\section{Introduction}
The nature of the quantum vacuum is fundamental to the study of quantum information. By coupling the vacuum field to particle detectors, it is possible to gain understanding of the quantum vacuum and its response to the structure of spacetime. This
is particularly interesting if topological features or horizons are present.
These detectors are often taken to be simple two-level quantum systems (known as Unruh-DeWitt detectors) that interact with an underlying scalar field, a model that captures the essential features of the light-matter interaction \cite{MartinMartinez:2012th,Funai:2021jpc}.  Such detectors have been employed to study the structure of spacetime \cite{Smith2016,Ng2017}, black holes \cite{Henderson2018,Tjoa2020}, and the thermality of de Sitter spacetime \cite{Steeg2009,Huang2017}.

Consider a detector with uniform acceleration in flat spacetime. The detector will experience the Unruh effect, in which the temperature increases in proportion to acceleration \cite{Fulling1973,Davies1975,Unruh1976}, and  heat up.  This effect arises because the temperature of the vacuum of one set of modes is different than the vacuum temperature of a second set of modes.
Though highly idealized in its original assumptions (such as that of an eternally uniformly accelerating detector), a model-independent derivation of the Unruh effect has been given  in the context of axiomatic quantum field theory \cite{sewell1982quantum}, with the field temperature (given by the Kubo-Martin-Schwinger (KMS) condition \cite{Kubo1957,Martin1959,Haag1967}) and the temperature measured by the detector being the same. There have since been many demonstrations that detectors undergoing other forms of acceleration (non-uniform, circular) get hot \cite{Bell:1986ir,Costa:1987vv,Percoco:1991iv,Villalba:1999uj,Ostapchuk:2011ud,Hu:2012jr,Doukas2013}.
 In these more general situations the field temperature is  positively correlated with the detector temperature, with the latter a monotonically increasing function of the former.

In the last few years it has been shown that some physical situations exhibit the so-called \textit{anti}-Unruh effect instead, in which  the temperature of the field is no longer  positively correlated with that measured by the detector \cite{Brenna2016,Garay2016}.
 This anti-Unruh effect can be split into two cases: a  weak Anti-Unruh effect (as the temperature of the field increases, the detector clicks less often) and a strong Anti-Unruh effect (the field temperature and detector temperature are inversely related) \cite{Garay2016}.

When considering black holes, the Hawking effect is the analogue of the Unruh effect \cite{Birrell1982}.
While in general detector temperatures are positively correlated with the field temperature outside a black hole
\cite{Hodgkinson2012,Smith2014,Hodgkinson:2014iua,Ng:2014kha}, recently an anti-Hawking effect was shown to also exist, in which a static Unruh-DeWitt (UdW) detector exhibited both strong and weak versions of the phenomenon \cite{Henderson:2019uqo}.  This was explicitly demonstrated for  the (2+1) dimensional static Banados-Teitelboim-Zanelli (BTZ) black hole.
For sufficiently small black holes, the temperature measured by the detector would decrease as the Kubo-Martin Schwinger (KMS) field temperature of the Hawking radiation increased. The anti-Hawking effect has since been observed  for a
broader range of boundary conditions \cite{Campos:2020twd}, though its weak version is not observed for massless topological black holes in four spacetime dimensions \cite{Campos:2020lpt}.

The effects of spacetime dragging due to rotation on these phenomena are much less understood.
The quantum vacuum around a rotating black hole
is known to exhibit features significantly different from  its non-rotating counterpart \cite{Winstanley_2001,Casals_2008,Ottewill:2000yr}.  Several investigations of the behaviour of quantum scalar fields in
the background of a rotating BTZ black hole have been carried out \cite{Mann:1996ze,Singh:2011gd,Bussola:2017wki,Meitei:2018mgo}, and studies   investigating the response of Unruh-DeWitt detectors   in such spacetimes have also been undertaken \cite{Hodgkinson2012}.  Recently it was shown
that rotation has very significant effects on the entanglement harvesting abilities of   UdW  detectors, with
the harvested entanglement being considerably amplified at intermediate distances (about 20-50 horizon radii) from the
black hole \cite{Robbins2020}.

Motivated by this, we study here  the implications of rotation for the anti-Hawking effect.   We find that  rotation  increases the intensity of the weak anti-Hawking effect, but has   a negligible influence on its threshold critical temperature. However for the strong anti-Hawking effect,  we find that there is a strong dependence on the angular momenta, with the effect becoming stronger or weaker depending on the boundary conditions.  The  influence of AdS length on the strong and weak versions of the effect is likewise distinct:  the weak anti-Hawking effect is independent of AdS length whereas the strong version sees an increased temperature range.

\section{Unruh-DeWitt Detectors}

To model the interaction between the detectors and the field, we take the detectors to be two-level quantum systems with ground state $\ket{0}_D$ and excited state $\ket{1}_D$, separated by an energy gap $\Omega_D$. We shall assume that these detectors have a spacetime trajectory $x_D(\tau)$. The interaction Hamiltonian is
\begin{align}
H_D=\lambda\chi_D(\tau)\left(e^{i\Omega_D\tau}\sigma^+ + e^{-i\Omega_D\tau}\sigma^-\right)\otimes\phi[x_D(\tau)]\,,
\end{align}
where the switching function dictating the duration of the interaction between the detector and field is $\chi_D(\tau)$. $\lambda\ll1$ is the field-detector coupling constant, and the ladder operators that raise and lower the energy levels of the detectors are $\sigma^+= \ket{1}_D\bra{0}_D$, $\sigma^-= \ket{0}_D\bra{1}_D$, respectively.

Let the initial state of the detector-field system be $\ket{\Psi_i}=\ket{0}_D\ket{0}$. The final state after a time $t$ is then $\ket{\Psi_f}=U(t,0)\ket{\psi_i}$, where $U(t,0)=\mathcal{T}e^{-i\int dt\left[\frac{d\tau_D}{dt}H_D(\tau_D))\right]}$, with $\mathcal{T}$ being the time-ordering operator. With the reduced density operator   $\rho=\text{Tr}_\phi\ket{\Psi_f}\bra{\Psi_f}$ being the state of the system after integrating over the field's degrees of freedom, we have \cite{Smith2016,Smith2017}
\begin{align}
\rho_{AB}=\begin{pmatrix}
1-P_D &0\\
0&P_D
\end{pmatrix}
+\mathcal{O}(\lambda^4)\,,
\end{align}
where
\begin{widetext}
\begin{align}
P_D=\lambda^2\int d\tau_Dd\tau_D'\chi_D(\tau_D)\chi_D(\tau_D')e^{-i\Omega_D(\tau_D-\tau_D')}W(x_D(\tau_D),x_D(\tau_D'))
\label{eq: PD}
\end{align}
\end{widetext}
 is the detector's transition probability. We note that this quantity depends on the two-point correlation function, $W(x,x')=\braket{0|\phi(x)\phi(x')|0}$ (also called the Wightman function) of the vacuum. From the transition probability, we can then define the response function,
 \begin{align}
 \mathcal{F}=\frac{P_D}{\lambda^2\sigma}\,,
 \end{align}
 where $\sigma$ describes the timescale of interaction between the field and the detector. In this paper, we shall focus on a Gaussian switching function, $\chi_D(\tau)=e^{-\frac{\tau^2}{2\sigma^2}}$.

 We will  consider fields whose Wightman functions   obey the relation
 \begin{equation}\label{KMScond}
 W(\tau-i/T_{KMS},\tau')=W(\tau',\tau)
 \end{equation}
 known as the Kubo-Martin-Schwinger (KMS) condition  \cite{Kubo1957,Martin1959,Haag1967}.  The quantity $T_{KMS}$
 in \eqref{KMScond} can be regarded as the temperature of the quantum field in the spacetime.  It depends only on
 the nature of the quantum field and the spacetime background.

Correspondingly, we can also define the detector's temperature from its excitation to de-excitation ratio (EDR). Let
 \begin{align}
 \mathcal{R}=\frac{\mathcal{F}(\Omega)}{\mathcal{F}(-\Omega)}\,,
 \end{align}
 such that there exists a temperature, $T$, that obeys the same form of the KMS condition \cite{Fewster2016},
 \begin{align}
  \mathcal{R}=e^{-\Omega/T}\,.
 \end{align}
Labelling the temperature that obeys this condition by $T_{EDR}$, we have
\begin{align}
T_{EDR}=-\frac{\Omega}{\log\mathcal{R}}
\end{align}
The quantity $T_{EDR}$ can be regarded as the temperature that the UdW detector registers in the spacetime.

Normally we expect $T_{EDR}$ and $T_{KMS}$ to be positively correlated: as the black hole gets hotter, the field temperature  increases
and the temperature registered by the UdW detector likewise increases.   This is indeed the case for most situations
in black hole physics. As noted above, it was recently shown that this is not always the case  \cite{Henderson:2019uqo}, and
that sometimes the contrary situation, known as the anti-Hawking effect, occurs.  As with the anti-Unruh effect \cite{Brenna2016,Garay2016}, we define
\begin{align}\label{weakAH}
&\frac{d\mathcal{F}(\Omega)}{dT_{KMS}}<0  \quad \textrm{weak}\\
&  \frac{\partial T_{EDR}}{\partial T_{KMS}} < 0
\quad
\textrm{strong}
\label{strongAH}
\end{align}
for the weak and strong anti-Hawking effects respectively.

 \section{Rotating BTZ Black holes}

 We can write the action of our system as $S=S_{EH}+S_{\phi}$, where
 \begin{align}
 S_{EH}=\frac{1}{16\pi}\int R\sqrt{-g}d^3x
 \end{align}
 is the Einstein-Hilbert action ($R$ is the Ricci scalar and $g$ is the determinant of the metric tensor $g_{\mu\nu}$) and
 \begin{align}
 S_{\phi}=-\int\left(\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{16}R\phi^2\right)\sqrt{-g}d^3x
 \end{align}
 is the action for a conformally-coupled scalar field $\phi$. We are interested in analyzing both the KMS temperature of the field and the EDR temperature of a detector near a rotating BTZ black hole, whose line element is  \cite{Banados1992}
\begin{equation}\label{btzmet}
ds^2=-(N^\perp)^2dt^2+f^{-2}dr^2+r^2(d\phi+N^{\phi}dt)^2
\end{equation}
where, $N^\perp=f=\sqrt{-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}}$ and $N^\phi =-\frac{J}{2r^2}$ with $M=\frac{r_+^2+r_-^2}{\ell^2}$ the mass of the black hole  and $J=\frac{2r_+ r_-}{\ell}$ its angular momentum. The Hawking temperature is
\begin{align}
T_H=\frac{1}{2\pi\ell}\left(\frac{r_+^2-r_-^2}{r_+}\right)\,,
\end{align}
where the inner and outer horizon radii are denoted by $r_-$ and $r_+$, and $\ell$ is the AdS length. Note that $|J|\leq M\ell$, with extremality occurring when $r_+=r_-$ (i.e. $J=M\ell$).
\begin{figure*}[t!]
\centering
\begin{minipage}{\textwidth}
  \centering
    \includegraphics[width=0.6\textwidth]{PDLegendConstantM.pdf}
\end{minipage}\\
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.1_Omega=0.1_Zeta=1_Inset_M=100}.pdf}
    \caption*{(a) $\zeta=1$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.1_Omega=0.1_Zeta=0_Inset_M=100}.pdf}
    \caption*{(b) $\zeta=0$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.1_Omega=0.1_Zeta=-1_Inset_M=100}.pdf}
    \caption*{(c) $\zeta=-1$}
\end{minipage}
\caption{Response functions for a black hole of mass $M=1/10$ for Dirichlet,  transparent, and Neumann boundary conditions and an energy gap of $\Omega\ell=1/10$. The inset plots correspond to $M=100$. As expected, the rotation of the black hole has a smaller effect for larger masses. As the mass of the black hole increases, the weak anti-Hawking effect goes away for $\zeta=1$ and $\zeta=0$.
Note that for $\zeta=-1$, the weak anti-Hawking effect is still present even for large mass black holes, with the distinctions
between the different rotation parameters  so tiny that the curves effectively all overlap.
}
\label{fig: FJ_LargerMass}
\end{figure*}

\begin{figure*}[t!]
\centering
\begin{minipage}{\textwidth}
  \centering
    \includegraphics[width=0.6\textwidth]{PDLegendConstantM.pdf}
\end{minipage}\\
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.001_Omega=1_Zeta=1}.pdf}
    \caption*{(a) $\Omega\ell=1$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.001_Omega=0.1_Zeta=1}.pdf}
    \caption*{(b) $\Omega\ell=1/10$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.001_Omega=0.01_Zeta=1}.pdf}
    \caption*{(c) $\Omega\ell=1/100$}
\end{minipage}
\caption{Response of a rotating BTZ black hole with mass $M=1/1000$ and Dirichlet boundary conditions ($\zeta=1$). We note that for transparent and Neumann boundary conditions, qualitatively similar results are obtained.
}
\label{fig: FJ}
\end{figure*}

