\section{Introduction}
Quasiparticles (QPs) are confined and form quantized energy levels in a vortex core~\cite{Caroli1964}.
The energies of the quantized levels are expressed as $E_N=\Delta E(N+1/2)$, where $\Delta E\equiv\hbar\omega_0=\hbar \Delta_0^2/E_F$ ($\Delta_0$ and $E_F$ are the superconducting gap and Fermi energy) is the level spacing, and $N$ is an integer.
Because each level has some width $\delta E=\hbar/\tau$, the ratio of the energy spacing to its width, $r\equiv\Delta E/\delta E=\omega_0\tau$, represents the degree of quantized nature of the vortex core.
In the limit $r\gg 1$, the quantized nature inside the core is distinct, while in the opposite limit $r\ll 1$, the core can be regarded as a normal metal.
For an $r$ value between the two limits, the vortex core is called moderately clean, where the quantized nature is marginal.
As the quantized core is expected to represent many novel features, it is extremely important to know the $r$ value. 
In fact, parameter $r$ is deeply correlated with the dynamics of vortices.
Based on the conventional understanding~\cite{Blatter1994}, when an external driving current $\bm{J}=(J_0,0)$ is applied in the $x$-$y$ plane to a superconductor under a magnetic field in the $z$ direction, the driving force $\Phi_0\bm{J}\times\hat{z}$ forces a vortex, whose displacement is $\bm{u} = (x, y)$, to move in a certain direction with an angle $\phi=\tan^{-1}(\dot{y}/\dot{x}$), generating an electric field $\bm{E}$, where $\Phi_0=h/2e$ is the flux quantum, and $\hat{z}$ is a unit vector in the $z$ direction.
When we can neglect the influence of pinning, the equation of motion of a vortex is as follows:
\begin{equation}
\Phi_0\bm{J}\times\hat{z}=\eta \dot{\bm{u}}+\alpha_H\dot{\bm{u}}\times\hat{z},
\end{equation}
where $\eta=\pi\hbar nr/(1+r^2)$ is the viscous drag coefficient in the longitudinal direction, $\alpha_H=\pi\hbar nr^2/(1+r^2)$ is the viscous drag coefficient in the transvers direction, and $n$ is the density of QPs.
Solving the equation of motion yields the flux-flow Hall angle: 
\begin{equation}
\tan\theta\equiv\tan(\phi-\frac{\pi}{2})=\frac{\alpha_H}{\eta}=r.
\end{equation}
On the other hand, under a constant current, the flux-flow resistivity $\rho_{f}$ relates to the effective viscous drag coefficient as $\eta_{eff}
=\Phi_0B/ \rho_{f}~$\cite{Golosovsky1996, Tsuchiya2001}, and 
$r$ can also be represented in terms of $\eta_{eff}$ as
\begin{equation}
\eta_{eff}\equiv\eta\left(1+\frac{\alpha_H^2}{\eta^2}\right)=\pi\hbar n r.
\end{equation}

Both the measurements of the flux-flow resistivity $\rho_{f}$ and flux-flow Hall angle $\theta$ can provide information about the QPs state in the vortex core.
Thus, we can choose either of the two methods: (1) the flux-flow Hall angle measurement or (2) the effective viscous drag coefficient measurement.
So far, the latter approach has been commonly considered owing to the ease of experimentation~\cite{Klein1993,Golosovsky1994}.
To obtain information about $r$, we need the values of viscous drag coefficients at sufficiently low temperatures, well below the superconducting transition temperature $T_c$, which is  unachievable by an ordinary DC resistivity measurement because of pinning.
Note that we measured the DC Hall resistivity of FeSe$_{x}$Te$_{1-x}$ films in the mixed state near $T_c$ and found that the sign reversal of the DC Hall voltage appears often possibly due to pinning~\cite{Ogawa2018}.
Therefore, even near $T_c$, the effect of pinning cannot be neglected in DC measurements.
Thus, high frequency measurements (typically microwave)~\cite{Gittleman1968} and/or the analysis of the data by a reliable model that can obtain the value of viscous drag coefficients from the measured data~\cite{Coffey1991} are necessary.

