# Quantum Kinetic Theory of Nonlinear Optical Currents: Finite Fermi surface
and Fermi sea contributions

Pankaj Bhalla<EMAIL_ADDRESS>Department of Physics, School of
Engineering and Sciences, SRM University AP, Amaravati, 522240, India  ARC
Centre of Excellence in Future Low-Energy Electronics Technologies, Australia
Kamal Das<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kanpur, Kanpur-208016, India  Amit Agarwal<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Kanpur, Kanpur-208016,
India  Dimitrie Culcer<EMAIL_ADDRESS>School of Physics, The University
of New South Wales, Sydney 2052, Australia  ARC Centre of Excellence in Future
Low-Energy Electronics Technologies, The University of New South Wales, Sydney
2052, Australia

###### Abstract

The quantum kinetic framework provides a versatile method for investigating
the dynamical optical and transport currents of crystalline solids. In this
paper, starting from the density-matrix equations of motion, we present a
general theoretical path to obtain the nonlinear optical response in an
elegant and transparent manner. We devise an extensive kinetic theory that can
be applied to materials with arbitrary band structures and captures intraband
and interband coherence effects, finite Fermi surfaces, and disorder effects.
We present a classification of the nonlinear optical currents arising from the
interference of the interband and intraband components of the density matrix
with distinct symmetry and quantum geometrical origin for each contribution.
In this context, we report the following four primary findings. (i) The Fermi
Golden Rule approach is insufficient to derive the correct expression for the
injection current, a shortcoming that we remedy in our theory while
associating the injection current with the intraband-interband contribution to
the second-order density matrix. (ii) The interband-intraband contribution
yields a resonant current that survives irrespective of any symmetry
constraint in addition to the well-known anomalous nonlinear current (non-
resonant), which requires time-reversal symmetry. (iii) Quite generally, the
nonlinear current is significantly enhanced by contributions arising from the
finite Fermi surface. (iv) The finite Fermi surface and Fermi sea additionally
lead to sizable novel nonlinear effects via contributions we term double
resonant and higher-order pole. We investigate such effects in sum frequency
and difference frequency generation. As an illustration, we compute the
nonlinear response of the topological antiferromagnet CuMnAs and thin film
tilted Weyl semimetals as model systems dominated by interband coherence
contributions. We find that the nonlinear response of CuMnAs is responsive to
the direction of the finite magnetization field and the response of Weyl
semimetal to the tilt. In addition, the choice of the polarization angle of
the beam is crucial to have a nonlinear current in CuMnAs, while it is not the
case for Weyl semimetals.

##  I Introduction

Probing novel nonlinear optical effects due to the light-matter interaction
has recently become a subject of great interest due to the cutting-edge
advancement in fields such as ultra-fast phenomenon and optoelectronics Boyd
(2008); Braunstein (1962); Morimoto and Nagaosa (2016); Orenstein  _et al._
(2021). In particular, nonlinear optical responses are essential to
understanding the symmetry and geometry of the electron wave function.
Nonlinear phenomena attracting recent interest include second and third
harmonic generation, rectification, and photocurrents, all of which are
intimately tied to the nature of the Bloch wave functions Fiebig  _et al._
(2005); Dean and van Driel (2009, 2010); Mikhailov (2011); Kaminski  _et al._
(2009); McIver  _et al._ (2012); You  _et al._ (2019); Takasan  _et al._
(2021); Li  _et al._ (2018); Hipolito and Pereira (2017); Sipe and Ghahramani
(1993); Sipe and Shkrebtii (2000); Zhang  _et al._ (2019); Gao and Zhang
(2021); Fei  _et al._ (2020); Tokura and Nagaosa (2018); Bhalla and Rostami
(2022). Specifically, these effects are mainly determined by the momentum-
space quantum geometric quantities such as Berry curvature, quantum metric,
and metric connection, which tend to be large in systems having broken
inversion symmetry, time-reversal symmetry, or both Xiao  _et al._ (2010); Ahn
_et al._ (2022); Yu  _et al._ (2019); Ahn  _et al._ (2020a).

One famous example that has been studied for decades is the anomalous Hall
effect due to the finite Berry curvature in time-reversal symmetry broken
systems Nagaosa  _et al._ (2010); Culcer (2022); Culcer  _et al._ (2003). Such
responses are robust in gapped systems, which reflect the topology of the
system Yu  _et al._ (2010); Chang  _et al._ (2013); Cullen  _et al._ (2021).
In addition to these, recently, the local quantum geometrical quantity – the
Berry curvature – has been shown to be a key driver of second-order responses
such as the linear photo-currents, quantized circular photo-currents,
injection currents, and the nonlinear Hall effect Sipe and Shkrebtii (2000);
Hosur (2011); Tan and Rappe (2019); Sodemann and Fu (2015); Morimoto  _et al._
(2016); Rostami and Polini (2018); Xiao  _et al._ (2019); Ahn  _et al._
(2020b); Watanabe and Yanase (2021); Bhalla  _et al._ (2021); Kaplan  _et al._
(2022a); Glazov and Golub (2020); Gao and Zhang (2021); Zeng  _et al._ (2021);
Sinha  _et al._ (2022); Chakraborty  _et al._ (2022); Kumar  _et al._ (2020);
Lahiri  _et al._ (2022a, b); Varshney  _et al._ (2022); Bhalla (2021); Zhang
_et al._ (2022); Kaplan  _et al._ (2022b); Golub and Glazov (2022); Leppenen
and Golub (2022). An example of a linear photo-current is the shift current,
which reflects the shift in the position of the electron wave packet upon
excitation from the valence to the conduction band, and is observed in bismuth
telluride, where it is related to Fermi surface anisotropy induced by warping
effects Kim  _et al._ (2017). In contrast, in centrosymmetric systems, the
shift current is contributed by photon drag processes arising from non-
vertical transitions Shi  _et al._ (2021). Second harmonic generation (SHG) is
another example of a second-order response with a geometrical origin:
experimentally it has been observed in inversion symmetry breaking TaAs Weyl
semimetals, whose Weyl nodes are monopoles of the Berry curvature Wu  _et al._
(2017); Patankar  _et al._ (2018); Sirica  _et al._ (2019). All these second-
order photocurrents and SHG phenomena come under the roof of most general
second-order frequency conversion phenomena such as sum frequency generation
(SFG), and difference frequency generation (DFG) Shen (2016); Morita (2018);
de Juan  _et al._ (2020). Conventionally, these processes arise due to the
interference of two incoming beams having frequencies $\omega_{i}$ and
$\omega_{j}$, which generate one outcoming radiation with net frequency
$\omega_{\delta}=\omega_{i}\pm\omega_{j}$. The signal associated with the SFG
is governed by third rank tensor
$\sigma_{abc}(\omega_{\delta};\omega_{j},\omega_{l})$, and the DFG by
$\sigma_{abc}(\omega_{\delta};\omega_{j},-\omega_{l})$. We explore both of
these in the present study.

Remarkably, most studies have been concerned with nonlinear processes specific
to particular materials and in a narrow region of applicability –
overwhelmingly focussing on clean, undoped materials Sipe and Shkrebtii
(2000); Sipe and Ghahramani (1993); Aversa and Sipe (1995). Nevertheless, in
doped systems, the limit $1/\tau\rightarrow 0$, with $\tau$ the relaxation
time scale, is unrealistic. Although nonlinear currents stemming from the
Fermi sea, such as shift, and injection, give finite results, recently
explored nonlinear currents induced by finite Fermi surface contributions,
such as Drude, resonant photogalvanic, and double resonant currents diverge
Bhalla  _et al._ (2020, 2022). Thus to understand the overall behavior of
quantum geometry-driven currents in a consistent manner, it is important to
develop a formalism that comprehensively accounts for nonlinear optical
currents with distinct physical origins within both clean and dirty limits,
which in our opinion, is lacking in the literature.

