# Dissipation and geometry in nonlinear quantum transports of multiband
electronic systems

Yoshihiro Michishita<EMAIL_ADDRESS>RIKEN Center for Emergent
Matter Science (CEMS), Wako, Saitama 351-0198, Japan  Naoto Nagaosa
<EMAIL_ADDRESS>RIKEN Center for Emergent Matter Science (CEMS),
Wako, Saitama 351-0198, Japan  Department of Applied Physics, The University
of Tokyo, Bunkyo, Tokyo 113-8656, Japan

###### Abstract

Nonlinear responses in condensed matters attract recent intensive interest
because they provide rich information about the materials and hold the
possibility of being applied in diodes or high-frequency optical devices.
Nonlinear responses are often closely related to the multiband nature of the
system which can be taken into account by the geometric quantities such as the
Berry curvature as shown in the nonlinear Hall effect. Theoretically, the
semi-classical Boltzmann treatment or the reduced density matrix method have
been often employed, in which the effect of dissipation is included through
the relaxation time approximation. In the diagrammatic method, the relaxation
is treated through the imaginary part of the self-energy of the Green function
and the consequent broadening of the spectral function for the integration
over the real frequency. Therefore, the poles of the Green function do not
play explicit pole when there is finite dissipation. In this paper, in stark
contrast to this conventional picture, we show that the poles of the Green
function determine mostly the nonlinear response functions with dissipation,
which leads to the terms with the Fermi distribution function of complex
argument and results in the dissipation-induced geometric term. We elucidate
the geometric origin of the nonreciprocal conductivity, which is related to
the Berry curvature generalized to the higher derivative. Finally, we derive
the analytical results on the geometric terms of the nonlinear conductivities
in the type-I and type-II Weyl Hamiltonian to demonstrate their crucial roles.

††preprint: APS

##  I Introduction

Recently, nonlinear response in the bulk systems in condensed matter physics
has been intensively studied, especially in the context of the higher-harmonic
generationsPetersen  _et al._ (2006); Zhao  _et al._ (2016); Harter  _et al._
(2017); Sipe and Ghahramani (1993); Morimoto and Nagaosa (2016); Wu  _et al._
(2017), nonlinear Hall effectSodemann and Fu (2015); Ma  _et al._ (2019); Kang
_et al._ (2019); Dzsaber  _et al._ (2021); Zhou  _et al._ (2020), photovoltaic
effectMcIver  _et al._ (2012); Kastl  _et al._ (2015); Chan  _et al._ (2017);
Cook  _et al._ (2017); Isobe  _et al._ (2020), and non-reciprocal
transportZhang  _et al._ (1992); Morimoto and Nagaosa (2018); Tokura and
Nagaosa (2018); Nakai and Nagaosa (2019); Wakatsuki  _et al._ (2017); Itahashi
_et al._ (2020); Ando  _et al._ (2020); Yuan and Fu (2021); Daido  _et al._
(2021); He  _et al._ (2021) because they have the information on the symmetry
of material and have the possibility of the application to devices. For
example, we can detect the parity-breaking of the bulk through the detection
of the second harmonic generationPetersen  _et al._ (2006); Zhao  _et al._
(2016); Harter  _et al._ (2017). The photovoltaic effect in bulk has the
possibility to the high-frequency rectification devicesIsobe  _et al._ (2020).
Moreover, the large non-reciprocality was found in the
superconductorsWakatsuki  _et al._ (2017); Itahashi  _et al._ (2020); Ando
_et al._ (2020) and can be applied to the diode devices.

In the linear conductivity, the effect of dissipation and multi-band
contribution are usually separated, such as the anomalous quantum Hall effect,
which can be described by the Berry curvature and is not affected by
dissipation. On the other hand, in the nonlinear conductivity, both of
dissipation and multi-band contribution intertwine the novel transport. For
examples, the Berry curvature dipole term is proportional to the lifetime,
which is the inverse of the strength of dissipation, while it is also
proportional to the Berry curvatureDu  _et al._ (2019); Nandy and Sodemann
(2019); Du  _et al._ (2021). Moreover, it has been pointed out in Ref.Morimoto
and Nagaosa (2018) that, for the nonreciprocal current under the time-reversal
symmetry (TRS), both dissipation and multi-band are necessary. Therefore, it
is essential to analyze properly the effect of dissipation on the multi-band
contribution in the nonlinear conductivities.

In most previous studies, the nonlinear response has been commonly studied by
the semi-classical Boltzmann (SCB) treatmentSodemann and Fu (2015); Nandy and
Sodemann (2019); Morimoto  _et al._ (2016); Du  _et al._ (2019) or the reduced
density matrix (RDM) methodSipe and Ghahramani (1993); Ventura  _et al._
(2017); Passos  _et al._ (2018); Watanabe and Yanase (2021); Watanabe  _et
al._ (2021a, b). In these methods, it is not easy to consider the effect of
dissipation rigorously, and therefore, we usually use the relaxation time
approximation (RTA) to include the dissipation, or calculate each relaxation
time for various scattering such as the side-jump and skew scatteringNandy and
Sodemann (2019); Du  _et al._ (2019). RTA has the problem that it breaks the
gauge invariance between the velocity gauge and the length gaugeVentura  _et
al._ (2017); Michishita and Peters (2021), and it cannot describe the proper
relaxation when considering the finite input frequencyMichishita and Peters
(2021), while it well describes the relaxation when considering the DC
inputMichishita and Peters (2021).

In the microscopic diagrammatic theory, the relaxation is treated by the
imaginary part of the self-energy of the Green’s function and the vertex
corrections. The former also leads to the broadening of the spectral function,
which appears in the integration over the real frequency. The reason for this
integration path is to avoid the poles of the Fermi distribution function at
Matsubara frequencies, but those contributions are small for nonlinear
responses as will be shown below. Therefore, the poles of the Green’s with
imaginary parts and the Fermi distribution function with complex argument
plays an important role. In the previous studies, while the relaxation of the
non-equilibrium states can be considered through the RTA, the effect of the
broadening of the distribution function cannot be captured.

In this paper, we analyze the effect of the broadening of the spectral
function on the nonlinear transport. We elucidate that the broadening of the
spectral function results in the shift of the Fermi distribution function (DF)
to the imaginary direction and the Matsubara term, which cannot be described
by the SCB treatment or the RDM method. The shift of the Fermi FD to the
imaginary direction gives the novel dissipation-induced geometric terms. For
example, the Christoffel symbol term appears in second-order nonlinear
transport, and this term also gives the multi-band correction to the nonlinear
Drude term. We note that this Christoffel symbol term is completely different
from Ref.Gao  _et al._ (2014) and Ref.Gao and Xiao (2019), in which it appears
under the magnetic field or the nonuniform electric field. We also elucidate
the geometric origin of the non-reciprocal conductivity, which is related to
the Berry curvature generalized to higher derivative and is also a
dissipation-induced geometric term. Moreover, we analytically derive the
geometric term in the Weyl Hamiltonian both for the type-I and type-II cases.
We show that the chemical potential dependence of the nonlinear Hall
conductivity for each case is completely different, and therefore, the
observation can be the detection of the Weyl points and its type. Especially
for the type-I case, we also show that we can estimate the relaxation time in
the material from this observation.

In the following, we derive the shift of the Fermi DF to the imaginary
direction and the broadening terms from the Green function methodsParker  _et
al._ (2019); Michishita and Peters (2021). First, we derive them in the linear
conductivity to illustrate the formulation in section II. We show the shift of
the Fermi DF results in the quantum metric term at the Fermi surface. In
section III, we extend the results in the linear response to the nonlinear
transport, and derive the geometrical terms, such as the Christoffel symbol
term and the generalized Berry curvature term. Then, we numerically calculate
it in a model for transition metal dichalcogenides and show its dissipation-
strength and chemical potential dependence. In section IV, we derive the
analytical results of the geometric terms such as the Berry curvature dipole
term and the Christoffel symbol term for the type-I and type-II Weyl
Hamiltonian. In section V, we summarize our results.

##  II Dissipative geometry in linear conductivity

Before considering nonlinear conductivity, we first analyze dissipation effect
in linear conductivity. Although the methods we use in this paper is not so
effective in linear conductivity, the analysis in linear conductivity is
pedagogical and helps us to understand the results in nonlinear conductivity.

###  II.1 Formulation

In this paper, we include the dissipation effect via the imaginary part of the
single-particle self-energy and calculate the conductivity with the Green
function methods. Throughout this paper, we ignore the momentum and frequency
dependence of the dissipation and suppose the dissipation strength is same for
all bands. This assumption and approximation are justified when we consider
the impurity scattering, which is independent of the momentum transfer under
the Born approximation. It is also justified to ignore the vertex correction
because here we ignore the momentum dependence of the self-energy and satisfy
the Ward-Takahashi equation. Under these approximations and assumptions, the
single-particle Green function has the same eigenstate as the Hermitian part
of the effective Hamiltonian
$\mathcal{H}_{\mathrm{eff}}=\mathcal{H}_{0}+\mathrm{Re}\Sigma^{R}$. We also
set $e=k_{B}=\bar{h}=1$ throughout this paper.

First, we analyze the dissipation effect through the distribution function in
the linear conductivity, and focus on the symmetric part of the linear DC
conductivity
$\sigma^{\alpha\beta}_{DC}=(\sigma^{\alpha;\beta}+\sigma^{\beta;\alpha})/2$
for simplicity. $\alpha(\beta)$ in $\sigma^{\alpha;\beta}$ represents the
output(input) direction. $\sigma^{\alpha\beta}_{DC}$ can be written in the
Green function methods with the band-indices $n,m$ as,

$\displaystyle\sigma^{\alpha\beta}_{\mathrm{DC}}$
$\displaystyle=\sum_{\bm{k}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\mathrm{Re}\sum_{nm}\mathcal{J}^{\alpha}_{nm}G^{R}_{m}\mathcal{J}^{\beta}_{mn}\bigl{(}G^{R}_{n}-G^{A}_{n}\bigr{)}\frac{\partial
f}{\partial\omega}$ (1)

where $\mathcal{J}^{\alpha}=\partial^{\alpha}\mathcal{H}_{\mathrm{eff}}$,
$\partial^{\alpha}=\partial/\partial k^{\alpha}$,
$\mathcal{O}_{nm}=\braket{n}{\mathcal{O}}{m}$,
$\mathcal{O}_{n}=\mathcal{O}_{nn}$, $\ket{n}$ is the eigenstates of
$\mathcal{H}_{\mathrm{eff}}$, and $f(\omega)$ is the Fermi distribution
function. Throughout this paper, we omit writing the momentum dependence of
the function, such as $\mathcal{J},\epsilon_{n},G^{R(A)}_{n}$, and the
frequency dependence of the Green function. In the limit
$|\omega|\rightarrow\infty$, the integrand is proportional to
$1/|\omega|^{3}$, and therefore, the integration
$\int_{-\infty}^{\infty}d\omega/2\pi$ is equivalent to the contour integral
$\oint_{C}d\omega/2\pi$.(along the closed loop $C$ in Fig. 1.) For this
integral, we should consider the poles of the advanced Green function (green
cross marks in Fig. 1.) and the Matsubara frequencies from the Fermi
distribution function (red cross marks in Fig. 1.). Then, Eq. (1) can be
written as,