High-$T_c$ cuprates are among the most promising candidate materials for the quantized core due to their large energy gap and small Fermi energy.  
Thus, we investigated flux flow by using  microwaves and obtained several interesting but puzzling features.
In the $\eta_{eff}$ measurement, it was found that the core in motion was moderately clean ($\omega_0\tau\sim0.1-0.3$) for a wide range of materials, independent of the cleanness of the core~\cite{Tsuchiya2001,Hanaguri1999,Maeda2007,Maeda2007a}.
However, during the direct Hall angle measurement using the microwave techniques  developed by us~\cite{Ogawa2021}, we found that $\omega_0\tau\sim1-3$ for both  Bi$_2$Sr$_2$CaCu$_2$O$_y$ and YBa$_2$Cu$_3$O$_y$, which are larger by an order of magnitude than those evaluated using the $\eta_{eff}$ measurement, and rather close to the originally expected value of $r$~\cite{Ogawa2021b}.
We explain the origin of the discrepancy in terms of the non-linearity in the viscous drag coefficient~\cite{Larkin1976a} and other possible dissipation mechanisms for the moving vortex that are not represented by the basic equation of motion of the vortex, including those recently proposed ~\cite{Hofmann1998,Hayashi1998,Smith2020a,Kogan2021a}.
However, it is not known whether such a large value and the discrepancy between the two methods will be observed in other clean superconductors.

FeSe is another promising candidate for the quantized core because of its very small Fermi energy, comparable to the superconducting gap $\Delta_0\sim E_F$~\cite{Kasahara2014}, and long QP-lifetime $\tau$ in the superconducting state~\cite{Okada2021}.
Indeed, scanning tunneling spectroscopy (STS) measurements of FeSe single crystals show Friedel-like oscillations, which shows the quantized levels in the core~\cite{Hanaguri2019}, and we expect that features specific to superclean cores will be observed.
The $\eta_{eff}$ measurement of a pure single crystal of FeSe yields   $r=1\pm0.5$~\cite{Okada2021}.
Although this is the largest value of  $r$ measured thus far among various superconductors, it is still in the moderate clean region.
We expect a much larger value in flux-flow Hall angle measurements.
Another interesting viewpoint is that FeSe is a multi-band superconductor, where we expect to observe novel effects. 
Indeed, in multi-band superconductors, the dissociation of electron and hole vortices has been  proposed theoretically~\cite{Lin2013}.
Although there have been experimental studies to investigate flux flow in multi-band superconductors~\cite{Shibata2003,Akutagawa2008,Okada2012,Takahashi2012a,Okada2013a,Okada2013b,Okada2014,Okada2015,Okada2021}, no such novel feature has been reported.

In this study, we investigated the microwave flux-flow Hall effect in FeSe pure single crystals using the cross-shaped bimodal cavity technique.
We find that the flux-flow Hall angle of FeSe is about 0.5 at low temperatures, which is equal to or smaller than that evaluated by the $\eta_{eff}$ measurement.
In addition, the Hall angle behaves in a different manner as a function of the magnetic field from what was obtained from the $\eta_{eff}$ measurement. 
These features contrast the cuprate superconductors.
We argue that these might be characteristic of multi-band superconductors  due to the cancellation of contributions from the hole and the electron.
Therefore, our study is the first demonstration of novel feature characteristics of flux flow in multi-band superconductors.