In this paper, we systematically elaborate the dynamics of the nonlinear
optical current in response to the light-matter interaction in length gauge
within the density matrix formalism while accounting for the disorder and
Fermi surface and sea effects in doped systems. We provide the general
framework of the quantum kinetic theory for distinct second-order optical
processes in response to the external electric or laser fields. We focus on
the second-order currents arising from the mutual interference of the
intraband and interband effects. This allows us to classify the nonlinear
current, according to its origins, into four types, namely intraband-
intraband, intraband-interband, interband-intraband, and interband-interband,
as shown schematically in the tree map of currents in Fig. 1. The Drude
current or the intraband-intraband current is generated due to the momentum
derivative of the non-equilibrium distribution function and shows a divergence
in the clean limit. The intraband-interband arises through the velocity
difference between the distinct bands and is known as the injection current.
The contribution of the intraband response to the interband part of the
density matrix leads to the nonlinear anomalous current proportional to the
Berry curvature, which survives only for a time-reversal symmetric system and
shows a non-resonant structure. However, the resonant counterpart is
proportional to the quantum metric and contributes to $\mathcal{P}\mathcal{T}$
symmetric systems, which will constitute an essential segment of this paper.
The complete interband coherence effect yields the shift current due to the
shift of the wave packet, double resonant due to the asymmetric Fermi surface
and higher-order pole by the momentum displaced joint density of states. In
addition, we observe that the finite Fermi surface generates strong absorption
peaks in the resonant (part of interband-intraband) and double resonant (part
of interband-interband) components, thus enhancing the total nonlinear
current.

Within the quantum kinetic approach, our study reveals that the finite Fermi
surface contribution is key to the resonant nature of nonlinear responses in
doped systems. This originates from distinct quantum geometric quantities. In
CuMnAs, for example, the nonlinear response is sensitive to the direction of
the magnetization field. Likewise, the nonlinear current due to linearly
polarized light depends on the propagation direction of the incident beams.
The second-order current with the magnetization direction along $\hat{x}$-axis
varies as $j_{y}^{(2)}\sim\cos^{2}\gamma$ and $j_{x}^{(2)}\sim\sin 2\gamma$,
where $\gamma$ is the polarization angle along $\hat{x}$-axis. On the other
hand, in thin films of tilted Weyl semimetals, the quantum geometric
quantities are insensitive to the tilt, which is the parameter breaking time-
reversal. At the same time, the finite tilt makes the response more
pronounced. In addition, the nonlinear current is generated here, irrespective
of the choice of the polarization angles. Moreover, we discuss in detail how
the different nonlinear response components contribute to optical currents
using the symmetry properties of quantum geometric quantities and measurement
geometry.

The paper is organized as follows. In Sec. II, the general theoretical kinetic
framework to compute the optical currents is presented. Here, we solve the
kinetic equations in momentum space to calculate the components of the density
matrix, both diagonal and off-diagonal, in the band index for the linear and
nonlinear cases. Next, the optical nonlinear currents are calculated by
employing the diagonal and off-diagonal density matrix solutions. In Sec.
III.1, the symmetry analysis of different quantum geometric quantities is
given, and the spatial geometrical analysis for various optical conductivity
tensor components is discussed. Later, the theory is tested for materials such
as topological antiferromagnetic CuMnAs and thin film tilted Weyl semimetal in
Sec. IV. Finally, in Sec. V, we conclude with future perspectives.

Figure 1: A schematic tree for the generation of different contributions of
the first-order and second-order density matrix $\rho$ that leads to distinct
forms of nonlinear currents. Here the subscripts $d$ and $o$ stand for the
diagonal and off-diagonal parts of the density matrix. In the double
subscripts such as $dd$, $do$, $od$ and $oo$, the first letter indicates the
diagonal and off-diagonal part of the second-order density matrix and later
letter corresponds to the dependence of the relevant part of the first-order
density matrix on the second-order.

##  II Optical Currents and Quantum Kinetic Approach

In this section, we determine the general form of the nonlinear currents in
response to the optical electric field. Specifically, we derive the
polarization or dipole moment per unit volume of a system depending on the
strength of the optical field. Phenomenologically, in response to an optical
field, the time-dependent polarization for a lossless medium can be expressed
in the form Boyd (2008); Bloembergen (1996)

$\tilde{P}_{a}(t)=\sum_{b}\chi_{ab}\tilde{E}_{b}(t)+\sum_{bc}\chi_{abc}\tilde{E}_{b}(t)\tilde{E}_{c}(t)+\cdots,$
(1)

where $\chi_{ab}$, and $\chi_{abc}$ are optical susceptibilities of second-
rank and third-rank respectively and $\tilde{\bm{E}}(t)$ is the optical field.
Further, using the relation between the polarization and the current
${\bm{j}}=d{\bm{P}}/dt$, the optical current is written in the form

$\tilde{\bm{j}}(t)=\tilde{\bm{j}}^{(1)}(t)+\tilde{\bm{j}}^{(2)}(t)+\cdots$ (2)

Here, $\tilde{\bm{j}}^{(1)}$ and $\tilde{\bm{j}}^{(2)}$ are proportional to
the first and second power of the optical field, respectively. For an optical
field of the form
$\tilde{\bm{E}}(t)=\sum_{j}{\bm{E}}^{\omega_{j}}e^{-i\omega_{j}t}$ where
${\bm{E}}^{-\omega_{j}}={\bm{E}}^{*\omega_{j}}$ having ${\bm{E}}^{*}$ as the
complex conjugate of the field ${\bm{E}}$, the second-order optical current
can be written as

$\tilde{j}_{a}^{(2)}(t)=\sum_{bc}\sigma_{abc}\tilde{E}_{b}(t)\tilde{E}_{c}(t),$
(3)

where $\sigma_{abc}$ denotes the second-order optical conductivity of the
system. Further, depending on the frequency components of the optical field,
the nonlinear optical current is given by

$\tilde{j}_{a}^{(2)}(t)=\sum_{\delta}j_{a}^{(2)}(\omega_{\delta})e^{-i\omega_{\delta}t},$
(4)

where the summation runs over distinct components of the frequency. The
different combinations of frequency components lead to various nonlinear
phenomena, namely sum frequency summation (SFG)
$j_{a}^{(2)}(\omega_{j}+\omega_{l})$, second harmonic generation (SHG)
$j_{a}^{(2)}(2\omega_{j})$, difference frequency generation (DFG)
$j_{a}^{(2)}(\omega_{j}-\omega_{l})$, and optical rectification (OR)
$j_{a}^{(2)}(0)$. We employ the density matrix approach to evaluate the
corresponding nonlinear response functions, which is discussed in the
following subsection.

###  II.1 Density matrix approach

We start from the quantum Liouville equation for the time-dependent single-
particle density matrix $\rho({\bm{k}},t)$ in the momentum space Culcer  _et
al._ (2017),

$\begin{array}[]{rl}&\displaystyle\frac{\partial\rho({\bm{k}},t)}{\partial
t}+\frac{i}{\hbar}[\mathcal{H}({\bm{k}},t),\rho({\bm{k}},t)]=0.\end{array}$
(5)

Here $\mathcal{H}({\bm{k}},t)$ is the full Hamiltonian of the system,
including the light-matter interaction, and $[\cdot,\cdot]$ refers to the
commutator bracket. In the length gauge, the perturbed Hamiltonian for the
spatially uniform and time-varying optical field reduces in the form

$\mathcal{H}({\bm{k}},t)=\mathcal{H}_{0}({\bm{k}})+\mathcal{H}_{E}(t)+U,$ (6)

where $\mathcal{H}_{0}({\bm{k}})$ is the unperturbed and Bloch Hamiltonian of
the system, $\mathcal{H}_{E}(t)=e{\bm{r}}\cdot{\bm{E}}(t)$ represents the
interaction with the electric field, ‘$-e$’ the electronic charge and $U$ the
disorder potential. Considering Eq. (6), the quantum kinetic equation (5) on
averaging over disorder configurations takes the form

