{2\times 2}).$
(65)

This form is employed to compute the discriminant for the following examples.

(a) We start with a $2\times 2$ off-diagonal Hamiltonian

$\mathcal{H}({\bm{k}})=\begin{pmatrix}0&1\\\ \cos k_{x}+\cos k_{y}+i(\cos
k_{x}-\cos k_{y})&0\end{pmatrix}.$ (66)

The Hamiltonian preserves the following symmetries

$\displaystyle\bar{C}_{4}\mathcal{H}^{*}(k_{x},k_{y})\bar{C}_{4}^{-1}$
$\displaystyle=\mathcal{H}(k_{y},-k_{x}),$ (67) $\displaystyle
M_{x}\mathcal{H}(k_{x},k_{y})M_{x}^{-1}$
$\displaystyle=\mathcal{H}(-k_{x},k_{y}),$ (68)

where $\bar{C}_{4}=\sigma_{0}$ and $M_{x}=\sigma_{0}$. Table 6 shows the
minimal configuration of WG$\\#11$ is either 4 EPs at the mirror lines or 8
EPs at general points.

To find EPs in this model, we obtain the discriminant
$\operatorname{Disc}_{E}[\mathcal{H}]({\bm{k}})=4[\cos k_{x}+\cos k_{y}+i(\cos
k_{x}-\cos k_{y})]$ by using Eq. 65. As the discriminant vanishes, the
Hamiltonian becomes a Jordan canonical form and the two eigenstates coalesce
as an EP. In this regard, the four EPs are located at $(\pm\pi/2,\pm\pi/2)$
and $(\pm\pi/2,\mp\pi/2)$ in the mirror lines. To compute the discriminant
number of the EPs, we particularly consider the EP at $(\pi/2,\pi/2)$. Near
the EP, the leading order of the discriminant is given by

$\operatorname{Disc}_{E}[\mathcal{H}]({\bm{k}})=-(1+i)dk_{x}-(1-i)dk_{y}+O(d{\bm{k}}^{2}),$
(69)

where $dk_{x/y}=k_{x/y}-\pi/2$. By the definition (54) of the discriminant
number, we have $\nu_{\rm disc}=+1$ for the EP at $(\pi/2,\pi/2)$. The
$\bar{C}_{4}$ rotation symmetry extends to this EP to the remaining three by
implementing the discriminant number relation (63). Hence, the EPs at
$(\pm\pi/2,\pm\pi/2)$/$(\pm\pi/2,\mp\pi/2)$ possess $+1/-1$ charge.

(b) Consider a non-Hermitian Hamiltonian preserving $C_{4}$ rotation symmetry
and $\bar{M}_{x}$ reflection symmetry

$\mathcal{H}({\bm{k}})=\begin{pmatrix}0&1\\\
(h({\bm{k}})-m)(h({\bm{k}})+m)&0\end{pmatrix},$ (70)

where $h({\bm{k}})=\cos k_{x}-\cos k_{y}+i\sin k_{x}\sin k_{y}$. We can check
that the Hamiltonian obeys

$\displaystyle C_{4}\mathcal{H}(k_{x},k_{y})C_{4}^{-1}$
$\displaystyle=\mathcal{H}(k_{y},-k_{x}),$ (71)
$\displaystyle\bar{M}_{x}\mathcal{H}^{*}(k_{x},k_{y})\bar{M}_{x}^{-1}$
$\displaystyle=\mathcal{H}(-k_{x},k_{y}),$ (72)

where $C_{4}=\sigma_{0}$ and $\bar{M}_{x}=\sigma_{0}$. According to Table 6,
for WG$\\#11$ there are three types of the EP minimal configurations — two EPs
at $\Gamma,\ M$ with $\nu_{\rm{disc}}=\pm 4$, eight EPs at mirror lines with
$\nu_{\rm{disc}}=\pm 1$, and sixteen EPs at general points with
$\nu_{\rm{disc}}=\pm 1$.

To study EPs in this model, we use Eq. 65 and then obtain the discriminant
$\operatorname{Disc}_{E}[\mathcal{H}]({\bm{k}})=4(h({\bm{k}})-m)(h({\bm{k}})+m)$.
First, starting with $m=0$, we find two EPs are located at $\Gamma$ and $M$.
The discriminant near these two points can be written as

$\displaystyle\operatorname{Disc}_{E}[\mathcal{H}]({\bm{k}})=$
$\displaystyle(k_{x}-ik_{y})^{4}+O({\bm{k}}^{5})$ (73) $\displaystyle=$
$\displaystyle(dk_{x\pi}+idk_{y\pi})^{4}+O(d{\bm{k}}_{\pi}^{5}),$ (74)

where $dk_{x\pi}=k_{x}-\pi,dk_{y\pi}=k_{y}-\pi$. Therefore, by using the
definition (54) of the discriminant number, $\nu_{\rm disc}=+4$ for the EP at
$\Gamma$ and $\nu_{\rm disc}=-4$ for the EP at $M$. This minimal configuration
possesses the EPs located at the $C_{4}$ rotation centers. The $C_{4}$
rotation symmetry (61) forces the discriminant numbers to be a multiple of
$4$.

When we increase $m$ from $0$, the EP at $\Gamma$ is split into four EPs at
$(0,\pm\cos^{-1}(1-m))$ and $(\pm\cos^{-1}(1-m),0)$ in the mirror lines. Since
the original EP at $\Gamma$ possesses $+4$ discriminant number and the four
EPs connected by the $C_{4}$ rotation symmetry share the same charge, each EP
has $+1$ discriminant number. Similarly, the EP at $M$ is split into four EPs
with $-1$ discriminant number at $(\pm\cos^{-1}(-1+m),\pi)$ and
$(\pi,\pm\cos^{-1}(-1+m))$.

Figure 16:  The spectrum of the dissipative Chern insulator shows the
locations of the EPs. (a) Eight EPs with $\nu_{\rm abs}=\pm 1$ appear as
$\sqrt{3}/2<\gamma_{z}<\sqrt{15/8}$. (b,c) The eight EPs merge to four EPs
with $\nu_{\rm abs}=0$ as $\gamma=\sqrt{15/8}$ and $\gamma=\sqrt{3}/2$
respectively. (d) In the region of $|\gamma_{z}|<\sqrt{3}/2$, the non-
Hermitian phase with a line gap is topological.

(c) Being a fundamental 2D example with non-trivial band topology, the Chern
insulator has been widely studied in Hermitian topological band theory. One of
the simplest Hermitian Hamiltonians describing the Chern insulator Qi  _et
al._ (2008) is given by

$\displaystyle\begin{aligned} \mathcal{H}_{\rm C}(\bm{k})&=\sin
k_{x}\sigma_{x}+\sin k_{y}\sigma_{y}\\\ &+(\cos k_{x}+\cos
k_{y}+M)\sigma_{z}.\end{aligned}$ (75)

In the region $0<|M|<2$, the nonzero Chern number leads to the presence of the
chiral edge states and the quantized conductance. Then we add on-site
dissipations uniformly to the Chern insulator so that the Hamiltonian becomes
non-Hermitian

$\displaystyle\begin{aligned} \mathcal{H}_{\rm nC}(\bm{k})=\mathcal{H}_{\rm
C}(\bm{k})-i\gamma_{z}\sigma_{z}-i\gamma_{0}\sigma_{0},\end{aligned}$ (76)

where $\gamma_{0}\pm\gamma_{z}$ are the on-site dissipations for the different
orbitals. Due to the dissipative nature, $\gamma_{0}\pm\gamma_{z}$ must be
greater than zero, namely $\gamma_{0}\pm\gamma_{z}\geq 0$. The reason is that
due to the dissipative energies $-i(\gamma_{0}\pm\gamma_{z})$, the
corresponding time evolution operator on each site exhibits exponential decay
trend ($e^{-(\gamma_{0}\pm\gamma_{z})t}$) as time evolves.

In the presence of the on-site dissipation, this non-Hermitian Chern insulator
preserves the following two symmetries $C_{4}\mathcal{H}_{\rm
nC}(k_{x},k_{y})C_{4}^{-1}=\mathcal{H}_{\rm nC}(k_{y},-k_{x})$ and
$M_{x}\mathcal{H}^{t}_{\rm nC}(k_{x},k_{y})M_{x}^{-1}=\mathcal{H}_{\rm
nC}(-k_{x},k_{y})$, where $C_{4}=e^{i\pi\sigma_{z}/4}$ and $M_{x}=\sigma_{z}$,
which belongs to WG$\\#11$. That is, the minimal number of EPs with charge
$\pm 1$ is given by 8 according to Table 6.

By Eq. 65, the discriminant can be written in the explicit form

$\displaystyle\operatorname{Disc}_{E}[\mathcal{H}]({\bm{k}})=$ (77)
$\displaystyle-4[\sin^{2}k_{x}+\sin^{2}k_{y}$ $\displaystyle+(\cos k_{x}+\cos
k_{y}+M-i\gamma_{z})^{2}].$

To demonstrate the evolution of the EPs, we particularly choose $\gamma_{0}=0$
and $M=1/2$ and start with $\gamma_{z}=\sqrt{7}/2$. The vanishing discriminant
determines eight EPs in the 2D BZ — ($\pm 2\pi/3,\pm\pi/2$), ($\pm
2\pi/3,\mp\pi/2$), ($\pm\pi/2,\pm 2\pi/3$), ($\pm\pi/2,\mp 2\pi/3$) as shown
in Fig. 16(a). Knowing one of the EP discriminant numbers, we can obtain the
other discriminant numbers by using $C_{4}$ rotation symmetry and $M_{x}$
transpose reflection symmetry, because the symmetry relations (59) connect the
discriminant numbers of the eight EPs. The function of the discriminant is
expanded to the linear order near ${\bm{k}}_{1}=(2\pi/3,\pi/2)$

$\displaystyle\operatorname{Disc}_{E}[\mathcal{H}]({\bm{k}})=$
$\displaystyle-2\sqrt{3}\left(1-i\sqrt{7}\right)(k_{x}-\frac{2\pi}{3})+4i\sqrt{7}\left(k_{y}-\frac{\pi}{2}\right)$
$\displaystyle+O\big{(}({\bm{k}}-{\bm{k}}_{1})^{2}\big{)}$ (78)

Using the definition (54) of the discriminant number, we have
$\nu_{\rm{disc}}({\bm{k}}_{1})=1$. In this regard, the $C_{4}$ rotation
symmetry duplicates the three EPs at ($-2\pi/3,-\pi/2$), ($\pm\pi/2,\pm
2\pi/3$) with $\nu_{\rm{disc}}({\bm{k}}_{1})=1$, while $M_{x}$ transpose
reflection symmetry leads to the four remaining EPs with
$\nu_{\rm{disc}}({\bm{k}}_{1})=-1$. The presence of the eight EPs is the
minimal configuration consistent with Table 6.