In the Hartle-Hawking vacuum, a conformally coupled-scalar field has a Wightman function that can be written as the image sum over the Wightman functions for AdS${}_3$ \cite{Lifschytz1994,Carlip1998},
\begin{align}
W_{BTZ}(x,x')=\sum_{n=-\infty}^\infty \eta^nW_{AdS_3}(x,\Gamma^nx')
\end{align}
where $\Gamma x'$ takes $(t,r,\phi)\to(t,r,\phi+2\pi)$, $\eta=1$ is an untwisted scalar field and $\eta=-1$ is a twisted scalar field. This yields \cite{Hodgkinson2012,Smith2014}
\begin{widetext}
\begin{equation}
\begin{aligned}
W_{BTZ}&=\frac{1}{4\pi}\frac{1}{2\sqrt{\ell}}\sum_{n=-\infty}^{n=\infty}\eta^n\left(\frac{1}{\sqrt{\sigma_{\epsilon}(x,\Gamma^nx')}} -\frac{\zeta}{\sqrt{\sigma_{\epsilon}(x,\Gamma^nx')+2}}\right)
\label{eq: sum}
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
\sigma_\epsilon(x,\Gamma^nx')^2=&-1+\sqrt{\alpha(r)\alpha(r')}\cosh\left[\frac{r_+}{\ell}(\Delta\phi-2\pi n)-\frac{r_-}{\ell^2}(t-t')\right]\\
&-\sqrt{(\alpha(r)-1)(\alpha(r')-1)}\cosh\left[\frac{r_+}{\ell^2}(t-t')-\frac{r_-}{\ell}(\Delta\phi-2\pi n)\right]
\label{eq: sigma 0}
\end{aligned}
\end{equation}
and
\begin{align}
\alpha(r)&=\frac{r^2-r_-^2}{r_+^2-r_-^2}\qquad
\Delta\phi=\phi-\phi'
\end{align}
\end{widetext}
with the respective boundary conditions as $\zeta=1$ (Dirichlet), $\zeta=0$ (transparent), and $\zeta=-1$ (Neumann). We shall take the detector to have switching function $\chi_D(\tau_D)=e^{-\tau_D^2/2\sigma^2}$ and only consider untwisted scalar fields with $\eta=1$.

To calculate the transition probabilities, we will work in the co-rotating frame of the detectors  \cite{Hodgkinson2012}: \begin{align}
t&=\frac{\ell r_+\tau}{\sqrt{r^2-r_+^2}\sqrt{r_+^2-r_-^2}} \label{eq: CRM t}\\
\phi&=\frac{r_-\tau}{\sqrt{r^2-r_+^2}\sqrt{r_+^2-r_-^2}} \label{eq: CRM phi}\ .
\end{align}
in which case \cite{Vagenas2002}
\begin{align}\label{temprel}
T_{KMS}=T_{H}/\gamma\,,
\end{align}
where
\begin{align}\label{lorfac}
\gamma=\frac{\sqrt{r^2-r_+^2}\sqrt{r_+^2-r_-^2}}{r_+}
\end{align}
is the Lorentz factor.
Straightforward calculations show that we can rewrite equation \eqref{eq: PD} as $P_D=\sum_{n=-\infty}^\infty\eta^n\left\{I_n^--\zeta I_n^+\right\}$, where
\begin{align}
I_n^\pm =K_P\int_{-\infty}^{\infty}dz\frac{e^{-a\left(z-\frac{2\pi nr_-}{\ell}\right)^2}e^{-i\beta\left(z-\frac{2\pi nr_-}{\ell}\right)}}{\sqrt{\left(\cosh(\alpha_n^\pm)-\cosh\left(z\right)\right)}}
\label{eq: In}
\end{align}
\begin{widetext}
and
\begin{align}
K_P&=\frac{\lambda ^2 \sigma_D}{4 \sqrt{2 \pi }}\\
a&=\frac{1}{(4\pi T_{KMS}\sigma_D)^2} \qquad
\beta =\frac{\Omega_D}{2\pi T_{KMS}}\\
\cosh(\alpha_n^\pm)&= \pm4\ell^2\pi^2T_{KMS}^2+(1+4\ell^2\pi^2T_{KMS}^2)\cosh\frac{2\pi n r_+}{\ell}
\end{align}

In the limit of an infinite interaction time (i.e. $\sigma\to\infty$), we note that we can write the $n=0$ term (corresponding to AdS spacetime) analytically as
\begin{align}
\lim_{\sigma\to\infty}P_{D,n=0}=\lim_{\sigma\to\infty}I_0=\frac{\sqrt{\pi}}{4}\left[1-\tanh\frac{\Omega_D}{2T_{KMS}}\right]\left[1-\zeta P_{-1/2+i\beta}\left(\cosh\alpha_0^+\right)\right]
\label{eq: I0}
\end{align}
\end{widetext}

To investigate the influence of rotation (and AdS length) on the weak and strong anti-Hawking effects, we must determine the dependence of the response function and EDR temperature on the KMS temperature.  To vary the latter, we locate the detector
at   $r=R_D$ in the co-rotating frame and solve \eqref{temprel} and \eqref{lorfac} for $R_D$ in terms of $T_{KMS}$ and the
other parameters.  The response $P_D$ in \eqref{eq: PD} and the EDR temperature  are then  functions of $T_{KMS}$.

\section{Weak Anti-Hawking Effect for Rotating BTZ Black Holes}

 To set the context for our investigation, we first compare the situation for large and intermediate mass black holes with a detector energy gap of $\Omega_D\ell=1/10$ (with other energy gaps yielding qualitatively similar results). This is shown in   Figure~\ref{fig: FJ_LargerMass}, where we plot detector response as a function of $T_{KMS}$.  The general trend is that for both masses and all boundary conditions the response is suppressed at small KMS temperatures  and asymptotes to a constant value at a large ones. Apart from these two general features, there is notably distinct behaviour as these parameters are varied.
In the main figure, we depict the situation for an intermediate mass  $M=1/10$, where we see that rotation has marginal impact at small and large KMS temperatures, but  significantly amplifies  detector response at  intermediate KMS temperatures.  An even larger influence is due to boundary conditions,
where we observe that the weak anti-Hawking effect is absent for Dirichlet boundary conditions, but present for the other two. The
negative slope to the right of the peak is steeper for Neumann boundary conditions, indicative of increasing strength of the weak effect as $\zeta$ decreases.
In the inset of each subfigure, we consider the large mass $M=100$ case, where we see that the weak anti-Hawking effect disappears for $\zeta=1,0$ boundary conditions, yet remains for $\zeta=-1$, recovering earlier results
 \cite{Henderson:2019uqo} for large-mass black holes.  We see that there is negligible dependence of the response on angular momentum in this large mass case for all KMS temperatures.

\begin{figure*}[t!]
\centering
\begin{minipage}{\textwidth}
  \centering
    \includegraphics[width=0.6\textwidth]{PDLegendConstantM.pdf}
\end{minipage}\\
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.001_Omega=1_Zeta=1_Derivative}.pdf}
    \caption*{(a) $\Omega\ell=1$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.001_Omega=0.1_Zeta=1_Derivative}.pdf}
    \caption*{(b) $\Omega\ell=1/10$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
  \centering
    \includegraphics[width=\textwidth]{{PD_M=0.001_Omega=0.01_Zeta=1_Derivative}.pdf}
    \caption*{(c) $\Omega\ell=1/100$}
\end{minipage}
\caption{ Derivative of the response with respect to the KMS temperature \eqref{weakAH} of a rotating BTZ black hole with mass $M=1/1000$, and Dirichlet boundary conditions ($\zeta=1$). We note that for transparent and Neumann boundary conditions, qualitatively similar results are obtained.}
\label{fig: FJ derivative}
\end{figure*}
 It is clear from this that rotational effects are more pronounced for smaller mass black holes, and so in Figure~\ref{fig: FJ}, we plot the dependence of the response function on  KMS temperature for  $M=1/1000$.   The weak effect is now clearly evident for  Dirichlet boundary conditions, with the peak response enhanced as much as sevenfold, compared to the left diagram in  Figure~\ref{fig: FJ_LargerMass}.  Similar results hold for the other boundary conditions.   As before, as rotation increases, the response is amplified for all values of $T_{KMS}$.
For all values of rotation and all gaps, the response asymptotes to a value of $P_D=\frac{\sqrt\pi}{4}$, in accord with equation \eqref{eq: I0}, noting from equation  \eqref{eq: In}  that
  $I_n\to0$ for $n\neq0$.

 The strength of the weak anti-Hawking effect depends on the magnitude of the
 negative value of the slope of equation \eqref{weakAH}  after the peak.  We see that this
increases with decreasing gap, showing that smaller gap enhances the weak anti-Hawking effect, which we illustrate for Dirichlet boundary conditions in Figure \ref{fig: FJ derivative}.  The slope peaks at $\frac{d\mathcal{F}}{d T_{KMS}\ell}\approx-0.15$ for the large energy gap and $\frac{d\mathcal{F}}{d T_{KMS}\ell}\approx-0.6$ for the small gap. For each gap, we also see that the weak anti-Hawking effect is amplified with increasing rotation, by
as much as 50\% for near-extremal black holes,  for all gaps in the figure.

Furthermore, we find that the weak anti-Hawking effect occurs after a critical value of $T_{KMS}$   that depends on the detector's energy gap, but not on the rotation of the black hole, again evident from  Figure \ref{fig: FJ derivative}.  Though we have only
illustrated results for  Dirichlet boundary conditions $\zeta=1$, we emphasize that this critical value depends on $\zeta$,  with the critical temperature becoming smaller as $\zeta\to\-1$.

Finally, we note that changing the AdS length will not change the strength of the weak effect. Physically, this is because the AdS length is the only length scale present (as $\sigma\to\infty$), and everything is calibrated against this length.

\section{Strong Anti-Hawking Effect for Rotating BTZ Black Holes}

Let us now turn our attention to the strong anti-Hawking effect. In Figures~\ref{fig: TEDRKMS} and~\ref{fig: strong anti-Hawking Omega=1/100}, we plot the relationship between the EDR   and KMS temperatures for $M=1/1000$ and different boundary conditions, with $\Omega\sigma=1$ and $\Omega\sigma=1/10$, respectively. We note several interesting features.
\begin{figure*}[t!]
\centering
\begin{minipage}{\textwidth}
  \centering
    \includegraphics[width=0.6\textwidth]{PDLegendConstantM.pdf}
\end{minipage}\\
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{TEDR_M=0.001_Omega=1_Zeta=1}.pdf}
    \caption*{(a) $\zeta=1$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{TEDR_M=0.001_Omega=1_Zeta=0}.pdf}
    \caption*{(b) $\zeta=0$}
\end{minipage}
\begin{minipage}{0.3\textwidth}
  \centering
    \includegraphics[width=\textwidth]{{TEDR_M=0.001_Omega=1_Zeta=-1}.pdf}
    \caption*{(c) $\zeta=-1$}
\end{minipage}
\caption{EDR temperature for a black hole of mass $M=1/1000$ and energy gap of $\Omega\sigma=1$. We plot KMS temperature down to $T_{KMS}\ell=10^{-5}$. The insets show the relation between the EDR temperature and KMS temperature for larger values of $T_{KMS}$.}
\label{fig: TEDRKMS}
\end{figure*}
\begin{figure}
  \centering
    \includegraphics[width=\columnwidth]{PDLegendConstantM.pdf}
    \begin{minipage}{0.45\textwidth}
  \centering
    \includegraphics[width=\columnwidth]{{TEDRConstantM_M=0.001_Omega=1_Zeta=1}.pdf}
        \caption*{(a) $\Omega\sigma=1$}
    \end{minipage}
    \begin{minipage}{0.45\textwidth}
  \centering
\includegraphics[width=\columnwidth]{{TEDRConstantM_M=0.001_Omega=0.01_Zeta=1}.pdf}
    \caption*{(a) $\Omega\sigma=1/100$}
\end{minipage}
\caption{EDR temperature for a black hole of mass $M=1/1000$. The results are similar for $\Omega\sigma=1/100$.  The insets show the EDR temperature for our first three values of the angular momentum, plotted on a linear scale (rather than a log scale, as is the case for the main plots).
}
\label{fig: strong anti-Hawking Omega=1/100}
\end{figure}

 First, it is evident from Figure~\ref{fig: TEDRKMS} that a strong anti-Hawking effect is  present for all three boundary conditions.
This is clear at low values of $T_{KMS}$, where we see a negative slope indicative of the strong effect. Eventually a minimum
is reached and $T_{EDR}$ begins to increase with $T_{KMS}$.
The insets indicate the behaviour at large $T_{KMS}$, where we see that this quantity is indeed positively correlated with
$T_{EDR}$, and there is minimal dependence on angular momentum, even for small $M$.
 Unlike the weak effect, we see that the maximum decreases with increasing values of angular momentum for all boundary conditions. Consequently the strength of the strong effect (the slope in equation \eqref{strongAH})
 likewise diminishes.
 As extremality is approached, the strong effect essentially vanishes.