$\displaystyle\sigma^{\alpha\beta}_{\mathrm{DC}}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha\beta}_{\mathrm{M}}+\sigma^{\alpha\beta}_{\mathrm{G}}$
(2) $\displaystyle\sigma^{\alpha\beta}_{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\mathrm{Re}\sum_{\bm{k}}\frac{i}{2\beta}\sum_{nm}\sum_{\omega_{M}>0}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{mn}+\mathcal{J}^{\beta}_{nm}\mathcal{J}^{\alpha}_{mn})$
(3) $\displaystyle\ \ \ \ \ \ \ \ \ \ \
\times\Bigl{[}\frac{\partial}{\partial\omega}\Bigl{(}G^{R}_{m}(G^{R}_{n}-G^{A}_{n})\Bigr{)}\Bigr{]}_{i\omega_{M}}$
$\displaystyle\sigma^{\alpha;\beta}_{\mathrm{G}}$ $\displaystyle=$
$\displaystyle\mathrm{Re}\sum_{\bm{k}}\sum_{n}\Bigl{[}\mathcal{J}^{\alpha}_{n}\mathcal{J}^{\beta}_{n}\tau$
$\displaystyle\ +i\sum_{m\neq
n}\frac{(\mathcal{Q}^{\alpha\beta}_{D;n,m}\\!+\\!\mathcal{Q}^{\beta\alpha}_{D;n,m})}{2}(\epsilon_{nm}\\!+\\!2i\eta)\Bigr{]}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}+i\eta},$

Figure 1: Path integration
Path integration in the DC limit.

with

$\displaystyle\mathcal{Q}^{\alpha\beta}_{D;n,m}=\frac{\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{mn}}{(\epsilon_{nm}+2i\eta)^{2}},$
(5)

where $\omega_{M}=(2M+1)\pi/\beta$ is the Fermionic Matsubara frequency,
$\beta$ is the inverse of temperature, $\epsilon_{n}$ is the eigenvalue of
$\mathcal{H}_{\mathrm{eff}}$,$\eta=-\mathrm{Im}\Sigma^{R}$, $\tau=1/(2\eta)$,
$\epsilon_{nm}=\epsilon_{n}-\epsilon_{m}$ and
$\mathcal{Q}^{\alpha\beta}_{D;n}\equiv\sum_{m\neq
n}\mathcal{Q}^{\alpha\beta}_{D;n,m}$ is the dissipative quantum geometric
tensor, which correspond to $\braket{\partial^{\alpha}n}{\partial^{\beta}n}$
in the limit of $\eta\rightarrow 0$. We call the first term
$\sigma^{\alpha;\beta}_{M}$ as “ Matsubara term” because it represents the
contribution from the poles of the Fermi distribution functions at the
Fermionic Matsubara frequencies. While the contribution from the poles of the
advanced Green function $\sigma_{G}$ corresponds to the results by the SCB
treatment or the RDM method in $\eta\rightarrow 0$ limit, the Matsubara term
$\sigma_{M}$ is proportional to $\eta$111We can see $\sigma_{M}\propto\eta$
from
$(G^{R}_{n}-G^{A}_{n})_{i\omega_{M}}=2i\eta/((i\omega_{M}-\epsilon_{n})^{2}+4\eta^{2})$.
We can also see $\sigma_{M}\propto\eta$ in figure 9. and therefore cannot be
condsidered in the SCB treatment or the RDM method. Next, we analyze the
second term in Eq. (LABEL:polesA). We can understand the dissipative geometric
term as the multi-band correction to the Drude term, which reads,

$\displaystyle\tilde{\sigma}^{\alpha\beta}_{\mathrm{Drude}}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha\beta}_{\mathrm{Drude}}+\sigma^{\alpha\beta}_{\mathrm{QM:re}}$
(6) $\displaystyle\sigma^{\alpha\beta}_{\mathrm{QM:re}}$ $\displaystyle\sim$
$\displaystyle\sum_{\bm{k}}\sum_{n}\frac{g^{\alpha\beta}_{S;n}}{\tau}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}$ (7) $\displaystyle
g^{\alpha\beta}_{S;n}$ $\displaystyle=$
$\displaystyle\sum_{m}g^{\alpha\beta}_{S;n,m},\ \
g^{\alpha\beta}_{S;n,m}=\frac{(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{mn}+\mathcal{J}^{\beta}_{nm}\mathcal{J}^{\alpha}_{mn})}{2(\epsilon^{2}_{nm}+4\eta^{2})}.$

The detail derivation is written in appendix A. We can derive
$\sigma^{\alpha\beta}_{\mathrm{QM:re}}$ from the dissipative geometric term.
We call $g^{\alpha\beta}_{S;n}$ as the “smeared quantum metric,” which becomes
the quantum metric in $\eta\rightarrow 0$ limit, and
$\sigma^{\alpha\beta}_{\mathrm{QM:re}}$ as the “(smeared) quantum metric
term”. At the band degeneracy where $\epsilon_{nm}=0$, the smeared quantum
metric is proportional to $\tau^{2}$, and therefore, the quantum metric term
becomes proportional to $\tau$. This means the quantum metric term describes
the multi-band correction to the Drude term which is proportional to $\tau$.
We note that we can also derive this correction from the reduced density
matrix methods under the RTA, by changing $f(\epsilon_{n})\rightarrow
f(\epsilon_{n}+i\eta)$ and approximating
$\mathrm{Im}f(\epsilon_{n}+i\eta)\simeq\eta(\partial
f/\partial\omega)_{\epsilon_{n}}$. Therefore, we can consider the quantum
metric term stems from the imaginary part of the Fermi DF. We also numerically
check how large these terms and show that, the quantum metric term are
dominant when the band degeneracy exists at the Fermi surface in appendix C.
We also show in appendix C that the Matsubara term is also finite, and
therefore, the treatment in this section is not so effective in linear
response cases. On the other hand, as we will show in the next section, the
Matsubara term is small and the description by the shift of the Fermi DF to
the imaginary direction works well in the nonlinear responses.

##  III Dissipation-induced geometry in nonlinear response

###  III.1 formulation

Next, we consider the effect of the broadening of DF on second-order nonlinear
DC conductivity by the same procedure as the linear case. In nonlinear
conductivity, a lot of terms emerge from the dissipation effect and the
formula becomes so complicated. Therefore, we write the detail derivation in
appendix D, and here write down the final and summarized results, which read,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{DC}}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{M}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{Drude}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC}}+\mathcal{O}(\tau^{-2}).$

$\sigma^{\alpha;\beta\gamma}_{\mathrm{M}}$,
$\sigma^{\alpha;\beta\gamma}_{\mathrm{Drude}}$,
$\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD}}$,
$\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS}}$, and
$\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC}}$ represent respectively the
Matsubara term in nonlinear conductivity, the nonlinear Drude term, the Berry
curvature dipole (BCD) term, the Christoffel symbol term, and the generalized
Berry curvature term. Here, we newly elucidate last two terms,
$\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS}}$ and
$\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC}}$, by considering the effect of
dissipation. Under $\mathcal{T}$-symmetry, $\sigma_{\mathrm{Drude}}$ and
$\sigma_{\mathrm{ChS}}$ must be zero, while under $\mathcal{PT}$-symmetry,
$\sigma_{\mathrm{BCD}}$ and $\sigma_{\mathrm{gBC}}$ must be zero. We summarize
$\tau$-dependence of the dominant contribution of each term in Table 1. Below,
we describe the detailed analysis of each term.

Table 1: Symmetry classification and $\tau$-dependence term |  $\mathcal{T}$ |  $\mathcal{PT}$ |  $\tau$-dependence
---|---|---|---
$\sigma_{\mathrm{M}}$ |  - |  - | $\mathcal{O}(\tau^{-1})$
$\sigma_{\mathrm{Drude}}$ |  $\times$ |  ✓ | $\mathcal{O}(\tau^{2})$
$\sigma_{\mathrm{BCD}}$ |  ✓ |  $\times$ | $\mathcal{O}(\tau)$
$\sigma_{\mathrm{ChS}}$ |  $\times$ |  ✓ | $\mathcal{O}(\tau^{0})\ (\mathcal{O}(\tau^{2}))$
$\sigma_{\mathrm{gBC}}$ |  ✓ |  $\times$ | $\mathcal{O}(\tau^{-1})$

###  III.2 analysis of each term

In this subsection, we analyze each term in Eq. (LABEL:NLR). Each term, except
for the Matsubara term, is so complex, and therefore, we write the simple
forms under some approximations, which are
$\mathrm{Im}f(\epsilon_{n}+i\eta)\simeq\eta(\partial
f/\partial\omega)_{\epsilon_{n}}$ and $\epsilon^{2}_{nm}\tau^{2}\gg 1$. We
write the detail derivation of each term and the full terms without
approximation in appendix D. We also note that the approximation
$\mathrm{Im}f(\epsilon_{n}+i\eta)\simeq\eta(\partial
f/\partial\omega)_{\epsilon_{n}}$ can be justified even when $\beta\eta\sim
0.5$.(see appendix A.)

####  III.2.1 Matsubara term in nonlinear conductivity

The Matsubara term in nonlinear coductivity is the contribution from the poles
at Fermionic Matsubara frequencies as in the case of linear conductivity:

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\sum_{nml}\sum_{\omega_{M}>0}\mathrm{Re}\Bigl{[}\frac{\partial}{\partial\omega}\Bigl{\\{}2\mathcal{J}^{\alpha}_{nm}\frac{\partial
G^{R}_{m}}{\partial\omega}\mathcal{J}^{\beta}_{ml}G^{R}_{l}\mathcal{J}^{\gamma}_{ln}(G^{R}_{n}\\!-\\!G^{A}_{n})$
(10) $\displaystyle+\mathcal{J}^{\alpha}_{nm}\frac{\partial
G^{R}_{m}}{\partial\omega}\mathcal{J}^{\beta\gamma}_{mn}(G^{R}_{n}\\!-\\!G^{A}_{n})\Bigr{\\}}+\Bigl{\\{}\beta\leftrightarrow\gamma\Bigr{\\}}\Bigr{]}_{\omega=i\omega_{M}},$

where
$\mathcal{J}^{\beta\gamma}=\partial^{\beta\gamma}\mathcal{H}_{\mathrm{eff}}$
and $\partial^{\beta\gamma}=\partial^{\beta}\partial^{\gamma}$. As we will
numerically show, in the second-order conductivity under the time reversal
symmetry and the condition $\pi k_{B}T>\eta$, the Matsubara term is small
enough to be ignored, compared with the other finite terms, and therefore, the
description by the Fermi DF of complex argument works well.

####  III.2.2 nonlinear Drude term

The nonlinear Drude term can be written as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{Drude}}$
$\displaystyle\simeq$ $\displaystyle
2\sum_{\bm{k}}\sum_{n}\tau^{2}\mathcal{J}^{\alpha}_{n}\partial^{\beta}\partial^{\gamma}f(\epsilon_{n}).$
(11)

Because the nonlinear Drude term is proportional to $\tau^{2}$, this term is
most dominant in clean metals without TRS. We note that, if there is the band
degeneracy at the Fermi surface, the Christoffel symbol term can be also
dominant as we will show later.

####  III.2.3 Berry curvature dipole term

The BCD termSipe and Shkrebtii (2000); Sodemann and Fu (2015) can be written
as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD}}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD:re}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD:im}}$
(12) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD:re}}$
$\displaystyle\simeq$ $\displaystyle
2\tau\sum_{\bm{k}}\sum_{nm}\partial^{\gamma}(\Omega^{\alpha\beta}_{S;n,m})f(\epsilon_{n})$
(13) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD:im}}$
$\displaystyle\simeq$
$\displaystyle\sum_{\bm{k}}\sum_{nm}\frac{\Omega^{\alpha\beta}_{S;n}\mathcal{J}^{\gamma}_{n}}{\epsilon_{nm}\tau}\Bigl{(}\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}_{\epsilon_{n}}$
(14) $\displaystyle\Omega^{\alpha\beta}_{S;n,m}$ $\displaystyle=$
$\displaystyle\frac{-i(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{mn}-\mathcal{J}^{\beta}_{nm}\mathcal{J}^{\alpha}_{mn})}{\epsilon^{2}_{nm}+4\eta^{2}}$
(15)

Under the TRS, the nonlinear Drude term must be zero and the BCD term is
dominant. Because $\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD:im}}$, which stems
from the imaginary part of the Fermi DF, is proportional to $\eta$, it is not
so large in clean systems. Here we can describe the BCD term by the smeared
Berry curvature $\Omega^{\alpha\beta}_{S;n}$, which secures the convergence of
the BCD term at the band-crossing points.