$\frac{\partial\rho({\bm{k}},t)}{\partial
t}+\frac{i}{\hbar}[\mathcal{H}_{0},\rho({\bm{k}},t)]+J(\rho({\bm{k}},t))=-\frac{i}{\hbar}[\mathcal{H}_{E},\rho({\bm{k}},t)].$
(7)

Here $J(\rho)$ represents the scattering term that takes into account the
impact of the disorder potential. In the present framework, we treat the
scattering term under the relaxation time approximation and approximate the
term as $\rho({\bm{k}},t)/\tau$, with $\tau$ being a parameter specifying the
time taken to relax the system towards the equilibrium state or the relaxation
time scale. For simplicity, we consider the relaxation time scale $\tau$ as a
constant parameter across the Fermi surface. Therefore, the kinetic equation
becomes

$\frac{\partial\rho}{\partial
t}+\frac{i}{\hbar}[\mathcal{H}_{0},\rho]+\frac{\rho-\rho^{(0)}}{\tau}=-\frac{i}{\hbar}[\mathcal{H}_{E},\rho].$
(8)

We have written $\rho({\bm{k}},t)$ as $\rho$ to simplify the notation, while
$\rho^{(0)}$ is the equilibrium density matrix. To find the solution for the
kinetic equation, we expand the density matrix perturbatively in the powers of
the time-dependent and space-homogeneous optical field.

$\rho=\rho^{(0)}+\rho^{(1)}+\rho^{(2)}+\cdots,$ (9)

where $\rho^{(N)}\propto E^{(N)}$ having the superscript ‘$N$’ for an order of
the field.

In the band basis representation, the density matrix for the most simple two-
band model can be represented as

$\rho=\begin{pmatrix}\rho_{mm}&\rho_{mp}\\\ \rho_{pm}&\rho_{pp}\end{pmatrix},$
(10)

where $m$ and $p$ refer to band indices. Within this band basis
representation, Eq. (8) for the $N^{\text{th}}$ order density matrix can be
written

$\frac{\partial\rho_{mp}^{(N)}}{\partial
t}+\frac{i}{\hbar}[\mathcal{H}_{0},\rho^{(N)}]_{mp}+\frac{\rho^{(N)}_{mp}}{\tau}=\frac{e{\bm{E}}(t)}{\hbar}\cdot\left[D_{\bm{k}}\rho^{(N-1)}\right]_{mp}.$
(11)

The covariant derivative Nagaosa and Morimoto (2017)
$[D_{\bm{k}}\rho]_{mp}=\partial_{\bm{k}}\rho_{mp}-i[\mathcal{R}_{\bm{k}},\rho]_{mp}$
where $\mathcal{R}_{mp}({\bm{k}})=\langle
u_{\bm{k}}^{m}|i\partial_{\bm{k}}u_{{\bm{k}}}^{p}\rangle$ is the momentum
space Berry connection with $|u_{\bm{k}}^{m}\rangle$ the periodic part of the
Bloch wave-function, and $\partial_{\bm{k}}$ represents the momentum
derivative. This is obtained by inserting the expression for $\mathcal{H}_{E}$
in the commutator $[\mathcal{H}_{E},\rho]$ and using the relation
$|m,{\bm{k}}\rangle=e^{-i{\bm{k}}\cdot{\bf r}}|u_{\bm{k}}^{m}\rangle$ and
$\hat{{\bm{r}}}|m,{\bm{k}}\rangle=i[\partial_{\bm{k}}e^{-i{\bm{k}}\cdot{\bf
r}}]|u_{\bm{k}}^{m}\rangle$. Further, the corresponding right-hand side term
serves as the driving term that generates a response in the system. It is also
a fully intrinsic term and is determined by the electronic structure of the
system. Note that the right-hand side of the equation contains the
$(N-1)^{th}$-order density matrix due to the presence of the field factor. It
is also evident from this expression that to find the solution of the density
matrix of order $N\geq 1$, one requires the solution for the proceeding order
due to $(N-1)$ order term in the right-hand side of the Eq. (11). However, for
$N=0$ case the right side of Eq. (11) approaches to zero which gives
$\rho_{mp}^{(0)}=f^{0}(\varepsilon_{\bm{k}}^{m})\delta_{mp}$ the equilibrium
Fermi-Dirac distribution function and is defined as
$f^{0}(\varepsilon_{\bm{k}}^{m})=[e^{\beta(\varepsilon_{\bm{k}}^{m}-\mu)}+1]^{-1}$
having $\beta=[k_{B}T]^{-1}$ with $k_{B}$ the Boltzmann constant, $T$ the
electron temperature, $\mu$ represents the chemical potential, and
$\varepsilon_{\bm{k}}^{m}$ corresponds to the electron dispersion for
$m^{\text{th}}$ band. To study the dynamics of the linear and nonlinear
currents, we calculate the diagonal ($m=p$) or the intraband and the off-
diagonal ($m\neq p$) or the interband part of the density matrix correspond to
the linear and the quadratic power of the electric field in the following
subsections.

####  II.1.1 Linear order density matrix

To recover the linear response, we set $N=1$ and solve the kinetic equation by
splitting the density matrix into diagonal and off-diagonal components in the
band index such as $\rho=\rho_{mm}\delta_{mp}+\rho_{mp}$. Here the first term
is the diagonal or intraband part of the density matrix owing to the Dirac
delta function, which vanishes for insulators. The second term refers to the
off-diagonal or interband coherence part of the density matrix.

Firstly for the intraband contribution to the density matrix or $m=p$ case,
Eq. (11) reduces to

$\frac{\partial\rho_{mm}^{(1)}}{\partial
t}+\frac{\rho_{mm}^{(1)}}{\tau_{\alpha}}=\frac{e{\bm{E}}(t)}{\hbar}\cdot\frac{\partial
f^{0}(\varepsilon_{\bm{k}}^{m})}{\partial{\bm{k}}},$ (12)

where $\tau_{\alpha}$ is the time scale for the intraband transitions. It is
to be noted that the commutator between the Bloch state Hamiltonian
$\mathcal{H}_{0}$ and the zeroth order density matrix $\rho^{(0)}$ is zero. On
solving the linear order differential equation by taking an integrating
factor, the intraband time-dependent density matrix takes the form

$\rho_{mm}^{(1)}=\frac{e}{\hbar}\sum_{j}\partial_{c}f_{m}^{0}g_{0;\alpha}^{\omega_{j}}E_{c}^{\omega_{j}}e^{-i\omega_{j}t}.$
(13)

Here, we define $g_{0;\alpha}^{\omega_{j}}=[1/\tau_{\alpha}-i\omega_{j}]^{-1}$
and $\partial_{k_{c}}\equiv\partial_{c}$ for brevity. Similarly, the interband
component ($m\neq p$) of the density matrix comes out to be

$\frac{\partial\rho_{mp}^{(1)}}{\partial
t}+i\omega_{mp}\rho_{mp}^{(1)}+\dfrac{\rho_{mp}^{(1)}}{\tau_{\gamma}}=i\frac{eE_{c}(t)}{\hbar}{\mathcal{R}}_{mp}^{c}F_{mp}.$
(14)

With $\hbar\omega_{mp}=\varepsilon_{m,{\bm{k}}}-\varepsilon_{p,{\bm{k}}}$ as
the interband transition energy at momentum ${\bf k}$, $\tau_{\gamma}$ as the
time scale corresponding to the interband transitions and
$F_{mp}=f^{0}(\varepsilon_{\bm{k}}^{m})-f^{0}(\varepsilon_{\bm{k}}^{p})$ is
the difference in the occupation between two distinct bands. Here we use the
relation $D_{\bm{k}}\rho^{(0)}=-i[\mathcal{R}_{\bm{k}},\rho^{(0)}]$ due to
vanishing $\partial_{\bm{k}}\rho_{mp}^{(0)}$ as the equilibrium part contains
only the diagonal elements in the band index. The solution of Eq. (14) for the
off-diagonal part of the density matrix gives