The exception is for Dirichlet boundary conditions.  In Figure~\ref{fig: strong anti-Hawking Omega=1/100}, we plot the EDR temperature for Dirichlet boundary conditions for $\Omega\sigma=1$ and $\Omega\sigma=1/100$. In the insets, we see that there is a small strong anti-Hawking effect for non-rotating black holes, similar to what was found in \cite{Henderson:2019uqo}. As the angular momentum increases,  the peak in Figure~\ref{fig: strong anti-Hawking Omega=1/100} moves rightward, and the threshold value of $T_{KMS}$ at which the strong effect appears increases.   This threshold value
will reach a maximum for some $J/M\ell$, after which
 it decreases with increasing  $J/M\ell$ as extremality is approached, as is clear from Figure~\ref{fig: strong anti-Hawking derivative}.
We also qualitatively see that the magnitude of the slope becomes larger for larger angular momenta, indicating that the strong anti-Hawking effect becomes stronger. For smaller values of the energy gap, we see that the strong anti-Hawking effect increases in strength, with the $\Omega\sigma=1/10$ case being similar to $\Omega\sigma=1/100$.  This can be seen by noting that $J/M\ell=0.9$ does not exhibit the strong anti-Hawking effect for $\Omega\sigma=1$, but there is a small effect present at this value of the angular momentum for $\Omega\sigma=1/100$.

For transparent and Neumann boundary conditions, we find that the strong anti-Hawking effect is much more pronounced, as can be seen from Figure~\ref{fig: TEDRKMS}.
Furthermore, as $\zeta$ increases from $-1$ to $1$ in Figure \ref{fig: TEDRKMS}, the range of $T_{KMS} \ell$ over which the strong effect is present also decreases, and is very small for   Dirichlet boundary conditions.

This range decreases with increasing angular momentum for transparent and Neumann boundary conditions. There is a minimal value of $T_{EDR}\ell$ as function of $T_{KMS} \ell$, and this minimal value decreases as the angular momentum of the black hole increases.

  For Dirichlet boundary conditions and sufficiently small angular momentum, larger angular momenta also yields a decreasing range of $T_{KMS}$ temperatures for which the strong effect holds, up to a critical value of $J/M\ell$. Beyond this value, increasing angular momenta results in a greater range of $T_{KMS}$ temperatures  for which the effect is present. As a result, the anti-Hawking effect appears to nearly disappear for near-extremal black holes $J \geq 0.9999 M \ell$ for transparent and Neumann boundary conditions, yet is still present for Dirichlet boundary conditions. In Figure \ref{fig: strong anti-Hawking Omega=1/100}, we see that there is a strong anti-Hawking effect for a non-rotating black hole, but the effect disappears (or almost disappears, depending on the energy gap) as the angular momentum increases to $J/M\ell=0.9$.

Beyond this, however, as we continue to approach extremality in the Dirichlet case, the strong effect emerges at lower KMS temperatures, as shown in   Figure~\ref{fig: strong anti-Hawking derivative}.  Indeed, its range and strength both get larger
as $J/M\ell$ gets very close to unity, as evidenced by the curve for $J/M\ell=0.9999$.   It is quite remarkable that there is such a strong dependence on boundary conditions.

Finally,  inspection of Figures~\ref{fig: TEDRKMS} and \ref{fig: strong anti-Hawking Omega=1/100} indicate that the strong
effect does not monotonically depend on $J/M\ell$.   Indeed, we observe a `crossover' effect at small $T_{KMS}$,
in which the values of
$T_{EDR}$ decrease with increasing angular momentum as   $T_{KMS} \to 0$, whereas at sufficiently large $T_{KMS}$ the rate of change of  $T_{EDR}$  with respect to $T_{KMS}$ increases with increasing angular momentum such that the higher-$J$ curves cross over the lower-$J$ curves.  At large $T_{KMS}$, we see that $T_{EDR}\sim T_{KMS}$, with the largest $T_{EDR}$ corresponding to the largest $J$ for fixed $T_{KMS}$, and the smallest $T_{EDR}$ corresponding to the smallest $J$.
This is clearly evident in Figure~ \ref{fig: strong anti-Hawking Omega=1/100}.
We have also verified that this effect is also present for Neumann and transparent boundary conditions, though the `crossover' occurs at larger KMS temperatures than the Dirichlet case.

 By comparing Figures \ref{fig: FJ}-\ref{fig: strong anti-Hawking Omega=1/100}, we see that there is no range of $T_{KMS}$ for which
 the strong anti-Hawking effect   overlaps
 with the weak anti-Hawking effect,  with the weak anti-Hawking effect appearing for $T_{KMS}\ell\gtrsim1$, while the strong anti-Hawking effect appears for $T_{KMS}\ell\lesssim0.1-0.5$. The exact temperature range is dependent on the boundary conditions, energy gap, and in the case of the strong anti-Hawking effect, the angular momentum.
 Furthermore, the critical KMS temperature at which the strong anti-Hawking effect disappears becomes smaller for larger angular momentum, in contrast to the weak anti-Hawking effect where the critical temperature
at which this  effect appears has minimal dependence on angular momentum.  In addition, we again note that the location of this critical temperature for the strong effect is highly dependent on the boundary conditions.

\begin{figure}
  \centering
    \includegraphics[width=\columnwidth]{StrongAntiHawkingLegend.pdf}
    \begin{minipage}{0.45\textwidth}
  \centering
    \includegraphics[width=\columnwidth]{{StrongAntiHawkingDerivative1}.pdf}
        \caption*{(a) $\Omega\sigma=1$}
    \end{minipage}
    \begin{minipage}{0.45\textwidth}
  \centering
\includegraphics[width=\columnwidth]{{StrongAntiHawkingDerivative0.01}.pdf}
    \caption*{(a) $\Omega\sigma=1/100$}
\end{minipage}
\caption{ Strong anti-Hawking effect for a near-extremal black hole of mass $M=1/1000$ and Dirchlet boundary conditions. The results are similar for $\Omega\sigma=1/10$.
}
\label{fig: strong anti-Hawking derivative}
\end{figure}

Our last consideration is that of  the impact of changing the AdS length on the strong  effect. Here the situation differs from the
weak anti-Hawking effect  as now there is a second length scale present ($\sigma$, the width of the switching function). In Figure~\ref{fig: strong anti hawking changing AdS length}, we consider the effect of changing the AdS length for a non-rotating BTZ black hole, compared to a near-extremal BTZ black hole. In the  non-rotating case,   increasing AdS length increases the range of $T_{KMS}$ temperatures where the strong anti-Hawking effect holds. However, the marginal effect of increasing $\ell$ is reduced for larger and larger values of the AdS length.
We also see that for small $T_{KMS}\ell$, a larger AdS length will  broaden
 the initial peak. In the case of a near-extremal black hole, the situation is similar. As we saw in Figure \ref{fig: TEDRKMS}a, there was a tiny strong anti-Hawking effect present for near-extremal black holes. For larger AdS lengths, we similarly see that the temperature range of the strong anti-Hawking effect increases in size. However, we note that this effect is still relatively weak and only becomes noticeable for larger values of $\ell$.

\begin{figure}
  \centering
    \includegraphics[scale=0.5]{AdSLegend.pdf}
        \begin{minipage}{0.45\textwidth}
  \centering
\includegraphics[width=\columnwidth]{{M=0.001_J=0_Omega=0.1_Zeta=1_VaryingAdSLengths}.pdf}
        \caption*{(a) $J/M\ell=0$}
\end{minipage}
    \begin{minipage}{0.45\textwidth}
  \centering
\includegraphics[width=\columnwidth]{{M=0.001_J=0.9999_Omega=0.1_Zeta=1_VaryingAdSLengths}.pdf}
        \caption*{(b) $J/M\ell=0.9999$}
\end{minipage}
\caption{Changing AdS lengths for the strong anti-Hawking effect for a black hole of mass $M=1/1000$, Dirichlet boundary conditions, and energy gap of $\Omega\sigma=1/10$.  We plot KMS temperature down to $T_{KMS}\ell=10^{-5}$. The insets show the effect of changing AdS length on the EDR temperature for larger values of $T_{KMS}$.
}
\label{fig: strong anti hawking changing AdS length}
\end{figure}

\section{Conclusion}

 As with entanglement harvesting  \cite{Robbins2020},
rotation can have a significant impact on the anti-Hawking effect. For large-mass black holes with Dirichlet and transparent boundary conditions, the weak anti-Hawking effect vanishes, as expected; it is present for Neumann boundary conditions
\cite{Henderson:2019uqo}. In all cases the effects of rotation are negligible. But as the mass of the black hole decreases, rotation
 significantly amplifies the weak version of the effect.

 The impact of rotation on the strong effect is somewhat inverted.  We find that rotation tends to weaken the strength of the strong anti-Hawking effect for transparent and Neumann boundary conditions, with it nearly vanishing for near-extremal black holes. In contrast, for Dirichlet boundary conditions, larger angular momenta causes the strong anti-Hawking effect to be reduced before being amplified again.

Furthermore, for the strong anti-Hawking effect, the relationship between angular momentum and detector temperature is
non-monotonic  for each boundary condition, leading to a `crossover' phenomenon that is most prominent for Dirichlet boundary conditions. For small $T_{KMS}L$, larger angular momenta yield smaller $T_{EDR}L$, whereas for larger values of  $T_{KMS}L$,   larger angular momenta yield \textit{larger}  $T_{EDR}$.  More work is needed to better understand how this cross-over effect comes about and its dependence on the  boundary conditions.
It would also be interesting to consider whether $3+1$ dimensional rotating black holes also exhibit these same findings as the rotating BTZ black hole.

While the weak anti-Hawking effect is independent of  the AdS length $\ell$,
increasing AdS length increases the range of $T_{KMS}$ temperatures where the strong anti-Hawking effect holds.
A larger AdS length will also broaden the initial peak  for small $T_{KMS}\ell$.
However as  $\ell$ continues to increase, its impact on the strong effect becomes increasingly marginal.

In summary, our results indicate that the effects of spacetime dragging on the quantum vacuum can significantly modify
detector response as small field KMS field temperatures, as exemplified by the anti-Hawking effect(s). The role of boundary
conditions is very important; indeed, it is surprising that there is such a strong dependence of the strong anti-Hawking effect on boundary conditions for  small-mass rotating black holes.  The origin of this effect merits further study.
\bigskip

{\it Acknowledgements}
$\quad$
MR was funded by an Ontario Graduate Scholarship. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada, Asian Office of Aerospace Research \& Development Grant FA2386-19-1-4077,  and the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.

\bibliographystyle{unsrt}


\section{Introduction}
Onsager reciprocity relation indicates that for anomalous Hall effect in linear response regime, selected material is required to break time-reversal symmetry since the Berry curvature is odd in momentum space.\cite{PhysRevResearch.2.032066}\cite{PhysRevLett.124.087402}\cite{PhysRevLett.123.246602} Therefore, the integral of Berry curvature over momentum will vanish with Fermi distribution of electrons in equilibrium. This is given by $\Omega_{a}(-k)=-\Omega_{a}(k)$. The Kramer pairs of $k$ and $-k$ are both occupied. However, according to recent research, nonlinear Hall conductivity can be still remained in time-reversal symmetric crystals. What we need is only inversion-symmetry breaking. In this case, the energy gap emerges at each Dirac node or Weyl node. More importantly, it has been found out the Berry curvature dipole is responsible for nonlinear Hall response in quantum transport by both experimental and theoretical study. \cite{PhysRevLett.115.216806}\cite{PhysRevLett.105.026805}\cite{PhysRevB.100.195117}\cite{PhysRevLett.121.246403} Indeed, there are two types
of materials creating non-trivial Berry curvature dipole. The first
kind is topological crystalline insulator SnTe, which will
undergo a ferroelectric distrotion at low temperature~\cite{PhysRevLett.122.186801},
time-reversal symmetric Weyl semimetals in the TaAs material class\cite{PhysRevLett.115.216806}
and Rashba material BiTeI\cite{PhysRevLett.121.246403} They all have strong spin-orbit
coupling contributing to their tilted Dirac cone. These tilted Dirac
cones will not change their Berry curvature but crucial to non-vanishing
dipole term. The second type is two-dimensional Dirac material without
spin-orbit coupling. Their inversion symmetry breaking attributes
to external field and substrate. Even more importantly, the appearance of a finite dipole can only be captured taking explicitly into account the terms accounting for the warping of the Fermi surface\cite{PhysRevLett.123.196403}. This new phenomena has been already studied in quantum nonlinear Hall effect\cite{PhysRevLett.115.216806} and thermal Hall effect\cite{PhysRevResearch.2.032066} with semi-classical Boltzmann equation. Inspired by the two studies, I expect to explore more on nonlinear Nernst effect. Without conventional Boltzmann equation and semi-classical approximation, I begin with generalization of the quantum kinetic theory which is more fundamental to us. With the theory in temperature field, I will investigate the nonlinear response theory in thermoelectric transport.