####  III.2.4 Christoffel symbol term

The Christoffel symbol term can be described as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS}}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:I}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:II}}$
(16) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:I}}$
$\displaystyle=$ $\displaystyle
2\sum_{\bm{k}}\sum_{n}\Gamma^{\alpha;\beta\gamma}_{S;n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}$ (17)
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:II}}$
$\displaystyle\simeq$ $\displaystyle
2\sum_{\bm{k}}\sum_{n}\Gamma^{\alpha;\beta\gamma}_{S^{\prime};n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}$ (18)

where $\Gamma^{\alpha;\beta\gamma}_{S^{(^{\prime})};n}$ is the smeared
Christoffel symbol of the first kindGao  _et al._ (2014); Ahn  _et al._
(2020). Starting from the conventional Christoffel symbol
$\Gamma^{\alpha;\beta\gamma}_{n}$, which reads,

$\displaystyle\Gamma^{\alpha;\beta\gamma}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{(}\partial^{\gamma}g^{\alpha\beta}_{n}+\partial^{\beta}g^{\gamma\alpha}_{n}-\partial^{\alpha}g^{\beta\gamma}_{n}\Bigr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{(}\braket{\partial^{\alpha}n}{\partial^{\beta\gamma}n}+\braket{\partial^{\beta\gamma}n}{\partial^{\alpha}n}\Bigr{)}$
$\displaystyle=$
$\displaystyle\sum_{m(\neq)n}\mathrm{Re}\Bigl{[}\frac{\mathcal{J}^{\alpha}_{nm}}{\epsilon^{2}_{nm}}\Bigl{(}\sum_{l(\neq
n)}\frac{\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln}+(\beta\\!\leftrightarrow\\!\gamma)}{\epsilon_{nl}}\\!+\\!\mathcal{J}^{\beta\gamma}_{mn}\Bigr{)}\Bigr{]},$

we define $\Gamma^{\alpha;\beta\gamma}_{S;n}$ by substituing
$g^{\alpha\beta}_{n}\rightarrow g^{\alpha\beta}_{S;n}$, and
$\Gamma^{\alpha;\beta\gamma}_{S^{\prime};n}$ by substituting
$1/\epsilon_{nl}\rightarrow\epsilon_{nl}/(\epsilon^{2}_{nl}+4\eta^{2})$ and
$1/\epsilon^{2}_{nm}\rightarrow 1/(\epsilon^{2}_{nm}+4\eta^{2})$.
$\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:I}}$ stems from the imaginary part
of the Fermi FD in the term which is originally the BCD terms and the
nonlinear Drude term, while $\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:II}}$
stems from the full interband contribution. Interestingly, in this regime,
even though $\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:I}}$ stems from the
imaginary part of the Fermi FD and the dissipation, it seems not to depend the
dissipation strength. On the other hand, when we consider the band degeneracy
at the Fermi surface $\epsilon_{n}=\epsilon_{m}\simeq 0$, the Christoffel
symbol term is proportional to $\tau^{2}$ because $g^{\alpha\beta}_{S;n}$ or
$1/(\epsilon^{2}_{nm}+4\eta^{2})$ is proportional to $\tau^{2}$, and
therefore, the Christoffel symbol term also gives the multi-band correction to
the nonlinear Drude term. We note that the difference between
$\Gamma^{\alpha;\beta\gamma}_{S;n}$ and
$\Gamma^{\alpha;\beta\gamma}_{S^{\prime};n}$ appears when focusing on the
band-degeneracy of the two-band systems with linear dispersion. In that case,
$\Gamma^{\alpha;\beta\gamma}_{S;n}$ is finite and gives the correction to the
nonlinear Drude term while $\Gamma^{\alpha;\beta\gamma}_{S^{\prime};n}$ is
zero. We also note that this Christoffel symbol term is different from the one
under magnetic field derived by the SCB treatment.Gao  _et al._ (2014)

####  III.2.5 generalized Berry curvature term

The generalized Berry curvature term can be written as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC}}=\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:re}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:im}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:add}}$
(21) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:re}}$
$\displaystyle\simeq$ $\displaystyle\sum_{\bm{k}}\sum_{n,m(\neq
n)}\frac{\Omega^{\alpha,\beta\gamma}_{S^{\prime};n,m}}{\epsilon_{nm}\tau}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}$ (22)
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:add}}$
$\displaystyle\simeq$ $\displaystyle 2\sum_{\bm{k}}\sum_{n,m,l(\neq
n)}\Bigl{\\{}\frac{\mathrm{Im}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln})}{\epsilon^{2}_{nm}\epsilon_{nl}}\times$
(23) $\displaystyle\
\Bigl{(}\frac{1}{\epsilon_{nm}\tau}\\!-\\!\frac{1}{\epsilon_{nl}\tau}\Bigr{)}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}+(\beta\\!\leftrightarrow\\!\gamma)\Bigr{\\}},$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:im}}$
$\displaystyle\simeq$
$\displaystyle-\sum_{\bm{k}}\sum_{n}\frac{\Omega^{\alpha,\beta\gamma}_{S^{\prime};n}}{\tau}\Bigl{(}-\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}_{\epsilon_{n}},$
(24)

where $\Omega^{\alpha,\beta\gamma}_{S;n}$ is the smeared Berry curvature
generalized to the second-order derivative and we define
$\Omega^{\alpha;\beta\gamma}_{S^{\prime};n,m}$ as derived from
$\Omega^{\alpha;\beta\gamma}_{n,m}$, which reads,

$\displaystyle\Omega^{\alpha,\beta\gamma}_{n,m}$ $\displaystyle=$
$\displaystyle
2\mathrm{Im}\Bigl{[}\braket{\partial^{\alpha}n}{m}\braket{m}{\partial^{\beta\gamma}n}\Bigr{]}$
$\displaystyle=$
$\displaystyle\sum_{m(\neq)n}\mathrm{Im}\Bigl{[}\frac{\mathcal{J}^{\alpha}_{nm}}{\epsilon^{2}_{nm}}\Bigl{(}\sum_{l(\neq
n)}\frac{\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln}+(\beta\\!\leftrightarrow\\!\gamma)}{\epsilon_{nl}}\\!+\\!\mathcal{J}^{\beta\gamma}_{mn}\Bigr{)}\Bigr{]},$

by substituting
$1/\epsilon_{nl}\rightarrow\epsilon_{nl}/(\epsilon^{2}_{nl}+4\eta^{2})$ and
$1/\epsilon^{2}_{nm}\rightarrow 1/(\epsilon^{2}_{nm}+4\eta^{2})$.
$\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:add}}$ is zero when considering the
two-band model and therefore it represent the more than two-band correction to
$\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:re}}$. When we consider the
nonreciprocal transport $\alpha=\beta=\gamma$ under the TRS, only this
generalized Berry curvature term can be finite. In that case, the
nonreciprocal conductivity is proportional to $\eta$ and the dissipation is
essential for the nonreciprocal conductivity as pointed out in Ref.Morimoto
and Nagaosa (2018). Moreover, we find that, when we focus on two-band models,
higher order terms in momentum, such as $k^{2}$ term, in the Hamiltonian is
also necessary so that
$J^{\alpha\alpha}_{mn}=\bra{m}(\partial^{\alpha\alpha}\mathcal{H}_{eff})\ket{m}$
is nonzero, for the diagonal part of the finite generalized Berry curvature
with $\alpha=\beta=\gamma$. It is because, $l$ is $m$ in Eq. (LABEL:gBC_f) for
the two-band model, and hence the diagonal part of the generalized Berry
curvature vanishes as
$\Omega^{\alpha;\alpha\alpha}_{n,m}=2\Omega^{\alpha\alpha}_{n,m}\mathcal{J}^{\alpha}/\epsilon_{nm}=0$.222One
can check
$\Omega^{\alpha,\beta\gamma}_{n,m}=(\Omega^{\alpha\beta}_{n,m}\mathcal{J}^{\gamma}_{m}+(\beta\leftrightarrow\gamma))/\epsilon_{nm}$
by substituting $l=m$ in Eq. (LABEL:gBC_f) This means that the simple linear
Weyl Hamiltonian cannot generate the nonreciprocal transport under the TRS.
(see the detail in appendix D.)

###  III.3 Model calculation

Now we estimate how large the dissipation-induced geometric terms are in the
model calculations. We use the model which effectively describes 2D
transition-metal dichalcogenides with uniaxial strain, such as MX2 (M = Mo, W
and X = S, Te)Wilson and Yoffe (1969); Liu  _et al._ (2015); Kanasugi and
Yanase (2020), which reads,

$\displaystyle\mathcal{H}_{eff}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k},s,s^{\prime}}\Bigl{(}\left(\epsilon(\bm{k})\\!-\\!\mu\right)\sigma^{0}+(\bm{h}\\!+\\!\bm{g}(\bm{k}))\cdot\bm{\sigma}\Bigr{)}_{ss^{\prime}}c_{\bm{k},s}^{{\dagger}}c_{\bm{k},s^{\prime}}$
(27) $\displaystyle\epsilon(\bm{k})$ $\displaystyle=$ $\displaystyle
2t\Bigl{(}(1-p)\cos(\bm{k}\cdot\bm{a_{1}})$ (28) $\displaystyle\ \ \ \ \ \ \
+\\!\cos(\bm{k}\cdot\bm{a_{2}})\\!+\\!\cos(\bm{k}\cdot(\bm{a_{1}}\\!+\\!\bm{a_{2}}))\Bigr{)}$
$\displaystyle g^{x}(\bm{k})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}}{2}\Bigl{[}\sin(\bm{k}\cdot(\bm{a_{1}}+\bm{a_{2}}))+\sin(\bm{k}\cdot\bm{a_{2}})\Bigr{]}$
(29) $\displaystyle g^{y}(\bm{k})$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{1}}{\sqrt{3}}\Bigl{[}\sin(\bm{k}\cdot\bm{a_{1}})\\!+\\!\frac{\sin(\bm{k}\cdot(\bm{a_{1}}\\!+\\!\bm{a_{2}}))\\!-\\!\sin(\bm{k}\cdot\bm{a_{2}})}{2}\Bigr{]}$
$\displaystyle g^{z}(\bm{k})$ $\displaystyle=$
$\displaystyle\frac{2\alpha_{2}}{3\sqrt{3}}\Bigl{[}\sin(\bm{k}\cdot\bm{a_{1}})\\!+\\!\sin(\bm{k}\cdot\bm{a_{2}})\\!-\\!\sin(\bm{k}\cdot\bm{a_{1}}\\!+\\!\bm{a_{2}})\Bigr{]},$

where $\mu$ is the chemical potential, $\bm{h}$ is the magnetic filed, $t$ is
the hopping, $\bm{a_{1}}=(1,0)$, $\bm{a_{2}}=(-0.5,\sqrt{3}/2)$, $p$
represents the effect of the uniaxial strain, and $\alpha_{1(2)}$ is the spin-
orbit coupling. When $\bm{h}=0$, this model holds the time-reversal symmetry,
while mirror symmetry of $y$-direction is broken due to finite $p$. In this
section, we set the parameters as
$t=0.5,p=0.3,\alpha_{1}=0.08,\alpha_{2}=0.06$ in the numerical calculation for
figure 2, figure 3, 4, and (5. Figure 2 shows the energy dispersion of the
model when $\mu=0$. It has the band degeneracy $M$-, $M^{\prime}$-, and
$\Gamma$\- point and their energy levels are $\epsilon=-0.7$, $\epsilon=-1.3$,
$\epsilon=2.7$. The density of states is large near $\epsilon=-1.3$
corresponding to the nearly flat dispersion along $K^{\prime}$-$M^{\prime}$.

Figure 2: Dispersion of the model on the high symmetric line.
we set $\mu=0$. The band degeneracy exists at $\epsilon=-0.7$,
$\epsilon=-1.3$, and $\epsilon=2.7$ at $M$-, $M^{\prime}$ and $\Gamma$-point.