$\rho_{mp}^{(1)}=i\frac{e}{\hbar}\sum_{j}\mathcal{R}_{mp}^{c}F_{mp}g_{mp;\gamma}^{\omega_{j}}E_{c}^{\omega_{j}}e^{-i\omega_{j}t},$
(15)

where
$g_{mp;\gamma}^{\omega_{j}}=[1/\tau_{\gamma}-i(\omega_{j}-\omega_{mp})]^{-1}$
relates to the joint density of states broadened by the interband relaxation
time scale. From this, we find that the linear order $\rho_{mp}^{(1)}$ depends
on the shift of the Fermi function $F_{mp}$ and this contribution survives
only for $m\neq p$. In a compact form, the complete solution of the first-
order density matrix can be written as

$\rho_{mp}^{(1)}=\dfrac{e}{\hbar}\sum_{j}\tilde{\rho}_{mp;j}^{(1),c}E_{c}^{\omega_{j}}e^{-i\omega_{j}t},$
(16)

where

$\begin{array}[]{rl}\tilde{\rho}_{mp;j}^{(1),c}=\tilde{\rho}_{mp}^{(1),c}(\omega_{j})&\displaystyle=\partial_{c}\rho_{mp}^{(0)}g_{0;\alpha}^{\omega_{j}}\delta_{mp}+i{\mathcal{R}}_{mp}^{c}F_{mp}g_{mp;\gamma}^{\omega_{j}}.\end{array}$
(17)

This expression can be further simplified in the low temperature region by
replacing the Fermi distribution function and its energy derivative with the
Heaviside step function $\Theta(\varepsilon_{\bf k}^{m}-\mu)$ and Dirac delta
function $-\delta(\varepsilon_{\bf k}^{m}-\mu)$.

####  II.1.2 Density matrix to second-order in the electric field

Substituting $N=2$ in Eq. (11), we get

$\frac{\partial\rho_{mp}^{(2)}}{\partial
t}+\dfrac{i}{\hbar}[\mathcal{H}_{0},\rho^{(2)}]_{mp}+\dfrac{\rho_{mp}^{(2)}}{\tau}=\frac{eE_{b}(t)}{\hbar}[D_{b}\rho^{(1)}]_{mp}.$
(18)

Using the solution for the linear order density matrix Eq. (16), we obtain

$\begin{array}[]{rl}&\displaystyle\frac{\partial\rho_{mp}^{(2)}}{\partial
t}+i\omega_{mp}\rho^{(2)}_{mp}+\dfrac{\rho_{mp}^{(2)}}{\tau_{\alpha}}\\\\[12.91663pt]
&\displaystyle=\frac{e^{2}}{\hbar^{2}}\sum_{P}\sum_{j,l}\bigg{\\{}\left[\partial_{b}\tilde{\rho}_{mp;l}^{(1),c}-i\sum_{n}\left(\mathcal{R}^{b}_{mn}\tilde{\rho}_{np;l}^{(1),c}-\tilde{\rho}_{mn;l}^{(1),c}\mathcal{R}^{b}_{np}\right)\right]\\\\[12.91663pt]
&\displaystyle~{}~{}~{}~{}E_{b}^{\omega_{l}}E_{c}^{\omega_{j}}e^{-i(\omega_{j}+\omega_{l})t}\bigg{\\}}.\end{array}$
(19)

Here, the sum over $P$ refers to the intrinsic permutation symmetry
$(b,\omega_{l}\leftrightarrow c,\omega_{j})$. For $m=p$ case,
$\rho_{mm}^{(2)}$ can be written as a sum of two terms,
$\rho_{mm}^{(2)}=\rho_{mm}^{(2),dd}+\rho_{mm}^{(2),do}$ where the first term
refers to the intraband-intraband (or diagonal-diagonal) part and stems from
the diagonal component of the first-order density matrix. The second term
corresponds to the intraband-interband (or diagonal-off diagonal) part by the
off-diagonal component of the first-order density matrix. The first term
$\rho_{mm}^{(2),dd}$ is defined as

$\rho_{mm}^{(2),dd}=\dfrac{e^{2}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{j,l}\tilde{\rho}_{mm}^{(2),dd}(\omega_{\delta};\omega_{j},\omega_{l})E_{b}^{\omega_{l}}E_{c}^{\omega_{j}}e^{-i\omega_{\delta}t},$

having

$\tilde{\rho}_{mm}^{(2),dd}(\omega_{\delta};\omega_{j},\omega_{l})=\partial_{b}\tilde{\rho}_{mm;l}^{(1),c}g_{0;\alpha}^{\omega_{\delta}}.$
(20)

Here, we define $\omega_{\delta}=\omega_{j}+\omega_{l}$. Similarly, the
intraband-interband part of the density matrix $\rho_{mm}^{(2),do}$ is given
by

$\begin{array}[]{rl}\tilde{\rho}_{mm}^{(2),do}(\omega_{\delta};\omega_{j},\omega_{l})=&\displaystyle-i\sum_{n}\left(\mathcal{R}^{b}_{mn}\tilde{\rho}_{nm;l}^{(1),c}-\tilde{\rho}_{mn;l}^{(1),c}\mathcal{R}^{b}_{nm}\right)g_{0;\alpha}^{\omega_{\delta}}.\end{array}$
(21)

In the same spirit, the density matrix for $m\neq p$ case can be expressed in
the form $\rho_{mp}^{(2)}=\rho_{mp}^{(2),od}+\rho_{mp}^{(2),oo}$. Firstly, the
interband-intraband part $\rho_{mp}^{(2),od}$ is defined in the form

$\tilde{\rho}_{mp}^{(2)od}(\omega_{\delta};\omega_{j},\omega_{l})=-i\mathcal{R}^{b}_{mp}\left(\tilde{\rho}_{pp;l}^{(1),c}-\tilde{\rho}_{mm;l}^{(1),c}\right)g_{mp;\gamma}^{\omega_{\delta}}.$
(22)

Secondly, the interband-interband part of the density matrix is given by

$\begin{array}[]{rl}\tilde{\rho}_{mp}^{(2),oo}(\omega_{\delta};\omega_{j},\omega_{l})=&\displaystyle
g_{mp;\gamma}^{\omega_{\delta}}\Big{[}\left\\{\partial_{b}-i\left(\mathcal{R}^{b}_{mm}-\mathcal{R}^{b}_{pp}\right)\right\\}\tilde{\rho}_{mp;l}^{(1),c}\\\\[12.91663pt]
&\displaystyle-i\sum_{n\neq(m,p)}\left(\mathcal{R}^{b}_{mn}\tilde{\rho}_{np;l}^{(1),c}-\tilde{\rho}_{mn;l}^{(1),c}\mathcal{R}^{b}_{np}\right)\Big{]}.\end{array}$
(23)

Here, the second term in the open brackets contributes to multi-band systems.

###  II.2 Nonlinear Optical Currents

To obtain the expressions for nonlinear (i.e., second-order in the field)
optical currents, we use the general definition of the current, which is the
trace of the velocity operator with the density matrix. Accordingly, the time-
dependent optical current for $d$-dimensional momentum space in the band index
basis can be expressed in the form

${\bm{j}}(t)=-e\sum_{p}\int\frac{d^{d}{\bm{k}}}{(2\pi)^{d}}\langle
p|{\bm{v}}({\bm{k}})\rho({\bm{k}},t)|p\rangle.$ (24)

Here, ${\bm{v}}_{pm}({\bm{k}})$ the velocity matrix in the band basis in the
Bloch representation is
${v}^{a}_{pm}({\bm{k}})=\hbar^{-1}(\delta_{pm}\partial_{a}\varepsilon_{\bm{k}}^{m}+i\mathcal{R}^{a}_{pm}\hbar\omega_{pm})$.
The first part refers to the group velocity of Bloch electrons along the
spatial direction $a$, and the second part corresponds to the interband
velocity component. Using this, the current in an arbitrary direction can be
expressed as $j^{a}(t)=-e\sum_{m,p}\sum_{\bm{k}}v_{pm}^{a}\rho_{mp}^{(2)}(t)$.
Below we make use of the different interband and intraband components of the
density matrix, such as $\rho^{dd}$, $\rho^{do}$, $\rho^{od}$ and $\rho^{oo}$
to obtain the nonlinear currents and classified them accordingly.