On the one hand, as we increase $\gamma_{z}$ to $\sqrt{15/8}$, the two EPs
from ($2\pi/3,\pi/2$) and $(\pi/2,2\pi/3)$ are merged into one EP at
$(\cos^{-1}(-1/4),\cos^{-1}(-1/4))$ as shown in Fig. 16(b). The charge at the
merging point is neutralized. The $C_{4}$ rotation symmetry duplicates the
merging to the six remaining EPs; there are four EPs with zero discriminant
numbers located at $(\pm\cos^{-1}(-1/4),\pm\cos^{-1}(-1/4))$ and
$(\pm\cos^{-1}(-1/4),\mp\cos^{-1}(-1/4))$. When $\gamma_{z}>\sqrt{15/8}$, the
four neutralized EPs vanish and the non-Hermitian system has a line gap.

On the other hand, as $\gamma_{z}$ is decreased to $\sqrt{3}/2$, the two EPs
at ($\pm 2\pi/3,\pi/2$) merged into one EP at ($\pi,\pi/3$) and the merge
neutralizes the charge. Similarly, by merging the six remaining EP are reduced
to three neutralized EPs separately located at ($\pi,-\pi/3$) and
($\pm\pi/3,\pi$) as shown in Fig. 16(c). As $\gamma_{z}<\sqrt{3}/2$, the four
EPs vanish and the bulk spectrum has a line gap as shown in panel (d). In the
open boundary condition the robust edge states connect the two separate bulk
region in the complex plane; hence, this gapped phase is topologically non-
trivial Shen  _et al._ (2018a).

In short, $|\gamma_{z}|<\sqrt{3}/2$ is the topological region, while
$\gamma_{z}>\sqrt{15/8}$ is the trivial region. As
$\sqrt{3}/2<\gamma_{z}<\sqrt{15/8}$, the 8 EPs with $\nu_{\rm{disc}}=\pm 1$
represent the topological phase transition region. Therefore, the EP minimal
configurations listed in Table 6 can indicate the topological phase transition
for the non-Hermitian line-gap phases.

###  V.2 Non-Hermitian FPs

We briefly introduce the topological protection for FPs in 2D non-Hermitian
models, which was recently studied in the literature Yang  _et al._ (2021a);
Kawabata  _et al._ (2019a). Next, we directly extend our new no-go theorem of
the 17 WGs to the FPs.

The FPs are the direct generalization of Fermi surface from Hermitian to non-
Hermitian systems. We first recall the Fermi surface defined for a Hermitian
Hamiltonian $\mathcal{H}_{\rm{H}}(\bm{k})$. In the 2D BZ the Fermi surface,
which is determined by $\det[\mathcal{H}_{\rm{H}}(\bm{k})-\mu]=0$, is 1D
closed lines. Given a complex chemical potential $\mu$ and a non-Hermitian
Hamiltonian $\mathcal{H}_{\rm{nH}}(\bm{k})$, the non-Hermitian Fermi surface
is located at ${\bm{k}}$ satisfying $E_{n}(\bm{k})-\mu=0$. That is, the two
constraints of the equation in the real and imaginary parts form two separate
closed lines in the BZ and the non-Hermitian Fermi surface is the crossing
points of the two lines; hence, we call Fermi points (FPs) for non-Hermitian
Fermi surface in 2D non-Hermitian systems. This is different from 1D Fermi
lines in 2D Hermitian systems, since there exists a well-defined invariant
characterizing any non-Hermitian FP in the BZ. To identify all of the FPs
${\bm{K}}_{\rm{FP}}$ in the BZ, we use determinant

$\det[\mathcal{H}_{\rm{nH}}({\bm{K}}_{\rm{FP}})-\mu]=\prod_{n}(E_{n}({\bm{K}}_{\rm{FP}})-\mu)=0.$
(79)

These FPs ${\bm{K}}_{\rm{FP}}$ can be characterized by winding number as an
invariant

$\nu_{\rm{FP}}\left({\bm{K}}_{\rm{FP}}\right)=\frac{i}{2\pi}\oint_{\Gamma\left({\bm{K}}_{\rm{FP}}\right)}\nabla_{{\bm{k}}}(\ln\det[\mathcal{H}_{\rm{nH}}(\bm{k})-\mu])\cdot
d{\bm{k}},$ (80)

where $\Gamma({\bm{K}}_{\rm{FP}})$ is a loop counterclockwise encircling the
FP at ${\bm{K}}_{\rm{FP}}$. Since in a proper basis the Hamiltonian
$\mathcal{H}_{nH}(\bm{k})$ is single-valued in the entire BZ and the integrand
is a singularity at the FP, the winding number $\nu_{\rm{FP}}$ of the FP is
always quantized as an integer. For any non-zero winding number, the integral
loop is not contractable so that the FP ${\bm{K}}_{\rm{FP}}$ inside the loop
is a stable singularity point, which cannot be removed. Furthermore, only FPs
with non-zero winding numbers are singularities in the entire BZ.

Here we point out that the (non-Hermitian) FPs and the (Hermitian) Dirac
points share the identical mathematical structure. In order to see this, we
map the non-Hermitian Hamiltonian $\mathcal{H}_{\rm{nH}}(\bm{k})$ to a
Hermitian Hamiltonian with chiral symmetry Gong  _et al._ (2018)

$\mathcal{H}_{\rm{H}}(\bm{k})=\left(\begin{array}[]{cc}{0}&{\mathcal{H}_{\rm{nH}}(\bm{k})-\mu}\\\
{\mathcal{H}_{\rm{nH}}^{\dagger}(\bm{k})-\mu^{*}}&{0}\end{array}\right).$ (81)

The zero energy nodes of the Hermitian Hamiltonian
$\mathcal{H}_{\rm{H}}(\bm{k})$ appear when the determinant of the off-diagonal
matrix vanishes $\det[\mathcal{H}_{\rm{nH}}(\bm{k})-\mu]=0$. The equation is
exactly identical to Eq. 79 determining the non-Hermitian FPs. Furthermore,
the definitions of the topological invariants (14) for the Hermitian nodes and
the non-Hermitian FPs are the same. Hence, these non-Hermitian FPs in
$\mathcal{H}_{nH}(\bm{k})$ are mathematically equivalent to the nodal points
in $\mathcal{H}_{H}(\bm{k})$ with the same winding numbers. The no-go theorem
for chiral symmetry can be directly extended to the non-Hermitian FP no-go
theorem Yang  _et al._ (2021a) described by

$\sum_{j}\nu_{\rm{FP}}({\bm{K}}_{\rm{FP}}^{j})=0$ (82)

This charge neutralization manifests that in the BZ an FP with non-zero
winding number must be accompanied by another FP with the opposite winding
number.

####  V.2.1 Conditions from crystalline symmetries

To study the relations of FPs connected by symmetries, we consider four types
of the combination crystalline symmetries — conventional crystalline,
transpose crystalline, complex-conjugation, and conjugate-transpose
symmetries. We use $g$ to indicate the generators of the first two symmetries
($U_{g},U_{g}\mathcal{K}^{t}$), while $\bar{g}$ is used to indicate the
generators of the last two symmetries
($U_{g}^{*},U_{g}\mathcal{K}^{\dagger}$). Transferring from the EP no-go
theorem for the 17 WGs to the FP cases, we simply replace the discriminant in
the integrand of the discriminant number (54) by determinant for the FP
winding number (80)

$\operatorname{Disc}_{E}[\mathcal{H}_{\rm{nH}}]({\bm{k}})\rightarrow\det[\mathcal{H}_{\rm{nH}}(\bm{k})-\mu].$
(83)

It is straightforward that the FP no-go theorem follows the same rules of the
EPs as listed in Table 6 and we use the absolute number
$\nu_{\rm{abs}}=\sum_{{\bm{k}}_{\rm{FP}}^{i}\in\rm{BZ}}|\nu_{\rm{FP}}({\bm{k}}_{\rm{FP}}^{i})|$
to express the FP no-go theorem. The reason of the identical rules is that the
relations between two FPs (${\bm{K}}_{\rm{FP}}$ and $g{\bm{K}}_{\rm{FP}}$)
connected by the symmetries are identical to the EP cases (58,63)

$\displaystyle\nu_{\rm{FP}}(g\bm{K}_{\rm{FP}})=$
$\displaystyle\det(g)\nu({\bm{K}}_{\rm{FP}})$ (84)
$\displaystyle\nu_{\rm{FP}}(\bar{g}\bm{K}_{\rm{FP}})=$
$\displaystyle-\det(\bar{g})\nu({\bm{K}}_{\rm{PF}})$ (85)

and for $C_{n}$ the charges of the FPs at rotation centers
($\bm{K}_{\rm{FP}}^{r}$) are limited by the same constraint (61)

$\displaystyle\nu_{\rm{FP}}(\bm{K}_{\rm{FP}}^{r})=nj.$ (86)

The only difference between the FPs and the EPs is that the chemical potential
has to be included in the symmetry equations (55) to build the FP connections.
For $g$,

$\displaystyle\det[\mathcal{H}_{nH}(g\bm{k})-\mu]$
$\displaystyle=\det[U_{g}\mathcal{H}_{nH}^{1/t}(\bm{k})U^{-1}_{g}-\mu]$
$\displaystyle=\det[\mathcal{H}_{nH}(\bm{k})-\mu].$ (87)

The chemical potential $\mu$ can be any complex number, then the FP relation
(84) still holds. On the other hand, for $\bar{g}$, there is an additional
complex conjugation so that

$\displaystyle\det[\mathcal{H}_{nH}(\bar{g}\bm{k})-\mu]$
$\displaystyle=\det[U_{\bar{g}}\mathcal{H}^{*/{\dagger}}_{nH}(\bm{k})U^{-1}_{\bar{g}}-\mu]$
$\displaystyle=\det[\mathcal{H}_{nH}(\bm{k})-\mu^{*}]^{*},$ (88)

To satisfy the FP relation (85), the chemical potential must be real; hence,
the Fermi energy of the FP has to be a real number.

####  V.2.2 Examples

We use three examples to demonstrate that our no-go theorem in table 6 covers
the minimal configurations of the non-Hermitian FPs.

Figure 17:  FPs with non-zero charges ($\nu_{\rm{abs}}$) are located at the
crossings of the red lines (${\rm Re}(\det(H({\bm{k}})-\mu))=0$) and the blue
lines (${\rm Im}(\det(H({\bm{k}})-\mu))=0$). (a) Two FPs with $\nu_{\rm
abs}=\pm 2$ at the rotation centers are split into four FPs with $\nu_{\rm
abs}=\pm 1$ in the mirror lines. Finally, these FPs merge to one FP with
$\nu_{\rm abs}=0$. (b) A FP with $\nu_{\rm abs}=0$ is split into four FPs with
$\nu_{\rm abs}=\pm 1$ in the mirror lines. (c) Eight FPs $\nu_{\rm abs}=\pm 1$
are the general points.