####  III.3.1 Cases with the time-reversal symmetry

Figure 3: $\eta$-dependence and $\mu$-dependence of each contribution in
nonlinear Hall conductivity.
The top panel shows the dissipation-strength dependence of the nonlinear Hall
conductivity $\sigma^{y;xx}$, and the bottom panel shows that the chemical
potential dependence of $\sigma^{y;xx}$. We set $\mu=-0.9$ in the top panel
and $\eta=0.02$ in the bottom panel. We perform the momentum integration by
$1000\times 1000$ and frequency integration by $1000$. The blue, orange,
green, red, and brown plots respectively represent the generalized Berry
curvature term, $\sigma_{\mathrm{BCD:re}}$, $\sigma_{\mathrm{BCD:im}}$, the
Matsubara term, and the results by the Green function methods which coincides
with the sum of all the terms.

In this subsection, we calculate the nonlinear Hall conductivity
$\sigma^{y;xx}$ and the non-reciprocal conductivity $\sigma^{y;yy}$ in the
model introduced above. We note that the model holds the mirror symmetry in
$x$-direction and therefore $\sigma^{x;yy}=\sigma^{y;yx}=\sigma^{x;xx}=0$. In
Figure. 3, we calculate $\eta$-dependence, and $\mu$-dependence of the
nonlinear Hall conductivity $\sigma^{y;xx}$. Under the time-reversal symmetry,
the dominant contribution are the BCD term in Eq. (12)) and the generalized
Berry curvature (gBC) term in Eq. (21). In the top panel of figure.3, as we
have shown theoretically, $\sigma_{\mathrm{BCD:re}}$ is proportional to
$1/\eta$, while $\sigma_{\mathrm{BCD:im}}$ and $\sigma_{\mathrm{gBC}}$ are
proportional to $\eta$ in the regime $\eta<T$. In the limit
$\eta\gg\epsilon_{nm}$, $\sigma_{\mathrm{BCD:im}}$ and $\sigma_{\mathrm{gBC}}$
are proportional to $1/\eta$, and therefore they both decrease in $\eta>0.04$.
At $\eta=\pi T$, due to the singular behavior of $\mathrm{Im}(\partial
f/\partial\omega)$, $\sigma_{\mathrm{BCD:im}}$ change its sign and the
Matsubara term become large as to compensate it. The bottom panel of figure.3
shows $\mu$-dependence. Around $\mu=-1.3$, there is quadratic dispersion at
the Fermi surface around at $M$\- and $M^{\prime}$-point, and the gBC term
becomes large.

Figure 4: $\eta$-dependence and $\mu$-dependence of each contribution in
nonreciprocal conductivity.
The top panel shows the dissipation-strength dependence of the non-reciprocal
conductivity $\sigma^{y;yy}$, and the bottom panel shows that the chemical
potential dependence of $\sigma^{y;yy}$. We set $\mu=-0.9$ in the top panel
and $\eta=0.02$ in the bottom panel. The blue, red, and orange plots
respectively represent the generalized Berry curvature term, the Matsubara
term, and the results by the Green function methods. We note that blue and
orange plots overlap in the upper panel.

Next, we calculate the non-reciprocal conductivity. As I have shown in the
previous section, under the time-reversal symmetry, only the gBC term is
finite. The top panel of figure.4 shows the $\eta$-dependence of the non-
reciprocal conductivity $\sigma^{y;yy}$. As in the case of the nonlinear Hall
conductivity, the gBC term is proportional to $\eta$ for $\eta<T$ and
proportional to $1/\eta$ for $\eta\gg\epsilon_{nm}$.

In the bottom panel of figure.4, the non-reciprocal conductivity (gBC term)
behaves as same as the gBC term in the nonlinear Hall conductivity. (see the
bottome panel of figure.3)

####  III.3.2 Cases without the time-reversal symmetry

Figure 5: $\eta$-dependence and $\mu$-dependence of each contribution in
nonreciprocal conductivity.
The top panel shows the dissipation-strength dependence of the non-reciprocal
conductivity $\sigma^{y;yy}$, and the bottom panel shows that the chemical
potential dependence of $\sigma^{y;yy}$. We set $\mu=-0.9$ in the top panel
and $\eta=0.02$ in the bottom panel. The blue, orange, red, and green plots
respectively represent the Drude term, the Christoffel symbol term, the
Matsubara term, and the results by the Green function methods.

Next, we consider the case $\bm{h}\neq 0$, in which time-reversal symmetry is
broken. Without the time-reversal symmetry, the Drude term, the Christoffel
symbol term can be finite. Here, we focus on the non-reciprocal conductivity
with the magnetic field in $x$-direction $h_{x}=0.05$ for figure 5.

In time-reversal symmetry broken systems, the (nonlinear) Drude term, which is
proportional to $\tau^{2}$, is dominant in the small dissipation regime. (See
the top panel of figure 5) The Christoffel symbol term seems also proportional
to $\tau^{2}$ and gives the multi-band correction to the Drude term, because
there is band degeneracy around at the Fermi surface.

The bottom panel of figure 5 shows the $\mu$-dependence of the non-reciprocal
conductivity, In the regime $\mu>-0.2$, the band velocity is large at the
Fermi surface, and therefore the Drude term is dominant. On the other hand, in
the regime $-0.85<\mu<-0.5$, the band velocity is not so large and there is
the degeneration near the Fermi surface at $M^{\prime}$-point, and therefore
the Christoffel symbol term is dominant.

##  IV nonlinear conductivity induced by dissipative quantum geometry in the
Weyl Hamiltonian

In this section, we derives the analytical results about the geometric terms
derived in the previous sections.

###  IV.1 Hamiltonian and its dissipative quantum geometry

Here, we consider the tilted Weyl Hamiltonian, which reads,

$\displaystyle\mathcal{H}(\bm{k})$ $\displaystyle=$
$\displaystyle(-\mu+\bm{t}\cdot\bm{k})\sigma^{0}+t_{0}\bm{k}\cdot\bm{\sigma},$
(32)

where $\sigma^{0}$ is the two by two unit matrix, $\mu$ is the chemical
potential, $t$ represents the tilting, $t_{0}$ is the Fermi velocity,
$\bm{\sigma}=\\{\sigma^{x},\sigma^{y},\sigma^{z}\\}$ is the Pauli matrix. When
$t<t_{0}$, this Hamiltonian describes the type-I Weyl Hamiltonian ,and when
$t>t_{0}$, it describes the type-II Weyl Hamiltonian. Here we set
$\bm{t}=\\{0,0,t\\}$ for the simplicity.

In this Hamiltonian, the eigenvalues and the smeared geometrical quantities
are,

$\displaystyle E_{\pm}$ $\displaystyle=$ $\displaystyle tk_{z}\pm
t_{0}k=t_{0}k(\lambda_{t}\cos\theta\pm 1)$ (33)
$\displaystyle\mathcal{J}^{\alpha}_{\pm}$ $\displaystyle=$ $\displaystyle
t_{0}k^{\alpha}(\delta_{\alpha z}\lambda_{t}\pm 1)$ (34)
$\displaystyle\Omega^{\alpha\beta}_{S;+-}$ $\displaystyle=$
$\displaystyle-\Omega^{\alpha\beta}_{S;-+}=-\Omega^{\beta\alpha}_{S;+-}=\frac{k_{\gamma}\epsilon_{\alpha\beta\gamma}}{8(k^{2}+\tilde{\eta}^{2})k}$
(35) $\displaystyle g^{\alpha\beta}_{S;+-}$ $\displaystyle=$ $\displaystyle
g^{\alpha\beta}_{S;-+}=g^{\beta\alpha}_{S;+-}=\frac{\delta_{\alpha\beta}(k^{2}-k^{2}_{\alpha})}{4(k^{2}+\tilde{\eta}^{2})k^{2}}$
(36)

where $k=|k|$, $\lambda_{t}=t/t_{0}$, $\cos\theta=k_{z}/k$, and
$\tilde{\eta}=\eta/t_{0}$ is the dissipation strength renormalized by the
Fermi velocity. We also define the geometric quantities, which appear in the
nonlinear conductivity, as follows:

$\displaystyle D^{\mu\nu}_{\pm}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\epsilon^{\mu\lambda\eta}\Omega^{\lambda\eta}_{S;\pm}\mathcal{J}^{\nu}_{\pm}=\pm\frac{k_{\mu}(\delta_{z\nu}\lambda_{t}\pm
k^{\nu}/k)}{8(k^{2}+\tilde{\eta}^{2})k}$ (37) $\displaystyle F^{\mu\nu}_{\pm}$
$\displaystyle\equiv$ $\displaystyle\partial^{\mu}g^{\nu\nu}_{S;\pm}$
$\displaystyle=$
$\displaystyle-\frac{k^{\mu}}{2(k^{2}+\tilde{\eta}^{2})^{2}k^{2}}\Bigl{\\{}(1-\frac{k^{2}_{\nu}}{k^{2}})(2k^{2}\\!+\\!\tilde{\eta}^{2})+\delta_{\mu\nu}(k^{2}\\!+\\!\tilde{\eta}^{2})\Bigr{\\}}$
$\displaystyle\Gamma^{\mu;\nu\eta}_{S;\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\delta_{\eta\mu}F^{\nu\eta}_{\pm}+\delta_{\mu\nu}F^{\eta\mu}_{\pm}-\delta_{\nu\eta}F^{\mu\nu}_{\pm})$
(39) $\displaystyle\Omega^{\alpha,\beta\gamma}_{S;\pm}$ $\displaystyle=$
$\displaystyle\frac{\pm
k(\Omega^{\alpha\beta}_{S;\pm}\mathcal{J}^{\gamma}_{\pm}+\Omega^{\alpha\gamma}_{S;\pm}\mathcal{J}^{\beta}_{\pm})}{2(k^{2}+\tilde{\eta}^{2})}$
(40)

In the following, we suppose that $\beta\eta<1$ and the broadening term can be
ignored. We also approximate $\mathrm{Re}(-\partial
f/\partial\omega)_{\epsilon_{n}+i\eta}\simeq(-\partial
f/\partial\omega)_{\epsilon_{n}}\simeq\delta(\epsilon_{n})$ and
$\mathrm{Im}f(\epsilon_{n}+i\eta)\simeq\eta(\partial
f/\partial\omega)_{\epsilon_{n}}$, and then, analyze the geometric terms in
the type-I case ($0<\lambda_{t}<1$) and the type-II case ($\lambda_{t}>1$).