In this work I study the quantum kinetic theory of nonlinear Nernst
effect (NNE) in thermoelectrical transport. There is a research before on transport properties of nonlinear Nernst effect beginning from Boltzmann equation with semi-classical approximation.\cite{PhysRevB.99.201410} I will derive expression and equation of the density matrix from the basic quantum Liouville equation. I develop theory
of the nonlinear electronic transport induced by temperature gradient
in the presence of disorder. I will also introduce the new type of dipole: $\textbf{thermoelectric Berry curvature dipole}$
instead of dipole before\cite{PhysRevLett.115.216806}. This new thermoelectric Berry curvature dipole will play an important role in thermoelectric transport. This theory is also crucial to experimental physicists since they
can measure the electric current in presence of temperature gradient. I here provide a theoretical prediction of the relationship between thermoelectric conductivity and chemical potential. We can also figure out that this thermoelectric current is also connected to a term with Berry curvature which is totally different from the electric Hall effect.

This paper is organized as follows. In the second section, I will
briefly introduce not only quantum kinetic equation for Bloch electrons in the
presence of disorder, temperature gradient but also the solution
of density matrix to the equation. In the third part, I give the general expression of the density matrix by solving the quantum kinetic equation and derive the second-order response. It will also be explained that why thermoelectric Berry curvature dipole is constructed and how it influences the transport. To show the adaptability and reliability of my generalized theory, I apply the quantum kinetic theory in the presence of electric field and compare my results with that in the research before. I take disorder effect into account by applying the scattering
theory as well. I prove the terms related to Berry curvature and Berry curvature dipole have no effects on the conductivity. In the fourth section, I employ the theory before to a specific model: topological crystalline insulator, which presents
the nontrivial thermoelectric Berry curvature dipole. I will show how its thermoelectric Berry curvature dipole and thermoelectric conductivity change with chemical potential of the valley numerically. Last but not the least, I will discuss the quantum kinetic theory in more general case: non-static solution and its application to derivation of optical conductivity. This is still unfamiliar to most of the researchers since all the previous research on quantum kinetic theory only focus on the case in DC limit. Our ambition is to discover the optical current and optical conductivity in any frequency. I will also check the theory with results in semi-classical approximation.

\section{Quantum Kinetic Theory}
Without external field, the Hamiltonian
of the system is: $H=H_{0}+U$, where $U$ represents disorder potential.
The free Hamiltonian satisfies:

\begin{equation}
	H_{0}|m,\textbf{k}\rangle=\epsilon_{k}^{m}|m,\textbf{k}\rangle
\end{equation}

Here m represents band index and $\epsilon_{\textbf{k}}^{m}$ are
dispersion relationship of m-th band. In the presence of disorder,
the quantum Liouville equation after integrating the disorder's coordinates
is our beginning point~\cite{Schmidt2001}

\begin{equation}
	\frac{\partial\langle\rho\rangle}{\partial t}+\frac{i}{\hbar}[H_{0},\langle\rho\rangle]+K(\langle\rho\rangle)=0
\end{equation}

where $\langle\rho\rangle$ is density matrix after integrating all the
disorder\cite{Schmidt2001,Liboff2003}: $\langle\rho\rangle=\frac{1}{V^{n}}\int dR_{1}...dR_{n}\rho(\textbf{r},R_{1},...,R_{n})$.
$R_{1},...,R_{n}$ are coordinates of disorder. For further step, the well-known Luttinger proposal\cite{PhysRev.135.A1505}\cite{PhysRevLett.114.196601} is introduced. To describe the thermal transport in material, I similarly add scalar potential $\psi$ which satisfies $\nabla\psi=\nabla T/T$. Therefore, the thermal field and the thermal driving term take the forms:
\begin{equation}
	\textbf{E}_{T}=-\frac{\partial\textbf{A}_{T}}{\partial t}\equiv-\frac{\nabla T}{T}
\end{equation}
\begin{equation}
	D_{T}(\langle\rho\rangle)=\frac{1}{2\hbar}\frac{\nabla T}{T}\frac{D(\{H_{0},\langle\rho\rangle\})}{D\textbf{k}}
\end{equation}
The covariant derivative is defined as\cite{PhysRevB.96.235134}:

\begin{equation}
	\frac{DX}{D\textbf{k}}=\nabla_{\textbf{k}}X-i[\mathcal{R}_{k},X]
\end{equation}

where X is a matrix and $\mathcal{R}$ is Berry connection: $\mathcal{R}_{\textbf{k}a}^{mn}=i\langle u_{\textbf{k}}^{m}|\partial_{k_{a}}u_{\textbf{k}}^{n}\rangle$,
$\mathcal{R}_{\textbf{k}}^{mn}=\sum_{a=1}^{3}\mathcal{R}_{\textbf{k}a}^{mn}e_{a}$
Then we can construct the kinetic equation in the presence of disorder and thermal field\cite{PhysRevB.101.155204}:
\begin{equation}
	\frac{\partial\langle\rho\rangle}{\partial t}+\frac{i}{\hbar}[H_{0},\langle\rho\rangle]+K(\langle\rho\rangle)=D_{T}(\langle\rho\rangle)
\end{equation}
Here we give the numerical result of scattering term:

\begin{widetext}
	\begin{equation}
		K(\langle\rho\rangle)=\frac{1}{\hbar^{2}}\langle\int_{0}^{\infty}dt'[U,[e^{-iH_{0}t'/\hbar}Ue^{iH_{0}t'/\hbar},e^{-iH_{0}t/\hbar}\langle\rho\rangle e^{iH_{0}t/\hbar}]]\rangle
	\end{equation}

This can be decomposed into two parts:\cite{PhysRevB.96.035106}

	\begin{equation}
		[I(\langle\rho\rangle)]_{\textbf{k}}^{mm}=\frac{2\pi}{\hbar}\sum_{m',\textbf{k}'}\langle U_{\textbf{k}\textbf{k}'}^{mm'}U_{\textbf{k}'\textbf{k}}^{m'm}\rangle(n_{\textbf{k}}^{m}-n_{\textbf{k}'}^{m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})
	\end{equation}

	\begin{equation}
		[J(\langle\rho\rangle)]_{\textbf{k}}^{mm''}=\frac{\pi}{\hbar}\sum_{m',\textbf{k}'}\langle U_{\textbf{k}\textbf{k}'}^{mm'}U_{\textbf{k}'\textbf{k}}^{m'm''}\rangle[(n_{\textbf{k}}^{m}-n_{\textbf{k}'}^{m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})+(n_{\textbf{k}}^{m''}-n_{\textbf{k}'}^{m'})\delta(\epsilon_{\textbf{k}}^{m''}-\epsilon_{\textbf{k}'}^{m'})],~(m\neq m'')
	\end{equation}
\end{widetext}

Especially, due to energy conservation law, the main contribution is only from band-diagonal part $\langle n\rangle$. Therefore,
the two disorder terms can be rewritten into: $I(\langle n\rangle)$ and $J(\langle n\rangle)$.
These results can also be found in \cite{Liboff2003}\cite{PhysRevB.96.235134}.

To solve the equation (6), we should sepearte the density matrix into two parts:  $\langle\rho\rangle=\langle\rho_{0}\rangle+\langle\rho_{T}\rangle$, where $\langle\rho_{0}\rangle=\sum_{m}f_{0}(\xi_{\textbf{k}})|m\rangle\langle m|$ represents the equilibrium-state distribution. In this passage we mainly focus on the nonequilibrium part: $\langle\rho_{T}\rangle$ induced by temperature gradient in the density matrix. The solution to this part yields:

\begin{equation}
	\langle n_{T}\rangle_{\textbf{k}}^{m}=\tau_{k}^{m}\frac{\nabla T}{T}\cdot\textbf{v}_{\textbf{k}}^{m}(\epsilon_{\textbf{k}}^{m}-\mu)\frac{\partial f_{0}(\epsilon_{k}^{m})}{\partial\epsilon_{k}^{m}}
\end{equation}
\begin{equation}
	\langle S_{T}\rangle_{\textbf{k}}^{mm'}=-i\hbar\frac{[D_{T}(\langle\rho_{0}\rangle)]_{\textbf{k}}^{mm'}-[J(\langle n_{T}\rangle)]_{\textbf{k}}^{mm'}}{\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}}^{m'}}
\end{equation}

Here $\textbf{v}_{\textbf{k}}^{m}=\frac{1}{\hbar}\nabla_{\textbf{k}}\epsilon_{\textbf{k}}^{m}$,
and $\tau_{k}^{m}$ represents relaxation time which takes the form:
$1/\tau_{k}^{m}=2\pi\sum_{m'}\langle U_{\textbf{kk}'}^{mm'}U_{\textbf{k'k}}^{m'm}\rangle\int\frac{d^{d}k}{(2\pi)^{d}}\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})$.
This will be figured out in specific material.

Ignoring the impurity term, we can calculate the electric current
induced by temperature gradient with intrinsic velocity operator \cite{PhysRevB.101.155204}:

\[
J_{x}=\frac{1}{2}Tr[(-e)\{v_{x},\langle S_{T}\rangle\}]
\]
\begin{equation}
	=(-e)\frac{\partial_{y}T}{T}\sum_{m}\int\frac{d^{d}k}{(2\pi)^{d}}\times\Omega_{\textbf{k},z}^{m}(\epsilon_{\textbf{k}}^{m}-\mu)f_{0}(\epsilon_{\textbf{k}}^{m})
\end{equation}

Here $\{,\}$ represents anti-commutation. $\{A,B\}=AB+BA$. It is used
to prove the hermiticity of the electric current. The reason why we only pay attention to the off-diagonal part of density matrix is
no contribution comes from the diagonal part. The integral of diagonal part
is proven to be zero because it is an odd function of momentum. In addition, we can see the current is directly connected to Berry curvature of bands: $\Omega_{\textbf{k},a}^{m}=i\epsilon_{abc}\langle\partial_{k_{b}}m|\partial_{k_{c}}m\rangle$

However, this current can only be measured in time-reversal symmetry
breaking material. For time-reversal symmetric crystals, (12) will contributes
nothing. We have to consider the nonlinear Nernst effect.

\section{General theory of Nonlinear Nernst effect}

According to study by Fu \cite{PhysRevLett.115.216806},
nonlinear Hall conductivity tensor in the second harmonic term is for material preserving the time-reversal symmetry. We will here prove its reasonability with quantum kinetic theory and develop theory into thermoelectric transport

Let's focus on quantum Liouville equation first. Instead of the form
like (6), we give the general expression for the equation with temperature
gradient.

\begin{equation}
	(\mathcal{L}-D_{T})\langle\rho\rangle_{F}=D_{T}\langle\rho_{0}\rangle
\end{equation}

Here we define an operator $\mathcal{L}=P+K$, where $P\langle\rho\rangle_{F}\equiv\frac{i}{\hbar}[H_{0},\langle\rho\rangle_{F}]$.
This is accurate for $\mathcal{L}\langle\rho_{0}\rangle=0$. So we
can give the direct solution of it.

\begin{equation}
	\langle\rho\rangle_{F}=\sum_{N=1}^{\infty}(\mathcal{L}^{-1}D_{T})^{N}\langle\rho_{0}\rangle
\end{equation}

(14) is a nontrivial result for the term of $N=2$ is the response in nonlinear regime which may be related to Berry curvature dipole. The result before is just
the simplest approximation of (13). Indeed, equation (14) is obtained by iteration. In linear response theory, we just consider the $N=1$ case. Now we turn to the quadratic term.