\section{Methods}
The magnitude of the flux-flow Hall angle is given by $|\tan\theta|=|E_y/E_x|$ from Faraday's law $\bm{E}=\dot{\bm{u}}\times\bm{B}$.
The definition of the surface impedance tensor  is $\tilde{Z}\equiv\bm{E}^{\parallel}(z=0)/\int^{\infty}_{0}\bm{J}dz$, which yields $\bm{E}\propto\tilde{Z}\bm{J}$, where the symbol $\parallel$ indicates that it is parallel to the surface of the sample.
Thus, the magnitude of the flux-flow Hall angle is represented as 
\begin{equation}
|\tan\theta|=\left|\frac{\sigma_{xy}}{\sigma_{xx}}\right|=\left|\frac{Z^H}{Z^L}\right|, 
\end{equation}
where $\sigma_{xx}$ and $\sigma_{xy}$ are the longitudinal and transverse conductivities, respectively, and $Z^L$ and $Z^H$ are the diagonal and off-diagonal components of the surface impedance tensor, respectively~\cite{Ogawa2021}.
Three types of measurements are necessary to obtain the magnitude of the flux-flow Hall angle:  the DC resistivity measurements, measurements of the longitudinal components of the surface impedance tensor, $Z^L$, and measurements of the transverse components of the surface impedance tensor, $Z^H$.
The DC resistivity tensor $\tilde{\rho}$ is measured using a Physical Property Measurement System (PPMS) from Quantum Design in a standard six-probe configuration. 
The longitudinal components of the surface impedances tensor $Z^L$ are measured using the conventional cavity perturbation method with a cylindrical cavity~\cite{Klein1993}, whereas
the transverse components of the surface impedances tensor $Z^H$ are measured using a cross-shaped bimodal cavity, in which the changes in the resonance characteristics and the surface impedance tensor of the sample are represented as
\begin{equation}
\Delta\left(\frac{1}{2Q}\right)=\Delta\left( G^LR^L+G^H|X^H|\right)
\end{equation}
and
\begin{equation}
\Delta\left(\frac{f}{f_0}\right)\equiv-\frac{f-f_0}{f_0}=\Delta\left( G^L X^L-G^H |R^H|\right),
\end{equation}
where $Q$ denotes the quality factor of the resonance, $f$ and $f_0$ denote the resonance frequencies with and without the sample, respectively, $R^L$ and $X^L$ are the real and imaginary parts of the longitudinal components of the surface impedance tensor ($Z^L\equiv R^L-iX^L$), respectively, $R^H$ and $X^H$ are the real and imaginary parts of the transverse components of the surface impedance tensor ($Z^H\equiv R^H-iX^H$), respectively, $G^L$ and $G^H$ represent the geometric constants in the longitudinal and transverse directions, respectively, which depend on the shape of the cavity and the sample and take the same value under the assumption that these geometries are symmetrical ($G^L=G^H$), and $\Delta$ represents the difference between the data of the same sample at different temperatures.
For explicit procedure, see ref.~\cite{Ogawa2021}.

FeSe pure single crystals were synthesized by the chemical vapor transport technique in a temperature gradient furnace using KCl/AlCl$_3$ flux~\cite{Bohmer2013}.
We performed the microwave measurements using the cross shaped bimodal cavity, operating in the two orthogonal TE$_{011}$ and TE$_{101}$ modes at 15.8 GHz, which have a quality factor $Q$ of approximately $3\times 10^3$.
The sample sizes  for the cross-shaped bimodal cavity measurements are typically $1.2\times1.2\times 0.1~ \mathrm{mm}^3$.
In all experiments, magnetic fields of up to 7 T were applied under field-cooled conditions.

\section{Results and Discussion}

Figures 1(a) and (b) show the temperature dependence of  the DC longitudinal resistivity $\rho_{xx}$ of FeSe.
$T_c^{zero}$ is about 9 K and residual resistivity ratio ($RRR=\rho_{xx}(300$~K$)/\rho_{xx}(T_{c}^{onset}=10$~K)) is about 29.
Figure 1(c) shows the magnetic field dependence of the DC Hall angle $\tan\theta_{dc}$ in the normal state. 
The DC Hall angle of FeSe single crystal above $T_c$ shows non-linearity with respect to the magnetic field due to its multi-carrier nature, as is well known~\cite{Lei2012}.
Figure 2 shows the temperature dependence of the longitudinal components of the surface impedance tensor $Z^L$ under the magnetic field.
$Z^L$ increases with an increasing magnetic field and decreases with a decreasing temperature.
This behavior is in accordance with flux-flow resistivity~\cite{Okada2021}.
Therefore, most of the observed behavior of  $Z_L$ can be attributed to flux-flow.
These results demonstrate that the sample used for experiments has the usual properties of FeSe single crystals.
Figure 3 shows the magnitude of the magnetic field dependence of the flux-flow Hall angle $|\tan\theta|$.
The uncertainty is mainly due to the small signal (change in the resonance characterisitics) of the sample.
It increases with decreasing temperature due to the development of a superconducting condensate in small magnetic fields.
We observed that the magnitude of the flux-flow Hall angle at 5.3 K is about 0.5.
It is equal to or slightly smaller than that evaluated from the effective viscous drag coefficient measurement ($r= 1.0\pm 0.5$)~\cite{Okada2021}.
The flux-flow Hall angle does not depend on magnetic field strongly, even showing a very weak magnetic field dependence.
This field dependence is rather different from what was observed in the $\eta_{eff}$ measurement, where it increases with increasing magnetic field as $\tan\theta\sim B^{0.2}$, corresponding to the sublinear increase of the flux-flow resistivity as a function of magnetic field.
\begin{figure}[htbp]
\centering
 	\includegraphics[bb=0 0 660 420,keepaspectratio,width=86mm]{Fig1.png}
 	\caption{ (a) The temperature dependence of the longitudinal resistivity of FeSe under a magnetic field near $T_c$ (b) The temperature dependence of the normalized longitudinal resistivity $\rho_{xx}/\rho_{xx}(300$~K$)$ at 0 T; $T_c^{zero}$ is about 9 K and $RRR=\rho _{xx}(300 $ K$)/\rho_{xx}(T_c^{onset}=10$ K$)$ is about 29 (c) The magnetic field dependence of the DC Hall angle $\tan\theta_{dc}$ in the normal state (20 K, 15 K, and 12 K, respectively). The Hall angle behaves non-monotonically with an increasing magnetic field due to its multi-carrier nature.
}
\end{figure}
\begin{figure}[htbp]
\centering
 	\includegraphics[bb=0 0 660 420, keepaspectratio,width=86mm]{Fig2.png}
 	\caption{Temperature dependence of the longitudinal components of the surface impedance tensor $Z^L$;  magnetic fields of up to 7 T are applied under field-cooled conditions.
}
\end{figure}
\begin{figure}[htbp]
\centering
 	\includegraphics[bb=0 0 900 600, keepaspectratio,width=86mm]{Fig3.png}
 	\caption{Magnetic field dependence of the magnitude of the flux-flow Hall angle $|\tan\theta|$; magnetic fields of up to 7 T are applied under field-cooled conditions.
Open circles indicate the flux-flow Hall angle calculated from the data of $\eta_{eff}$ measurements~\cite{Okada2021}.
The magnitude of flux-flow Hall angle  at low temperatures is equal to or smaller than that evaluated from the effective viscous drag coefficient and does not depend on magnetic-field strongly even showing a very weak field dependence, which are different from $\eta_{eff}$ measurements.}
\end{figure}