###  IV.2 Type-I case

We here consider the case where $\lambda_{t}<1$. We can analytically calculate
the BCD term and the Christffel symbol term in the limit $\eta\ll|\mu|$ and
$\eta\gg|\mu|$ as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD;I}}$
$\displaystyle\simeq\tau\int\frac{\bm{dk}}{(2\pi)^{3}}\Bigl{\\{}\Omega^{\alpha\beta}_{S;\pm}\mathcal{J}^{\gamma}_{\pm}+(\beta\leftrightarrow\gamma)\Bigr{\\}}\delta(\epsilon_{\pm})$
(41) $\displaystyle=\tau\int\frac{\bm{dk}}{(2\pi)^{3}}(\delta_{\gamma
z}-\delta_{\beta z})\frac{1}{2}\epsilon_{\alpha\beta\gamma}(D^{zz}_{\pm})$
(42) $\displaystyle=\frac{\tau}{2}\epsilon_{\alpha\beta\gamma}(\delta_{\gamma
z}-\delta_{\beta z})\times$ (43) $\displaystyle\ \
\begin{dcases}\frac{1}{8\lambda_{t}}\Bigl{\\{}2\lambda_{t}\\!-\\!\mathrm{ln}\Bigl{(}\frac{1\\!+\\!\lambda_{t}}{1\\!-\\!\lambda_{t}}\Bigr{)}\Bigr{\\}}\
\ \ (\eta\ll|\mu|),\\\
-\frac{1}{8\lambda_{t}^{3}}\frac{\mu^{2}}{\eta^{2}}\Bigl{\\{}2\lambda_{t}\\!-\\!\tanh^{-1}\Bigl{(}\frac{2\lambda_{t}}{1\\!+\\!\lambda_{t}^{2}}\Bigr{)}\Bigr{\\}}\
(\eta\gg|\mu|),\end{dcases}$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS;I}}$
$\displaystyle\simeq$
$\displaystyle\int\frac{\bm{dk}}{(2\pi)^{3}}\Gamma^{\alpha;\beta\gamma}_{S;\pm}\delta(\epsilon_{\pm})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{(}\delta_{\alpha\beta}\delta_{\gamma
z}\tilde{F}^{\gamma\alpha}_{\mathrm{I}}\\!+\\!\delta_{\gamma\alpha}\delta_{\beta
z}\tilde{F}^{\beta\gamma}_{\mathrm{I}}\\!-\\!\delta_{\beta\gamma}\delta_{\alpha
z}\tilde{F}^{\alpha\beta}_{\mathrm{I}}\Bigr{)}$

with

$\displaystyle\tilde{F}^{zz}_{\mathrm{I}}$
$\displaystyle=\begin{dcases}\frac{\mu}{t_{0}\lambda_{t}}\Bigl{\\{}-\frac{6}{\lambda_{t}^{2}}+\frac{3(2-\lambda_{t}^{2})}{2\lambda_{t}^{3}}\mathrm{ln}\Bigl{(}\frac{1+\lambda_{t}}{1-\lambda_{t}}\Bigr{)}+\frac{1}{1-\lambda_{t}^{2}}\Bigr{\\}}\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ (\eta\ll|\mu|),\\\
\frac{(3-\lambda_{t}^{2})\mu}{t_{0}\lambda_{t}^{3}}\Bigl{\\{}\frac{1}{2\lambda_{t}}\tanh^{-1}\Bigl{(}\frac{2\lambda_{t}}{1+\lambda_{t}^{2}}\Bigr{)}-\frac{1}{1-\lambda_{t}^{2}}\Bigr{\\}}\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ (\eta\gg|\mu|),\end{dcases}$ (46)
$\displaystyle\tilde{F}^{zx}_{\mathrm{I}}=\tilde{F}^{zy}_{\mathrm{I}}=\tilde{F}^{xz}_{\mathrm{I}}=\tilde{F}^{yz}_{\mathrm{I}}$
$\displaystyle=\begin{dcases}\frac{\mu}{t_{0}\lambda_{t}}\Bigl{\\{}-\frac{6}{\lambda_{t}^{2}}+\frac{3(2-\lambda_{t}^{2})}{2\lambda_{t}^{3}}\mathrm{ln}\Bigl{(}\frac{1+\lambda_{t}}{1-\lambda_{t}}\Bigr{)}+\frac{1}{1-\lambda_{t}^{2}}\Bigr{\\}}\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ (\eta\ll|\mu|),\\\
\frac{(3-\lambda_{t}^{2})\mu}{t_{0}\lambda_{t}^{3}}\Bigl{\\{}\frac{1}{2\lambda_{t}}\tanh^{-1}\Bigl{(}\frac{2\lambda_{t}}{1+\lambda_{t}^{2}}\Bigr{)}-\frac{1}{1-\lambda_{t}^{2}}\Bigr{\\}}\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ (\eta\gg|\mu|),\end{dcases}$ (47)

Here we omit the analysis of the gBC term, because here we approximate
$\mathrm{Re}(-\partial
f/\partial\omega)_{\epsilon_{n}+i\eta}\simeq\delta(\epsilon)$. This
approximation is justified when $T\rightarrow 0$ and $\eta\rightarrow 0$ with
$k_{B}T\gg\eta$, while the gBC term is proportional to $\eta$. Interestingly,
the nonlinear Hall conductivity by the BCD term is independent of the chemical
potential in the regime $|\mu|\gg\eta$, while it is proportional to
$\mu^{2}/\eta^{2}$ in the regime $|\mu|\ll\eta$, which results in the dip
around $|\mu|<\eta$. The Christoffel symbol term is proportional to $\mu$ in
both regime. We note that, although Eq. (43) and Eq. (46) appears to diverge
at $\lambda_{t}\rightarrow 0$, they becomes zero by appropriately expanding
the terms in $\\{\dots\\}$ in Eq. (43) and Eq. (46).

###  IV.3 Type-II case

Next, we analyze the type-II Weyl Hamiltonian where $\lambda_{t}>1$. Although
the only difference in the model from type-I is the magnitude of
$\lambda_{t}$, the behavior is completely different. Because the integral of
momentum space $\int_{0}^{\infty}dkk^{2}$ is not convergent. Therefore, we
introduce the cut-off scale $\Lambda$
($\int_{0}^{\infty}dk\rightarrow\int_{0}^{\Lambda}dk$), in which the
approximation to the Weyl Hamiltonian is justified. Then, we can derive the
BCD term and the Christoffel symbol term as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{BCD;II}}$ $\displaystyle=$
$\displaystyle-\frac{\tau\epsilon_{\alpha\beta\gamma}(\delta_{\gamma
z}-\delta_{\beta
z})}{32\pi^{2}\lambda_{t}}\mathrm{ln}\Bigl{(}\frac{\Lambda^{2}+\eta^{2}}{(\mu/(1+\lambda_{t}))^{2}+\eta^{2}}\Bigr{)}$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS;II}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{(}\delta_{\alpha\beta}\delta_{\gamma
z}\tilde{F}^{\gamma\alpha}_{\mathrm{II}}\\!+\\!\delta_{\gamma\alpha}\delta_{\beta
z}\tilde{F}^{\beta\gamma}_{\mathrm{II}}\\!-\\!\delta_{\beta\gamma}\delta_{\alpha
z}\tilde{F}^{\alpha\beta}_{\mathrm{II}}\Bigr{)}$ (49)
$\displaystyle\tilde{F}^{zz}_{\mathrm{II}}$ $\displaystyle=$
$\displaystyle\frac{\Lambda}{t_{0}}\Bigl{(}\frac{3}{2\lambda_{t}^{2}}-\frac{1}{\lambda_{t}^{4}}\Bigr{)}+\mathcal{O}(\mathrm{ln}\Lambda)$
(50) $\displaystyle\tilde{F}^{zx}_{\mathrm{II}}$ $\displaystyle=$
$\displaystyle\tilde{F}^{zy}_{\mathrm{II}}=\tilde{F}^{xz}_{\mathrm{II}}=\tilde{F}^{yz}_{\mathrm{II}}=\frac{\Lambda}{t_{0}}\Bigl{(}\frac{1}{\lambda_{t}^{2}}-\frac{1}{\lambda_{t}^{4}}\Bigr{)}+\mathcal{O}(\mathrm{ln}\Lambda)$

In the type-II case, when we consider the limit $\mu\rightarrow 0$ with
$\eta/\mu=\mathrm{const.}$, the BCD term shows a logarithmic divergence, while
it is convergence with finite $\eta$. This behavior is completely different
from the type-I case, where the BCD term has the dip at $\mu=0$. In the limit
$\Lambda\gg t_{0},\mu,\eta,\lambda_{t}$, the dominant term of the Christoffel
symbol term is independent of $\mu$ and $\eta$ in the type-II Weyl
Hamiltonian.

Figure 6: $\mu$-dependence of the nonlinear Hall conductivity in type-I and II
Weyl Hamiltonian.
The top (bottom) panel shows the nonlinear Hall conductivity in the type-I
(II) Weyl Hamiltonian. We set the parameter as $t_{0}=1.0,t=0.3$ for the
type-I Weyl Hamiltonian and $t_{0}=0.3,t=0.5$ for the type-II Weyl
Hamiltonian. We also set $k_{B}T=0.02$ and $\eta=0.01$ for each case. The
terms except the BCD term and the gBC term are almost zero.

###  IV.4 Numerical results

We also check the numerical calculation about the chemical potential
dependence of the nonlinear Hall effect $\sigma^{y;zx}=-\sigma^{x;yz}$.
Interestingly, in addition to the $\mu$-independence of the nonlinear Hall
conductivity at $|\mu|\gg\eta$ as we have analytically shown, the gBC term
compensates the dip of the BCD term around $\mu=0$, which results in complete
$\mu$-independence of the nonlinear Hall conductivity.(See the top panel of
figure 6.) The bottom panel of figure 6 shows that the peak behaviour of the
nonlinear Hall conductivity at $\mu=0$ as we have shown, and its order of the
conductivity is much larger than type-I Weyl systems. This result means that
the type-II Weyl materials can show large nonlinear Hall effect.

We again stress that this chemical potential independence or the peak behavior
of the nonlinear Hall conductivity can be highly utilized for the detection of
the Weyl points and their type.

##  V Outlook and Summary

In this paper, we have analyzed the dissipation effect on the linear and
nonlinear DC conductivity under the Markov approximation in multi-band
systems.

Starting from the Green function formalism, we elucidates the effect of the
dissipation: the shift of the distribution function in the imaginary direction
and the Matsubara term, which cannot be included by the conventional methods
such as the SCB treatment and the reduced density matrix methods. Moreover, we
clarify that the novel terms from the imaginary part of the distribution
function also have the geometric nature, as the quantum metric term in the
linear conductivity, and the Christoffel symbol term and the generalized Berry
curvature term in the nonlinear conductivity. These terms give the multi-band
correction to the (nonlinear) Drude term when there is the band degeneracy at
the Fermi surface. Although the Matsubara term is not small in the linear
response, it is small enough to be ignored in nonlinear response especially
under the TRS, and therefore, the description of the Fermi DF of complex
argument works well. Under the TRS, the inversion symmetry breaking is encoded
in the multi-band effect such as the BCD and the generalized Berry curvature.
At the pole of the Matsubara frequency, the large imaginary value of
$i\omega_{M}$ in denominator of the $G(i\omega_{M})$ cloaks such kind of the
geometric structure, and therefore, the Matsubara term becomes almost zero.
When $\pi k_{B}T\sim\eta$, $i\omega_{M}-i\eta$ in the denominator of
$G^{A}(i\omega_{M})$ becomes almost zero and the above cloaking is unveiled,
and the Matsubara term becomes finite. In the nonlinear conductivity without
TRS and in the linear conductivity, the Drude term, which is the intra-band
contribution and does not need the geometric structure, can be finite, and
therefore, the Matsubara term can be also finite.

We have also elucidated the geometric origin of the non-reciprocal
conductivity under the TRS, which can be described by the Berry curvature
generalized to the higher-order derivative. For two-band systems, we have
identified another condition of the non-reciprocal conductivity under the
time-reversal symmetry, that is the quadratic term, in addition to the
dissipation which is pointed out in Ref.Morimoto and Nagaosa (2018).

Then, we numerically calculate the $\eta$\- and $\mu$-dependence of each
contribution in the model which describes TMD materials. The results shows
that the Matsubara term becomes large at $\eta>\pi k_{B}T$ when the terms from
the imaginary part of the distribution function is not small. We have shown
that, in some regime, the novel geometric term become dominant.

Finally, we analyze each geometric terms in the Weyl Hamiltonian. We clarify
that the chemical potential dependence around at the Weyl point of nonlinear
Hall conductivity is completely different between the type-I and type-II case.
While the nonlinear Hall conductivity is comletely independent of teh chemical
potential in the type-I case, it shows the logarithmic divergence behavior at
the energy level where the Weyl points exist in the type-II case. This result
suggest that the detection of the chemical potential dependence of the
nonlinear Hall conductivity under the time-reversal symmetry can be utilized
to detect the existence of the Weyl points and their type. Moreover, we also
show that the type-II Weyl materials can show large nonolinear Hall
conductivity due to its divergent behaviour at the Weyl points.