Similarly, we can calculate the off-diagonal term(without impurity):

\begin{equation}
	\langle S_{T^{2}}\rangle=-i\hbar\frac{\partial_{y}T}{T}\sum_{nn'}\frac{\epsilon_{n'}\langle n_{T}\rangle_{\textbf{k}}^{n'}-\epsilon_{n}\langle n_{T}\rangle_{\textbf{k}}^{n}}{\epsilon_{n}-\epsilon_{n'}}\times|n\rangle\langle n|\partial_{k_{y}}n'\rangle\langle n'|
\end{equation}

Here $\epsilon_{m}=\epsilon_{k}^{m}-\mu$. In the following parts,
we note $\partial_{k_{a}}\rightarrow\partial_{a}$

We can also obtain the general expression of thermoelectric current for the quadratic term. Detailed calculation is displayed in Appendix A.
\begin{widetext}
	\begin{equation}
		J_{x}^{(2)}=\frac{1}{2}Tr[(-e){v_{x},\langle S_{T^{2}}\rangle}]=J_{x1}^{(2)}+J_{x2}^{(2)}
	\end{equation}
\begin{equation}
	J_{x1}^{(2)}=\frac{e}{2\hbar}(\frac{\partial_{y}T}{T})^{2}\sum_{m}\tau_{\textbf{k}}^{m}\epsilon_{m}^{2}f_{0}\partial_{y}\Omega_{\textbf{k}z}^{m}=e(\frac{\partial_{y}T}{T})^{2}D_{y}
\end{equation}
\begin{equation}
	J_{x2}^{(2)}=\frac{e}{\hbar}(\frac{\partial_{y}T}{T})^{2}\sum_{m}\tau_{\textbf{k}}^{m}\epsilon_{m}\partial_y\epsilon_{\textbf{k}}^{m}f_{0}\Omega_{\textbf{k}z}^{m}
\end{equation}
\end{widetext}

We can see this thermoelectric currect includes two terms, which will be expanded in the following context. For simplicity, we have $\sum_{m}\int\frac{d^{d}k}{(2\pi)^{d}}\rightarrow\sum_{m}$.
Here we define the thermoelectric Berry curvature dipole: $D_{y}=\frac{1}{2\hbar}\sum_{m}\int\frac{d^{d}k}{(2\pi)^{d}}\tau_{\textbf{k}}^{m}\epsilon_{m}^{2}f_{0}\partial_{y}\Omega_{\textbf{k}z}^{m}$
. Actually, when the temperature is low enough, the contribution can be considered only from the conduction band with assumption: $\mu>0$. Since we only consider the problem on the Fermi surface, the relaxation time can be replaced with $\tau_{k_{F}}^{+}$. In this case, only the electrons near the Fermi surface on the conduction band give rise to transport. So the thermoelectric Berry curvature can be rewritten into another form.
\begin{equation}
	D_{y}=\frac{1}{2\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{+}^{2}f_{0}\partial_{y}\Omega_{\textbf{k}z}^{+}
\end{equation}
Here we modify the dipole by removing the relaxation time from it. We can also see another exhilirating story that this current contains a term with Berry curvature, which does not show up in the electric Hall effect. This is different from one in \cite{PhysRevLett.115.216806}
since this is thermoelectric current instead of electric current induced
by electric field. After the calculation, we still remain a question: the kinetic theory developed here should be consistent with semi-classical wavepacket dynamics\cite{PhysRevLett.95.137204}, it is crucial to check whether the result in nonlinear Hall effect is the same as that before. When the external field is electric field, we have the form of density matrix:

\begin{equation}
	\langle S_{E^2}\rangle =-ieE_{y}\sum_{n,n'}\frac{\langle n_{E}\rangle^{n'}_{\textbf{k}}-\langle n_{E}\rangle^{n}_{\textbf{k}}}{\epsilon_{n}-\epsilon_{n'}}|n\rangle\langle n|\partial_{y}n'\rangle\langle n'|
\end{equation}
\[
	\langle n_{E}\rangle^{m}_{\textbf{k}}=e\textbf{E}\cdot\textbf{v}_{\textbf{k}}^{m}\frac{\partial f_{0}}{\partial\textbf{k}}
\]

Since we take the zero-temperature limit, there is only contribution from conductance band. With this approximation, we obtain the same result in \cite{PhysRevLett.115.216806} by repeating the same procedure.
\[
	J_{x}=\frac{1}{2}Tr[(-e){v_{x},\langle S_{E^2}\rangle}]=-\frac{1}{2\hbar}e^{3}E_{y}^{2}\sum_{n}\tau_{\textbf{k}}^{n}\frac{\partial f_{0}}{\partial k_{y}}\Omega_{\textbf{k}z}^{n}
\]
\begin{equation}
	=\frac{1}{2\hbar}e^{3}E_{y}^{2}\sum_{n}\tau_{k_{F}}^{n}\frac{\partial \Omega_{\textbf{k}z}^{n}}{\partial k_{y}}f_{0}
\end{equation}
\begin{equation}
	\chi=\frac{1}{2\hbar}e^{3}\tau_{k_{F}}^{+}\int\frac{d^{2}k}{(2\pi)^{2}}f_{0}\frac{\partial \Omega_{\textbf{k}z}^{+}}{\partial k_{y}}
\end{equation}

More details are presented in Appendix A. This also indicates that our theory is consistent with semi-classical apprximation.

The results (12) and (16-18) are derived without considering the impurity
scattering. With (11), we find the current corresponding to linear term which is induced by impurity takes the form of:
\begin{equation}
	J_{x1}^{i}=\pi e\frac{\partial_{y}T}{T}\sum_{m,m'}\langle U_{\textbf{k}\textbf{k'}}^{mm'}U_{\textbf{k'}\textbf{k}}^{m'm}\rangle(n_{k}^{m}-n_{k'}^{m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})\Omega_{kz}^{m}
\end{equation}

This is shown to be 0 after integral where $n_{\textbf{k}}^{m}=(\tau_{k_{F}}^{m})\epsilon_{m}f_{0}(\epsilon_{\textbf{k}}^{m})$.

The off-diagonal matrix elements of quadratic term induced by disorder
is given by:

\begin{equation}
	\langle S_{T^{2}}'\rangle_{k}^{mm'}=i\hbar\frac{[J(\langle n_{T^{2}}\rangle)]_{\textbf{k}}^{mm'}}{\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}}^{m'}}
\end{equation}

So the current term connected to the dipole can be shown as:

\begin{equation}
	J_{x2}^{i}=\pi e(\frac{\partial_{y}T}{T})^{2}\sum_{m,m'}\langle U_{\textbf{k}\textbf{k'}}^{mm'}U_{\textbf{k'}\textbf{k}}^{m'm}\rangle(N_{\textbf{k}}^{m}-N_{\textbf{k}'}^{m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})\partial_{y}\Omega_{\textbf{k}z}^{m}
\end{equation}

Where $N_{\textbf{k}}^{m}=(\tau_{k_{F}}^{m})^{2}\epsilon_{m}^{2}f_{0}(\epsilon_{\textbf{k}}^{m})$.

This current can also be proven to be 0 after integral, which tells
that current terms in impurity scattering which is linked to dipole contributes nothing to transport. More
details are illustrated in Appendix B. We have to pay attention that here we only consider the impurity-scattering terms related to Berry curvature and dipole. They will have no effects on transport. Besides, all above is about results in DC limit. We have also developed one in the AC limit which will be discussed in the discussion section.

Since then, we have developed quantum kinetic theory of nonlinear Nernst effect
in thermoelectric transport. For further step, we are intended to
apply our results to a specific system: topological crystalline insulator.

\section{Application}

We firstly consider the Dirac semimetal materials(DSM).
In many realistic DSM, Dirac cones are more or less distorted.
The tilted Dirac cone can be realized in a number of types of materials.
To be specific, we focus on topological crystalline insulators
such as SnTe. Experiments tell us there are tilted Dirac cones on (001) surface of it. Therefore, we can calculate the thermoelectric conductance induced by
thermoelectric Berry curvature dipole.

Firstly, the low-energy model of the (001) surface is given by

\begin{equation}
	H=\xi w_{y}k_{y}\sigma_{0}+v_{x}k_{x}\sigma_{x}+\xi v_{y}k_{y}\sigma_{y}+\frac{\Delta}{2}\sigma_{z}
\end{equation}

The energy bands will take the form: $\epsilon_{\textbf{k}}^{\pm}=w_{y}k_{y}\pm\sqrt{(v_{x}k_{x})^{2}+(v_{y}k_{y})^{2}+(\frac{\Delta}{2})^{2}}$.
Here $w_{y}$ is the tilted parameter, $\Delta$ is energy gap, $v_{x},v_{y}$
represent fermi velocity in different direction. Here $\xi=\pm1$
represents the freedom of valley, which conserves time reversal(TR) symmetry
of the system. Due to the ferroelectric distortion, the Dirac
cone are turned into gapped one. Meanwhile, form of energy bands is stable since if we take influence of disorder into account, this form is still invariant. To properly account for such a dynamically generated kinetic term, we add a term $\lambda\omega\sigma_{x}$ in free fermion action. Since that, we can correct the corresponding dispersion\cite{PhysRevB.98.195123}:
\begin{equation}
	||E-tv_{y}k_{y}-\lambda E\sigma_{x}-v_{x}k_{x}\sigma_{x}-v_{y}k_{y}\sigma_{y}||=0
\end{equation}
Here we use the convention: $tv_{y}=\xi w_{y}$. After solving the equation, we have the effect energy band.
\begin{equation}
	E_{\pm}=t^{eff}v_{y}^{eff}k_{y}\pm\sqrt{(v_{x}^{eff}k_{x})^{2}+(v_{y}^{eff}k_{y})^{2}}
\end{equation}
where
\begin{equation}
	t^{eff}=\frac{t+\lambda}{1+t\lambda}
\end{equation}

\begin{equation}
	v_{y}^{eff}=\frac{1+t\lambda}{1-\lambda^{2}}v_{y}
\end{equation}

\begin{equation}
	v_{x}^{eff}=\frac{1}{\sqrt{1-\lambda^{2}}}v_{x}
\end{equation}
This method has been used by Sikkenk and Fritz to study the disorder effect in 3D tilted Weyl semimetal(WSM)\cite{PhysRevB.96.155121}. With renormalization group(RG), this term is determined to be marginal one which can not be ignored simply. However, this perturbation does nothing to the form of energy band since we can turn coefficients into effective one compared with (24).

We begin with the topological band and Berry curvature. Although we introduce the tilted parameter, the corresponding eigenvectors are still invariant:

\begin{equation}
	|\pm,\textbf{k}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{1\pm\frac{\Delta}{2\epsilon_{\textbf{k}}}}\\
		\pm e^{i\theta}\sqrt{1\mp\frac{\Delta}{2\epsilon_{\textbf{k}}}}
	\end{pmatrix}
\end{equation}

The angle $\theta$ is defined by $e^{i\theta}=\frac{v_{x}k_{x}+iv_{y}k_{y}}{k_{\bot}},k_{\bot}=\sqrt{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}}$.
In this way, we also have the Berry curvature which is the same one in 2D WSM.

\begin{equation}
	\Omega_{\textbf{k},z}^{\pm}=i\epsilon_{zbc}\langle\partial_{b}\pm|\partial_{c}\pm\rangle=\mp\frac{\xi\Delta v_{x}v_{y}}{4\epsilon_{\textbf{k}}}
\end{equation}

Where $\epsilon_{\textbf{k}}=\sqrt{(v_{x}k_{x})^{2}+(v_{y}k_{y})^{2}+(\frac{\Delta}{2})^{2}}$.
In general case, this will contribute nothing to transport after sum
of $\xi$. However, we will obtain the nonlinear Hall conductivity
by taking the nonlinear Hall effect into accounnt. After calculating
the relaxation time, we will give the form of conductivity. Due to
the same contribution from the different valleys, we just calculate
one and multiply it by 2. Before approaching the final result, we just make some basic assumptions: firstly, we also consider the case with low-enough temperature. Further, we assume that warping of the Fermi surface can be ignored when calculating the relaxation time.

