\section{S1. Convergence of dynamic winding number\label{Sec1}}
According to Eq.(2) in the main text, it seems that the definition of dynamic winding number depends on the initial state, and it is unclear whether such number is convergent in the long time.
Here, we prove that the initial state can be rather general and the dynamic winding number is convergent.
The time-average of $\langle\sigma_j\rangle$ is given as
\begin{eqnarray}
\overline{\sigma_j}=\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^{T}\frac{\sum_{\mu,\mu'}c_{\mu}c_{\mu'}^*e^{-i(\varepsilon_{\mu}-\varepsilon_{\mu'}^*)t}\langle\chi_{\mu'}|\sigma_j|\varphi_{\mu}\rangle}
  {\sum_{\mu}|c_{\mu}|^2e^{-i(\varepsilon_{\mu}-\varepsilon_{\mu}^*)t}}dt.
  \label{osigmajt}
\end{eqnarray}
For Hermitian systems, $\varepsilon_{\mu}=\varepsilon_{\mu}^*$, and the periodic terms vanish in the long time average and only the diagonal terms preserve.
The above equation can be simplified as
\begin{eqnarray}
   \overline{\sigma_j}= \sum\limits_{\mu}|c_{\mu}|^2
  \langle\chi_{\mu}|\sigma_j|\varphi_{\mu}\rangle=\big(|c_{+}|^2-|c_{-}|^2\big)\frac{h_j}{\varepsilon_{+}}
\end{eqnarray}
For the non-Hermitian systems, we assume that the eigenenergy $\varepsilon_\mu=\mu(A+iB)$, where $A$ and $B$ are real numbers.
When $B>0$, Eq.~\eqref{osigmajt} is approximately given as
\begin{eqnarray}
    \overline{\sigma_j}&=&\langle\chi_{+}|\sigma_j|\varphi_{+}\rangle=\frac{h_j}{\varepsilon_{+}}. \label{sigmajplus}
\end{eqnarray}
Similarly, when  $B<0$, Eq.~\eqref{osigmajt} is approximately given as
\begin{eqnarray}
    \overline{\sigma_j}&=&\langle\chi_{-}|\sigma_j|\varphi_{-}\rangle=-\frac{h_j}{\varepsilon_{+}}. \label{sigmajminus}
\end{eqnarray}
Combining with Eqs.~\eqref{sigmajplus} and  \eqref{sigmajminus}, we can also obtain
\begin{eqnarray}
  \frac{\overline{\sigma_j}}{\overline{\sigma_i}}=\frac{h_j}{h_i},
  \label{osigmajnj}
\end{eqnarray}
in the conditions $|c_{+}|^2\ne|c_{-}|^2$ for Hermitian systems and  $c_{+}\ne 0$ for $B>0$ or $c_{-}\ne 0$ for $B<0$ in non-Hermitian systems.
It means that the dynamic winding number also converges in the long time limits when the initial state satisfies a few constraint.
According to the Eq.~\eqref{osigmajnj}, one can also obtain
\begin{eqnarray}
\eta_{ji}=\phi_{ji}=\arctan \left(\frac{\langle\chi_{\mu}|\sigma_j|\varphi_{\mu}\rangle}{\langle\chi_{\mu}|\sigma_i|\varphi_{\mu}\rangle}\right),
\label{osigmajnj_1}
\end{eqnarray}
where the azimuthal angle $\eta_{ji}=\arctan(\overline{\sigma_j}/\overline{\sigma_i})$ and $\phi_{ji}=\arctan(h_j/h_i)$.

\section{S2. Relation between dynamic winding number and time-averaged spin textures \label{Sec2}}
For the non-Hermitian case, the azimuthal angle $\eta_{ji}$ and $\phi_{ji}$ is generally a complex angle, so that they do not represent physical observables in the biorthogonal system.