Although we consider the DC conductivity in this paper, the similar analysis
can be applied to the photovoltaic effect, in which the dissipation holds the
important role. It is left for the future works.

## acknowledge

YM deeply appreciate to Hikaru Watanabe, Yoichi Yanase, and Kazuaki Takasan
for fruitful discussions. YM also thanks Masahiko G. Yamada, Atsuo Shitade,
and Hiroshi Shinaoka for their advice about the numerical calculations. We
performed all calculation with JuliaBezanson  _et al._ (2017). The code which
we used is uploaded to https://github.com/YoshihiroMichishita/julia. This work
is supported by the WISE program, MEXT. YM is supported by a JSPS research
fellowship and by JSPS KAKENHI (Grant No. 20J12265). Naoto Nagaosa is
supported by JSPS KAKENHI Grant (No. 18H03676), and by JST CREST Grant Number
JPMJCR1874, Japan.

##  Appendix A Detail derivation of the dissipation-induced geometric term in
linear conductivity

Contribution from the pole of the advanced Green functions to the linear
conductivity can be described as follows,

$\displaystyle\sigma^{\alpha\beta}_{\mathrm{G}}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha\beta}_{\mathrm{Drude}}+\sigma^{\alpha\beta}_{\mathrm{QM:re}}+\sigma^{\alpha\beta}_{\mathrm{QM:im}}$
(52) $\displaystyle\sigma^{\alpha\beta}_{\mathrm{Drude}}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k}}\sum_{n}\mathcal{J}^{\alpha}_{n}\mathcal{J}^{\beta}_{n}\tau\mathrm{Re}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}+i\eta}$ (53)
$\displaystyle\sigma^{\alpha\beta}_{\mathrm{QM:re}}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k}}\sum_{n}\frac{g^{\alpha\beta}_{S;n}}{\tau}\mathrm{Re}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}+i\eta}$ (54)
$\displaystyle\sigma^{\alpha\beta}_{\mathrm{QM:im}}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k}}\sum_{n}\sum_{m\neq
n}g^{\alpha\beta}_{S;n,m}\epsilon_{nm}\mathrm{Im}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}|_{\epsilon_{n}+i\eta}$ (55) $\displaystyle
g^{\alpha\beta}_{S;n,m}$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{mn}+\mathcal{J}^{\beta}_{nm}\mathcal{J}^{\alpha}_{mn})}{2(\epsilon^{2}_{nm}+4\eta^{2})}.$
(56)

We call $g^{\alpha\beta}_{S;n}$ as “the smeared quantum metric.” We note that
the smeared geometric quantities are well defined and finite at gapless points
where $\epsilon_{nm}=0$.
$\sigma^{\alpha\beta}_{\mathrm{QM:re}}+\sigma^{\alpha\beta}_{\mathrm{QM:im}}$
is the novel term, which we call “quantum metric term” at the Fermi surface.

We note that we can approximate $\mathrm{Re}f(\epsilon_{n}+i\eta)\simeq
f(\epsilon_{n})$ and $\mathrm{Im}f(\epsilon_{n}+i\eta)\simeq\eta\partial
f(\epsilon_{n})/\partial\omega$ when $\beta\eta\ll 1$. Figure. 7 shows that
this approximation is valid even when $\beta\eta\sim 0.5$. Under this
approximation, in the both limits $\epsilon_{nm}\gg\eta$ and
$\epsilon_{nm}\rightarrow 0$, the terms proportional to the imaginary part of
the distribution function can be written as,

$\displaystyle
g^{\alpha\beta}_{S;n,m}\epsilon_{nm}\mathrm{Im}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}|_{\epsilon_{n}\\!+\\!i\eta}\simeq\frac{\epsilon_{nm}}{2\tau}g^{\alpha\beta}_{S;n,m}\Bigl{(}-\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}|_{\epsilon_{n}}$
(57)

When we see the frequency derivative of the Fermi distribution function as
$\sim\mathcal{O}(\beta)$, the imaginary part contributions are the order
$\sim\mathcal{O}(\epsilon_{nm}\beta)$ compared to the real part of the
contribution. This means that the contribution from the imaginary part of the
distribution function can be dominant at low temperature. We note that, under
the approximation $\mathrm{Re}f(\epsilon_{n}+i\eta)\simeq f(\epsilon_{n})$ and
$\mathrm{Im}f(\epsilon_{n}+i\eta)\simeq\eta\partial
f(\epsilon_{n})/\partial\omega$, Eq. (54) can be derived from the RDM methods
by substituting $f(\epsilon_{n})\rightarrow f(\epsilon_{n}+i\eta)$ in the
anomalous quantum Hall term and considering the imaginary part of the Fermi
distribution function. From this point of view,
$\sigma^{\alpha\beta}_{\mathrm{QM:re}}$ also originates from the imaginary
part of the distribution function in the anomalous quantum Hall term.

When considering the band degeneracy at the Fermi surface which means
$\epsilon_{n}=\epsilon_{m}$, the smeared quantum metric
$g^{\alpha\beta}_{S;n}$ is proportional to $\tau^{2}$ and the qunatum metric
term in Eq. (7) is proportional to $\tau$, which means the quantum metric term
gives the multi-band correction to the Drude term in Eq. (53). Therefore, the
quantum metric term becomes important in the materials which have the large
band degeneracy at the Fermi surface or the strongly-dissipative system.

Figure 7: Approximation about the imaginary part of the distribution function.
The blue(orange) plots show the real(imaginary) part of $f(\omega+i\eta)$. The
green plots show $f(\omega)$ and the red plots show $\eta\partial
f/\partial\omega$. The parameters are $T=0.02$, and $\eta=0.01$ on the left
panel and $\eta=0.02$ on the right panel. This figure shows that the
approximation $\mathrm{Re}f(\omega+i\eta)\simeq f(\omega)$ and
$\mathrm{Im}f(\omega+i\eta)\simeq\eta\partial f(\omega)/\partial\omega$ is
enough good even when $\beta\eta\sim 0.5$.

##  Appendix B Intuitive understanding about the broadening term

In this part, we give the intuitive understanding about the shift of the Fermi
DF and the broadening term. In the conventional band representation, the
occupancy of the band is determined by its energy and the Fermi DF
($f(\epsilon_{n})$), and it is true in the limit of $\eta\rightarrow 0$.
However, when $\eta$ is finite, there is the broadening of the spectral
function and the occupancy of the band cannot be determined just by the band
energy and the Fermi DF. In the small dissipation regime $\eta\ll T$, the
broadening is small and the bands, who have the energy level where
$(df/d\omega)_{\epsilon_{n}}$ is finite, contribute to the transport. For an
example, in Figure. (8), the band, whose energy is $|\epsilon_{n}|>T$, almost
does not contribute the transport. In this case, the description by the Fermi
DF is accurate, and the shift of the Fermi DF to the imaginary direction well
approximates the broadening of the spectral function. On the other hand, when
we consider the highly dissipative case $\eta>T$, even though the band energy
is far away from the Fermi energy $|\epsilon_{n}|\gg T$ and
$(df/d\omega)_{\epsilon_{n}}\simeq 0$, there is the overlap between
$df(\omega)/d\omega$ and $A(\omega)$, and this band can contribute to the
transport. In this case, the description by the Fermi DF becomes bad, and the
Matsubara term, which is not described by the Fermi DF, can become large.
These understanding should also hold true in nonlinear conductivity.

Figure 8: broadening of the spectral function v.s. derivative of the Fermi DF.
The blue(orange) plots show the $\omega$-derivative of the Fermi FD
$(-\partial f/\partial\omega)$ (the spectral function $A(\omega)$). Here we
set the temperature $T=0.02$, and $\epsilon_{n}=0.1$. $\eta=0.01(0.1)$ in the
left(right) panel.

##  Appendix C model calculation

We numerically calculate these contribution to the linear conductivity in the
nodal-line semimetals, which is described by the following Hamiltonian
asMatsushita  _et al._ (2020),

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mu\tau^{0}+t(2+\cos{k_{0}}-\cos{k_{x}}-\cos{k_{y}}-\cos{k_{z}})\tau^{z}$
(58) $\displaystyle\ \ +v\sin{k_{z}}\tau^{y}+\Delta\tau^{x}.$

Figure 9: Linear conductivity in a nodal-line semimetal.
The parameters are $T=0.02,\eta=0.04$ on the top panel and $T=0.02,\mu=0$ on
the bottom panel. We perform the momentum integration by $400\times 400\times
400$ and frequency integration by $1000$. The blue, orange, green, red plots
respectively describe the Drude term, the quantum metric term
($\sigma^{zz}_{\mathrm{QM:re}}+\sigma^{zz}_{\mathrm{QM:im}}$), the total
conductivity calculated by the Green function methods, and the Matsubara term.

$\mu,t,v,\Delta$ represent the chemical potential, the hopping amplitude, the
hybridization, and the TRS breaking parameter. We note that this tight-binding
Hamiltonian describes the low-energy dispersion near the Fermi level of CaAgP
and Ca3P2Yamakage  _et al._ (2016); Chan  _et al._ (2016). For the upper panel
of figure 9, the chemical potential ($\mu$) dependence of the linear
conductivity is shown. Here the temperature $k_{B}T=0.02$, $\eta=0.04$, and
$\Delta=0$. This model has the nodal line at the Fermi surface when
$\mu=0,\Delta=0$ and the quantum metric term becomes large. We note that, in
this model, $\mathcal{J}^{z}_{nn}$ is zero on the nodal line and therefore the
Drude term becomes almost zero at $\mu=0$.

In this case, the quantum metric term is dominant at $\mu=0$ and the Matsubara
term is less than 0 to compensate the overestimated quantum metric term in the
description of the Fermi distribution function. Even when the system has a
quadratic band-touching at the Fermi surface and the Drude term is zero, if
these band are constructed by the hybridization, conductivity can be finite
due to the quantum metric term.

The bottom panel of figure 9 shows that $\eta$-dependence of the linear
conductivity. Here the temperature $T=0.02$, $\mu=0$, and $\Delta=0$. In the
small dissipation regime, the quantum metric term and the Matsubara term is
proportional to $\eta$, and therefore, the Drude term is dominant. In our
formulation, $\eta=\pi T$, where the Fermi distribution function
$f(\epsilon_{n}+i\eta)$ behaves as the Bose distribution function
$f_{B}(\epsilon_{n})$, is a singular. In the large dissipation regime where
$\eta>\pi T$, contribution from the Drude term becomes almost zero. This large
dissipation regime at low temperature can be realized in the quantum critical
regime, because the large quantum fluctuation behaves as the large dissipation
from the single-particle points of view. In such regime, the SCB treatment and
also the reduced density matrix method is not appropriate. Therefore, we must
use the Green function methods in such regime.

The relaxation time $\tau$ in usual materials is about $1\sim 100$
picosecondDu  _et al._ (2019), which equals to $\eta\simeq 2.1*10^{-5}\sim
2.1*10^{-3}$[eV]. In this case, the temperature which satisfies $\pi
k_{B}T>\eta$ is about $T>7.5*10^{-2}\sim 7.5$[K], and this condition is often
not satisfied in the condensed matter physics. Therefore, our result shows
that we should consider the broadening of the spectral function or to say the
quantum dissipation at low temperature, and the SCB treatment or the reduced
density matrix formalism is not appropriate especially when we consider the
transport at a degenerated Fermi surface.