The result of the flux-flow Hall effect in FeSe is remarkable in two senses.
(1)  Unlike cuprate superconductors, the magnitude of the flux-flow Hall angle is equal to or smaller than that evaluated from $\eta_{eff}$.
(2) There is a discrepancy in the magnetic field dependence of the Hall angle between the direct Hall angle measurement and the $\eta_{eff}$ measurement in the same material.
To understand the first feature, we consider that the multi-band nature of FeSe is essentially important.
FeSe has two hole bands at the $\Gamma$ point and two electron bands at the $M$ point~\cite{Subedi2008}.
Both holes and electrons contribute to DC transport properties in the normal state, leading to a cancellation of the Hall voltage~\cite{Lei2012}.
We believe that a similar effect can take place even in vortex dynamics.
However, except for DC measurements in the vicinity of $T_c$~\cite{Jin2001,Lei2010,Wang2011,Ogawa2018},
no direct observation of the flux-flow Hall effect in multi-band superconductors well below $T_c$ has  been reported in experiments.
On the other hand, theoretical studies of vortex dynamics in multi-band superconductors have been undertaken~\cite{Babaev2002,Lin2013,Silaev2016}.
In particular, a microscopic theory~\cite{Kopnin2001} shows that the flux-flow conductivity tensor has contributions from holes as well as electrons.
We believe that the competition of these contributions can explain the difference in the estimated number of Hall angle between the two methods in both the magnitude.
Below we will discuss this scenario more explicitly.