(a) We first consider a one-band model with the Hamiltonian

$H({\bm{k}})-\mu=\cos k_{x}-\cos k_{y}-\mu+i\sin k_{x}\sin k_{y},$ (89)

where $\mu$ is the chemical potential and imposed to be real to preserve
$\bar{M}$ symmetry. By checking the symmetries, the Hamiltonian obeys

$C_{2}(H(k_{x},k_{y})-\mu)C_{2}^{-1}=H(-k_{x},-k_{y})-\mu,$ (90)

where $C_{2}=1$, and

$\bar{M}_{x}(H(k_{x},k_{y})-\mu)\bar{M}_{x}^{-1}=H(-k_{x},k_{y})-\mu,,$ (91)

where $\bar{M}_{x}=\mathcal{K}^{*}$. Hence, the system belongs to WG$\\#6$
with $C_{2}$ and $\bar{M}_{x}$. When $\mu=0$, two FPs are located at $(0,0)$
and $(\pi,\pi)$ separately. By expanding the energy near these points, we have

$\displaystyle E({\bm{k}})=$
$\displaystyle-(k_{x}-ik_{y})^{2}/2+O({\bm{k}}^{2})$ (92) $\displaystyle=$
$\displaystyle-(dk_{x\pi}+idk_{y\pi})^{2}/2+O(d{\bm{k}}_{\pi}^{2}),$ (93)

where $dk_{x\pi}=k_{x}-\pi,dk_{y\pi}=k_{y}-\pi$. Hence, by using the winding
number equation (80) of the FP, the FP at $(0,0)$ possesses $+2$ charge, while
the one at $(\pi,\pi)$ possesses $-2$ charge. The even numbers of the charges
at the $C_{2}$ rotation centers is consistent with the rotation constraint
(86).

Now we adjust the value of the chemical potential to $0<\mu<2$. The two FPs at
$\mu=0$ are split into four FPs located at
$(0,\pm\cos^{-1}(1-\mu)),(\pm\cos^{-1}(-1+\mu),\pi)$ as shown in Fig. 17(a).
The FPs at $(0,\pm\cos^{-1}(1-\mu))$, which inherit the charge from the FP at
$(0,0)$, both have $+1$ winding number. Similarly, the FPs at
$(\pm\cos^{-1}(-1+\mu),\pi)$ have $-1$ winding number. The distribution of the
four FPs at the mirror lines is consistent with WG$\\#6$ with $C_{2}$ and
$\bar{M}_{x}$ in Table 6. Finally, as $\mu=2$, the four FPs merge at $(0,\pi)$
and the total charge of this single FP is neutralized.

(b) For the second example, we change the one-band model to

$H({\bm{k}})-\mu=\cos k_{x}+\cos k_{y}-\mu+i(\cos k_{x}-\cos k_{y})$ (94)

The Hamiltonian with real chemical potential $\mu$ obeys

$\displaystyle\bar{C}_{4}(H(k_{x},k_{y})-\mu)\bar{C}_{4}^{-1}$
$\displaystyle=H(k_{y},-k_{x})-\mu,$ (95) $\displaystyle
M_{x}(H(k_{x},k_{y})-\mu)M_{x}^{-1}$ $\displaystyle=H(-k_{x},k_{y})-\mu,$ (96)

where $C_{4}^{*}=\mathcal{K}^{*}$ and $M_{x}=1$. Hence, the system belongs to
WG$\\#10$ with $\bar{C}_{4}$ and $M_{x}$.

We start with the chemical potential $\mu=2$. The FP satisfying
$E({\bm{K}}_{\rm{PF}})=\mu$ is located at $(0,0)$. Because this is the only FP
in the entire system, to preserve the charge neutralization (82) the winding
number of this FP must be zero. Then as $\mu$ is decreased, the FP at $(0,0)$
is split into four FPs located at
$(\pm\cos^{-1}\frac{\mu}{2},\pm\cos^{-1}\frac{\mu}{2})$ and
$(\pm\cos^{-1}\frac{\mu}{2},\mp\cos^{-1}\frac{\mu}{2})$ as shown in Fig.
17(b). To identify the charge of the FP, we particularly consider the FP at
$(\pi/2,\pi/2)$ for $\mu=0$. The leading order of the energy with respect to
the chemical potential can be written as

$E({\bm{k}})-\mu=-(1+i)dk_{x}-(1-i)dk_{y}+O(d{\bm{k}}^{2}),$ (97)

where $dk_{x/y}=k_{x/y}-\pi/2$. By using the definition (80) of the winding
number, the charge of the FP at $(\pi/2,\pi/2)$ is given by $+1$. The
$\bar{C}_{4}$ rotation and reflection symmetries extend to this FP to the
remaining three with the winding number relation (85). Hence, the FPs at
$(\pm\cos^{-1}\frac{\mu}{2},\pm\cos^{-1}\frac{\mu}{2})$/$(\pm\cos^{-1}\frac{\mu}{2},\mp\cos^{-1}\frac{\mu}{2})$
possess $+1/-1$ charge.

(c) Consider a non-Hermitian Chern insulator and its Hamiltonian
$H_{C}({\bm{k}})$ is written in Eq. 76. It has been shown in EP example (c)
that the Chern insulator Hamiltonian preserves $C_{4}$ rotation symmetry and
$M_{x}$ mirror symmetry with $C_{4}=e^{i\pi\sigma_{z}/4}$ and
$M_{x}=\sigma_{z}$. Even in the presence of the complex chemical potential
$\mu$, the symmetries are still preserved. According to Table 6, the minimal
number of FPs with charge $\pm 1$ is given by 8. To show the distribution of
the FPs concretely, consider $\gamma_{0}=0$, $\gamma_{z}=\sqrt{3}/2$, $M=1/2$,
and $\mu=1$. Therefore, the Fermi momentum ${\bm{K}}_{\rm{FP}}$ is determined
by $\det(\mathcal{H}_{\rm C}({\bm{K}}_{\rm{PF}})-1)=0$. One of the FPs is
located at ${\bm{K}}^{++}_{\rm{FP}}=(\pi/2,2\pi/3)$. Near this point, we
expand the determinant to the leading order

$\det\big{(}\mathcal{H}_{\rm
C}({\bm{k}})-\mu\big{)}=-i\frac{\sqrt{3}}{2}dk_{x}+\frac{\sqrt{3}-3i}{2}dk_{y}+O(d\bm{k}^{2}),$
(98)

where $dk_{x}=k_{x}-\pi/2$ and $dk_{y}=k_{y}-2\pi/3$. The linear terms
determine the winding number $\nu({\bm{K}}^{++}_{\rm{FP}})=-1$. The $C_{4}$
rotation symmetry generates other three FPs with $\nu=-1$ located at
$(-\pi/2,-2\pi/3),(\mp 2\pi/3,\pm\pi/2)$. Furthermore, the transpose
reflection symmetry extends 4 additional FPs with $\nu=+1$ located at
$(\pm\pi/2,\mp 2\pi/3)$ and $(\pm 2\pi/3,\pm\pi/2)$ as shown in Fig. 17(c).

##  VI Applications

The generalized no-go theorem can lead to three useful applications in
condensed matter physics — layer groups, anomalous topological surface, and
twisted bilayer physics.

###  VI.1 Layer groups

LG$\\#$ | $1,4,5$ | $2,3,6,7$ | $8-13,27-36$ | $14-26,37-48$ | 49-52 | 53-64 | 65,74 | 67,69,78 | 68,70,79 | 66,73,75 | 71,72,76,77,80
---|---|---|---|---|---|---|---|---|---|---|---
WG$\\#$ | $1$ | $2$ | $3,4,5$ | $6,7,8,9$ | $10$ | $11,12$ | $13$ | $14$ | $15$ | $16$ | $17$
Table 7: The table shows the corresponding wallpaper groups from the
projection of the 80 layer groups. Each layer follows the no-go theorem of its
wallpaper group projection.

In chiral-symmetric, space-time-inversion symmetric and non-Hermitian systems,
the no-go theorem for the 17 wallpaper groups can be directly extended to the
80 layer groups. Compared with wallpaper groups, layer groups have an
additional degree of freedom in $z$ direction without translational symmetry.
With this additional direction, some layer groups can preserve $z$-flipping
symmetries de la Flor  _et al._ (2021), which can be reflection symmetry
$M_{z}$, rotation symmetries ($C_{2x},C_{2y}$), 3D inversion symmetry
($\mathcal{I}$). Each layer group can be formed by one of the wallpaper groups
with/without one $z$-flipping symmetry generator Hitzer and Ichikawa (2013).

The $z$-flipping symmetry operation contains $M_{z}$ operation. Here we assume
$M_{z}$ commutes with chiral operator ($S$) and 2D space-time inversion
operator ($C_{2z}T$). For the discussion associated with the no-go theorem of
the layer groups, for any related $z$-flipping symmetry operation, the effect
of the $z$-flipping can be neglected, and the remaining operation in the other
directions are kept. That is, $M_{z}\rightarrow$ no change;
$C_{2x},C_{2y}\rightarrow M_{y},M_{x}$; $\mathcal{I}\rightarrow C_{2z}$. In
this regard, the layer group can be projected to one of the wallpaper groups.
Table 7 shows the wallpaper group projections from all the 80 layer groups.
The no-go theorem of the layer group follows the corresponding wallpaper group
projection; this projection can be used for nodal points protected by chiral
symmetry and space-time inversion symmetry as well as non-Hermitian
exceptional points and Fermi points.

By treating the layer structure as the 2D lattice, we can project any layer
group to one of the wallpaper groups. Its no-go theorem inherits the theorem
of the corresponding wallpaper group. Let us use 2 layer groups as examples to
demonstrate the 2D projection.

$LG\\#48$ has four symmetry generators — $C_{2z}$ rotation, $M_{x}$
reflection, $T_{\frac{y}{2}}M_{z}$ glide reflection, $T_{\frac{x+y}{2}}$ half-
lattice-constant translation. Global symmetry operations are usually
independent of crystalline symmetry operations. Therefore, we can assume
$M_{z}T_{\frac{y}{2}}$ and $T_{\frac{x+y}{2}}$ commute with global symmetry
operations so that $M_{z}T_{\frac{y}{2}}$ and $T_{\frac{x+y}{2}}$ do not
affect the constraints of the nodal points. This layer group can be projected
into WG$\\#9$ with $C_{2z},\ M_{x},\ T_{\frac{x+y}{2}}$ and its no-go theorem
is based on WG$\\#9$.

$LG\\#54$ possesses two symmetry generators — $C_{4z}$ rotation,
$T_{\frac{x+y}{2}}C_{2y}$ screw rotation. Although $T_{\frac{x+y}{2}}C_{2y}$
flips the $z$ direction, this operator can be projected to
$T_{\frac{x+y}{2}}M_{x}$. Therefore, the projection group is $WG\\#12$.