####  II.2.1 Intraband-Intraband Current

For the intraband-intraband ($dd$) contribution of the density matrix, the
current is given by

$\tilde{j}_{a}^{(2),dd}(t)=-e\sum_{m}\sum_{\bm{k}}v_{mm}^{0a}\rho_{mm}^{(2),dd}(t),$
(25)

where $v_{mm}^{0a}=\hbar^{-1}\partial_{a}\varepsilon_{\bm{k}}^{m}$ is the
diagonal component of the velocity in the band basis. Using Eq. (II.1.2) in
the above equation, the second-order conductivity $\sigma_{dd}$ can be
expressed in the form

$\sigma^{dd}_{abc}(\omega_{\delta};\omega_{j},\omega_{l})=-\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m}v_{mm}^{0a}\partial_{b}\partial_{c}f_{m}^{(0)}g_{0;\alpha}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}.$
(26)

It is clear from the expression that $\sigma^{dd}$ is the Fermi surface
contribution due to the presence of the momentum derivative of the Fermi
function. In addition, it is entirely the intraband contribution and stems
from the single band only. Note that this contribution to the current is only
finite for metals and semimetals and can be completely ignored for insulators.

Moreover, the frequency dependence of this optical current is captured by the
product of two $g_{0}$ factors having the following form

$g_{0;\alpha}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}=\dfrac{1}{1/\tau_{\alpha}-i\omega_{l}}\times\dfrac{1}{1/\tau_{\alpha}-i\omega_{\delta}}.$
(27)

It is helpful to expand the product of $g_{0}$ factors by partial separation
below as

$g_{0;\alpha}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}=\frac{\tau_{\alpha}}{i(\omega_{l}-\omega_{\delta})}\left(\frac{1-i\omega_{l}\tau_{\alpha}}{1+\omega_{l}^{2}\tau_{\alpha}^{2}}-\frac{1-i\omega_{\delta}\tau_{\alpha}}{1+\omega_{\delta}^{2}\tau_{\alpha}^{2}}\right),$
(28)

where we have considered $(\omega_{l}-\omega_{\delta})\neq 0$. We emphasize
that one can not reproduce the dc limit from the above equation. In the low
frequency limit, quantitatively defined as $\omega\tau\ll 1$, Eq. (27) reduces
as
$g_{0;\alpha}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}=\tau_{\alpha}^{2}$,
which has the Drude nonlinear current like relaxation time dependence. In the
opposite frequency regime $\omega\tau\gg 1$, Eq. (27), the product of
$g$-factors varies with the inverse quadratic dependence of the frequency like
$g_{0;\alpha}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}=1/\omega_{\delta}\omega_{l}$.
This generates a similar expression as quoted in Ref. [Watanabe and Yanase,
2021] and is referred to as the nonlinear Drude term. Note that the nonlinear
Drude current does not rely on band geometric quantities.

####  II.2.2 Intraband-Interband Current

We consider the contribution of the intraband-interband ($do$) part of the
density matrix $\rho_{mm}^{(2),do}$ to the current. Here, the response arises
from the band geometrical quantity and the change in group velocity of the
carrier on transiting from one band to another band. The resulting nonlinear
current is termed as the injection current. It is given by

$j_{a}^{(2),do}(t)=-e\sum_{m}\sum_{\bm{k}}v_{mm}^{0a}\rho_{mm}^{(2),do}(t).$
(29)

Using the nonlinear density matrix calculated in Eq. (21), we have the
following expression for the nonlinear conductivity

$\sigma_{abc}^{do}(\omega_{\delta};\omega_{l},\omega_{j})=\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\Delta^{a}_{mp}\mathcal{Q}_{mp}^{bc}F_{mp}g_{mp;\gamma}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}.$
(30)

Here, we have defined $\Delta^{a}_{mp}=(v_{mm}^{0a}-v_{pp}^{0a})$, the
difference between the group velocities of two bands. We have also defined the
geometrical quantity, the band-resolved quantum geometric tensor (QGT) Ma  _et
al._ (2010) as
${\mathcal{Q}}_{mp}^{bc}=\mathcal{R}^{b}_{pm}\mathcal{R}^{c}_{mp}=(G_{mp}^{bc}-i/2\Omega_{mp}^{bc})$.
Here, the quantum metric is
$G_{mp}^{bc}=\\{{\mathcal{R}}^{b}_{pm},{\mathcal{R}}_{mp}^{c}\\}/2$ which is
symmetric under the exchange of spatial and band indices. On the other hand,
the Berry curvature
$\Omega_{pm}^{bc}=i[{\mathcal{R}}^{b}_{pm},{\mathcal{R}}_{mp}^{c}]/2$ is
antisymmetric under the exchange of band and spatial indices. The optical
frequency dependence of this part of conductivity comes from

$g_{mp;\gamma}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}=\dfrac{1}{1/\tau_{\gamma}-i(\omega_{l}-\omega_{mp})}\times\dfrac{1}{1/\tau_{\alpha}-i\omega_{\delta}}.$
(31)

On expanding the product of $g$-factors, one ends up with four terms having
two real and two imaginary terms. Here, the leading contributing terms to the
nonlinear current are

$\displaystyle g_{mp;\gamma}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}$
$\displaystyle\approx\frac{1/\tau_{\alpha}}{1/\tau_{\alpha}^{2}+\omega_{\delta}^{2}}\frac{1/\tau_{\gamma}}{1/\tau_{\gamma}^{2}+(\omega_{l}-\omega_{mp})^{2}}$
$\displaystyle+i\frac{\omega_{\delta}}{1/\tau_{\alpha}^{2}+\omega_{\delta}^{2}}\frac{1/\tau_{\gamma}}{1/\tau_{\gamma}^{2}+(\omega_{l}-\omega_{mp})^{2}}.$
(32)

This yields two terms to the interband-intraband conductivity. However, when
we take the limit $\omega_{\delta}\rightarrow 0$ in the above expression and
then consider $1/\tau_{\gamma}\rightarrow 0$, the second term vanishes. The
resulting expression is

$\displaystyle\sigma_{abc}^{do}(\omega_{\delta};\omega_{l},\omega_{j})=$
$\displaystyle-\frac{2\pi
e^{3}}{\hbar^{2}}\tau_{\alpha}\sum_{b,c}\sum_{m,p}\Delta_{mp}^{a}\mathcal{Q}_{mp}^{bc}F_{mp}\delta(\omega_{mp}-\omega_{l}).$
(33)

Here we use
$g_{mp;\gamma}^{\omega_{l}}g_{0;\alpha}^{\omega_{\delta}}=-\tau_{\alpha}\pi\delta(\omega_{mp}-\omega_{l})$.
This expression is similar to the expression given by Ahn et al.Ahn  _et al._
(2020a) using the Fermi golden rule. Notably, the injection conductivity is
linearly proportional to the relaxation time and is finite only in the dirty
limit. In addition, this is the Fermi sea response, and the corresponding real
part of the response is proportional to the quantum metric. On the other hand,
taking limits such as $1/\tau_{\gamma}\rightarrow 0$ and then
$\omega_{\delta}\rightarrow 0$ the first term of Eq. (II.2.2) becomes zero and
the nonlinear conductivity becomes

$\displaystyle\sigma_{abc}^{do}(\omega_{\delta};\omega_{l},\omega_{j})=$
$\displaystyle-i\lim_{\omega_{\delta}\rightarrow 0}\frac{2\pi
e^{3}}{\omega_{\delta}\hbar^{2}}\sum_{b,c}\sum_{m,p}\Delta_{mp}^{a}\mathcal{Q}_{mp}^{bc}F_{mn}\delta(\omega_{mp}-\omega_{l}).$
(34)

This expression diverges at $\omega_{\delta}\rightarrow 0$ and is consistent
with the calculation quoted recently using the kinetic approach Watanabe and
Yanase (2021). However, such divergence can be avoided by inserting the
relaxation time factor. Further, here the real part of the response depends on
the Berry curvature due to the imaginary factor in the expression.