According to the microscopic theory\cite{Kopnin2001}, in the flux-flow regime, vortex dynamics is dominated by the force balance equation $f_L+f_{env}=0$, where $f_L$ is the Lorentz force due to the external transport current and $f_{env}$ is the environmental force.
The environmental force originates from the environment on a vortex (e.g., Magnus force, spectral force, {\it etc.}) and is expressed as a linear superposition of terms contributed by the different bands, $k$ such that $f_{env}=\sum_kf_k$.
Therefore, for FeSe where both hole bands and electron bands contribute, the flux-flow conductivity tensor also has contributions from holes as well as electrons, and these contributions cause cancellations in the off-diagonal components (Hall components).
On the other hand, in the diagonal components, both contribute additively. 
Explicitly, the expressions for flux-flow conductivity in the longitudinal and transverse directions at low temperatures are as follows.
\begin{eqnarray}
\sigma_{xx}&=&\frac{2e}{(2\pi)^3B}\sum_n(\int_e+\int_h)S(p_z)dp_z\nonumber \\
&&\times\int\frac{db}{2}\frac{\partial\epsilon_n}{\partial b}\frac{df^{(0)}(\epsilon_n)}{d\epsilon}\gamma_{xx}(\epsilon_n),
\label{sigmaxx}
\end{eqnarray}
\begin{eqnarray}
\sigma_{xy}&=&\frac{2e}{(2\pi)^3B}\sum_n(\int_e-\int_h)S(p_z)dp_z\nonumber \\
&&\times\int\frac{db}{2}\frac{\partial\epsilon_n}{\partial b}\frac{df^{(0)}(\epsilon_n)}{d\epsilon}\gamma_{xy}(\epsilon_n), 
\label{sigmaxy}
\end{eqnarray}
where $p_z$ denotes the momentum along the $z$ axis , $S(p_z)$ denotes the area of the cross section of the Fermi surface cut by the plane $p_z=const$, $b$ denotes the impact factor, $\epsilon_n$ denotes the energy spectrum, $n$ denotes an index for the energy spectrum, $f^{(0)}$ denotes the equilibrium distribution function, and $\gamma_{xx(xy)}$ are the factors defined by $\omega_n$ and $\tau_n$, which are an energy-interval spacing and relaxation time, as $\gamma_{xx}(\epsilon_n)=\omega_n\tau_n/(1+\omega_n^2\tau_n^2)$ and $\gamma_{xy}(\epsilon_n)=\omega_n^2\tau_n^2/(1+\omega_n^2\tau_n^2)$, respectively.
We write these equations as $\sigma_{xx}= e\Gamma_x(N_e+N_h)/B $ and $\sigma_{xy}=e\Gamma_y(N_e-N_h)/B$, where $N_e$ and $N_h$ are the density of electrons and holes, respectively, and $\Gamma_{x}$ and $\Gamma_{y}$ are coefficients which represent other factors, respectively. 
By defining the ratio of the contributions of holes to those of electrons as $\beta\equiv N_h/N_e$ and $e\Gamma_{x(y)}\equiv D_{x(y)}$, we obtain that $\sigma_{xx}=D_xN_e(1+\beta)/B$ and $\sigma_{xy}=D_yN_e(1-\beta)B$.
With these notations, the experimentally evaluated value of the degree of quantization (DQ) by the $\tan\theta$ measurement, $r_H$, and that by the $\eta_{eff}$ measurement, $r_{\eta}$, are expressed as 
\begin{equation}
r_H=\left|\frac{\sigma_{xy}}{\sigma_{xx}}\right|=\left|\frac{D_y(1-\beta)}{D_x(1+\beta)}\right|
\end{equation}
 and
\begin{eqnarray}
r_{\eta}&=&\frac{B\Phi_0}{\pi\hbar n}\cdot\sigma_{xx}\left[1+\left(\frac{\sigma_{xy}}{\sigma_{xx}}\right)^2\right] \nonumber \\ 
&=&\frac{\Phi_0}{\pi\hbar}\cdot D_x\left[1+\left(\frac{D_y(1-\beta)}{D_x(1+\beta)}\right)^2\right],
\end{eqnarray}
respectively.

To see the ``complicated'' interaction of electron bands and hole bands, we change $\beta$ and investigate how $r_H$ ands $r_{\eta}$ behave explicitly, which is represented in figure 4(a). 
Note that the data is normalized by the value at $\beta=0$ (corresponding to a single-band case).
In figure 4(a), $\tilde{r}_H-\tilde{r}_{\eta}\equiv r_H(\beta)/r_H(0)-r_{\eta}(\beta)/r_{\eta}(0)$ are shown for the various value of $D_y/D_x$. 
We find  that $\tilde{r}_H-\tilde{r}_{\eta}$ is negative in wide range of regions, except in the vicinity of $\beta=0$.
This means that the DQs evaluated by $\tan\theta$ measurements are smaller than those evaluated by $\eta_{eff}$ measurements, rather generally for materials having both electron bands and hole bands.
This is in good agreement with what we observed in FeSe. 
\begin{figure}[htbp]
\centering
 	\includegraphics[bb=0 0 1150 400, keepaspectratio,width=86mm]{Fig4.png}
 	\caption{(a) Contours of $\tilde{r}_H-\tilde{r}_{\eta}$. In the blue colored region, $\tilde{r}_H$ is smaller than $\tilde{r}_{\eta}$. (b) The $\beta$ dependence of the normalized DQs $\tilde{r}_H$ and $\tilde{r}_{\eta}$ when $D_{y}/D_{x}=5/8$; the red lines represent values obtained from $\tan\theta$ measurements $\tilde{r}_H$, and the blue lines represent those obtained from $\eta_{eff}$ measurements $\tilde{r}_{\eta}$. Fig. 4(b) corresponds to the cross-section of Fig. 4(a) at $D_{y}/D_{x}=5/8$. $\tilde{r}_H$ is always smaller than $\tilde{r}_{\eta}$ except for $\beta$ is 0.}
\end{figure}