###  VI.2 Anomalous topological surface

The no-go theorem of the wallpaper groups also serves as clear guidance to
search for 3D non-trivial topological phases with anomalous surfaces. This
anomalous surface stems from the breaking of the 2D surface no-go theorem. The
reason is that the surface BZ (SBZ) can have boundaries and does not cover the
entire 2D BZ without boundaries, as shown in Fig. 18. Therefore, including all
charges in the SBZ, the integral path can not vanish due to the emergence of
the SBZ boundaries

$\sum_{{\bm{k}}_{i}\in\rm{SBZ}}C({\bm{k}}_{i})\neq 0.$ (99)

The violation of the charge neutralization can occur in the SBZ and brings
anomaly Hsieh  _et al._ (2016); Witten (2016). However, symmetries in the
wallpaper group might impose additional restrictions to remove the anomaly.
First, in the generalized no-go theorem tables (3, 4, 6, 5), “0” indicates the
absence of the topological points for the corresponding WG symmetry classes;
hence, the anomalous surface is forbidden so that the bulk is trivial. Second,
$\sim$ indicates all nodal points connected by symmetries in the minimal
configuration; meanwhile, the area of the SBZ also respects the symmetries.
The SBZ either includes or excludes all the nodal points connected by the
symmetries, as illustrated in Fig. 18(a). Thus, the charges are always
neutralized, and the anomalous surface is absent.

Figure 18: (color online) Topological points (nodal points, Fermi points, and
exceptional points) are located on the top surface. (a) Because two
topological points are connected by $\bar{C}_{2}$ rotation symmetry, the total
charges on the surface are neutralized. (b) Although in the entire 3D system
the total charges is neutralized in the presence of the green topological
points in the bulk, charge $+2$ on the top surface violates the no-go theorem.
(c) A topological point with charge $+1$ is located at a TRI point. (d) A
topological point (quadratic-dispersion nodal point, FP, and EP) with charge
$+2$ is located at a TRI point.

The way to realize the anomalous surface is to have all nodal points in the
minimal configuration NOT connected by symmetries. In the tables, the non-zero
numbers without $\sim$ indicate this case. Although the entire 3D system obeys
the 2D no-go theorem, some of the topological points can be outside of the SBZ
without breaking any symmetries, as illustrated in Fig. 18(b). In the SBZ, the
charges of the remaining topological points can violate the neutralization,
and the anomalous surface leads to the non-trivial bulk. The 3D bulk
topological invariant is given by the sum of the nodal-point charges on the
surface

$\nu_{\rm{3D}}=\sum_{{\bm{k}}_{i}\in{\rm SBZ}}C({\bm{k}}_{i})$ (100)

Let us investigate the non-trivial bulks by studying the violation of our
generalized no-go theorem on the surfaces for Hermitian and non-Hermitian
systems.

(a) For chiral symmetry, there are five AZ symmetry classes possessing stable
nodal points at zero energy. Since the absolute numbers in class BDI and CII
are either $0$ or number labeled by $\sim$, the anomalous surface cannot exist
in these classes. The trivial 3D topology in these two symmetry classes is
consistent with the ten-fold classification Kitaev (2009); Schnyder  _et al._
(2008). On the other hand, for class AIII, DIII, and CI, to realize the
anomalous surface, for a WG preserving rotation symmetry or reflection
symmetry, it is required that rotation operator $C_{n}^{+}$ commutes with $S$
and reflection operator $M_{i}^{-}$ anticommutes with $S$. Otherwise, the
crystalline symmetries always keep either the charge-neutralization or the
trivialization on the surface. On the anomalous surface, the sum
$\nu_{\rm{3D}}$ of the nodal-point charges is equal to the 3D winding number
quantized by chiral symmetry Ryu  _et al._ (2010). Hence, for any WGs with
$C_{n}^{+}$ symmetry or/and $M_{i}^{-}$ symmetry, chiral-symmetric topological
insulators/superconductors in class AIII, DIII, or CI are characterized by
this $\mathbb{Z}$ winding number. Particularly, $C_{2}^{+}$-inversion-
symmetric and $M_{i}^{+}$-refection-symmetric phases with the winding number
have been studied in the literature Shiozaki and Sato (2014); Chiu  _et al._
(2013).

Since nodal points off TRI points are connected by symmetries, to realize the
minimal unneutralized charges, the nodal point has to be located at a TRI
point, as shown in Fig. 18(c). For class AIII and DIII, the charge can be $\pm
1$ so that the $\mathbb{Z}$ winding number characterizes their bulk topology.
On the other hand, since Table 4 shows that for class CI the charge at a TRI
must be an even number as illustrated in Fig. 18(d), the $2\mathbb{Z}$ winding
number for the bulk topology is consistent with the ten-fold classification
Chiu  _et al._ (2016).

(b) For space-time inversion symmetry, Dirac nodes off TRI points for the 10
WGs preserving inversion symmetry are always connected by symmetries. Hence,
on the SBZ the total charges are neutralized. On the other hand, while for
class AI time-reversal symmetry forbids any Dirac node at any TRI points, for
class AII Dirac nodes can be present at TRI points; the SBZ can include only
one TRI point without breaking symmetries as shown in Fig. 18(c), so the
anomalous surface can exist only in class AII. Furthermore, since the Dirac
cone is characterized by the $\mathbb{Z}_{2}$ invariant (Berry phase), the
bulk invariant in class AII with $C_{2}$ rotation symmetry has
$\mathbb{Z}_{2}$ invariant, which is consistent with the literature Shiozaki
and Sato (2014).

(c) For non-Hermitian systems, to realize unneutralized exceptional points and
Fermi points on the surface, WG generators must be one or two of $C_{n}$ and
$\bar{M}_{x}$, which correspond to the absolute numbers without $\sim$ in
Table 6. The reason is that $\bar{C}_{n}$ or/and $M_{x}$ connect all of the
topological points in the minimal configuration or trivialize the points. This
is in agreement with that the non-trivial 3D bulk with the non-Hermitian
anomalous surface is forbidden by reflection symmetry Yang  _et al._ (2021b).
Back to the anomalous surface, $C_{n}$ rotation symmetry leads to a
topological point with $n\mathbb{Z}$ charge at the rotation center or $n$
symmetry-connected topological points with $\mathbb{Z}$ charges off the
rotation center as illustrated in Fig. 18 (b) and (d). The total charge on the
surface is always $n\mathbb{Z}$, and the 3D topological invariant is a
multiple of $n$

$\nu_{\rm{3D}}=nj,\ j\in\mathbb{Z}$ (101)

due to $C_{n}$ rotation symmetry.

The no-go theorem violations of Fermi points and exceptional points bring
distinct non-trivial bulks. To keep simplicity, we assume the absence of the
skin effect. First, for the anomalous Fermi-point surface, the bulk
Hamiltonian $H(\vec{k})$ must have a point gap where unneutralized Fermi
points with energy $E$ are located. The topology of the bulk can be described
by a 3D $\mathbb{Z}$ topological invariant Denner  _et al._ (2021); Kawabata
_et al._ (2019b); Gong  _et al._ (2018)

$W_{3D}=-\int_{\rm{3DBZ}}\frac{d^{3}\vec{k}}{(2\pi)^{3}}\epsilon_{ijl}{\rm{Tr}}[Q_{i}(\vec{k})Q_{j}(\vec{k})Q_{l}(\vec{k})]$
(102)

where $Q_{i}(\vec{k})=(H(\vec{k})-E)^{-1}\partial_{k_{i}}(H(\vec{k})-E)$. The
sum of the Fermi-point charges on the anomalous surface corresponds to the
integer value of the 3D topological invariant. On the other hand, different
from Fermi points at fixed energy, the exceptional points on the surface in
the entire energy spectrum are counted to examine the violation of the no-go
theorem as an anomaly. The violation leads to the non-trivial topological
bulk. The sum of the exceptional-point charges on the anomalous surface
corresponds to another $\mathbb{Z}$ bulk topological invariant, which is
unknown in the literature. The hint of the no-go theorem leads to a new
direction to study this new non-Hermitian topological phase.

##  VII Twisted Bilayer Physics

Two graphene-monolayers with a small twisted angle form Moiré superlattice. At
the magic angle, the emergence of the flat bands in the Moiré BZ has become an
important playground to explore strongly correlated physics. In the two
monolayers, the Dirac nodes located at $K,\ K^{\prime}$ evolve to complete
flat bands in the entire Moiré BZ through the interlayer coupling. The
presence of the Dirac nodes is one of the key elements for the band flatness.
Our generalized no-go theorem can guide us to find other platforms hosting the
similar Dirac nodes sorted by symmetries and can extend to square lattice and
non-Hermitian platforms.

As discussed previously in sec. III.5.3, graphene without spin-orbital
coupling can be simplified to an effective spinless system, which preserves
time-reversal symmetry with $T^{2}=1$ and sublattice (chiral) symmetry, and
this monolayer system belongs to WG$\\#17$ in class BDI with $C_{6}^{-}$ and
$M_{x}^{-}$. To realize alternative similar twisted bilayer systems, it is
worth investigating if other platforms can host only two Dirac nodes at $K,\
K^{\prime}$. Since the symmetry connection of the two nodes ($\sim$) is not a
factor affecting the Dirac node locations, we are looking for these two
configurations labeled by $2(K,K^{\prime})$ and $\tilde{2}(K,K^{\prime})$.
Table 3 shows that the same WG in class AIII, BDI and CII also can host the
two Dirac nodes at $K,\ K^{\prime}$ as the minimal configuration. In this
regard, the subgroups of these three symmetry groups possessing the identical
Dirac physics are listed as WG$\\#16$ with $C_{6}^{-}$, WG$\\#15$ with
$C_{3}^{+},\ M_{x}^{-}$, WG$\\#14$ with $C_{3}^{+},\ M_{y}^{+}$, and WG$\\#13$
with $C_{3}^{+}$ in class AIII, BDI, and CII. Thus, based on those listed
symmetry groups, we can search for graphene-like platforms.

Our generalized no-go theorem also shows that in some symmetry classes the
Dirac nodes at $K,\ K^{\prime}$ must be accompanied by other Dirac nodes due
to the charge neutralization. Two Dirac nodes with $+1$ charges located at
$K,\ K^{\prime}$ can be neutralized by two approaches as illustrated in Fig. 7
(d) and (b) — 1. a quadratic dispersion node with $-2$ charge at $\Gamma$:
$4^{*}(K,K^{\prime},\Gamma)$, 2. a Dirac node with $+1$ charge at $\Gamma$ and
three Dirac nodes with $-1$ charges at $M_{1/2/3}$ separately:
$6(K,K^{\prime},\Gamma,M_{1/2/3})$. According to Table 3 and Table 4, the
corresponding symmetry classes are given by WG$\\#17$ with $C_{6}^{+},\
M_{x}^{-}$, WG$\\#16$ with $C_{6}^{+}$, WG$\\#15$ with $C_{3}^{+},\
M_{x}^{-}$, WG$\\#14$ with $C_{3}^{+},\ M_{y}^{-}$, and WG$\\#13$ in class
AIII, DIII, and CI. These symmetry classes provide new directions to search
for alternative Dirac platforms.