Note that Eqs. (II.2.2) and (II.2.2) are obtained by taking limiting cases and
yield different results for the injection current. One needs to be cautious
before taking the limits as it may lead to different results. To avoid such
confusion, we thoroughly provide the complete expression (Eq. 36) of the
injection conductivity without taking any limits which captures both the cases
shown in the literature Ahn  _et al._ (2020a); Watanabe and Yanase (2021).
Further, we observe that the resonance features survive only for the doped
system, and the resulting peak is controlled by the relaxation time scale.
This is ultimately a Fermi sea effect determined by the joint density of
states and the band velocity difference.

####  II.2.3 Interband-Intraband Current

We consider the current due to the off-diagonal contribution of the second-
order density matrix depending on the diagonal part of the first-order density
matrix. It reads

$j_{a}^{(2),od}(t)=-e\sum_{m}\sum_{\bm{k}}v_{pm}^{a}\rho_{mp}^{(2),od}(t).$
(35)

Substituting the expression for the interband-intraband density matrix
component $\rho_{mp}^{(2),od}$ from Eq. (22), the interband-intraband
nonlinear response becomes

$\sigma^{od}_{abc}(\omega_{\delta};\omega_{j},\omega_{l})=-\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\omega_{mp}\mathcal{Q}_{mp}^{ab}g_{0;\alpha}^{\omega_{l}}g_{mp;\gamma}^{\omega_{\delta}}\partial_{c}F_{mp}.$
(36)

The presence of the momentum derivative of the difference in the band
occupation, $\partial_{c}F_{mp}$ dictates that Eq. (36) is a Fermi surface
effect. Although this current is a second-order effect, it does not
simultaneously generate one- and two-photon absorption processes. Instead, it
only gives either absorption process depending on the incident energy of two
optical beams. To separate the resonant and non-resonant parts of the
conductivity, it is helpful to employ the identity

$\dfrac{\omega_{mp}}{1/\tau_{\gamma}-i(\omega-\omega_{mp})}=-i\left[1-\dfrac{1/\tau_{\gamma}-i\omega}{1/\tau_{\gamma}-i(\omega-\omega_{mp})}\right].$
(37)

Using the above relation we separate Eq. (36) into two parts as
$\sigma^{od}_{abc}(\omega_{\delta};\omega_{j},\omega_{l})=\sigma^{od,I}_{abc}(\omega_{\delta};\omega_{j},\omega_{l})+\sigma^{od,II}_{abc}(\omega_{\delta};\omega_{j},\omega_{l})$.
The first (non-resonant) part is to be

$\sigma^{od,I}_{abc}(\omega_{\delta};\omega_{j},\omega_{l})=\frac{e^{3}}{2\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\Omega_{mp}^{ab}g_{0;\alpha}^{\omega_{l}}\partial_{c}F_{mp}.$
(38)

We emphasize that when writing Eq. (38) we used the fact that $\Omega_{mp}$ is
anti-symmetric and ${\mathcal{G}}_{mp}$ is symmetric in band index. Such
simplification based on symmetry and anti-symmetry is only possible since
there are no resonance factors with the band index ($g_{mp}$). This non-
resonant part is known as the anomalous nonlinear response Rostami and Polini
(2018). The other (resonant) part of the conductivity is given by

$\sigma^{od,II}_{abc}(\omega_{\delta};\omega_{j},\omega_{l})=-i\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\mathcal{Q}_{mp}^{ab}\partial_{c}F_{mp}\frac{g_{0;\alpha}^{\omega_{l}}g_{mp;\gamma}^{\omega_{\delta}}}{g_{0;\gamma}^{\omega_{\delta}}}.$
(39)

Eq. (39) is one of the central results of this paper. This part of the current
will show resonance behavior near the Fermi energy, which can be inferred from
the derivative of the Fermi function. The optical field-dependent part is

$\frac{g_{0;\alpha}^{\omega_{l}}g_{mp;\gamma}^{\omega_{\delta}}}{g_{0;\gamma}^{\omega_{\delta}}}=\dfrac{1}{1/\tau_{\alpha}-i\omega_{l}}\times\dfrac{1/\tau_{\gamma}-i\omega_{\delta}}{1/\tau_{\gamma}-i(\omega_{\delta}-\omega_{mp})}.$
(40)

In the clean limit, one obtains

$g_{0;\alpha}^{\omega_{l}}g_{mp;\gamma}^{\omega_{\delta}}=i\dfrac{1}{\omega_{l}}\times\dfrac{\omega_{\delta}}{\omega_{\delta}-\omega_{mp}}.$
(41)

and in such a case, this current diverges at the lower energy scale and
approaches zero at higher energy. We emphasize that it is not right to
consider a dc field limit of Eq. (41) after applying the clean limit. Instead,
it is preferable to consider the dc limit from Eq. (40) and then consider the
clean or dirty limit. Notably, the resonant feature here will be observed at
$\omega_{\delta}=\omega_{mp}$.

####  II.2.4 Interband-Interband Current

Finally, the substitution of the interband part of the first-order density
matrix into the interband part of second-order $\rho_{mp}^{(2)}$ yields the
interband-interband current. It is defined as

$j_{a}^{(2),oo}(t)=-e\sum_{m,p}v_{pm}^{a}\rho_{mp}^{(2),oo}(t).$ (42)

We may refer to this current as purely an interband coherence current as it
arises from interband geometric quantities, off-diagonal components of the
velocity, and the joint density of states. Now, with [Eq. (23)] for
$\rho_{mp}^{(2),oo}$ and using the definition
$\mathcal{D}_{mp}^{b}=\partial_{b}-i\left(\mathcal{R}^{b}_{mm}-\mathcal{R}^{b}_{pp}\right)$,
the corresponding nonlinear conductivity part comes out to be

$\begin{array}[]{rl}&\displaystyle\sigma_{abc}^{oo}(\omega_{\delta};\omega_{j},\omega_{l})\\\\[12.91663pt]
&\displaystyle=-\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\mathcal{R}^{a}_{pm}\omega_{mp}g_{mp;\gamma}^{\omega_{\delta}}\Big{[}{\mathcal{D}}_{mp}^{b}{\mathcal{R}}_{mp}^{c}F_{mp}g_{mp;\gamma}^{\omega_{l}}\\\\[12.91663pt]
&\displaystyle-i\sum_{n\neq(m,p)}\left(\mathcal{R}^{b}_{mn}{\mathcal{R}}_{np}^{c}F_{np}g_{np;\gamma}^{\omega_{l}}-{\mathcal{R}}_{mn}^{c}\mathcal{R}^{b}_{np}F_{mn}g_{mn;\gamma}^{\omega_{l}}\right)\Big{]}.\end{array}$
(43)

Further with the help of Eq. (37) the above mentioned conductivity can be
written as
$\sigma_{abc}^{oo}(\omega_{\delta},\omega_{j},\omega_{l})=\sigma_{abc}^{oo,I}(\omega_{\delta},\omega_{j},\omega_{l})+\sigma_{abc}^{oo,II}(\omega_{\delta},\omega_{j},\omega_{l})+\sigma_{abc}^{oo,III}(\omega_{\delta},\omega_{j},\omega_{l})$
by applying the covariant derivative separately to distinct factors. The last
term of the above equation (Eq. II.2.4) contributes only to the multi-band
(more than two bands) systems. Here, the first part is given by