 %
This problem can be fixed by  decomposing the azimuthal angle into two parts, $\phi_{ji}=\Re(\phi_{ji})+\Im(\phi_{ji})$, where $\Re(\phi_{ji})$ and $\Im(\phi_{ji})$ represents the real part and image part of $\phi_{ji}$.
The azimuthal angle satisfies
\begin{eqnarray}
&&e^{i2\phi_{ji}}=e^{i2\Re(\phi_{ji})}e^{-2\Im(\phi_{ji})}=\frac{1+i\tan(\phi_{ji})}{1-i\tan(\phi_{ji})}=\frac{h_i+ih_j}{h_i-ih_j},
\nonumber \\
&&e^{-2\Im(\phi_{ji})}=\big|\frac{h_i+ih_j}{h_i-ih_j}\big|,
\label{osigmajnj_2}
\end{eqnarray}
$\Re(\phi_{ji})$ and $\Im(\phi_{ji})$ contribute to the argument and amplitude, respectively.
$\Im(\phi_{ji})$ is a real continuous periodic function of $\bm{k}$, so that $\oint_{S} \partial_{\bm k}\Im[\phi_{ji}(\bm k)]d\bm{k}=\oint_{S} \partial_{\bm k}\Im[\eta_{ji}(\bm k)]d\bm{k}=0$. It means that only the real part of azimuthal angle contributes to the dynamic winding number,
\begin{eqnarray}
w_d=\frac{1}{2\pi}\oint_{S} \partial_{\bm k}\Re(\eta_{ji}) d\bm{k}=\frac{1}{2\pi}\oint_{S} \partial_{\bm k}\Re(\phi_{ji}) d\bm{k},
\label{dynwinding_20190628}
\end{eqnarray}
Next, we will show that the real part of azimuthal angle is a physical observable.
According to the Eq.~\eqref{osigmajnj_2}, the real part of azimuthal angle satisfies
\begin{eqnarray}
\tan(2\Re(\phi_{ji}))=\frac{\Im\big(\frac{\pounds h_i+i\pounds h_j}{\pounds h_i-i\pounds h_j}\big)}{\Re\big(\frac{\pounds h_i+i\pounds h_j}{\pounds h_i-i\pounds h_j}\big)},
\end{eqnarray}
where $\pounds$ is a nonzero arbitrary constant.
After some algebras, one can rewrite the above relation as
\begin{eqnarray}
\tan(2\Re(\phi_{ji}))=\frac{\tan(\phi_{ji}^{RR})+\tan(\phi_{ji}^{LL})}{1-\tan(\phi_{ji}^{RR})\tan(\phi_{ji}^{LL})}=\tan(\phi_{ji}^{RR}+\phi_{ji}^{LL}), \nonumber
\end{eqnarray}
where
\begin{eqnarray}
\tan(\phi_{ji}^{RR})=\frac{\Re(\pounds h_j)+\Im(\pounds h_i)}{\Re(\pounds h_i)-\Im(\pounds h_j)}, \nonumber \\
\tan(\phi_{ji}^{LL})=\frac{\Re(\pounds h_j)-\Im(\pounds h_i)}{\Re(\pounds h_i)+\Im(\pounds h_j)},
\label{phi_ab}
\end{eqnarray}
which define two real angles $\phi_{ji}^{RR}$ and $\phi_{ji}^{LL}$, respectively.
It is worth noting that the two real angles $\phi_{ji}^{RR}$ and $\phi_{ji}^{LL}$ will be changed by different parameters $\pounds$, but $\Re(\phi_{ji})$ still keeps the same.
The relation between $\Re(\phi_{ji})$ and $\phi_{ji}^{RR}, \phi_{ji}^{LL}$ satisfies
\begin{eqnarray}
\Re(\phi_{ji})=\Re(\eta_{ji})=\frac{1}{2}(\phi_{ji}^{RR}+ \phi_{ji}^{LL})+n \frac{\pi}{2},
\label{phi_ab17}
\end{eqnarray}
where $n$ is an integer. It means the dynamic winding number $w_d=\frac{1}{2}(w_d^{RR}+w_d^{LL})$.