Figure 19: (color online) Momentum-space geometry of twisted bilayers shows
the similarity between twisted graphene (hexagonal) and square lattices. While
green and red Dirac cones are separately located in the different layers, the
blue lines indicate the boundaries of the Moiré BZs and the dashed lines
indicate the momentum hoppings. (a) The momentum hopping network connecting
the Dirac ones captures the the effective physics of the twisted bilayer
graphene. (b) The momentum hoppings $\bm{q}_{n}$ in the three directions stem
from the momentum difference between the two Dirac cones. (c) The twisted
square lattice with Dirac points at $M$ point can also be described by the
hopping network. (d) Similarly, the momentum difference between the two Dirac
cones leads to the hoppings $\bm{p}_{m}$ in the four directions.

To extend the twisted physics to square lattices, we consider the low-energy
effective continuum Hamiltonian of the twisted bilayer graphene Bistritzer and
MacDonald (2011) describes the single-valley physics of the two Dirac cones at
$K$ separately in the different layers with the interlayer coupling and can be
written in the real space with twisted angle $\theta$ and lattice constant
$a_{0}$ Tarnopolsky  _et al._ (2019)

$H_{K}=\begin{pmatrix}-iv_{F}\bm{\sigma_{\theta/2}\cdot\nabla}&\mathcal{T}_{3}(\bm{r})\\\
\mathcal{T}_{3}^{\dagger}(\bm{r})&-iv_{F}\bm{\sigma_{-\theta/2}\cdot\nabla}\end{pmatrix}$
(103)

where $\bm{\sigma_{\pm\theta/2}}=e^{\mp
i\theta\sigma_{z}/4}(\sigma_{x},\sigma_{y})e^{\pm i\theta\sigma_{z}/4}$,
$\bm{\nabla}=(\partial_{x},\partial_{y})$, and
$\mathcal{T}_{3}(\bm{r})=\sum_{n=1}^{3}T_{n}e^{-i\bm{q}_{n}\cdot\bm{r}}$. The
two diagonal terms represent the two Dirac cones located at the different
layers and the interlayer coupling $\mathcal{T}_{3}(\bm{r})$ is composed of
three momentum hoppings differed by angle $\psi=2\pi/3$

$\displaystyle\bm{q}_{n+1}$ $\displaystyle=k_{\theta}(\sin n\psi,-\cos
n\psi),$ (104) $\displaystyle T_{n+1}$
$\displaystyle=w_{\rm{AA}}\sigma_{0}+w_{\rm{AB}}(\sigma_{x}\cos
n\psi+\sigma_{y}\sin n\psi),$ (105)

where $k_{\theta}=\frac{8\pi\sin(\theta/2)}{3a_{0}}$ and $w_{\rm{AA}/\rm{BB}}$
indicates the strength of the AA/AB-basis coupling. Fig. 19(a,b) illustrate
the effective physics is captured by the network of the two Dirac cones with
the extension of the momentum hoppings. The platform for the twisted physics
can be naturally extended to a Dirac node at $M$ point in a square lattice
with lattice constant $b_{0}$ Luo  _et al._ (2021); Kariyado and Vishwanath
(2019). Consider the low-energy physics of the twisted square-lattice bilayer
and the Hamiltonian, which shares a similar form of the twisted bilayer
graphene (103), reads

$H_{M}=\begin{pmatrix}-iv_{F}\bm{\sigma_{\theta/2}\cdot\nabla}&\mathcal{S}_{4}(\bm{r})\\\
\mathcal{S}_{4}^{\dagger}(\bm{r})&-iv_{F}\bm{\sigma_{-\theta/2}\cdot\nabla}\end{pmatrix}.$
(106)

The physics describes the coupling between the two Dirac nodes at $M$
separately located in the two layers. The interlayer coupling has momentum
hoppings in four directions
$\mathcal{S}_{4}(\bm{r})=\sum_{m=1}^{4}S_{m}e^{-i\bm{p}_{m}\cdot\bm{r}}$;
Using the form of $\mathcal{T}_{3}(\bm{r})$ in the graphene, we can write the
details of the hoppings as

$\displaystyle\bm{p}_{m+1}$
$\displaystyle=p_{\theta}(\sin(m\phi+\phi_{0}),-\cos(m\phi+\phi_{0})),$ (107)
$\displaystyle S_{m+1}$
$\displaystyle=d_{0}+d_{1}(\sigma_{x}\cos(m\phi+\phi_{0})+\sigma_{y}\sin(m\phi+\phi_{0})),$
(108)

where $p_{\theta}=\pi\sin(\theta/2)/b_{0}$, $\phi=\pi/2$, and
$\phi_{0}=\pi/4$. Fig. 19(c,d) illustrate the expansion of the momentum
hoppings form the network of the two Dirac cones from the different layers.

Now we are looking for symmetry classes hosting a Dirac node at $M$. Due to
the no-go theorem, the Dirac node is always accompanied with other nodes.
According to Table 3 and Table 4, in class AIII and DIII the minimal
configuration $4(\Gamma,M,X,Y)$ can be realized in WG$\\#11,12$ with
$C_{4}^{+},\ M_{x}^{-}$ and WG$\\#10$ with $C_{4}^{+}$ as new platforms for
twisted bilayer physics.

Graphene hosts Dirac nodes at $K$ and $K^{\prime}$ points with $\pm 1$ charges
quantized by chiral (sublattice) symmetry. It is natural to seek a similar
non-Hermitian platform possessing topological points (EPs and FPs) at the same
locations. However, since $K$ and $K^{\prime}$ points are the $C_{3}$ rotation
center, different from the Hermitian nodal phase, the values of the non-
Hermitian charges must be a multiple of 3 (86,61). On the other hand, for the
square BZ, the charge of the topological point at $M$ point, which is the
$C_{4}$ rotation center, must be a multiple of 4. Thus, nodal points for
twisted Hermitian and non-Hermitian bilayers are different. The charge of the
nodal point at a $C_{n}$ rotation center in a non-Hermitian layer must be a
multiple of $n$, while there is no restriction for Hermitian layers.

##  VIII Conclusion

The stable topological points in 2D lattices arise in chiral-symmetric, space-
time-inversion-symmetric, and non-Hermitian systems. In the presence of
crystalline symmetries and time-reversal symmetry, the minimal number of the
topological points might be more than two beyond the original no-go theorem
(the fermion doubling theorem Nielsen and Ninomiya (1981a, b, c); Yang  _et
al._ (2021a)). We exhaustively study the generalized no-go theorem for the
three distinct types of lattice systems and list the minimal numbers of the
topological points with the configurations for given wallpaper groups. This
theorem provides an alternative scheme to classify the nodal topological
materials and includes the nodal topological points located at general points
in BZ, which might be overlooked in the symmetry indicator schemes Tang  _et
al._ (2019); Zhang  _et al._ (2019a); Vergniory  _et al._ (2019); Ono  _et
al._ (2019); Geier  _et al._ (2019).

This generalized no-go theorem is a unified principle and a fundamental rule
for forming Dirac semimetals, nodal time-reversal-symmetric topological
superconductors, and non-Hermitian lattices with exceptional points or Fermi
points. We start with chiral-symmetric nodal systems for the 17 wallpaper
groups. Space-time-inversion-symmetric nodal semimetals and non-Hermitian
nodal systems inherit the principle from the chiral-symmetric system with
small modifications. The examples in the manuscript show that the topological
materials in the literature must obey this ground rule, such as graphene and
d-wave superconductors. In addition, the theorem predicts several new Dirac
semimetals, such as six Dirac nodes in a hexagonal lattice (39) and a
quadratic node with two Dirac nodes (42). Following this guidance, we can
reveal and realize new types of novel nodal topological materials with
topological nodal points. In particular, these Dirac semimetals in our
prediction can potentially possess Majorana bound states with
superconductivity Ahn and Yang (2020). On the other hand, the non-Hermitian
topology of Fermi points and exceptional points is captured by $\mathbb{Z}$
invariants. Unlike the Hermitian Dirac nodes, the charges of those non-
Hermitian topological points at the rotation center, say $C_{n}$ rotation,
must be multiple of $n$. Although $\mathbb{Z}$ invariants also protect Dirac
nodes in chiral-symmetric lattices, the non-Hermitian no-go theorem is
distinct from the Hermitian one.

The generalization of the no-go theorem also leads to several applications in
condensed matter physics. The no-go theorem for the 17 wallpaper groups can be
directly extended to the 80 layer groups through the projection. Our
classification can determine if the no-go theorem on the surface can be
violated; since the violation implies non-trivial bulks, this approach
indirectly classifies Hermitian and non-Hermitian 3D topological bulk with
$\mathbb{Z}$ and $\mathbb{Z}_{2}$ invariants. Twisted bilayer graphene brings
fruitfully interesting physics from the coupling of the Dirac cones. New
predicted Dirac semimetals from the generalized no-go theorem can be new
platforms of twisted bilayers for strong correlations.

##  IX acknowledgement

We thank Chen Fang, Akira Furusaki, Guang Bian, and Qiang Zhang for the
helpful discussions and comments.

##  Appendix A Hamiltonian properties

In certain basis, each entry of the Hamiltonian can be a single-valued
function of momentum ${\bm{k}}$; this single-valued property is the key to
prove the fermion no-go theorem. On the other hand, the symmetry equations for
the Hamiltonian through crystalline symmetry play an essential role to
determine the minimal number of the topological points in the BZ. To set up
these foundations of the generalized no-go theorem, we prove these two
features above for 2D lattice systems in this appendix.

We intentionally choose the annihilation operator in the momentum basis in the
unit cell convention

$\phi_{\alpha
a}(\bm{k})=\frac{1}{\sqrt{N}}\sum_{\bm{R}_{n}}e^{-i\bm{k}\cdot\bm{R}_{n}}\psi_{\alpha}(\bm{R}_{n}+\bm{r}_{a}),$
(109)

where $\bm{R}_{n}$ is the central position of the $n$-th unit cell, $N$ is the
number of the unit cell. Label $a$ indicates different spatial locations of
atoms and $\bm{r}_{a}$ is the location of the atom, whereas label $\alpha$
indicates non-spatial dependent index, such as spin. In this convention, the
phase in the Fourier transformation does not have $\bm{r}_{a}$-dependence and
the location-dependence of the annihilation operator $\phi_{\alpha a}$ is
absorbed in index $a$. By the definition, the annihilation operator has
periodicity

$\phi_{\alpha a}(\bm{k}+m_{1}\bm{G}_{1}+m_{2}\bm{G}_{2})=\phi_{\alpha
a}(\bm{k}),$ (110)

where $\bm{G}_{i}$ is the reciprocal lattice vector and $m_{i}$ is an integer;
therefore, $\phi_{\alpha a}(\bm{k})$ is single-valued in the BZ. On the other
hand, the Hamiltonian in the form of the second quantization in momentum basis
is written as

$\hat{H}=\sum_{{\bm{k}}}\phi_{\beta b}^{\dagger}(\bm{k})H_{\beta b,\alpha
a}({\bm{k}})\phi_{\alpha a}(\bm{k}).$ (111)

Thus, each entry $H_{\beta b,\alpha a}({\bm{k}})$ in the free fermion
Hamiltonian is a single-valued function of momentum in the BZ.