Next, we will discuss the above mentioned scenario more quantitatively, and try to extract some numbers for parameters shown up above.
For that purpose, it should be noted that both $\Gamma_x$ and $\Gamma_y$ depend on purity of superconductors.
At low temperatures and for small magnetic fields limit, when $\omega_0\tau\approx 1$, explicit calculation of eqs.~(\ref{sigmaxx}) and (\ref{sigmaxy}) leads to $\Gamma_x\approx 0.8$ and $\Gamma_y\approx 0.5$~\cite{Kopnin2001}.
Soon below, we will reestimate $\Gamma_x$ and $\Gamma_y$ from the experimental data, and check the consistency of the choice, $\Gamma_y/\Gamma_x=D_y/D_x=5/8$.
With $D_y/D_x=5/8$,
\begin{equation}
r_H=\frac{5}{8}\cdot\left|\frac{1-\beta}{1+\beta}\right|
\end{equation}
 and
\begin{eqnarray}
r_{\eta}=\frac{\Phi_0}{\pi\hbar}\cdot D_x\left[1+\left(\frac{5}{8}\cdot\frac{1-\beta}{1+\beta}\right)^2\right].
\end{eqnarray}
Figure 4(b) shows the $\beta$ dependence of the normalized DQs $\tilde{r}_H$ and $\tilde{r}_{\eta}$ when $D_x/D_y$ equals to 5/8.
In particular, for $\beta=1$, the magnitudes of the electron and hole contributions are exactly the same, so $\tilde{r}_H=0$ due to the oppositely acting contribution of electrons and holes.
From eq. (11), we can calculate that $\beta\approx 0.11$  since $r_H\approx 0.5$ as shown in figure 3.
This means that the magnitude of the contribution from, for instance, electrons  is about one tenth of holes. (Since in our measurement, sign of the Hall coefficient cannot be obtained, the role of electrons and hole can be reversed).
Substituting this value into eq. (12), we find that $r_{\eta}/D_x=7.5\times10^{18}~\mathrm{C^{-1}}$, leading to $D_x$ is approximately $1.3\times 10^{-19}~\mathrm{C}$ with $r_{\eta}=1.0$.
This means $\Gamma_x\approx0.83$ and $\Gamma_y\approx0.52$, which is comparable to theoretically estimated value just above.

As for the magnetic field dependence of two DQs, $\beta$ is considered not to show the strong magnetic field dependence since it only depends on the carrier density.
The definition in eqs.~(\ref{sigmaxx}) and (\ref{sigmaxy}) suggest that $D_{x(y)}$, namely $\Gamma_{x(y)}$, is considered not to show strong magnetic field dependence either.
However, it should be noted that these equations were derived under the assumption that the order parameter is uniform (isotropic).
Experiments found that suprconducting gap of FeSe is highly anisotropic~\cite{Kasahara2014,Liu2018}.
Since in Fe-based superconductors including FeSe, the sub-linear magnetic field dependence of the flux-flow resistivity obtained by $\eta_{eff}$ study is closely related to the fact that the superconducting gap function is anisotropic~\cite{Okada2014,Okada2020}, we may expect that the anisotropy of superconducting gaps explains the difference in the magnetic-field dependence between $r_H$ and $r_{\eta}$.


\section{Conclusion}
In summary, we measured the flux-flow Hall effect in a multi-band superconductor FeSe pure single crystal to investigate the nature of the vortex core state by means of the cross-shaped bimodal cavity technique.
We found that the flux-flow Hall angle of FeSe is about 0.5 at low temperatures, which is equal to or smaller than that evaluated by the $\eta_{eff}$ measurements.
We consider that the observed feature is related to the multi-band nature of FeSe.
Our calculations for the flux-flow conductivity tensor for multi-band superconductors that has contributions from holes as well as electrons show partial cancellation of the flux-flow Hall voltage.
Therefore, our study is the first demonstration of a novel feature characteristic of flux flow in multi-band superconductors.