Here, $w_d^{\tau}=\frac{1}{2\pi}\oint_S \partial_{\bm{k}}\phi_{jl}^{\tau} d\bm{k}$, $\tau\in RR, LL$.
Interestingly, the two real angles $\phi_{ji}^{RR}$ and $\phi_{ji}^{LL}$ can be respectively replaced by time-averaged spin textures corresponding to $|\psi_{\bm{k}}(t)\rangle$ and $|\tilde{\psi}_{\bm{k}}(t)\rangle$,
\begin{eqnarray}
&&\phi_{ji}^{RR}=\arctan\left(\frac{\overline{\langle\psi_{\bm{k}}(t) |\sigma_j|\psi_{\bm{k}}(t)\rangle}}{\overline{\langle\psi_{\bm{k}}(t) |\sigma_i|\psi_{\bm{k}}(t)\rangle}}\right),\nonumber \\
&&\phi_{ji}^{LL}=\arctan\left(\frac{\overline{\langle\tilde{\psi}_{\bm{k}}(t) |\sigma_j|\tilde{\psi}_{\bm{k}}(t)\rangle}}{\overline{\langle\tilde{\psi}_{\bm{k}}(t) |\sigma_i|\tilde{\psi}_{\bm{k}}(t)\rangle}}\right),
\label{guanxishi_20190630}
\end{eqnarray}
where $\overline{\langle \bullet \rangle}= \lim_{T \rightarrow \infty} \frac{1}{T}\int_0^{T} \langle \bullet \rangle dt$.
The Eqs.~\eqref{guanxishi_20190630} indicates that the two real angles $\phi_{ji}^{RR}$ and $\phi_{ji}^{LL}$ are physical observables.
For simplicity, we prove the relation with two real angles, $\phi_{yx}^{RR}$ and $\phi_{yx}^{LL}$ in the case of $B>0$.
The right-right spin textures defined with $|\psi_{\bm{k}}(t)\rangle$ satisfies
\begin{eqnarray}
&&\frac{\overline{\langle\psi_{\bm{k}}(t) |\sigma_y|\psi_{\bm{k}}(t)\rangle}}{\overline{\langle\psi_{\bm{k}}(t) |\sigma_x|\psi_{\bm{k}}(t)\rangle}}=\frac{\langle\varphi_{+}|\sigma_y|\varphi_{+}\rangle}{\langle\varphi_{+}|\sigma_x|\varphi_{+}\rangle},
\label{spin polarization_20190630}
\end{eqnarray}
and the left-left spin textures defined with $|\tilde{\psi}_{\bm{k}}(t)\rangle$ satisfies
\begin{eqnarray}
\frac{\overline{\langle\tilde{\psi}_{\bm{k}}(t)|\sigma_y|\tilde{\psi}_{\bm{k}}(t)\rangle}}{\overline{\langle\tilde{\psi}_{\bm{k}}(t) |\sigma_x|\tilde{\psi}_{\bm{k}}(t)\rangle}}=\frac{\langle\chi_{+}|\sigma_y|\chi_{+}\rangle}{\langle\chi_{+}|\sigma_x|\chi_{+}\rangle}.
\label{spin polarization_20190631}
\end{eqnarray}
According to the Hamiltonian (1) in main text, neither the eigenstates $|\varphi_{\mu}\rangle$ nor $\langle\chi_{\mu}|$ are orthogonal in the non-Hermitian system.