Consider crystalline symmetry operator with translation $\tau$ acts on the
annihilation operator

$\displaystyle g_{\tau}$ $\displaystyle\psi_{\alpha a}(\bm{R}_{n}+\bm{r}_{a})$
$\displaystyle=$
$\displaystyle\sum_{\beta}W_{\alpha\beta}\psi_{\beta}(g_{\tau}(\bm{R}_{n}+\bm{r}_{a}))$
$\displaystyle=$
$\displaystyle\sum_{\beta}W_{\alpha\beta}\psi_{\beta}(\bm{R}_{n}^{\prime
ab}+\sum_{b}V_{ab}\bm{r}_{b})$ (112) $\displaystyle=$
$\displaystyle\sum_{\beta,b}W_{\alpha\beta}V_{ab}\psi_{\beta}(g\bm{R}_{n}+\bm{r}_{b}+n_{1}^{b}g\bm{a}_{1}+n_{2}^{b}g\bm{a}_{2}),$

where $W_{\alpha\beta}$ is a constant transformation unitary matrix for
crystalline operation, such as spin rotation and the orbital phase changes. We
note that the crystalline operator also alters the location of the
annihilation operator

$\displaystyle
g_{\tau}(\bm{R}_{n}+\bm{r}_{a})=\bm{R}_{n}^{\prime}+\sum_{b}V_{ab}\bm{r}_{b},$
(113)

where $\bm{R}_{n}^{\prime}$ is the central location of another unit cell.
Matrix $V_{ab}$ directly switches the original atom location to another and
does not mix other atoms locations. That is, each entry in $V_{ab}$ is either
0 or 1. In this regard, $V_{ab}$ and the summation of $b$ in Eq. 112 can be
moved out of the parenthesis. In general, $\bm{R}_{n}^{\prime ab}\neq
g\bm{R}_{n}$, where crystalline operator for position vector $g$ acts on
$R_{n}$ without translation and the invariant point of the operation (rotation
center or reflection) is in the original point; hence, we have

$\bm{R}_{n}^{\prime
ab}=g\bm{R}_{n}+n_{1}^{ab}g\bm{a}_{1}+n_{2}^{ab}g\bm{a}_{2},$ (114)

where integer $n_{i}^{ab}$, which depends on the atom location, indicates the
number of the unit cell shift after the crystalline symmetry operation and
$\bm{a}_{i}$ is the primitive vector obeying

${\bm{k}}\cdot(n_{1}\bm{a}_{1}+n_{2}\bm{a}_{2})=k_{1}n_{1}+k_{2}n_{2},$ (115)

where ${\bm{k}}=(k_{1}\bm{G}_{1}+k_{2}\bm{G}_{2})/2\pi$. Since the crystalline
symmetry is preserved, $g\bm{a}_{i}$ is still the primitive vector; the last
two terms in the equation above represent the shift of the unit cell.

We can discuss the $g$ acts on the annihilation operator in the momentum space

$\displaystyle g_{\tau}$ $\displaystyle\phi_{\alpha a}(\bm{k})$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\bm{R}_{n}}e^{-i\bm{k}\cdot\bm{R}_{n}}g_{\tau}\psi_{\alpha}(\bm{R}_{n}+\bm{r}_{a})$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\bm{R}_{n},\beta,b}e^{-i\bm{k}\cdot\bm{R}_{n}}W_{\alpha\beta}V_{ab}\psi_{\beta}(\bm{R}_{n}^{\prime
ab}+\bm{r}_{b})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\bm{R}_{n},\beta,b}e^{-ig\bm{k}\cdot
g\bm{R}_{n}}W_{\alpha\beta}V_{ab}\psi_{\beta}(\bm{R}_{n}^{\prime
ab}+\bm{r}_{b})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\bm{R}_{n},\beta,b}e^{-ig\bm{k}\cdot(\bm{R}_{n}^{\prime
ab}-n_{1}^{ab}g\bm{a}_{1}-n_{2}^{ab}g\bm{a}_{2})}W_{\alpha\beta}V_{ab}\psi_{\beta}(\bm{R}_{n}^{\prime
ab}+\bm{r}_{b})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\bm{R}_{n},\beta,b}e^{ig\bm{k}\cdot(n_{1}^{ab}g\bm{a}_{1}+n_{2}^{ab}g\bm{a}_{2})}W_{\alpha\beta}V_{ab}e^{-ig\bm{k}\cdot\bm{R}_{n}^{\prime
ab}}\psi_{\beta}(\bm{R}_{n}^{\prime ab}+\bm{r}_{b})$ $\displaystyle=$
$\displaystyle\sum_{\beta,b}e^{i(k_{1}n_{1}^{ab}+k_{2}n_{2}^{ab})}W_{\alpha\beta}V_{ab}\phi_{\beta
b}(g{\bm{k}})$ (116)

To simplify the expression of the crystalline symmetry operation, we can
define the single-valued ${\bm{k}}$-dependent unitary matrix

$\displaystyle U_{\alpha a,\beta
b}^{\dagger}({\bm{k}})=e^{i(k_{1}n_{1}^{ab}+k_{2}n_{2}^{ab})}W_{\alpha\beta}V_{ab}.$
(117)

The annihilation operator and creation operator can be rewritten in the
economic form

$g_{\tau}\phi({\bm{k}})=U^{\dagger}({\bm{k}})\phi(g{\bm{k}}),\
g_{\tau}\phi^{\dagger}({\bm{k}})=\phi^{\dagger}(g{\bm{k}})U({\bm{k}})$ (118)

Due to the symmetry, the second quantization Hamiltonian (111) is invariant
under the crystalline symmetry operation

$\displaystyle\sum_{{\bm{k}}}\phi^{\dagger}({\bm{k}})H({\bm{k}})\phi({\bm{k}})=g_{\tau}\sum_{{\bm{k}}}\phi^{\dagger}({\bm{k}})H({\bm{k}})\phi({\bm{k}})$
$\displaystyle=$
$\displaystyle\sum_{{\bm{k}}}\phi^{\dagger}(g{\bm{k}})U({\bm{k}})H({\bm{k}})U^{\dagger}({\bm{k}})\phi(g{\bm{k}})$
(119)

Hence, the symmetry equation for the free fermion Hamiltonian is given by

$H(g{\bm{k}})=U({\bm{k}})H({\bm{k}})U^{\dagger}({\bm{k}}).$ (120)

We will frequently use this symmetry equation through the manuscript for the
generalized no-go theorem.

##  Appendix B The winding number connection between ${\bm{K}}_{0}$ and
$g{\bm{K}}_{0}$

We provide the detailed deviations for the relations of the two winding
numbers at ${\bm{K}}_{0}$ and $g{\bm{K}}_{0}$. First, let us show that the
definition (8) of the winding number is equivalent to the one using the
flattened Hamiltonian. The flattened Hamiltonian

$\displaystyle
Q(\bm{k})=\sum_{0<E_{\alpha}}^{N_{+}}|\Psi^{\alpha}(\bm{k})\rangle\langle\Psi^{\alpha}(\bm{k})|-\sum_{0>E_{\beta}}^{N_{-}}|\Psi^{\beta}(\bm{k})\rangle\langle\Psi^{\beta}(\bm{k})|,$
(121)

where $|\Psi^{\gamma}(\bm{k})\rangle$ is the eigenstate of the
$\mathcal{H}({{\bm{k}}})$ with energy $E_{\gamma}$. Assuming the absence of
zero-energy states, chiral symmetry leads to $N_{+}=N_{-}$. Since the
flattened Hamiltonian inherits chiral symmetry (4) from the original
Hamiltonian by choosing a proper basis, chiral symmetry operator can be
written as $S=\tau_{z}\otimes\mathbb{I}_{n\times n}$ and the flattened
Hamiltonian has this off-diagonal form

$Q({\bm{k}})=\begin{pmatrix}0&q({\bm{k}})\\\
q^{\dagger}({\bm{k}})&0\end{pmatrix}.$ (122)

Due to $Q^{2}=\mathbb{I}_{2n\times 2n}$, $q({\bm{k}})$ is a unitary matrix and
invertible due to the absence of the zero-energy state. Using $q({\bm{k}})$,
the winding number is defined in a closed loop integral path in the BZ

$\displaystyle\nu$ $\displaystyle=\frac{i}{2\pi}\oint d\bm{k}\cdot
Tr[q^{-1}(\bm{k})\partial_{\bm{k}}q(\bm{k})]$ (123)
$\displaystyle=\frac{i}{2\pi}\oint d\bm{k}\cdot\partial_{\bm{k}}Tr[\ln
q(\bm{k})]$ $\displaystyle=\frac{i}{2\pi}\oint
d\big{(}\ln\det[q(\bm{k})]\big{)}$

This definition of the winding number is regularly used in the literature
Schnyder  _et al._ (2008); Chiu  _et al._ (2016).

To show that $h({\bm{k}})$ and $q({\bm{k}})$ lead to the same winding number,
we choose the eigenstates as the basis and then prove that
$\mathcal{H}({\bm{k}})$ commutes with $Q({\bm{k}})$ since

$\displaystyle\mathcal{H}(\bm{k})Q(\bm{k})$ (124) $\displaystyle=$
$\displaystyle\sum_{0<E_{\alpha}}^{N_{+}}E_{\alpha}(\bm{k})|\Psi^{\alpha}(\bm{k})\rangle\langle\Psi^{\alpha}(\bm{k})|-\sum_{0>E_{\beta}}^{N_{-}}E_{\beta}(\bm{k})|\Psi^{\beta}(\bm{k})\rangle\langle\Psi^{\beta}(\bm{k})|$
$\displaystyle=$ $\displaystyle Q(\bm{k})\mathcal{H}(\bm{k})$ (125)

The commutation relation can be rewritten in the proper basis

$\displaystyle 0=[\mathcal{H}(\bm{k}),Q(\bm{k})]$ (126)
$\displaystyle=\left(\begin{array}[]{cc}h(\bm{k})q^{\dagger}(\bm{k})-q(\bm{k})h^{\dagger}(\bm{k})&0\\\
0&h^{\dagger}(\bm{k})q(\bm{k})-q^{\dagger}(\bm{k})h(\bm{k})\\\
\end{array}\right).$ (129)

One can obtain

$\displaystyle\det[h(\bm{k})q^{\dagger}(\bm{k})]=\det[{q(\bm{k})}h^{\dagger}(\bm{k})]$
$\displaystyle=\det[(h(\bm{k}){q^{\dagger}(\bm{k})})^{\dagger}]=\det[h(\bm{k})q^{\dagger}(\bm{k})]^{*}.$
(130)

The above equation implies that $\det[h(\bm{k})q^{\dagger}(\bm{k})]$ is a real
number so that
$\ln\det(h({\bm{k}}))+\ln\det(q^{*}({\bm{k}}))=\gamma({\bm{k}})$ is a single-
valued function, which leads to $\oint d\gamma({\bm{k}})=0$. Using the
property of the unitary matrix $q({\bm{k}})$, we have
$\ln\det(q^{*})+\ln\det(q)=0$. Hence, the winding number with the closed
integral path $\Gamma({\bm{K}}_{0})$ can be directly written in form of
$h({\bm{k}})$

$\displaystyle\nu(\bm{K}_{0})$
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}d\big{(}\ln\det[q(\bm{k})]\big{)}$
(131)
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}d\big{(}\ln\det[h(\bm{k})]\big{)}$

To connect the relation between $\nu({\bm{K}}_{0})$ and
$\nu(\Gamma{\bm{K}}_{0})$, we first consider $g_{\bm{\tau}}=\\{g|\bm{\tau}\\}$
with spatial translation $\bm{\tau}$ as a $2D$ crystalline symmetry operator.