$\begin{array}[]{rl}&\displaystyle\sigma_{abc}^{oo,I}(\omega_{\delta};\omega_{j},\omega_{l})\\\
&\displaystyle=-\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\mathcal{R}^{a}_{pm}\omega_{mp}g_{mp;\gamma}^{\omega_{\delta}}g_{mp;\gamma}^{\omega_{l}}F_{mp}{\mathcal{D}}_{mp}^{b}{\mathcal{R}}_{mp}^{c}.\end{array}$
(44)

More simplifications are made using the sum rule Cook  _et al._ (2017) for the
covariant derivative on the Berry connection. Accordingly,

$\begin{array}[]{rl}{\cal
D}^{b}_{mp}{\mathcal{R}}^{c}_{mp}=&\displaystyle-\dfrac{1}{i\omega_{mp}}\left[\dfrac{v_{mp}^{b}\varDelta_{mp}^{c}+v_{mp}^{c}\varDelta_{mp}^{b}}{\omega_{mp}}\right.-w_{mp}^{bc}\\\\[12.91663pt]
&\displaystyle\left.+\sum_{n\neq(p,m)}\left(\dfrac{v_{mn}^{c}v_{np}^{b}}{\omega_{np}}-\dfrac{v_{mn}^{b}v_{np}^{c}}{\omega_{mn}}\right)\right].\end{array}$
(45)

Here, we have defined $w_{mp}^{bc}=\langle m|\partial_{b}\partial_{c}{\cal
H}|p\rangle$. From Eq. (45), it is straightforward that the last term does not
contribute to the two-band model. More precisely, the corresponding
conductivity for two band model can be written in the form as

$\begin{array}[]{rl}\sigma^{oo,I}_{abc}&\displaystyle=-\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\omega_{mp}\mathcal{C}_{mp}^{abc}g_{mp}^{\omega_{\delta}}g_{mp}^{\omega_{l}}F_{mp},\end{array}$
(46)

where
$\mathcal{C}_{mp}^{abc}=\mathcal{R}_{pm}^{a}\mathcal{D}^{b}_{mp}\mathcal{R}^{c}_{mp}=\Gamma_{mp}^{abc}+i\tilde{\Gamma}_{mp}^{abc}$
is the quantum geometric connection which is the sum of the quantum geometric
quantities namely metric connection $\Gamma_{mp}^{abc}$ and symplectic
connection $\tilde{\Gamma}_{mp}^{abc}$. Originally, the quantum geometric
connection $\mathcal{C}_{mp}^{abc}$ in the tangent subspace spanned by basis
vectors $\hat{e}_{mn}^{b}$ stems from the inner product of the tangent basis
vector $\hat{e}_{mp}^{b}$ and the derivative of such vectors
$\nabla_{b}\hat{e}_{mp}^{c}$. Further, $\tilde{\Gamma}_{mp}^{abc}$ is directly
related to the shift vector. To elaborate, consider the case of $a=c$ and
write Berry connection as
$\mathcal{R}_{mp}^{a}=|\mathcal{R}_{mp}^{a}|e^{i\phi_{mp}}$ with $\phi_{mp}$ a
phase factor. With this, we get

$\displaystyle\tilde{\Gamma}_{mp}^{abc}$
$\displaystyle=|{\mathcal{R}}^{a}_{mp}|^{2}\partial_{b}\phi_{mp}-\left(\mathcal{R}^{b}_{mm}-\mathcal{R}^{b}_{pp}\right)|{\mathcal{R}}^{a}_{mp}|^{2}~{}.$
(47)

This expression is consistent with the shift vector definition Morimoto and
Nagaosa (2016). The difference between Berry connections indicates the
difference of shifted Bloch wave functions between conduction and valence
band, and the momentum derivative of the phase factor $\partial_{b}\phi_{mp}$
maintains the gauge invariance. Further, using Eq. (37) and expressing
$g_{mp}^{\omega_{l}}$ with Sokhotski–Plemelj relation, it is straightforward
to express the interband-interband response in the form of the shift response
as shown in the literature Ahn  _et al._ (2020a); Kim  _et al._ (2017);
Watanabe and Yanase (2021). The second part of $\sigma_{abc}^{oo}$ is

$\sigma_{abc}^{oo,II}(\omega_{\delta};\omega_{j},\omega_{l})=-\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\omega_{mp}g_{mp}^{\omega_{\delta}}g_{mp}^{\omega_{l}}{\mathcal{Q}}^{ac}_{mp}\partial_{b}F_{mp}.$
(48)

This is the Fermi surface effect and arises due to the asymmetric Fermi
surface in the momentum space and the band geometric quantities. To illustrate
more, let us consider the case of low temperature. In this limit, the momentum
derivative of the Fermi function approaches to
$-\delta(\omega_{mp}-\mu)\partial\varepsilon/\partial{\bm{k}}$ having $\mu$
the chemical potential. If we perform the partial separation of
$g_{mp}^{\omega_{\delta}}g_{mp}^{\omega_{l}}$ and then solve the $k$ integral,
the response gives two resonant peaks at energy $\omega_{\delta}=2\mu$ and
$\omega_{l}=2\mu$. Note that the corresponding response persists irrespective
of geometry. Due to this feature, the associated current is known as the
double resonant current. However, in the case of $\omega_{\delta}\rightarrow
0$ (the optical rectification process), one of the peaks disappears, and the
induced effect is known as the resonant photovoltaic effect Bhalla  _et al._
(2020). Here, the integrand becomes proportional to
$2\tau\partial\varepsilon/\partial k_{a}\delta(\omega_{mp}-\varepsilon_{F})$
at energy scale $\omega_{l}=2\mu$. Clearly, the effect originates due to the
Fermi surface displacement and is finite for the doped systems.

The third part of the interband-interband response, due to the momentum
derivative of the joint density of states, is

$\displaystyle\sigma_{abc}^{oo,III}(\omega_{\delta};\omega_{j},\omega_{l})=$
$\displaystyle-\frac{e^{3}}{\hbar^{2}}\sum_{\mathcal{P}}\sum_{m,p}\omega_{mp}\mathcal{Q}^{ac}_{pm}g_{mp;\gamma}^{\omega_{\delta}}[\partial_{b}g_{mp;\gamma}^{\omega_{l}}]F_{mp}.$
(49)

The associated current is known as the higher-order pole current. This is
finite only for the finite scattering time and vanishes at $1/\tau\rightarrow
0$. To elaborate on the point, we consider the imaginary part of
$g_{mp;\gamma}^{\omega_{\delta}}[\partial_{b}g_{mp;\gamma}^{\omega_{l}}]$
which gives

$\begin{array}[]{rl}&\displaystyle{\rm
Im}[g_{mp;\gamma}^{\omega_{\delta}}\partial_{b}g_{mp;\gamma}^{\omega}]=\\\\[8.61108pt]
&\displaystyle\frac{(\partial_{b}\omega_{mp})1/\tau_{\gamma}}{1/\tau_{\gamma}^{2}+(\omega_{\delta}-\omega_{mp})^{2}}\bigg{[}\frac{2(\omega_{l}-\omega_{mp})(\omega_{\delta}-\omega_{mp})}{[1/\tau_{\gamma}^{2}+(\omega_{l}-\omega_{mp})^{2}]^{2}}\\\\[12.91663pt]
&\displaystyle+\left(\dfrac{2(\omega_{l}-\omega_{mp})^{2}}{[1/\tau_{\gamma}^{2}+(\omega_{l}-\omega_{mp})^{2}]^{2}}-\dfrac{1}{1/\tau_{\gamma}^{2}+(\omega_{l}-\omega_{mp})^{2}}\right)\bigg{]}.\end{array}$
(50)

Clearly, the non-zero imaginary part of the resonance factor arises only
through the incorporation of a finite scattering timescale due to disorder.