In this way, The form of the Berry curvature dipole and conductivity are taken as:

\begin{equation}		D_{y}=\frac{1}{2}\int\frac{d^{2}k}{(2\pi)^{2}}\epsilon_{+}^{2}f_{0}(\epsilon_{\textbf{k}}^{+})\partial_{y}\Omega_{\textbf{k}z}^{+}
\end{equation}

\begin{equation}
	\chi_1=\frac{4ev_{x}v_{y}}{n_{imp}U_{0}^{2}\mu(1+3\frac{\Delta^{2}}{4\mu^{2}})}D_{y}
\end{equation}
\begin{equation}
	\chi_2=\frac{4 ev_{x}v_{y}}{n_{imp}U_{0}^{2}\mu(1+3\frac{\Delta^{2}}{4\mu^{2}})}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{+}\frac{\partial\epsilon_{\textbf{k}}^+}{{\partial k_{y}}}f_{0}(\epsilon_{k}^{+})\Omega_{kz}^{+}
\end{equation}
Therefore, the total conductance is considered as: $\chi=\chi_1+\chi_2$. More details will be displayed in Appendix C. With $\partial_{y}\Omega_{\textbf{k},z}^{\pm}=\pm\frac{3\xi v_{x}v_{y}^{3}\Delta k_{y}}{4\epsilon_{\textbf{k}}^{5}}$,
we can also see that the integral will vanish if the Dirac cone is
not tilted. Although the untilted Dirac cone gives finite Berry curvature,
the Berry curvature dipole comes to zero since $\partial_{y}\Omega_{z}$
is odd function under Fermi surface.
The parameter set is $v_{x}\approx v_{y}\approx2.6328eV\cdot\mathring{A},\Delta=20meV,w_{y}=0.026328eV\cdot\mathring{A}$.

\begin{figure}[t]
	\centering
	\includegraphics[width=7.2 cm]{dipole.png}\\
	\caption{thermoelectric Berry curvature dipole(after rescaling) of SnTe as a function of $2\mu/\Delta$}
	\label{fig1}
\end{figure}

\begin{figure}[t]
	\centering
	\includegraphics[width=7.2 cm]{conductance.png}\\
	\caption{thermoelectric conductivity(after rescaling) of SnTe as a function of $2\mu/\Delta$}
	\label{fig2}
\end{figure}

\section{Discussion}

In summary, we begin with the quantum Liouville
equation and its solution in the presence of disorder and temperature
gradient. Further, we develop quantum kinetic theory of nonlinear Nernst effect with general solution to density matrix. We prove the persistence of Nernst coefficient in nonlinear regime with calculation. It also establishes a new concept of thermoelectric Berry curvature dipole dominant in the quadratic term and electric current in TR-invariant systems. The Berry curavture giving rise to linear response does not contribute to electric current without breaking time-reversal symmetry. Meanwhile, we have also proved that the main impurity scattering contributes nothing to thermoelectric transport. Finally, we apply our theory to SnTe, a topological crystalline insulator with time-reversal symmetry which has been intensively studied by recent experiments, and give numerical result of the thermoelectric Berry curvature dipole and thermoelectric conductivity. However, this theory also remains us some problems: if the external field is intensive, could we also expand the formula (13) as (14)? When the external field is in the DC limit, how could we solve the kinetic equation (6)?

Since we only care about second-order response, we can solve equation (13) iteratively. In this way, $\langle\rho_{T}\rangle=\mathcal{L}^{-1}D_{T}\langle\rho_{0}\rangle$, $\langle\rho_{T^{2}}\rangle=\mathcal{L}^{-1}D_{T}\langle\rho_{T}\rangle=(\mathcal{L}^{-1}D_{T})^{2}\langle\rho_{0}\rangle$, $\langle\rho_{T^{n}}\rangle=\mathcal{L}^{-1}D_{T}\langle\rho_{T^{n-1}}\rangle=(\mathcal{L}^{-1}D_{T})^{n}\langle\rho_{0}\rangle$. In this way, we can derive any-order response iteratively. We can always derive the nonlinear response with (14) however intensive the external field is. In the AC limit, when the external field is replaced with an oscillating one $E(t)=E_{0}e^{i\omega t}$, we are still unfamiliar with the solution of the density matrix. We can solve the kinetic equation by replacing the operator $\mathcal{L}$ with: $\mathcal{M}=\mathcal{L}-i\omega=P+K-i\omega$ for we only consider the distribution in frequency space instead of time space. Then we can similarly derive the unsteady-state kinetic equation as: $\langle\rho\rangle_{F}(\omega)=\sum_{N=1}^{\infty}(\mathcal{M}^{-1}D_{T})^{N}\langle\rho_{0}\rangle$. However, what does the unsteady-state stand for? To further consider this problem, we firstly write down the corresponding conductivity $\sigma_{\mu\nu}(\omega)=Tr[(-e)v^{\mu}\langle\rho\rangle_{F}(\omega)]/E_{0}^{\nu}$. This indicates conductivity when the external is oscillating one. In other words, this conductivity corresponds to optical conductivity in the experiments.

To clarify our quantum kinetic theory can be developed into one in AC limit, we shall calculate the second-order response in oscillating external electric field and check it with one in semi-classical approximation. We firstly ignore the scattering term to focus on the effects of $i\omega$. We firstly write down the formula for the first-order response:
\begin{equation}
	\frac{eE_{y}}{\hbar}\frac{\partial f_{0}(\epsilon_{\textbf{k}}^{m})}{\partial k_{y}}-i\omega\langle n\rangle_\textbf{k}^{m}=\frac{\langle n\rangle_\textbf{k}^{m}}{\tau_{k_F}^{m}}
\end{equation}
\begin{equation}
	\langle n\rangle_{k}^{m}=\frac{\frac{eE_{y}}{\hbar}\frac{\partial f_{0}(\epsilon_{\textbf{k}}^{m})}{\partial k_{y}}\tau_{k_F}^{m}}{1+i\omega\tau_{k_F}^{m}}
\end{equation}
Here we figure out the relationship between the diagonal part in AC limit and that in  DC limit:$\langle n\rangle_{k}^{m}(\omega)=\frac{\langle n\rangle_{k}^{m}} {1+i\omega\tau_{\textbf{k}}^{m}}$. Hence we come to the conclusion that by replacing the diagonal part in (18) with generalized one, we can get the nonlinear optical conductivity. (we also assume $\mu>0$ which indicates $m=+$)
\begin{equation}
	\chi=\frac{e^{3}\tau_{k_{F}}^{+}}{2(1+i\omega\tau_{k_{F}}^{+})}\int\frac{d^{2}k}{(2\pi)^{2}}f_{0}\frac{\partial \Omega_{\textbf{k}z}^{n}}{\partial k_{y}}
\end{equation}
This is consisent with the result in \cite{PhysRevLett.115.216806}.
\section{Acknowledgement}
I acknowledge helpful discussion with Yonghao Gao at Fudan University and professor Gang Chen at Hong Kong University. My work is also supported by Physics Departement of University of Science and Technology of China. I also acknowledge my advisor Prof. Shaolong Wan in USTC and discussions on detailed calculation with other people in the group.

\begin{widetext}
	\begin{appendix}
		\section{Deriviation of (16-18)}

		Firstly, let's focus on the derivation of off-diagonal part of linear response$\langle S_{T}\rangle$.
		According to (14), the lowest order should be:

		\begin{equation}
			\langle\rho_{T}\rangle=\mathcal{L}^{-1}D_{T}(\langle\rho_{0}\rangle)
		\end{equation}

		Therefore, the matrix elements should be:

		\begin{equation}
			\langle n_{T}\rangle_{\textbf{k}}^{m}=\tau_{\textbf{k}}^{m}[D_{T}(\langle\rho_{0}\rangle)]_{\textbf{k}}^{mm}
		\end{equation}

		\begin{equation}
			\langle S_{T}\rangle_{\textbf{k}}^{nn'}=-i\hbar\frac{D_{T}(\langle\rho_{0}\rangle)_{\textbf{k}}^{nn'}-J(\langle n_{T}\rangle)_{\textbf{k}}^{nn'}}{\epsilon_{n}-\epsilon_{n'}}
		\end{equation}

		To be specific, the off-diagonal one can take another form:

		\begin{equation}
			\langle S_{T}\rangle=\sum_{nn'}\frac{\partial_{y}T}{T}\frac{\epsilon_{n'}f_{0n'}-\epsilon_{n}f_{0n}}{\epsilon_{n}-\epsilon_{n'}}|n\rangle\langle n|\partial_{y}n'\rangle\langle n'|
		\end{equation}

		Similarly, compared with (A3), we can also derive both the off-diagonal part and the diagonal part of the quadratic term.

		\begin{equation}
			\langle S_{T^{2}}\rangle_{\textbf{k}}^{nn'}=-i\hbar\frac{D_{T}(\langle n_{T}\rangle)_{\textbf{k}}^{nn'}-J(\langle n_{T^{2}}\rangle)_{\textbf{k}}^{nn'}}{\epsilon_{n}-\epsilon_{n'}}
		\end{equation}

		\begin{equation}
			\langle n_{T^{2}}\rangle_{\textbf{k}}^{m}=\tau_{\textbf{k}}^{m}D_{T}(\langle n_{T}\rangle)_{\textbf{k}}^{m}=\tau_{\textbf{k}}^{m}\frac{1}{2\hbar}\frac{\partial_{y}T}{T}\frac{D(\{H_{0},\langle n_{T}\rangle\})}{Dk_{y}}
		\end{equation}

		These are the nonlinear responses in the presence of temperature gradient. Here I remain further discussion on influence caused by impurity scattering in Appendix B. We here just care about the part induced by temperature gradient. In this way, the off-diagonal term of(A5) can be obtained as

		\begin{equation}
			\langle S_{T^{2}}\rangle_{\textbf{k}}^{nn'}=-i\frac{\partial_{y}T}{T}\sum_{nn'}\frac{\epsilon_{n'}\langle n_{T}\rangle_{k}^{n'}-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}}{\epsilon_{n}-\epsilon_{n'}}|n\rangle\langle n|\partial_{y}n'\rangle\langle n'|
		\end{equation}

		So the electric current induced by thermal flow takes the form:

		\begin{equation}
			J_{x}^{(2)}=\frac{1}{2}Tr[(-e)\{v_{x},\langle S_{T^{2}}\rangle\}]
		\end{equation}

		This formula contains two parts: $\langle m|v_{x}\langle S_{T^{2}}\rangle|m\rangle$ and $\langle m|\langle S_{T^{2}}\rangle v_{x}|m\rangle$. Let's calculate the two parts separately. Since $\sum_{m}|\partial_{y}m\rangle\langle m|+|m\rangle\langle\partial_{y}m|=\partial_{y}(\sum_{m}|m\rangle\langle m|)=\partial_{y}I=0$,
		we can write down another form of the intrinsic velocity operator.

		\begin{equation}
			v_{x}=\sum_{m'}(\epsilon_{m'}-\epsilon_{n'})|\partial_{y}m'\rangle\langle m'|+|m'\rangle\langle\partial_{y}m'|=\sum_{m'}(\epsilon_{m'}-\epsilon_{n})|\partial_{y}m'\rangle\langle m'|+|m'\rangle\langle\partial_{y}m'|
		\end{equation}

		\[
		\langle m|(-e)v_{x}\langle S_{T^{2}}\rangle|m\rangle=ie\frac{\partial_{y}T}{T}\sum_{m'nn'}\frac{\epsilon_{n'}\langle n_{T}\rangle_{k}^{n'}-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}}{\epsilon_{n}-\epsilon_{n'}}(\epsilon_{m'}-\epsilon_{n'})\langle n|\partial_{y}n'\rangle\langle m|[|\partial_{x}m'\rangle\langle m'|+|m'\rangle\langle\partial_{x}m'|]|n\rangle\langle n'|m\rangle
		\]

		\[
		=-ie\frac{\partial_{y}T}{T}\sum_{m'nn'}[\epsilon_{n'}\langle n_{T}\rangle_{k}^{n'}-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}]\langle n|\partial_{y}m\rangle\langle m|\partial_{x}n\rangle\delta_{m'n}\delta_{n'm}
		\]

		\[
		=ie\frac{\partial_{y}T}{T}\sum_{n}[\epsilon_{n'}\langle n_{T}\rangle_{k}^{n'}-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}]\langle\partial_{x}m|n\rangle\langle n|\partial_{y}m\rangle
		\]

		\begin{equation}
			=-ie\frac{\partial_{y}T}{T}[\epsilon_{m}\langle n_{T}\rangle_{k}^{m}\langle\partial_{x}m|\partial_{y}m\rangle-\sum_{n}\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{x}m|n\rangle\langle n|\partial_{y}m\rangle]
		\end{equation}

		\[
		\langle m|(-e)\langle S_{T^{2}}\rangle v_{x}|m\rangle=ie\frac{\partial_{y}T}{T}\sum_{m'nn'}\frac{\epsilon_{n'}\langle n_{T}\rangle_{k}^{n'}-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}}{\epsilon_{n}-\epsilon_{n'}}(\epsilon_{m'}-\epsilon_{n})\langle\partial_{y}n|n'\rangle\langle m|n\rangle\langle n'|[|\partial_{x}m'\rangle\langle m'|+|m'\rangle\langle\partial_{x}m'|]|m\rangle
		\]

		\begin{equation}
			=ie\frac{\partial_{y}T}{T}[\epsilon_{m}\langle n_{T}\rangle_{k}^{m}\langle\partial_{y}m|\partial_{x}m\rangle-\sum_{n}\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{y}m|n\rangle\langle n|\partial_{x}m\rangle]
		\end{equation}