Since the matrix form of the symmetry operator commutes with chiral symmetry
operator $S=\tau_{z}\otimes 1$, the unitary symmetry operator in the single-
particle basis (or particle-hole basis for superconductors) is written in a
generic form of

$U_{g}^{+}({\bm{k}})=\begin{pmatrix}U_{g1}({\bm{k}})&0\\\
0&U_{g2}({\bm{k}})\end{pmatrix},$ (132)

where $U_{gi}({\bm{k}})$ is a momentum-dependent unitary matrix, which can
describe a nonsymmorphic symmetry operation. The crystalline symmetry equation
(15) connects the Hamiltonian at ${\bm{K}}_{0}$ and $g{\bm{K}}_{0}$

$\displaystyle\mathcal{H}(g{\bm{k}})=$ $\displaystyle
U_{g}^{+}({\bm{k}})\mathcal{H}({\bm{k}})U_{g}^{+{\dagger}}({\bm{k}})$ (133)
$\displaystyle=$
$\displaystyle\begin{pmatrix}0&U_{g1}({\bm{k}})h({\bm{k}})U^{\dagger}_{g2}({\bm{k}})\\\
U_{g2}({\bm{k}})h^{\dagger}({\bm{k}})U^{\dagger}_{g1}({\bm{k}})&0\end{pmatrix}.$

We have $h(g{\bm{k}})=U_{g1}({\bm{k}})h({\bm{k}})U_{g2}^{\dagger}({\bm{k}})$.
The integral path $\Gamma(g{\bm{K}}_{0})$ is an infinitesimal close loop
encircling nodal point $g\bm{K}_{0}$ and then the winding number at
$\Gamma(g{\bm{K}}_{0})$ is given by

$\displaystyle\nu(g\bm{K}_{0})$
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(g\bm{K}_{0}\right)}d\big{(}\ln\det[h(\bm{k})]\big{)}$
(134)
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}\det(g)d\big{(}\ln\det[h(g\bm{k})]\big{)}$
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}\det(g)d\big{(}\ln\det[h(\bm{k})]\big{)}$
$\displaystyle=\det(g)\nu({\bm{K}}_{0})$

We use the integral $\oint_{\Gamma(K_{0})}d\big{(}\ln\det
U_{gi}({\bm{k}})\big{)}=0$ since $U_{gi}({\bm{k}})$ does not collect
additional $U(1)$ winding phase in the infinitesimal integral loop. The reason
is that $\det(U_{g\pm}({\bm{k}}))$ does not vanish to create a nodal point due
to the nature of unitary matrix.

Secondly, consider time reversal symmetry operator $T^{+}$ commutes with
chiral symmetry operator $S$

$\mathcal{H}(-{\bm{k}})=T\mathcal{H}({\bm{k}})T^{-1}=U_{T}\mathcal{H}^{*}({\bm{k}})U_{T}^{\dagger},$
(135)

where $T^{+}=U_{T}\mathcal{K}$ and $\mathcal{K}$ is the complex conjugate
operator. The commutation relation between $T^{+}$ and $S$ gives the form of

$T^{+}=\begin{pmatrix}U_{T1}&0\\\ 0&U_{T2}\end{pmatrix}\mathcal{K},$ (136)

where $U_{Ti}$ is momentum-independent since time reversal symmetry operator
is non-spatial. In 2D momentum space, time reversal operation is equivalent to
the combination of $C_{2}$ rotation and complex conjugation. Therefore,

$\displaystyle\nu(-\bm{K}_{0})$
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(-\bm{K}_{0}\right)}d\big{(}\ln\det[h(\bm{k})]\big{)}$
(137)
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}d\big{(}\ln\det[h^{*}(\bm{k})]\big{)}$
$\displaystyle=\frac{-i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}d\big{(}\ln\det[h(\bm{k})]\big{)}$
$\displaystyle=-\nu({\bm{K}}_{0})$

Hence, with the commutation relation between $T^{+}$ and $S$, time reversal
operation flips the sign of the winding number for the nodal point at
$-{\bm{K}}_{0}$.

On the other hand, when symmetry operator anticommutes with $S$, the relation
of the winding numbers between ${\bm{K}}_{0}$ and $g{\bm{K}}_{0}$ flips the
sign. Following the similar derivation, we have crystalline symmetry operator
$U_{g}^{-}({\bm{k}})$, which anticommutes with $S$, in the form of

$U_{g}^{-}({\bm{k}})=\begin{pmatrix}0&V_{g1}({\bm{k}})\\\
V_{g2}({\bm{k}})&0\end{pmatrix},$ (138)

where $V_{gi}$ is a unitary matrix. This symmetry operator links the
Hamiltonian at ${\bm{K}}_{0}$ and $g{\bm{K}}_{0}$

$\displaystyle\mathcal{H}(g{\bm{k}})=$ $\displaystyle
U_{g}^{-}({\bm{k}})\mathcal{H}({\bm{k}})U_{g}^{-{\dagger}}({\bm{k}})$ (139)
$\displaystyle=$
$\displaystyle\begin{pmatrix}0&V_{g1}({\bm{k}})h^{\dagger}({\bm{k}})V^{\dagger}_{g2}({\bm{k}})\\\
V_{g2}({\bm{k}})h({\bm{k}})V^{\dagger}_{g1}({\bm{k}})&0\end{pmatrix}.$

Using
$h(g{\bm{k}})=V_{g1}({\bm{k}})h^{\dagger}({\bm{k}})V_{g2}^{\dagger}({\bm{k}})$
and following the similar derivation with $U_{g}^{+}({\bm{k}})$, we have this
relation of the winding numbers at the two nodal points

$\displaystyle\nu(g\bm{K}_{0})$
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}\det(g)d\big{(}\ln\det[h(g\bm{k})]\big{)}$
(140)
$\displaystyle=\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}\det(g)d\big{(}\ln\det[h^{\dagger}(\bm{k})]\big{)}$
$\displaystyle=\frac{-i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}\right)}\det(g)d\big{(}\ln\det[h(\bm{k})]\big{)}$
$\displaystyle=-\det(g)\nu({\bm{K}}_{0})$

For time-reversal symmetry operator $T^{-}$ anticommuting with $S$, since the
complex conjugate operator in $T^{-}$ flips the sign of the winding number,
the relation between ${\bm{K}}_{0}$ and $-{\bm{K}}_{0}$ with time-reversal
symmetry is given by

$\nu(-{\bm{K}}_{0})=\nu({\bm{K}}_{0})$ (141)

The results of the winding number relations are summarized in Eqs. 17, 18 and
Table. 2.

##  Appendix C The algebra between chiral operator and time-reversal operator

We know that $T^{2}=C^{2}=\pm 1$ in class BDI, CII and $T^{2}=-C^{2}=\pm 1$ in
class CI, DIII. Let us consider $T^{2}=C^{2}=\pm 1$. We can choose a proper
basis and phase so that chiral operator $S$ is hermitian and $S^{2}=1$. By
sandwiching $S^{2}$ with $T$ and $C$, we have

$TC=TS^{2}C=T^{2}CTC^{2}=CT.$ (142)

This commutation relation leads to $0=T[T,C]=[T,TC]=[T,S]$. That is, class
BDI, CII preserve $T^{+}$ time-reversal symmetry. Following the similar
derivation, we have $T^{-}$ time-reversal symmetry for class CI, DIII.

##  Appendix D Neutralized winding number summation

Figure 20: (color online) The integral paths (red line) of the winding numbers
for the nodal points (black dots) evolve in the BZ. The black box represents
the entire boundary of the BZ. (a) The integral path of the winding number is
an infinitesimal loop encircling a nodal point. (b) The integral path of the
multiple loops grows without passing through any nodal point and through the
deformation two loops form one big loop. (c) The integral path for all the
nodal nodal is equivalent to the integral path along the boundary of the BZ,
since through the deformation the path does not pass any nodal points. By
using the periodic boundary condition of the BZ, this integral path vanishes.

Each nodal point in the BZ is characterized by winding number. The essence of
the no-go theorem is that the summation over winding numbers for all the nodal
points must vanish. To prove the neutralization we simply write the definition
of the winding number summation

$\displaystyle\sum_{i}\nu(\bm{K}_{0}^{i})$
$\displaystyle=\sum_{i}\frac{i}{2\pi}\oint_{\Gamma\left(\bm{K}_{0}^{i}\right)}d\big{(}\ln\det[q(\bm{k})]\big{)}$
(143)
$\displaystyle=\frac{i}{2\pi}\oint_{\partial\rm{BZ}}d\big{(}\ln\det[h(\bm{k})]\big{)}$
$\displaystyle=0,$

where $K_{0}^{i}$ represent all of the nodal points ($E=0$) in the BZ. The
integral path of the winding number for any nodal point is an infinitesimal
loop counterclockwise encircling that nodal point. The integral path for all
the nodal points can continuously deform to the boundary of the BZ
($\partial$BZ) as illustrated in Fig. 20. Importantly, the integral path never
passes any nodal points during the deformation so that the total winding
number is unchanged after the deformation; hence, the second line with
$\partial$BZ in the equation holds. Since $h(\bm{k})$ can be a single-valued
in the entire BZ, by using the periodic boundary, $\partial$BZ vanishes so
that the summation must be zero.