##  Appendix D Detail derivation of the dissipation-induced geometric terms in
nonlinear conductivity

The nonlinear conductivity is described with the Green function formalism as

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{DC}}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\mathrm{Im}\mathrm{Tr}\Bigl{[}\Bigl{\\{}2\mathcal{J}^{\alpha}_{nm}\frac{\partial
G^{R}_{m}}{\partial\omega}\mathcal{J}^{\beta}_{ml}G^{R}_{l}\mathcal{J}^{\gamma}_{ln}(G^{R}_{n}-G^{A}_{n})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ +\mathcal{J}^{\alpha}_{nm}\frac{\partial
G^{R}_{m}}{\partial\omega}\mathcal{J}^{\beta\gamma}_{mn}(G^{R}_{n}-G^{A}_{n})\Bigr{\\}}+\\{\beta\leftrightarrow\gamma\\}\Bigr{]},$
(59)

where
$\mathcal{J}^{\beta\gamma}=\partial^{\beta}\partial^{\gamma}\mathcal{H}_{\mathrm{eff}}$.
By starting from this Green function formalism and performing the same
procedure as in the linear case, we can derive the following equation as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{DC}}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k}}\Bigl{(}\sigma^{\alpha;\beta\gamma}_{\mathrm{M}}+\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Drude}}+\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{BCD}}++\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Inter}}\Bigr{)},$
(60) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{Th}}$ $\displaystyle=$
$\displaystyle\sum_{nml}\sum_{\omega_{M}>0}\mathrm{Re}\Bigl{[}\frac{\partial}{\partial\omega}\Bigl{\\{}2\mathcal{J}^{\alpha}_{nm}\frac{\partial
G^{R}_{m}}{\partial\omega}\mathcal{J}^{\beta}_{ml}G^{R}_{l}\mathcal{J}^{\gamma}_{ln}(G^{R}_{n}-G^{A}_{n})+\mathcal{J}^{\alpha}_{nm}\frac{\partial
G^{R}_{m}}{\partial\omega}\mathcal{J}^{\beta\gamma}_{mn}(G^{R}_{n}-G^{A}_{n})\Bigr{\\}}+\bigl{\\{}\beta\leftrightarrow\gamma\Bigr{\\}}\Bigr{]}_{\omega=i\omega_{M}}$
(61) $\displaystyle\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Drude}}$
$\displaystyle=$ $\displaystyle
2\sum_{n}\tau^{2}\mathrm{Re}\Bigl{[}\mathcal{J}^{\alpha}_{n}\mathcal{J}^{\beta}_{n}\mathcal{J}^{\gamma}_{n}\Bigl{(}-\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}+\mathcal{J}^{\alpha}_{n}\mathcal{J}^{\beta\gamma}_{n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}+\sum_{m(\neq
n)}\mathcal{J}^{\alpha}_{n}\frac{\mathcal{J}^{\beta}_{nm}\mathcal{J}^{\gamma}_{mn}+(\beta\leftrightarrow\gamma)}{\epsilon_{nm}+2i\eta}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}\Bigr{]}_{\omega=\epsilon_{n}+i\eta}$ (62)
$\displaystyle\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{BCD}}$
$\displaystyle=$ $\displaystyle
2\sum_{n}\tau\mathrm{Im}\Bigl{[}\mathcal{Q}^{\alpha\beta}_{D;n}\mathcal{J}^{\gamma}_{n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}+(\beta\leftrightarrow\gamma)\Bigr{]}_{\omega=\epsilon_{n}+i\eta}$
(63) $\displaystyle\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Inter}}$
$\displaystyle=$ $\displaystyle 2\sum_{nm(m\neq
n)}\mathrm{Re}\Bigl{[}\frac{\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta\gamma}_{mn}}{(\epsilon_{nm}+2i\eta)^{2}}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}+\sum_{l\neq
n}\frac{\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln}+(\beta\leftrightarrow\gamma)}{(\epsilon_{nm}\\!+\\!2i\eta)^{2}(\epsilon_{nl}\\!+\\!2i\eta)}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}\Bigr{]}_{\omega=\epsilon_{n}+i\eta}$ (64)

$\sigma^{\alpha;\beta\gamma}_{\mathrm{M}}$ is the Matsubara term for the
second order nonlinear response. Tilde above the terms means the
classification of terms in the limit $\eta\rightarrow 0$. First, we analyze
the ”Drude term” $\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Drude}}$, which
corresponds to the Drude term in the conventional analysis. While the first
term and second term in Eq. (62) only get the contribution from the real part
of the distribution function, the third term has the contribution from the
imaginary part. Under the approximation
$\mathrm{Re}f(\epsilon_{n}+i\eta)\simeq f(\epsilon_{n})$ and
$\mathrm{Im}f(\epsilon_{n}+i\eta)\simeq\eta\partial
f(\epsilon_{n})/\partial\omega$, we can derive the following form from the
third term in Eq. (62) as,

$\displaystyle\sum_{m(\neq
n)}\mathcal{J}^{\alpha}_{n}\frac{\mathcal{J}^{\beta}_{nm}\mathcal{J}^{\gamma}_{mn}+(\beta\leftrightarrow\gamma)}{\epsilon_{nm}+2i\eta}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}\Bigr{]}_{\omega=\epsilon_{n}+i\eta}$
$\displaystyle=2\sum_{n\neq
m}\tau^{2}\epsilon_{nm}\mathcal{J}^{\alpha}_{n}g^{\beta\gamma}_{S;n,m}\mathrm{Re}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\omega=\epsilon_{n}+i\eta}$ $\displaystyle\ \ \ \
+2\sum_{n}\tau\mathcal{J}^{\alpha}_{n}g^{\beta\gamma}_{S;n}\mathrm{Im}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\omega=\epsilon_{n}+i\eta}$ (65)
$\displaystyle\simeq 2\sum_{n\neq
m}\tau^{2}\epsilon_{nm}\mathcal{J}^{\alpha}_{n}g^{\beta\gamma}_{S;n,m}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\omega=\epsilon_{n}}$ $\displaystyle\ \ \ \
+\sum_{n}\mathcal{J}^{\alpha}_{n}g^{\beta\gamma}_{S;n}\Bigl{(}\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}_{\omega=\epsilon_{n}}$
(66)

The second term newly arise from the imaginary part of the distribution
function, which represents the broadening of the spectral function. We note
that the other terms in $\tilde{\sigma}_{\mathrm{Drude}}$ can be summarized
into the conventional Drude term as
$\tau^{2}\sum_{n}\mathcal{J}^{\alpha}_{n}\partial^{\beta}\partial^{\gamma}f(\epsilon_{n})$
in the limit of $\eta\rightarrow 0$. We can transform the second term into

$\displaystyle\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Drude}}$
$\displaystyle=$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{Drude}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{sQMD}}$
(67) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{Drude}}$
$\displaystyle=$ $\displaystyle
2\tau^{2}\sum_{n}\mathcal{J}^{\alpha}_{n}\Biggl{[}\mathcal{J}^{\beta}_{n}\mathcal{J}^{\gamma}_{n}\mathrm{Re}\Bigl{(}-\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}+$
(68) $\displaystyle\ \
\Bigl{(}\mathcal{J}^{\beta\gamma}_{n}+\sum_{m}\epsilon_{nm}g^{\beta\gamma}_{S;n,m}\Bigr{)}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}\Biggr{]}|_{\epsilon_{n}+i\eta}$
$\displaystyle\simeq$ $\displaystyle
2\tau^{2}\sum_{n}\mathcal{J}^{\alpha}_{n}\partial^{\beta}\partial^{\gamma}f(\epsilon_{n})$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{sQMD}}$ $\displaystyle=$
$\displaystyle
2\sum_{n}\tau\mathcal{J}^{\alpha}_{n}g^{\beta\gamma}_{S;n}\mathrm{Im}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\omega=\epsilon_{n}+i\eta}$ (69)
$\displaystyle\simeq$
$\displaystyle\sum_{n}\mathcal{J}^{\alpha}_{n}g^{\beta\gamma}_{S;n}\Bigl{(}\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}_{\omega=\epsilon_{n}}$
$\displaystyle=$
$\displaystyle\sum_{n}\partial^{\alpha}g^{\beta\gamma}_{S;n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\omega=\epsilon_{n}},$

and therefore, we call it the (smeared) quantum metric dipole term. Because
here it is the smeared quantum metric, when there is a band-degeneracy at the
Fermi surface, this term is proportional to $\tau^{2}$ (because
$g^{\beta\gamma}_{S;n}$ is proportional to $\tau^{2}$ when $\epsilon_{nm}=0$)
and gives the multi-band correction to the conventional nonlinear Drude term,
as same as the linear conductivity. On the other hand, in the regime
$|\epsilon_{n}-\epsilon_{m}|\tau\gg 1$, this term becomes almost independent
from the strength of the dissipation even through this term stems from the
dissipation.

Next, we analyze the originally Berry curvature dipole term. Under the
approximation, we can derive

$\displaystyle\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{BCD}}$
$\displaystyle=$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{dBCD}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{dQMD}}$
(70) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{dBCD}}$
$\displaystyle=$ $\displaystyle
2\sum_{n}\tau\Omega^{\alpha\beta}_{D;n}\mathcal{J}^{\gamma}_{n}\mathrm{Re}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}\\!+\\!i\eta}+(\beta\leftrightarrow\gamma)$
(71) $\displaystyle=$ $\displaystyle
4\tau\sum_{n}\partial^{\gamma}\Omega^{\alpha\beta}_{D;n}\mathrm{Re}f(\epsilon_{n}+i\eta)+(\beta\leftrightarrow\gamma)$
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{dQMD}}$ $\displaystyle=$
$\displaystyle-2\sum_{n}g^{\alpha\beta}_{D;n}\mathcal{J}^{\gamma}_{n}\mathrm{Im}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}+(\beta\leftrightarrow\gamma)$ (72)
$\displaystyle=$ $\displaystyle
4\tau\sum_{n}\partial^{\gamma}g^{\alpha\beta}_{D;n}\mathrm{Im}f(\epsilon_{n}+i\eta)+(\beta\leftrightarrow\gamma),$

where $\Omega^{\alpha\beta}_{D;n}=\mathrm{Im}\mathcal{Q}^{\alpha\beta}_{D;n}$
is the dissipative Berry curvature and
$\Omega^{\alpha\beta}_{D;n}=2\mathrm{Re}\mathcal{Q}^{\alpha\beta}_{D;n}$ is
the dissipative quantum metric. In the limit $\eta\rightarrow 0$, Eq. (71)
becomes the well known the Berry curvature dipole term. We note that the
dissipative quantum geometry is different from the smeared quantum geometry.
In this representation, because it is the dissipative geometry, the
(dissipative) Berry curvature dipole term is not necessarily the Hall
conductivity. Therefore, it is more convenient to convert the smeared
geometric terms as written as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{dBCD}}$
$\displaystyle=2\tau\sum_{n,m}\Biggl{[}\Bigl{(}\frac{\epsilon^{2}_{nm}\tau^{2}-1}{\epsilon^{2}_{nm}\tau^{2}+1}\Omega^{\alpha\beta}_{S;n,m}-\frac{2\epsilon_{nm}\tau}{\epsilon^{2}_{nm}\tau^{2}+1}g^{\alpha\beta}_{S;n,m}\Bigr{)}$
$\displaystyle\ \ \ \ \ \ \ \ \
\times\mathcal{J}^{\gamma}_{n}\mathrm{Re}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}\\!+\\!i\eta}+(\beta\leftrightarrow\gamma)\Biggr{]}$
(73) $\displaystyle\simeq
2\tau\sum_{n,m}\Biggl{[}\Bigl{(}\Omega^{\alpha\beta}_{S;n}-\frac{2g^{\alpha\beta}_{S;n,m}}{\epsilon_{nm}\tau}\Bigr{)}\mathcal{J}^{\gamma}_{n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}+(\beta\leftrightarrow\gamma)\Biggr{]}$
(74) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{dQMD}}$
$\displaystyle=2\tau\sum_{n,m}\Biggl{[}\Bigl{(}\frac{\epsilon^{2}_{nm}\tau^{2}-1}{\epsilon^{2}_{nm}\tau^{2}+1}g^{\alpha\beta}_{S;n,m}+\frac{2\epsilon_{nm}\tau}{\epsilon^{2}_{nm}\tau^{2}+1}\Omega^{\alpha\beta}_{S;n,m}\Bigr{)}$
$\displaystyle\ \ \ \ \ \ \ \ \
\times\mathcal{J}^{\gamma}_{n}\mathrm{Im}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}\\!+\\!i\eta}+(\beta\leftrightarrow\gamma)\Biggr{]}$
(75)
$\displaystyle\simeq\sum_{n}\Biggl{[}-\Bigl{(}\partial^{\gamma}g^{\alpha\beta}_{S;n}\Bigr{)}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}+\frac{2\Omega^{\alpha\beta}_{S;n,m}}{\epsilon_{nm}\tau}\mathcal{J}^{\gamma}_{n}\Bigl{(}-\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}_{\epsilon_{n}}$
$\displaystyle\ \ \ \ \ \ \ \ \ +(\beta\leftrightarrow\gamma)\Biggr{]}$ (76)