We adopt biorthogonal vectors which fulfill $\langle\chi_{\nu}| \varphi_{\mu}\rangle=\delta_{\nu,\mu}$ and $\sum_{{\mu}}|\varphi_{\mu} \rangle \langle \chi_{\mu} |=1$ by normalizing  $|\varphi_{\mu}\rangle=|\varphi_{\mu}\rangle/N_{\mu}$ and $\langle\chi_{\mu}|=\langle\chi_{\mu}|/N_{\mu}$ with $N_{\mu}=\sqrt{\langle\chi_{\mu}| \varphi_{\mu}\rangle}$, this is,
\begin{eqnarray} \label{equaion1a}
|\varphi_{\mu}\rangle&=&\frac{1}{\sqrt{2\varepsilon_{\mu}(\varepsilon_{\mu}-h_z)}}(h_x-ih_y,\varepsilon_{\mu}-h_z)^{\hat{T}}, \nonumber \\
\langle \chi_{\mu}|&=&\frac{1}{\sqrt{2\varepsilon_{\mu}(\varepsilon_{\mu}-h_z)}}(h_x+ih_y,\varepsilon_{\mu}-h_z),
\label{BZT}
\end{eqnarray}
where the superscript $\hat{T}$ is the transpose operation.
Combining with Eq.~\eqref{spin polarization_20190630}, \eqref{spin polarization_20190631} and \eqref{BZT}, we can immediately obtain,
\begin{eqnarray}
&&\frac{\langle\varphi_{+}|\sigma_y|\varphi_{+}\rangle}{\langle\varphi_{+}|\sigma_x|\varphi_{+}\rangle}= \frac{\Re(h_y\pounds_1)+\Im(h_x\pounds_1)}{\Re(h_x\pounds_1)-\Im(h_y\pounds_1)}, \nonumber \\
&&\frac{\langle\chi_{+}|\sigma_y|\chi_{+}\rangle}{\langle\chi_{+}|\sigma_x|\chi_{+}\rangle}= \frac{\Re(h_y\pounds_1)-\Im(h_x\pounds_1)}{\Re(h_x\pounds_1)+\Im(h_y\pounds_1)},
\label{zhengming_20190630}
\end{eqnarray}
where $\pounds_1=h_z^*+\varepsilon_{+}^*$. Similarly, one can obtain $\pounds_1=h_z^*-\varepsilon_{+}^*$ for the case of $B<0$.
Combining with Eq.~\eqref{phi_ab} and \eqref{zhengming_20190630}, one can easily obtain the relations of Eq.~\eqref{guanxishi_20190630}.

\section{S3. Dynamic winding number in the presence/absence of chiral symmetry \label{Sec3}}
Winding number has been widely used for characterizing the topology of Hermitian systems with chiral symmetry.
In one dimension, winding number can be applied to both Hermitian and non-Hermitian systems with or without chiral symmetry.
Here, we consider a 1D two-band topological system governed by the Hamiltonian,
\begin{eqnarray}
  H(k)=h_x(k)\sigma_x+h_y(k)\sigma_y+h_z(k)\sigma_z. \label{equaion1aa}
\end{eqnarray}
The conventional winding number for each band is defined as~\cite{syin2018geometrical, sjiang2018topological},
\begin{eqnarray}
w_\mu=\frac{1}{\pi}\oint_S dk \langle\chi_\mu|i\partial_k|\varphi_\mu\rangle
=\frac{1}{2\pi}\oint_S dk \frac{h_x\partial_k h_y-h_y\partial_k h_x}{\varepsilon_{\mu}(\varepsilon_{\mu}-h_z)}.
\end{eqnarray}
where $S$ is a closed loop with $k$ varying from $0$ to $2\pi$.
Next, we will build relation between conventional winding number to the dynamic winding number in different situations.

\subsection{A. Chiral symmetric systems} \label{ChiralSystem}
When $h_z=0$, the Hamiltonian~\eqref{equaion1aa} has chiral symmetry $\Gamma H(k) \Gamma =-H(k)$ with $\Gamma=i \sigma_x \sigma_y$.
The conventional winding numbers for different bands are the same, and we denote as
\begin{eqnarray} \label{equaion1b}
w_{\pm }=\frac{1}{2\pi}\oint_S dk \frac{h_x\partial_k h_y-h_y\partial_k h_x}{h_x^2+h_y^2}.