##  Appendix E Four Dirac nodes: WG$\\#17$ in class DIII

In all the WG symmetry classes, it is the most difficult task to find a tight-
binding model for the minimal configuration of a nodal point with $\nu=2$
located at $\Gamma$ and two nodal points with $\nu=-1$ at $K,K^{\prime}$ in
class DIII. In a 2-band model, time reversal symmetry $(T^{2}=-1)$ limits the
winding number of the nodal point at $\Gamma$ to an odd number. Hence, the
2-band model cannot host a nodal point at $\Gamma$ carrying an even winding
number. To find the minimal tight-binding model hosting this high-ordered
nodal point, we start with a 4-band model consisting of two 6-Dirac-point
Hamiltonians $H_{6}({\bm{k}})$ (39) in class DIII. Since $H_{6}({\bm{k}})$
possesses 6 Dirac nodes separately located at $\Gamma,M_{1,2,3},K^{\prime},K$,
two of the twelve Dirac nodes at $\Gamma$ carry $\nu=2$ winding number
together, two nodes at $M_{1,2,3}$ carry $\nu=-2$, and two nodes at
$K,K^{\prime}$ carry $\nu=2$. However, the nodal points at $M_{1,2,3}$ should
be removed and the signs of the winding numbers at $K,K^{\prime}$ should be
switched. To achieve this, we intensionally add $H_{4}({\bm{k}})$ (42) from
class CI with real coefficient $\alpha$ as a coupling between two
$H_{6}({\bm{k}})$’s. The $4\times 4$ Hamiltonian is written in the form of

$H_{4\times 4}({\bm{k}})=\begin{pmatrix}0&h_{6}({\bm{k}})&0&\alpha
h_{4}({\bm{k}})\\\ h_{6}^{*}({\bm{k}})&0&-\alpha h_{4}^{*}({\bm{k}})&0\\\
0&-\alpha h_{4}({\bm{k}})&0&-h_{6}({\bm{k}})\\\ \alpha
h^{*}_{4}({\bm{k}})&0&-h_{6}^{*}({\bm{k}})&0\\\ \end{pmatrix}.$ (144)

Since time reversal symmetry with $T=i\tau_{0}\sigma_{y}\mathcal{K}$ and
particle-hole symmetry $C=\tau_{0}\sigma_{x}\mathcal{K}$ are preserved, the
system belongs to class DIII. Hence, chiral symmetry operator is given by
$S=\tau_{0}\sigma_{z}$. Furthermore, this is WG$\\#17$ with $C_{6}$ rotation
and $M_{x}$ generators since the Hamiltonian obeys $M_{x}H_{4\times
4}(-k_{x},k_{y})M_{x}^{-1}=H_{4\times 4}(k_{x},k_{y})$ with
$M_{x}=\tau_{z}\sigma_{y}$ and $C_{6}H_{4\times
4}(\frac{k_{x}+\sqrt{3}k_{y}}{2},\frac{k_{y}-\sqrt{3}k_{x}}{2})C_{6}^{-1}=H_{4\times
4}(k_{x},k_{y})$ with
$C_{6}=\cos\pi/6\tau_{z}\sigma_{0}-i\sin\pi/6\tau_{z}\sigma_{z}$. The algebra
between the generators and the chiral symmetry operator is given by
$M_{x}^{-}$ and $C_{6}^{+}$. According to Table 4, this WG symmetry class can
host a nodal point at $\Gamma$ with $\nu=2$, two nodal points with $\nu=-1$ at
$K,K^{\prime}$ separately as the minimal configuration.

To realize the minimal configuration, we begin to increase $\alpha$ from $0$,
two Dirac points at each of $M_{1,2,3}$ points move toward $K,\ K^{\prime}$
along $\overline{MK},\ \overline{MK^{\prime}}$ separately as shown in Fig.
21(a). Since $h_{4}(0)$ vanishes, the node carrying $\nu=2$ at $\Gamma$ is
intact. As $\alpha=1/\sqrt{3}$, the moving Dirac points carrying $\nu=-1$ are
exactly located at $K,K^{\prime}$; Fig. 21(b) shows the 4-fold degenerate
point is located in $K$. Since originally each of $K^{\prime},K$ has $+2$
winding number and each of the Dirac nodes from $M_{1,2,3}$ carries $\nu=-1$,
after merging with the three Dirac points from $M_{1,2,3}$, the 4-fold
degenerate nodal point at $K$ or $K^{\prime}$ carries $\nu=-1$ winding number
in total. Thus, the Hamiltonian (144) with $\alpha=1/\sqrt{3}$ is the tight-
binding model hosting the 4-fold degenerate node at $\Gamma$ with $\nu=2$ and
the nodes at $K,K^{\prime}$ with $\nu=-1$. We note this is the minimal
configuration is unstable. That is, as $\alpha>1/\sqrt{3}$, from $K$ or
$K^{\prime}$ Dirac nodes separately move along $\overline{\Gamma K}$ or
$\overline{\Gamma K^{\prime}}$ toward $\Gamma$ as shown in Fig. 21(c).

Figure 21: The Dirac points move from $M_{1,2,3}$ as $\alpha$ varies from $0$.
(a) The Dirac node moves along $\overline{MK}$ as $0<\alpha<1/\sqrt{3}$. (b)
As $\alpha=1/\sqrt{3}$, the moving Dirac node is located at $K$. The total
winding number of the 4-fold degenerate nodal point becomes -1. (c) The Dirac
point moves away from $K$ toward $\Gamma$.

##  Appendix F $C_{2}T$ symmetry

Space-time inversion symmetry ($C_{2}T$) can protect 2D Dirac cones in the
presence of disorders, and quantizes the $\mathbb{Z}_{2}$ Berry phase in the
1D integral path. Here, we provide the detailed deviations; the Berry phase
with the closed integral path $\Gamma(\bm{K}_{0})$ at the nodal point
$\bm{K}_{0}$ can be written in Eq. 45.

Space-time inversion operator $C_{2}T$ is the combination of a unitary matrix
and complex conjugation $C_{2}T=V_{\bm{k}}\mathcal{K}$; the unitary matrix
$V_{{\bm{k}}}$ might be ${\bm{k}}$-dependent and does not have any singularity
point in the BZ. By assuming the absence of degenerate states, the relation of
wavefunctions under $C_{2}T$ symmetry is given by

$|u_{n}({\bm{k}})\rangle=e^{i\beta_{{\bm{k}}}^{j}}C_{2}T|u_{n}({\bm{k}})\rangle=e^{i\beta_{{\bm{k}}}^{j}}V_{\bm{k}}|u_{n}^{*}({\bm{k}})\rangle.$
(145)

The Berry phase with a non-contractible integral path can be written as

$\displaystyle\gamma(\bm{K}_{0})=$ $\displaystyle
i\sum_{n\in\rm{occ}}\oint\langle
u_{n}({\bm{k}})|\partial_{{\bm{k}}}|u_{n}({\bm{k}})\rangle d{\bm{k}}$
$\displaystyle=$ $\displaystyle i\sum_{n\in\rm{occ}}\oint\langle
u_{n}^{*}({\bm{k}})|V_{{\bm{k}}}^{\dagger}e^{-i\beta^{j}_{{\bm{k}}}}\partial_{{\bm{k}}}e^{i\beta^{j}_{{\bm{k}}}}V_{k}|u_{n}^{*}({\bm{k}})\rangle
d{\bm{k}}$ $\displaystyle=$
$\displaystyle-\sum_{n\in\rm{occ}}(\beta^{j}_{+}-\beta^{j}_{-})+i\sum_{n\in\rm{occ}}\oint\langle
u_{n}^{*}({\bm{k}})|\partial_{{\bm{k}}}|u_{n}^{*}({\bm{k}})\rangle d{\bm{k}}$
$\displaystyle+i\sum_{n\in\rm{occ}}\oint\langle
u_{n}({\bm{k}})|V_{k}^{\dagger}(\partial_{{\bm{k}}}V_{k})|u_{n}^{*}({\bm{k}})\rangle
d{\bm{k}},$ (146)

where $\beta^{j}_{\mp}$ represent the phases at the beginning and the end of
the integration path respectively; since the integration path is a closed
loop, the start point and the end point are identical. Hence, the first
summation is $2m\pi$, where $m$ is an integer. Furthermore, due to the absence
of any singularity in $V_{{\bm{k}}}$, the third term vanishes. After using the
identity

$\langle\phi_{{\bm{k}},j}^{*}|\partial_{{\bm{k}}}|\phi_{{\bm{k}},j}^{*}\rangle=\left\langle\partial_{{\bm{k}}}\phi_{{\bm{k}},j}|\phi_{{\bm{k}},j}\right\rangle=-\langle\phi_{{\bm{k}},j}|\partial_{{\bm{k}}}|\phi_{{\bm{k}},j}\rangle,$
(147)

we have the quantized Berry phase

$\gamma(\bm{K}_{0})=\sum_{n\in\rm{occ}}(\beta^{j}_{+}-\beta^{j}_{-})/2=0,\
\pi\ (\rm{mod}\ 2\pi)$ (148)

The modulo operation of $2\pi$ stems from gauge transform
$|u_{n}^{*}({\bm{k}})\rangle\rightarrow
e^{il\theta}|u_{n}^{*}({\bm{k}})\rangle$ by using $\bm{K}_{0}$ as the
singularity point for the transformation; Hence, the two Berry phases differed
by $2\pi l$ are equivalent.

The crystalline symmetry $\\{g|\bm{\tau}\\}$ connects two nodal points
$\bm{K}_{0}$ and $g\bm{K}_{0}$ so that the Hamiltonians at the two points
obeys

$\displaystyle\mathcal{H}(g{\bm{k}})=$ $\displaystyle
U_{g}({\bm{k}})\mathcal{H}({\bm{k}})U_{g}^{{\dagger}}({\bm{k}}),$ (149)

where the crystalline symmetry operator $U_{g}({\bm{k}})$ is a momentum-
dependent unitary matrix. Therefore, the wavefunctions at the two points are
connected by

$|u_{n}(g{\bm{k}})\rangle=U_{g}({\bm{k}})|u_{n}({\bm{k}})\rangle.$ (150)

With this relation the Berry phases at $g\bm{K}_{0}$ point is written in the
form of the Berry phase at $\bm{K}_{0}$

$\displaystyle\gamma(g\bm{K}_{0})$ (151) $\displaystyle=$ $\displaystyle
i\sum_{n\in\rm{occ}}\oint_{\Gamma(g\bm{K}_{0})}\langle
u_{n}(\bm{k})|\partial_{\bm{k}}|u_{n}(\bm{k})\rangle\cdot\mathrm{d}\bm{k}$
$\displaystyle=$ $\displaystyle
i\det(g)\sum_{n\in\rm{occ}}\oint_{\Gamma(\bm{K}_{0})}\langle
u_{n}(g\bm{k})|\partial_{g\bm{k}}|u_{n}(g\bm{k})\rangle\cdot\mathrm{d}g\bm{k}$
$\displaystyle=$ $\displaystyle
i\det(g)\sum_{n\in\rm{occ}}\oint_{\Gamma(\bm{K}_{0})}\langle
U_{g}(\bm{k})u_{n}(\bm{k})|\partial_{\bm{k}}|U_{g}(\bm{k})u_{n}(\bm{k})\rangle\cdot\mathrm{d}\bm{k}$
$\displaystyle=$ $\displaystyle
i\det(g)\sum_{n\in\rm{occ}}\oint_{\Gamma(\bm{K}_{0})}\langle
u_{n}(\bm{k})|\partial_{\bm{k}}|u_{n}(\bm{k})\rangle\cdot\mathrm{d}\bm{k}$
$\displaystyle=$ $\displaystyle\det(g)\gamma(\bm{K}_{0})$

Here we use that $U_{g}({\bm{k}})$ does not have any singularity in the BZ.
Since the Berry phase is either $0$ or $\pi$, the relation between
$\bm{K}_{0}$ and $g\bm{K}_{0}$ can be simplified to

$\gamma(g\bm{K}_{0})=\gamma(\bm{K}_{0}).$ (152)

Similarly, for time-reversal symmetry, the Berry phases at $\bm{K}_{0}$ and
$-\bm{K}_{0}$ are related by

$\gamma(\bm{K}_{0})=\gamma(-\bm{K}_{0})$ (153)

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