		The first term of (A10,A11) can be pointed out as Berry curvature:

		\begin{equation}
			-ie\frac{\partial_{y}T}{T}\sum_{m}[\epsilon_{m}\langle n_{T}\rangle_{k}^{m}\langle\partial_{x}m|\partial_{y}m\rangle-\epsilon_{m}\langle n_{T}\rangle_{k}^{m}\langle\partial_{y}m|\partial_{x}m\rangle]=-e\frac{\partial_{y}T}{T}\sum_{m}\epsilon_{m}\langle n_{T}\rangle_{k}^{m}\Omega_{\textbf{k},z}^{m}
		\end{equation}

		Due to the sum of index m, the second term of (A10,A11) can be combined
		together.

		\[
		ie\frac{\partial_{y}T}{T}\sum_{n,m}(\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{x}m|n\rangle\langle n|\partial_{y}m\rangle-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{y}m|n\rangle\langle n|\partial_{x}m\rangle)
		\]

		\[
		=ie\frac{\partial_{y}T}{T}\sum_{n,m}(\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{y}n|m\rangle\langle m|\partial_{x}n\rangle-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{x}n|m\rangle\langle m|\partial_{y}n\rangle)
		\]

		\[
		=ie\frac{\partial_{y}T}{T}\sum_{n}(\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{y}n|\partial_{x}n\rangle-\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\langle\partial_{x}n|\partial_{y}n\rangle)
		\]

		\begin{equation}
			=-e\frac{\partial_{y}T}{T}\sum_{n}\epsilon_{n}\langle n_{T}\rangle_{k}^{n}\Omega_{\textbf{k},z}^{m}
		\end{equation}

		Correspondingly, the current can be figured out:

		\begin{equation}
			J_{x}^{(2)}=-\frac{e}{2}(\frac{\partial_{y}T}{T})^{2}\sum_{m}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{m}^{2}\tau_{k}^{m}\frac{\partial f_{0}(\epsilon_{k}^{m})}{\partial k_{y}}\Omega_{kz}^{m}
		\end{equation}

		where $\Omega_{\textbf{k}z}^{m}=i(\langle\partial_{x}m|\partial_{y}m\rangle-\langle\partial_{y}m|\partial_{x}m\rangle)$
		represents Berry curvature.
		This can be separated into two parts if we basically assume only the conductance band contributes:
		\begin{equation}
			J_{x1}^{(2)}=\frac{e}{2}(\frac{\partial_{y}T}{T})^{2}\tau_{k_F}^{+}\sum_{m}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{m}^{2} f_{0}(\epsilon_{k}^{m})\frac{\partial\Omega_{kz}^{m}}{{\partial k_{y}}}
		\end{equation}
		\begin{equation}
		J_{x2}^{(2)}=e(\frac{\partial_{y}T}{T})^{2}\tau_{k_F}^{+}\sum_{m}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{m}\frac{\partial\epsilon_{\textbf{k}}^m}{{\partial k_{y}}} f_{0}(\epsilon_{k}^{m})\partial\Omega_{kz}^{m}
		\end{equation}
		In this way, we have the form of thermoelectric conductivity with (A15, A16)

		\begin{equation}
			\chi_1=\frac{e}{2}\tau_{k_{F}}^{+}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{+}^{2}f_{0}(\epsilon_{\textbf{k}}^{+})\partial_{y}\Omega_{\textbf{k},z}^{+}=e\tau_{k_{F}}^{+}D_{y}
		\end{equation}
		\begin{equation}
			\chi_2=e\tau_{k_F}^{+}\sum_{m}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{m}\frac{\partial\epsilon_{\textbf{k}}^m}{{\partial k_{y}}}f_{0}(\epsilon_{k}^{m})\Omega_{kz}^{m}
		\end{equation}
		Here $D_{y}=\frac{1}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{+}^{2}f_{0}(\epsilon_{\textbf{k}}^{+})\partial_{y}\Omega_{\textbf{k},z}^{+}$
		represents the new type of Berry curvature dipole. We can see there are two terms in the thermoelectric conductivity. The first one is connected to the thermoelectric Berry curvature dipole. The second one, which does not show up in electric Hall effect, is directly linked to the Berry curvature.

		\section{Derivation of (23,25)}

		At the beginning, let's focus on the impurity scattering of the linear
		term.

		\begin{equation}
			J_{x1}^{i}=\frac{1}{2}Tr[(-e){v_{x},\langle S_{T}'\rangle}]
		\end{equation}

		\begin{equation}
			\langle S_{T}'\rangle=i\hbar\sum_{nn'}\frac{[J(\langle n_{T}\rangle)]_{\textbf{k}}^{nn'}}{\epsilon_{n}-\epsilon_{n'}}|n\rangle\langle n'|
		\end{equation}

		With intrinsic velocity(A9), you can also obtain the current density(B1). To begin with, let's firstly consider the related two terms: $\langle m|v_{x}\langle S_{T}'\rangle|m\rangle$ and $\langle m|\langle S_{T}'\rangle v_{x}|m\rangle$:

		\[
		\langle m|v_{x}\langle S_{T}'\rangle|m\rangle=i\sum_{m'nn'}(\epsilon_{m'}-\epsilon_{n'})\frac{[J(\langle n_{T}\rangle)]_{\textbf{k}}^{nn'}}{\epsilon_{n}-\epsilon_{n'}}\langle m|[|\partial_{x}m'\rangle\langle m'|+|m'\rangle\langle\partial_{x}m'|]|n\rangle\langle n'|m\rangle
		\]

		\begin{equation}
			=i\sum_{n}[J(\langle n_{T}\rangle)]_{\textbf{k}}^{nm}\langle m|\partial_{x}n\rangle
		\end{equation}

		\[
		\langle m|\langle S_{T}'\rangle v_{x}|m\rangle=i\sum_{m'nn'}(\epsilon_{m'}-\epsilon_{n})\frac{[J(\langle n_{T}\rangle)]_{\textbf{k}}^{nn'}}{\epsilon_{n}-\epsilon_{n'}}\langle m|n\rangle\langle n'|[|\partial_{x}m'\rangle\langle m'|+|m'\rangle\langle\partial_{x}m'|]|m\rangle
		\]

		\begin{equation}
			=-i\sum_{n}[J(\langle n_{T}\rangle)]_{\textbf{k}}^{nm}\langle\partial_{x}n|m\rangle
		\end{equation}

		With (B3,B4) we can directly figure out the form of .

		\begin{equation}
			J_{x1}^{i}=e\sum_{n,m}\int\frac{d^{d}k}{(2\pi)^{d}}[J(\langle n_{T}\rangle)]_{\textbf{k}}^{nm}Im(\langle m|\partial_{x}n\rangle)
		\end{equation}

		If we take the zero-temperature approximation, the non-equilibrium
		distribution induced by temperature gradient will just take place
		on the conductance band. So this will contribute nothing after sum
		of band index. However, if we take the band-diagonal part into account,
		we can find the term related to Berry curvature.We assume the Born
		approximation: $\langle U(\textbf{r})U(\textbf{r'})\rangle=n_{imp}U_0^{2}\delta(\textbf{r}-\textbf{r'})$\cite{PhysRevB.97.201301}
		Then we have:
		\begin{equation}
			\langle U_{\textbf{k}\textbf{k'}}^{mm'}U_{\textbf{k'}\textbf{k}}^{m'm}\rangle=n_{imp}U_{0}^{2}\langle u_{\textbf{k}}^{m}|u_{\textbf{k'}}^{m'}\rangle\langle u_{\textbf{k'}}^{m'}|u_{\textbf{k}}^{m}\rangle
		\end{equation}
	In most cases, this is not a trivial result since we may have strong SOC in the material. We can not simply come to the general case. However, we can still make some approximation. Firstly, the temperature is low enough that we can still replace the relaxation time with one on the Fermi surface which is noted by $\tau_{k_{F}}^{m}$. Further, even though the Fermi surface is partly distorted, we still assume the diagonal part $\langle n_{T}\rangle_{\textbf{k}}^{m}$ and $\langle n_{T^{2}}\rangle_{\textbf{k}}^{m}$ are approximately considered as functions of $\epsilon_{\textbf{k}}^{m}$.
		\[
		J_{x1}^{i}=-i\pi e\sum_{m,m'}\langle U_{\textbf{k}\textbf{k}'}^{mm'}U_{\textbf{k}'\textbf{k}}^{m'm}\rangle(\langle n_{T}\rangle_{\textbf{k}}^{m}-\langle n_{T}\rangle_{\textbf{k}'}^{m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})(\langle m|\partial_{x}m'\rangle-\langle\partial_{x}m'|m\rangle)
		\]

		\begin{equation}
			=-i\pi e\sum_{m}\langle U_{\textbf{k}\textbf{k}'}^{mm}U_{\textbf{k}'\textbf{k}}^{mm}\rangle(\langle n_{T}\rangle_{\textbf{k}}^{m}-\langle n_{T}\rangle_{\textbf{k}'}^{m})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m})(\langle m|\partial_{x}m\rangle-\langle\partial_{x}m|m\rangle)
		\end{equation}

		\begin{equation}
			\rightarrow\pi e\frac{\partial_{y}T}{T}\sum_{m}\tau_{k_{F}}^{m}(n_{\textbf{k}}^{m}-n_{\textbf{k'}}^{m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m})\Omega_{\textbf{k},z}^{m}
		\end{equation}

		where $n_{\textbf{k}}^{m}=(\tau_{k_{F}}^{m})\epsilon_{m}f_{0}(\epsilon_{\textbf{k}}^{m})$,
		This is exactly the equation(21), which is only connected to $\epsilon_{\textbf{k}}^{m}$. Therefore,
		this current will vanish after integrating $\textbf{k'}$.

		We can clearly see that the term connected with Berry curvature has no effect on transport in linear regime. However, current from other terms is still unknown to us in general case. Hence, we can only consider other terms in specific models.

		Let's turn to the band-diagonal part of the quadratic term.

		\begin{equation}
			\langle n_{T^{2}}\rangle_{\textbf{k}}^{m}=\frac{\tau_{\textbf{k}}^{m}}{\hbar}\frac{\partial_{y}T}{T}[\frac{D(H_{0}\langle n_{T}\rangle)}{Dk_{y}}]_{\textbf{k}}^{m}=\frac{(\tau_{k_{F}}^{m})^{2}}{\hbar}(\frac{\partial_{y}T}{T})^{2}\partial_{y}(\epsilon_{m}^{2}\partial_{y}f_{0}(\epsilon_{\textbf{k}}^{m}))
		\end{equation}

		In this way, the impurity scattering part is derived as:

		\begin{equation}
			\langle S_{T^{2}}'\rangle_{k}^{mm'}=i\hbar\frac{[J(\langle n_{T^{2}}\rangle)]_{\textbf{k}}^{mm'}}{\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}}^{m'}}
		\end{equation}

		\begin{equation}
			\langle S_{T^{2}}'\rangle=i\pi\sum_{mm'm''}\sum_{\textbf{k}'}\langle U_{\textbf{kk'}}^{mm'}U_{\textbf{k'k}}^{m'm}\rangle\frac{(n_{\textbf{k}}^{(2)m}-n_{\textbf{k}'}^{(2)m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})+(n_{\textbf{k}}^{(2)m''}-n_{\textbf{k}'}^{(2)m'})\delta(\epsilon_{\textbf{k}}^{m''}-\epsilon_{\textbf{k}'}^{m'})}{\epsilon_{m}-\epsilon_{m''}}(|\partial_{y}m\rangle\langle m''|+|m\rangle\langle\partial_{y}m''|)
		\end{equation}

		Here $n_{\textbf{k}}^{(2)m}$ is not the one in quadratic term. It is
		given by:

		\begin{equation}
			n_{\textbf{k}}^{(2)m}=\frac{(\tau_{k_{F}}^{m})^{2}}{\hbar}(\frac{\partial_{y}T}{T})^{2}\epsilon_{m}^{2}\partial_{y}f_{0}(\epsilon_{\textbf{k}}^{m})
		\end{equation}

		So we can calculate each matrix element of it.(Here we also use the
		simpliest approximation)

		\begin{equation}
			\langle S_{T^{2}}'\rangle_{\textbf{k}}^{nn'}=i\pi \sum_{m,m',m''}\sum_{\textbf{k'}}\langle U_{\textbf{kk'}}^{mm'}U_{\textbf{k'k}}^{m'm}\rangle[\frac{g(m,m',n')}{\epsilon_{m}-\epsilon_{n'}}\langle n|\partial_{y}m\rangle\delta_{m''n'}+\frac{g(n,m',m'')}{\epsilon_{n}-\epsilon_{m''}}\langle\partial_{y}m''|n'\rangle\delta_{mn}]
		\end{equation}