\end{eqnarray}
The expression reduces to the Hermitian cases when $\langle\chi_{\mu}|=\langle\varphi_{\mu}|$.
If we define an azimuthal angle as $\phi_{yx}=\arctan(h_y/h_x)$, the above equation is given as
\begin{eqnarray}
 w_{\pm}=\frac{1}{2\pi}\oint_S \partial_k \phi_{yx} dk,
\end{eqnarray}
According to Eq.~\eqref{dynwinding_20190628}, we can immediately conclude that the conventional winding number is equal to the dynamic winding number,
\begin{eqnarray}
w_{\pm}=w_d,
\end{eqnarray}
under  a few constraints of initial state:  $|c_{+}|^2\ne|c_{-}|^2$ for Hermitian systems and $|c_{+}|^2\ne 0\bigcap |c_{-}|^2\ne 0$ for non-Hermitian systems.

\begin{figure}[htp]
\center
\includegraphics[width=4in]{figs4.eps}
  \caption{(a) The phase diagram of the 1D chiral-symmetric topological systems.
  The white, blue, green, dark-yellow and bright-yellow regions respectively share winding number as $w_{\pm}=0, 1/2, 1, 3/2$, and $2$.
  (b)-(f) $\Re(\eta_{y,x})$ as a function of $k$ in different parameters ($J_0,\delta$), which are ($1.5,0.2$), ($1.5,1$), ($0.5,1$), ($0.2,0.5$) and ($0.5,0.2$), corresponding to points $b, c, d, e$ and $f$ in (a), respectively. The other parameters are chosen as $J_1=1$, $J_2=1$ and  $h_z=0$.}\label{fig2b}
\end{figure}



To numerically verify our theory, we consider the systems with $h_x=J_0+J_1\cos(k)+J_2\cos(2k)$ and $h_y=J_1\sin(k)+J_2\sin(2k)-i\delta$.
In the non-Hermitian case with $\delta\neq0$, the conventional winding numbers $w_{\pm}$ can appear half integer values in some parameter ranges, different from the integer values in the Hermitian systems.
For simplicity, we take $J_1=1$, and $J_0$, $J_2$, $\delta$ are real.
The dispersion of this Hamiltonian is
\begin{eqnarray} \label{equaion10b}
\varepsilon_{\pm}(k)=\pm\sqrt{(J_0-\delta+e^{-ik}+J_2e^{-2ik})(J_0+\delta+e^{ik}+J_2e^{2ik})}.
\end{eqnarray}
The energy is symmetric about zero energy, which is ensured by the chiral symmetry.
Since the energy gap must close at phase transition points, we can determine the phase boundaries by the band-crossing condition $\varepsilon_{\pm}(k)=0$, which yields $J_0=\pm \delta+1-J_2$ and $J_0=\pm \delta-1-J_2$ for arbitrary $J_2$. Particularly, $J_0=J_2\pm\delta$ if $|J_2|>0.5$.
Fixing $J_1=1$, $J_2=1$ and $h_z=0$ and changing both $\delta$ and $J_0$, we calculate topological phase diagram distinguished by their winding numbers, see Fig.~\ref{fig2b} (a).
Here, the white, blue, green, dark-yellow and bright-yellow regions possess conventional winding number $w_{\pm}=0, 1/2, 1, 3/2$ and $2$, respectively.
In Figs.~\ref{fig2b} (b)-(f), we also give the angle $\Re(\eta_{yx})$ as a function of quasi-momentum $k$ with different parameters ($J_0,\delta$) marked as b, c, d, e, f in the Figs.~\ref{fig2b} (a).
 %
 The dynamic winding number are $0$, $1$, $1/2$, $3/2$ and $2$, respectively.
The numerical results are in well agreement with the theoretical prediction, which prove the validity for our dynamic approach once again.

\subsection{B. Non-chiral symmetric systems}
When $h_z\ne 0$, the Hamiltonian~\eqref{equaion1aa} breaks the chiral symmetry.