In the limit $\epsilon_{nm}\tau\gg 1$, up to the first order of $\eta$, the
first term in Eq. (73) is the conventional Berry curvature dipole term and the
second term is the terms pointed out in Ref.Gao  _et al._ (2014) (the second
term in Eq. (13) in Ref.Gao  _et al._ (2014)) Here, another QMD term emerges
from the original Berry curvature dipole term. Because it is the dissipative
quantum metric, when there is a band-degeneracy at the Fermi surface, this
term also gives the multi-band correction to the nonlinear Drude term. As same
as the smeared QMD term, in the limit $|\epsilon_{n}-\epsilon_{m}|\tau\gg 1$,
this behaves as is almost independent of the strength of the dissipation,
while it stems from the dissipation. When considering the limit
$\epsilon_{nm}\tau\gg 1$ and approximating
$\mathrm{Im}f(\epsilon_{n}+i\eta)=\eta(\partial
f/\partial\omega)|_{\epsilon_{n}}$, which is enough justified at
$\beta\eta<1$, the sum of the smeared QMD term and the first term in Eq. (76)
can be written as,

$\displaystyle\Bigl{(}\partial^{\gamma}g^{\alpha\beta}_{S;n}\\!+\\!\partial^{\beta}g^{\alpha\gamma}_{S;n}\\!-\\!\partial^{\alpha}g^{\beta\gamma}_{S;n}\Bigr{)}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}=2\Gamma^{\alpha;\beta\gamma}_{S;n}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}},$

where
$\Gamma^{\alpha;\beta\gamma}_{S;n}\equiv(\partial^{\gamma}g^{\alpha\beta}_{S;n}+\partial^{\beta}g^{\alpha\gamma}_{S;n}-\partial^{\alpha}g^{\beta\gamma}_{S;n})/2$
can be called the “smeared Christoffel symbol”, which is the generalization of
the Christoffel symbol in Ref.Ahn  _et al._ (2020) to the dissipative case.
Therefore, we can call the sum of those term as “the smeared Christoffel
symbol term” at the Fermi surface.

Finally, we analyze the interband term
$\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Inter}}$ in Eq. (64).

$\tilde{\sigma}^{\alpha;\beta\gamma}_{\mathrm{Inter}}$ can generate the
another Christoffel symbol term and the generalized Berry curvature term as,

$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{Inter;1}}=\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:II}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC}},$
(78) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{ChS:II}}$
$\displaystyle=2\sum_{\bm{k}}\sum_{n,m(\neq
n)}\Bigl{[}\mathrm{Re}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta\gamma}_{mn})\mathrm{Re}\Bigl{(}\frac{1}{(\epsilon_{nm}\\!+\\!2i\eta)^{2}}\Bigr{)}$
$\displaystyle\ \ +\Bigl{\\{}\sum_{l(\neq
n)}\mathrm{Re}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln})\mathrm{Re}\Bigl{(}\frac{1}{(\epsilon_{nm}\\!+\\!2i\eta)^{2}(\epsilon_{nl}\\!+\\!2i\eta)}\Bigr{)}\Bigr{\\}}$
$\displaystyle\ \
+\Bigl{\\{}\beta\leftrightarrow\gamma\Bigr{\\}}\Bigr{]}\mathrm{Re}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}+i\eta}+\mathcal{O}(\tau^{-2})$ (79)
$\displaystyle=2\sum_{\bm{k}}\sum_{n}g^{\alpha;\beta\gamma}_{S^{\prime};n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}=2\sum_{\bm{k}}\sum_{n}\Gamma^{\alpha;\beta\gamma}_{S^{\prime};n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}$ (80)
$\displaystyle\simeq\sum_{\bm{k}}\sum_{n}\Bigl{(}\braket{\partial^{\alpha}n}{\partial^{\beta\gamma}n}+\braket{\partial^{\beta\gamma}n}{\partial^{\alpha}n}\Bigr{)}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}$ (81)
$\displaystyle=2\sum_{\bm{k}}\sum_{n}\Gamma^{\alpha;\beta\gamma}_{n}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}$ (82)
$\displaystyle\Gamma^{\alpha;\beta\gamma}_{n}=\frac{1}{2}\Bigl{(}\partial^{\gamma}g^{\alpha\beta}_{n}+\partial^{\beta}g^{\gamma\alpha}_{n}-\partial^{\alpha}g^{\beta\gamma}_{n}\Bigr{)}$
$\displaystyle\ \ \ \ \ \ \ \
=\frac{1}{2}\Bigl{(}\braket{\partial^{\alpha}n}{\partial^{\beta\gamma}n}+\braket{\partial^{\beta\gamma}n}{\partial^{\alpha}n}\Bigr{)}$
(83)
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC}}=\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:re}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:im}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:add}}$
(84)
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:re}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:add}}$
$\displaystyle=2\sum_{\bm{k}}\sum_{n,m(\neq
n)}\Bigl{[}\mathrm{Im}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta\gamma}_{mn})\mathrm{Im}\Bigl{(}\frac{1}{(\epsilon_{nm}\\!+\\!2i\eta)^{2}}\Bigr{)}$
$\displaystyle\ \ +\Bigl{\\{}\sum_{l(\neq
n)}\mathrm{Im}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln})\mathrm{Im}\Bigl{(}\frac{1}{(\epsilon_{nm}\\!+\\!2i\eta)^{2}(\epsilon_{nl}\\!+\\!2i\eta)}\Bigr{)}\Bigr{\\}}$
$\displaystyle\ \
+\Bigl{\\{}\beta\leftrightarrow\gamma\Bigr{\\}}\Bigr{]}\mathrm{Re}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}+i\eta}$ (85)
$\displaystyle=\sum_{\bm{k}}\sum_{n,m(\neq
n)}\frac{\Omega^{\alpha,\beta\gamma}_{S^{\prime};n,m}}{\epsilon_{nm}\tau}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}}+\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:add}}+\mathcal{O}(\tau^{-2}),$
(86) $\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:add}}$
$\displaystyle\simeq 2\sum_{\bm{k}}\sum_{n,m,l(\neq
n)}\frac{\mathrm{Im}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln})}{\epsilon^{2}_{nm}\epsilon_{nl}}\Bigl{(}\frac{1}{\epsilon_{nm}\tau}-\frac{1}{\epsilon_{nl}\tau}\Bigr{)}\Bigl{(}-\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}},$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ +(\beta\leftrightarrow\gamma)$ (87)
$\displaystyle\sigma^{\alpha;\beta\gamma}_{\mathrm{gBC:im}}$
$\displaystyle=2\sum_{\bm{k}}\sum_{n,m(\neq
n)}\Bigl{[}\mathrm{Im}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta\gamma}_{mn})\mathrm{Re}\Bigl{(}\frac{1}{(\epsilon_{nm}\\!+\\!2i\eta)^{2}}\Bigr{)}$
$\displaystyle\ \ +\Bigl{\\{}\sum_{l(\neq
n)}\mathrm{Im}(\mathcal{J}^{\alpha}_{nm}\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln})\mathrm{Re}\Bigl{(}\frac{1}{(\epsilon_{nm}\\!+\\!2i\eta)^{2}(\epsilon_{nl}\\!+\\!2i\eta)}\Bigr{)}\Bigr{\\}}$
$\displaystyle\ \
+\Bigl{\\{}\beta\leftrightarrow\gamma\Bigr{\\}}\Bigr{]}\mathrm{Im}\Bigl{(}\frac{\partial
f}{\partial\omega}\Bigr{)}_{\epsilon_{n}+i\eta}$ (88)
$\displaystyle=-\sum_{\bm{k}}\sum_{n}\frac{\Omega^{\alpha,\beta\gamma}_{S^{\prime};n}}{\tau}\Bigl{(}-\frac{\partial^{2}f}{\partial\omega^{2}}\Bigr{)}_{\epsilon_{n}}+\mathcal{O}(\tau^{2}),$
(89)

where the
$\Omega^{\alpha,\beta\gamma}_{S^{\prime};n,m}(g^{\alpha,\beta\gamma}_{S^{\prime};n,m})$
is the smeared Berry curvature (quantum metric) generalized to the higher-
order derivative, which can be derived by the following substitution from
$\Omega^{\alpha,\beta\gamma}_{n,m}(g^{\alpha,\beta\gamma}_{n,m})$,

$\displaystyle\begin{dcases}\frac{1}{\epsilon^{2}_{nm}}\rightarrow\frac{1}{\epsilon^{2}_{nm}+4\eta^{2}}\\\
\frac{1}{\epsilon_{nm}}\rightarrow\frac{\epsilon_{nm}}{\epsilon^{2}_{nm}+4\eta^{2}},\end{dcases}$
(90) $\displaystyle g^{\alpha,\beta\gamma}_{n,m}$ $\displaystyle=$
$\displaystyle\mathrm{Re}\braket{\partial^{\alpha}n}{m}\braket{m}{\partial^{\beta\gamma}n}=\Gamma^{\alpha;\beta\gamma}_{n,m}$
$\displaystyle=$ $\displaystyle
2\mathrm{Re}\Bigl{[}\frac{\mathcal{J}^{\alpha}_{nm}}{\epsilon_{nm}}\frac{1}{\epsilon_{nm}}\Bigl{(}\mathcal{J}^{\beta\gamma}_{mn}+\frac{\mathcal{J}^{\beta}_{ml}\mathcal{J}^{\gamma}_{ln}+(\beta\leftrightarrow\gamma)}{\epsilon_{nl}}\Bigr{)}\Bigr{]}$
$\displaystyle\Omega^{\alpha,\beta\gamma}_{n,m}$ $\displaystyle=$
$\displaystyle
2\mathrm{Im}\braket{\partial^{\alpha}n}{m}\braket{m}{\partial^{\beta\gamma}n}$
(93)

If we consider $\partial^{\beta\gamma}$ as the derivative to the new
direction,
$\Omega^{\alpha,\beta\gamma}_{S^{\prime};n,m}(g^{\alpha,\beta\gamma}_{S^{\prime};n,m})$
is also the smeared Berry curvature (quantum metric) generalized to the
higher-order derivative. We note that, in two-band systems under the time-
reversal symmetry, only the first term in Eq. (85) can induce the non-
reciprocal conductivity $\sigma^{\alpha;\alpha\alpha}$. Under the time-
reversal symmetry, the Drude term, the smeared Christoffel term and the other
terms proportional to the smeared quantum metric are zero. Moreover, the terms
proportional to the conventional smeared Berry curvature is also zero because
$\alpha=\beta=\gamma$. However, the extended Berry curvature
$\Omega^{\alpha,\alpha\alpha}$ can be finite because $\partial^{\alpha\alpha}$
is different from $\partial^{\alpha}$. As we can see in Eq. (86), Eq. (87) and
Eq. (89), this non-reciprocal term under the time-reversal symmetry becomes
zero when $\eta\rightarrow 0$, which is pointed out in Ref.Morimoto and
Nagaosa (2018). Here we also find another necessary condition, which is that
the Hamiltonian must have the quadratic term, because
$\mathcal{J}^{\alpha\alpha}$ becomes zero without it. Therefore, non-
reciprocal transport under the time-reversal symmetry in DC limit is zero, for
example in the Weyl system which has the linear dispersion.

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