		With $g(m,m',n')=(n_{k}^{(2)m}-n_{k'}^{(2)m'})\delta(\epsilon_{k}^{m}-\epsilon_{k'}^{m'})+(n_{k}^{(2)m''}-n_{k'}^{(2)m'})\delta(\epsilon_{k}^{m''}-\epsilon_{k'}^{m'})$.
		The intrinsic contribution to velocity operator in the eigenstate
		basis is

		\begin{equation}
			v_{x}=\sum_{l'}(\epsilon_{l'}-\epsilon_{m''})[|\partial_{x}l'\rangle\langle l'|+|l'\rangle\langle\partial_{x}l'|]
		\end{equation}

		So the diagonal part is given by

		\[
		\langle l|v_{x}\langle S_{T^{2}}'\rangle|l\rangle=i\pi \sum_{l'n'n}\sum_{m,m',m''}\langle U_{\textbf{kk'}}^{mm'}U_{\textbf{k'k}}^{m'm}\rangle[\frac{g(m,m',n')}{\epsilon_{m}-\epsilon_{n'}}\langle n|\partial_{y}m\rangle\delta_{m''n'}+\frac{g(n,m',m'')}{\epsilon_{n}-\epsilon_{m''}}\langle\partial_{y}m''|n'\rangle\delta_{mn}]
		\]

		\begin{equation}
			\times(\epsilon_{l'}-\epsilon_{m''})\langle l|[|\partial_{x}l'\rangle\langle l'|+|l'\rangle\langle\partial_{x}l'|][|n\rangle\langle n'|\|l\rangle
		\end{equation}

		With tedious calculation, we obtain the only term seemingly connected
		to Berry curvature dipole.

		\begin{equation}
			i\pi\sum_{l,m,m',m''}\langle U_{\textbf{kk'}}^{mm'}U_{\textbf{k'k}}^{m'm}\rangle g(m,m',m'')\langle\partial_{y}m''|l\rangle\langle l|\partial_{x}m\rangle
		\end{equation}

		In this way, the current related to impurity scattering can be figured
		out:

		\begin{equation}
			J_{x2}^{i}=\frac{1}{2}Tr[(-e)\{v_{x},\langle S_{T^{2}}'\rangle\}]=\pi e\sum_{m,m'}\langle U_{\textbf{kk'}}^{mm'}U_{\textbf{k'k}}^{m'm}\rangle(N_{\textbf{k}}^{m}-N_{\textbf{k'}}^{m'})\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k'}}^{m'})\partial_{y}\Omega_{\textbf{k},z}
		\end{equation}

		where $N_{\textbf{k}}^{m}=(\tau_{k_{F}}^{m})^{2}\epsilon_{m}^{2}f_{0}(\epsilon_{\textbf{k}}^{m})$,
		which is only connected with $\epsilon_{\textbf{k}}^{m}$. Therefore,
		this current will vanish after integrating $\textbf{k'}$.

		To conclude, we have proved the current density induced by impurity scattering is zero in both linear regime and nonlinear regime. Similar results can be found in \cite{PhysRevB.96.235134}\cite{PhysRevB.101.155204}. However, here we only prove that the term related to the dipole contributes nothing to the transport in nonlinear regime. We do not consider all the terms in general case.

		\section{More details of calculation on topological crystalline insulator}

		After constructing the effective Hamiltonian of strained single-layer
		graphene, we can calculate the eigenstates $H|u_{\textbf{k}}^{\pm}\rangle=\pm\epsilon_{\textbf{k}}|u_{\textbf{k}}^{\pm}\rangle$
		and Berry curvature of the model.

		\begin{equation}
			|u_{\textbf{k}}^{\pm}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{1\pm\frac{\Delta}{2\epsilon_{\textbf{k}}}}\\
				\pm e^{i\theta}\sqrt{1\mp\frac{\Delta}{2\epsilon_{\textbf{k}}}}
			\end{pmatrix}
		\end{equation}
		where angle $\theta$ and $\epsilon_{\textbf{k}}$ is defined by $e^{i\theta}=\frac{A_{1}+iA_{2}}{\sqrt{A_{1}^{2}+A_2^{2}}}$ and $\epsilon_{\textbf{k}}=\sqrt{A_{1}^{2}+A_2^{2}+m^{2}}$.

		\begin{equation}
			\partial_{a}|u_{\textbf{k}}^{\pm}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\mp\frac{1}{2}\frac{1}{\sqrt{1\pm m/\epsilon_{k}}}\frac{\Delta}{2\epsilon_{\textbf{k}}^{2}}\frac{\partial\epsilon_{\textbf{k}}}{\partial k_{a}}\\
				\frac{1}{2}e^{i\theta}\frac{1}{\sqrt{1\mp m/\epsilon_{k}}}\frac{\Delta}{2\epsilon_{\textbf{k}}^{2}}\frac{\partial\epsilon_{\textbf{k}}}{\partial k_{a}}\pm\frac{i}{2}e^{i\theta}\frac{\partial\theta}{\partial k_{a}}\sqrt{1\mp\frac{\Delta}{2\epsilon_{\textbf{k}}}}
			\end{pmatrix}
		\end{equation}

		\[
		\Omega_{\textbf{k}z}^{\pm}=i(\langle\partial_{x}u_{\textbf{k}}^{\pm}|\partial_{y}u_{\textbf{k}}^{\pm}\rangle-\langle\partial_{y}u_{\textbf{k}}^{\pm}|\partial_{x}u_{\textbf{k}}^{\pm}\rangle)
		\]

		\begin{equation}
			=\mp\frac{\Delta}{4\epsilon_{\textbf{k}}^{3}}(\frac{\partial A_{1}}{\partial k_{x}}\frac{\partial A_{2}}{\partial k_{y}}-\frac{\partial A_{1}}{\partial k_{y}}\frac{\partial A_{2}}{\partial k_{x}})
		\end{equation}

		With the form of $A_{1}=v_{x}k_{x}$ and $A_{2}=v_{y}k_{y}$, you can obtain (28,29) automatically.

		For further step, we focus on the relaxation time.

		\begin{equation}
			\frac{1}{\tau_{\textbf{k}}^{m}}=\frac{2\pi }{\hbar}\int\frac{dk_{x}'dk_{y}'}{(2\pi)^{2}}\sum_{m,m'}\langle U_{\textbf{k}\textbf{k'}}^{mm'}U_{\textbf{k'}\textbf{k}}^{m'm}\rangle\delta(\epsilon_{\textbf{k}}^{m}-\epsilon_{\textbf{k}'}^{m'})
		\end{equation}
		We here take $m=m'=+$ since not only there is no crossover near each valley but also we just basically assume $\mu>0$ for simplicity. In this way, only electrons from the conduction band contribute. So the relaxation time is:
		\begin{equation}
			\frac{1}{\tau_{\textbf{k}}^{+}}=\frac{2\pi }{\hbar}\int\frac{dk_{x}'dk_{y}'}{(2\pi)^{2}}\langle U_{\textbf{k}\textbf{k'}}^{++}U_{\textbf{k'}\textbf{k}}^{++}\rangle\delta(\epsilon_{\textbf{k}}^{+}-\epsilon_{\textbf{k}'}^{+})
		\end{equation}
	With (C1), we have the Born approximation in the Bloch space\cite{PhysRevB.97.201301}:
	\begin{equation}
		 U_{\textbf{k}\textbf{k'}}^{++}=U\langle u_{k}^{+}|u_{k'}^{+}\rangle=\frac{U_{0}}{2}[\sqrt{(1+\frac{\Delta}{2\epsilon_{\textbf{k}}})(1+\frac{\Delta}{2\epsilon_{\textbf{k'}}})}+e^{i(\theta'-\theta)}\sqrt{(1-\frac{\Delta}{2\epsilon_{\textbf{k}}})(1-\frac{\Delta}{2\epsilon_{\textbf{k'}}})}]
	\end{equation}
	\begin{equation}
		\langle U_{\textbf{k}\textbf{k'}}^{++}U_{\textbf{k'}\textbf{k}}^{++}\rangle=\frac{n_{imp}U_{0}^{2}}{2}[1+\frac{\Delta^{2}}{4\epsilon_{\textbf{k}}\epsilon_{\textbf{k'}}}+cos(\theta'-\theta)\frac{\sqrt{v_x^{2}k_x^{2}+v_y^{2}k_y^{2}}\sqrt{v_x^{2}k_x'^{2}+v_y^{2}k_y'^{2}}}{\epsilon_{\textbf{k}}\epsilon_{\textbf{k'}}}]
	\end{equation}
	By neglecting the warping effect, we obtain the relaxation time:
	\begin{equation}
		\frac{1}{\tau_{\textbf{k}}^{+}}=\frac{n_{imp}U_{0}^{2}\mu}{4v_{x}v_{y}}(1+3\frac{\Delta^{2}}{4\mu^{2}})
	\end{equation}
		In SnTe, The energy bands are: $E=w_{y}k_{y}\pm\sqrt{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}+(\frac{\Delta}{2})^{2}}$,which
		can be calculated analytically. Energy surface is given by:

		\begin{equation}
			\frac{(k_{y}+k_{0})^{2}}{s_{y}^{2}}+\frac{k_{x}^{2}}{s_{x}^{2}}=1
		\end{equation}

		In the formula: $s_{y}^{2}=(\frac{v_{y}^{2}}{v_{y}^{2}-w_{y}^{2}}E^{2}-(\frac{\Delta}{2})^{2})/(v_{y}^{2}-w_{y}^{2}),s_{x}^{2}=(\frac{v_{y}^{2}}{v_{y}^{2}-w_{y}^{2}}E^{2}-(\frac{\Delta}{2})^{2})/v_{x}^{2},k_{0}=\frac{w_{y}E}{v_{y}^{2}-w_{y}^{2}}$.
		With $k_{y}=-k_{0}+s_{y}sin\theta,k_{x}=s_{x}cos\theta$,

		\begin{equation}
			dk_{x}dk_{y}=\begin{vmatrix}\frac{\partial k_{x}}{\partial E} & \frac{\partial k_{y}}{\partial E}\\
				\frac{\partial k_{x}}{\partial\theta} & \frac{\partial k_{y}}{\partial\theta}
			\end{vmatrix}dEd\theta=[\frac{v_{y}^{2}}{v_{x}(v_{y}^{2}-w_{y}^{2})^{3/2}}E-\frac{w_{y}}{v_{x}(v_{y}^{2}-w_{y}^{2})}\sqrt{\frac{v_{y}^{2}}{v_{y}^{2}-w_{y}^{2}}E^{2}-(\frac{\Delta}{2})^{2}}sin\theta]dEd\theta
		\end{equation}

		After preparation, let's focus on the conductivity and Berry curvature dipole. According to (19), we can derive $D_{y}$ first.

		\[
		D_{y}=\frac{1}{2}\int\frac{d^{2}k}{(2\pi)^{2}}\epsilon_{+}^{2}f_{0}(\epsilon_{\textbf{k}}^{+})\partial_{y}\Omega_{\textbf{k}z}^{+}
		\]

		\[
		=\frac{1}{2\hbar}\int_{E<\mu}\frac{dEd\theta}{(2\pi)^{2}}[\frac{v_{y}^{2}}{v_{x}(v_{y}^{2}-w_{y}^{2})^{3/2}}E-\frac{w_{y}}{v_{x}(v_{y}^{2}-w_{y}^{2})}\sqrt{\frac{v_{y}^{2}}{v_{x}(v_{y}^{2}-w_{y}^{2})}E^{2}-(\frac{\Delta}{2})^{2}}sin\theta](E-\mu)^{2}\frac{3\Delta v_{x}v_{y}^{3}(-k_{0}+s_{y}sin\theta)}{4(E+w_{y}k_{0}-w_{y}s_{y}sin\theta)^{5}}
		\]

		With calculation tools, we can give the result as Fig.1 in the context.
		Meanwhile, we can derive the conductivity with the relaxation time.
		\begin{equation}
			\chi_1=e\tau_{k_{F}}^{+}D_{y}=\frac{4 ev_{x}v_{y}}{n_{imp}U_{0}^{2}\mu(1+3\frac{\Delta^{2}}{4\mu^{2}})}D_{y}
		\end{equation}
	\[
	\chi_2=e\tau_{k_F}^{+}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{+}\frac{\partial\epsilon_{\textbf{k}}^+}{{\partial k_{y}}}f_{0}(\epsilon_{k}^{+})\Omega_{kz}^{+}
	\]
		\begin{equation}
		=\frac{4 ev_{x}v_{y}}{n_{imp}U_{0}^{2}\mu(1+3\frac{\Delta^{2}}{4\mu^{2}})}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon_{+}\frac{\partial\epsilon_{\textbf{k}}^+}{{\partial k_{y}}}f_{0}(\epsilon_{k}^{+})\Omega_{kz}^{+}
		\end{equation}
		We can numerically depict the result as Fig.2 above.
	\end{appendix}

\end{widetext}