Unlike the systems with chiral symmetry, the conventional winding number for each band is not a quantized number, which indicates that $w_{\pm}$ is no longer a topological invariant.
However, the sum of the winding numbers for different bands,
\begin{eqnarray} \label{equaion1d}
w_{t}=w_{+}+w_{-}=\frac{1}{\pi}\oint_S dk \frac{h_x\partial_k h_y-h_y\partial_k h_x}{h_x^2+h_y^2},
\end{eqnarray}
has been demonstrated to be a topological invariant~\cite{sjiang2018topological}.
The topological invariant $w_{t}$ is independent of $h_z$, although its definition is related to the eigenvector of $H(k)$.
The parameters $h_x$ and $h_y$ become very important for the definition of topological invariant.
Except for the exceptional point $h_x^2+h_y^2=0$, we introduce a complex angle $\phi_{yx}$ satisfying $\tan(\phi_{yx})=h_y/h_x$.
In terms of $\phi_{yx}$, $w_{t}$ can be represented as
\begin{eqnarray} \label{equaion1f}
&&w_{t}=\frac{1}{\pi}\oint_S \partial_k\phi_{yx} dk,
\end{eqnarray}
where the integral is also taken along a loop with $k$ from $0$ to $2\pi$.
According to Eq.~\eqref{sdynwinding_20190628}, we can relate the topological invariant $w_{t}$ to the dynamic winding number
\begin{eqnarray}
  w_{t}=w_++w_-= 2 w_d,
\end{eqnarray}
under a few constraints of initial state:  $|c_{+}|^2\ne|c_{-}|^2$ for Hermitian systems and $|c_{+}|^2\ne 0\bigcap |c_{-}|^2\ne 0$ for non-Hermitian systems.
Fixing $J_1=1$, $J_2=0$ and $h_z=0.5$ in the same model as that in Subsec.~\ref{sChiralSystem}, we calculate the topological invariant $w_{t}$ as a function of $\delta$ and $J_0$, see Fig.~\ref{fig1_nochiral_2019} (a).
Here, the white, green, and bright-yellow regions possess topological invariant $w_{t}=0, 1$ and $2$, respectively.
In Figs.~\ref{fig1_nochiral_2019} (b)-(f), we also give the angle $\Re(\eta_{yx})$ versus the quasi-momentum $k$ with different parameters ($J_0,\delta$) marked as b, c, d, e, f in the Fig.~\ref{fig1_nochiral_2019} (a).
 %
The dynamic winding number are $0$, $1/2$, $1$, $1/2$ and $0$, respectively.
The numerical results are also in well agreement with the theoretical prediction, which demonstrate the validity of our dynamic approach.

\begin{figure}[htp]
\center
\includegraphics[width=4in]{figs5.eps}
  \caption{(a) The phase diagram of the 1D non-chiral-symmetric topological systems.
  The white, green and yellow regions share winding number as $w_{t}=0, 1$,and $2$, respectively.
  (b)-(f) $\Re(\eta_{yx})$ as a function of $k$ in different parameters ($J_0,\delta$), which are ($1.7,0.3$), ($1,0.3$), ($0.3,0.3$), ($0.3,1$) and ($0.3,1.7$), corresponding to points $b, c, d, e$ and $f$ in (a), respectively. The other parameters are chosen as $J_1=1$, $J_2=0$ and $h_z=0.5$.}\label{fig1_nochiral_2019}
\end{figure}

\section{S4. Chern number in 2D systems  \label{Sec4}}
\subsection{A. Alternate choice of reference axis}
\begin{figure}[htp]
\center
\includegraphics[width=4.5in]{figs6.eps}
  \caption{Topologically nontrivial phase with Chern number $C=1$.
  Hermitian case: (a) Azimuthal angle $\phi_{yx}$ and $\phi_{zy}$ in parameter space ($k_x,k_y$),
  where blue and red points represent north and south poles of the Bloch spherical surface.
  Non-Hermitian case: (b) azimuthal angle $\Re(\phi_{yx})$ and $\Re(\phi_{zy})$ in parameter space ($k_x,k_y$).
  where blue and red points represent north and south EPs of the virtual Bloch spherical surface.
}\label{fig444}
\end{figure}
In our main text, we only consider $\theta=\theta_y=\arccos(h_y/|\vec{h}(\bm{k})|)$ and $\phi=\phi_{xz}=\arctan(h_x/h_z)$.
Alternatively, we can also take $\theta=\theta_z=\arccos(h_z/|\vec{h}(\bm{k})|)$  and $\phi=\phi_{yx}=\arctan(h_y/h_x)$, or $\theta=\theta_x=\arccos(h_x/|\vec{h}(\bm{k})|)$  and $\phi=\phi_{zy}=\arctan(h_z/h_y)$.
These two choices lead to distinct observations, but give the same Chern number.
In Fig.~\ref{fig444}(a) and (b), based on different choices of reference axis, we give the azimuthal angle $\phi_{yx}$ and $\phi_{zy}$ in Hermitian($\delta=0$) and non-Hermitain($\delta=0.5$) cases, where the other parameters are the same as the Fig. 2(c) of the main text.
Around the  singularity points, one can also easily obtain the dynamic winding number, and the Chern number $C=1$ in both the Hermitian and non-Hermitian cases, consistent with the ideal Chern number.
This is because the different references only differ from a gauge transformation and the Chern number do not depend on the choice of reference.

\subsection{B. Larger Chern number  \label{Sec5}}

\begin{figure}[!htp]
\center
\includegraphics[width=4in]{figs7.eps}
  \caption{Topologically nontrivial phase with Chern number $C=3$.
  Hermitian case at top: (a) and (b) correspond to the azimuthal angle $\phi_{xz}$ and $\eta_{xz}$ in parameter space ($k_x,k_y$).
  (c) Blue and red points represent north and south poles of the Bloch spherical surface.
  Non-Hermitian case at bottom: (d) and (e) respectively correspond to the azimuthal angle $\Re(\phi_{xz})$ and $\Re(\eta_{xz})$ in parameter space ($k_x,k_y$).
  (f) Blue and red points represent north and south EPs of the  virtual Bloch spherical surface.
}\label{fig5s8}
\end{figure}
The dynamic approach is clearly applicable to topological phases with larger Chern numbers.
To show this, we consider another two-band model which supports band structure with larger Chern number, this is,
\begin{eqnarray} \label{equaion10sa66}
h_x&=&J_x \sin(2k_x); h_y=J_y \sin(2k_y); h_z=m_z-J_z \cos(k_x)-J_z \cos(k_y)-i \delta.
\end{eqnarray}
For the Hermitian case $\delta=0$, the trivial phase is lying in $|m_z|>2J_z$, while the topological phases are distinguished as: ($i$) $J_z<m_z<2J_z$ with the Chern number $C=-1$; ($ii$) $0<m_z<J_z$ with $C=3$; ($iii$) $-J_z<m_z<0$ with $C=-3$; ($iv$) $-2J_z<m_z<-J_z$ with $C=1$.
Here we only verify topological phase with Chern number $C=3$, where the other parameters are chosen as $J_x=J_y=0.2,\ J_z=1$ and $m_z=0.5$.
In Figs.~\ref{fig5s8}(a) and \ref{fig5s8}(b), we give the azimuthal angle $\phi_{xz}$ and $\eta_{xz}$ in parameter space $(k_x,k_y)$, respectively.
One can find that more north and south poles appear in Fig.~\ref{fig5s8}(c), compared with Fig. 2 in the main text.
From the Eq.~(10) in main text, one can also easily obtain the Chern number $C=3$ via dynamic winding number.
For the non-Hermitian case, we consider $\delta=0.1$ and the other parameters are the same as those in the Hermitian case.
In Figs.~\ref{fig5s8}(d) and \ref{fig5s8}(e), we also give the azimuthal angle $\Re(\phi_{xz})$ and $\Re(\eta_{xz})$ in parameter space $(k_x,k_y)$, respectively.
The dynamic winding numbers around EPs become half, while the EPs become double as the Hermitian counterpart, see Fig.~\ref{fig5s8}(f).
Eventually, the Chern number keeps the same as that in the Hermitian case.


\section{S5. Experimental consideration \label{Sec6}}
One can immediately apply the dynamical approach for topological Hermitian systems.
Cold atom systems is an excellent platform to realize topological band models and detect topological invariants.
One and two dimensional spin-orbit couplings have been realized in a highly controllable Raman lattice \cite{squ2013observation, shamner2014dicke, ssun2018highly, ssun2018uncover}.
Initial states are quite easily prepared by loading the atoms into the lattices.
Here, the initial constraint $|c_{\bm{k}+}|^2\ne |c_{\bm{k}-}|^2$ may be not satisfied for some specific momentum $\bm k$, but the occurred probability is so small that the global dynamical azimuthal angle is not affected due to the topological nature.
The spin population $N_{\uparrow(\downarrow)}(\bm{k})$ with different momentum can be measured by spin-resolved time-of-flight (TOF)  absorption imaging\cite{ssun2018uncover}.
Thus, one can obtain the spin population difference $\langle\psi_{\bm{k}}(t)| \sigma_z | \psi_{\bm{k}}(t)\rangle=(N_{\uparrow}(\bm{k})-N_{\uparrow}(\bm{k}))/(N_{\uparrow}(\bm{k})+N_{\uparrow}(\bm{k}))$.
The spin textures $\langle\psi_{\bm{k}}(t)| \sigma_{x(y)} | \psi_{\bm{k}}(t)\rangle$ can be transferred to the spin population difference by applying $\pi/2$ pulse,
that is, $\langle\psi_{\bm{k}}(t)| \sigma_{x(y)} | \psi_{\bm{k}}(t)\rangle=\langle\psi_{\bm{k}}(t)|e^{-i\frac{\pi}{2}\frac{\sigma_{y(x)}}{2}} \sigma_{z}e^{i\frac{\pi}{2}\frac{\sigma_{y(x)}}{2}}| \psi_{\bm{k}}(t)\rangle$.
Because the cold atom systems have long coherent time, there is no obstacle to extract the dynamic winding number via long time average of the spin textures.

To apply the dynamical approach in topological non-Hermitian systems, we should first consider how to realize the topological non-Hermitian models in experiments.
Since two-level non-Hermitian models have been widely realized in optical systems, such as two coupled optical cavities~\cite{schang2014parity, sPeng2014}, optical waveguides~\cite{sgordon2000pmd, sruter2010observation, szeuner2015observation}, optomechanical cavity\cite{saspelmeyer2014cavity, sxu2016topological, sverhagen2017optomechanical} etc.
We mainly discuss how to extract dynamic winding number with two optical waveguides with tunable parameters.
A two-level non-Hermitian system can be realized by introducing gain and loss in the two waveguides.
The coupling strength can be tuned by the waveguide separation.
We regard the two different waveguides as two spin components.
The initial states can be prepared by randomly split the light injecting into the two waveguides.
One can obtain $\langle\psi(l)|\sigma_z|\psi(l)\rangle$ by measuring the intensity difference between two waveguides at propagating distance $l$. Here, the distance $l$ plays the role of time.
Actually, the final states will collapse into one of the eigenstate in the long distance.
Thus, the output intensity difference of the waveguides is sufficient and long distance average of the intensity difference is not necessary.
One can also obtain  $\langle\psi(l)|\sigma_{x(y)}|\psi(l)\rangle$ by insetting a beam splitter before intensity measurement.
By designing the waveguide separation and gain and loss rates, one can simulate the two-band model.
Repeating the above operations, one can finally construct the dynamic winding number.