###  II.3 Scattering time scale

Here we discuss the difference between the present and earlier treatment of
the scattering time in the response. In earlier works, $1/\tau$ was typically
added as an infinitesimally small imaginary part in the frequency
$\omega\rightarrow\omega+i\eta$ having $\eta=1/\tau$ to achieve convergence in
the low-frequency response Passos  _et al._ (2018). Here, $\eta$ originates
from causality via the slow switching of the perturbation and is generally set
to zero to obtain results in the clean limit. However, this approach cannot be
applied to systems where the interference of intraband and interband
transitions play a pivotal role in generating the different components of the
nonlinear response. In addition, the subtle difference between the two
approaches becomes more important while going beyond the linear response
regime to calculate the nonlinear response.

In the approach of the addition of a small imaginary term, the resonance
factor becomes

$\frac{1}{\omega_{l}+\omega_{j}-\varepsilon}\rightarrow\frac{1}{\omega_{l}+\omega_{j}-\varepsilon+i\eta}~{}.$
(51)

Nevertheless, one must be cautious while calculating the response where the
scattering term contributes twice. The qualitative aspects can be captured
correctly without taking the factor of $2$ in front of $\eta$ arising via the
addition of an imaginary factor to two frequencies. Still, it may
significantly affect the shape of resonances and yield different results
around the resonances with two approaches. Such subtle issues can be avoided
by considering the finite scattering term in the equation of motion as
incorporated in the present study.

##  III Symmetry and Geometrical analysis

###  III.1 Symmetry Analysis

In this section, we perform the symmetry analysis of the nonlinear optical
currents induced by the optical field. We describe how the properties of the
nonlinear currents are restricted by various symmetries such as the parity (or
space inversion) ($\mathcal{P}$), time reversal ($\mathcal{T}$), inversion-
time reversal ($\mathcal{P}\mathcal{T}$) symmetries. We begin by recalling
basic symmetry arguments. Firstly under $\mathcal{P}$ unitary transformation,
the position vector ${\bm{r}}$ changes sign to $-{\bm{r}}$; thus, momentum
changes sign. In this case, the Bloch Hamiltonian follows the eigenvalue
equation as

$\mathcal{P}\mathcal{H}({\bm{k}})|u_{{\bm{k}}}^{m}\rangle=\mathcal{H}(-{\bm{k}})\mathcal{P}|u_{{\bm{k}}}^{m}\rangle,$
(52)

which gives the energy eigenvalues of the Bloch Hamiltonian that remain
invariant on changing ${\bm{k}}$ to $-{\bm{k}}$:

$\varepsilon_{m}({\bm{k}})=\varepsilon_{m}(-{\bm{k}}).$ (53)

However, this symmetry is only preserved if the dispersion is an even function
of the momentum. Further, the Bloch eigen function follows the relation

$\mathcal{P}|u_{m,{\bm{k}}}\rangle=|u_{-{\bm{k}}}^{m}\rangle.$ (54)

The Berry connection $\mathcal{R}({\bm{k}})$, for two band system which
contains the momentum derivative of the eigen function satisfies the following
relation under parity inversion

$\mathcal{R}_{pm}^{b}({\bm{k}})=-\langle
u_{-{\bm{k}}}^{p}|i{\bm{\nabla}}|u_{-{\bm{k}}}^{m}\rangle=-\mathcal{R}_{pm}^{b}(-{\bm{k}}).$
(55)

However, the Berry curvature and the geometric tensor having the product of
two Berry connection factors remain invariant under $\mathcal{P}$ symmetry,

$\begin{array}[]{rl}\Omega_{pm}^{bc}({\bm{k}})&\displaystyle=\Omega_{pm}^{bc}(-{\bm{k}}),\quad
G_{pm}^{bc}({\bm{k}})=G_{pm}^{bc}(-{\bm{k}}).\end{array}$ (56)

Secondly, the time inversion symmetry ($\mathcal{T}$) is an anti-unitary
transformation in which complex number changes to its conjugate. Here the
momentum ${\bm{k}}\equiv i\partial_{\bm{r}}$ changes $-{\bm{k}}$ due to the
sign flip of $i$. Under $\mathcal{T}$ symmetry, the energy eigenvalues remain
invariant. However, the Bloch function follows

$|u_{-{\bm{k}}}^{n}\rangle^{*}=e^{i\phi(k)}|u_{{\bm{k}}}^{n}\rangle.$ (57)

Interestingly, the Berry connection, in addition to the sign change, also
reverses the band index order as dictated below

$\begin{array}[]{rl}\mathcal{R}_{pm}^{b}({\bm{k}})&\displaystyle=-\mathcal{R}_{mp}^{b}(-{\bm{k}}).\end{array}$
(58)

However, the band index criteria emerge only for multi-band systems. Further,
due to the complex conjugate condition under time-reversal transformation, the
geometric quantities satisfy

$\begin{array}[]{rl}\Omega_{pm}^{bc}({\bm{k}})&\displaystyle=-\Omega_{pm}^{bc}(-{\bm{k}}),\quad
G_{pm}^{bc}({\bm{k}})=G_{pm}^{bc}(-{\bm{k}}).\end{array}$ (59)

It is to be noted that the Berry curvature, which is equal to
$i[\mathcal{R}_{mp}^{b},\mathcal{R}_{mp}^{c}]/2$ reverses sign and the quantum
metric does not.

Thirdly, the combination of the parity and time-reversal symmetry
($\mathcal{P}\mathcal{T}$) properties yield

$\begin{array}[]{rl}\mathcal{R}_{pm}^{b}({\bm{k}})&\displaystyle=-\mathcal{R}_{mp}^{b}({\bm{k}})\quad\Omega_{pm}^{bc}({\bm{k}})=0,\quad
G_{pm}^{bc}({\bm{k}})=G_{pm}^{bc}({\bm{k}}).\end{array}$ (60)

Using these symmetries, we can understand the different contributions of the
current qualitatively. Under $\mathcal{P}$ symmetry, the electric field
reverses the sign as $E\rightarrow-E$, and the current also changes sign
$j\rightarrow-j$. In the nonlinear currents, the product of two electric
fields preserves the sign, which ensures that the only component of the
conductivity tensor should be zero, which changes the sign to fulfill the
condition for current.

First, the $\sigma_{abc}^{dd}$ contribution under $\mathcal{P}$ symmetry
depends on the following quantities

$v_{mm}^{0a}=-v_{mm}^{0a};~{}~{}\partial_{b}\partial_{c}f_{m}^{(0)}=\partial_{b}\partial_{c}f_{m}^{(0)};~{}~{}g_{0;\alpha}^{\omega_{l}}=g_{0;\alpha}^{\omega_{l}},$
(61)

which flips the sign of the intraband-intraband response as
$\sigma^{dd}_{abc}=-\sigma^{dd}_{abc}$. The other components of the response
follow as

$\sigma^{do}_{abc}=-\sigma^{do}_{abc};~{}~{}\sigma^{od}_{abc}=-\sigma^{od}_{abc};~{}~{}\sigma^{oo}_{abc}=-\sigma^{oo}_{abc},$
(62)

due to the sign reversal of the velocity change $\Delta_{mn}^{a}$, shift of
the Fermi function in the momentum space $\partial_{c}F_{mp}$, and Berry
connection respectively.

Second under time-reversal $\mathcal{T}$ symmetry, the electric field does not
change sign, but the current follows $j(=dP/dt)\rightarrow-j$. Thus, the
conductivity tensor must be an odd function under time-reversal symmetry.
Here,

$\mathcal{Q}_{mp}=\mathcal{Q}_{mp}^{*};~{}~{}g_{0;\alpha}^{\omega_{l}}g_{mp;\alpha}^{\omega_{\delta}}=[g_{0;\alpha}^{\omega_{l}}g_{mp;\alpha}^{\omega_{\delta}}]^{*}$
(63)

However, only contribution stemming from the Berry curvature (odd in nature)
in $\sigma_{od}$, $\sigma_{do}$, and $\sigma_{oo}$ will be non-zero.

Third under $\mathcal{P}\mathcal{T}$ symmetry, we have

$\mathcal{Q}_{mp}=\mathcal{Q}_{mp};~{}~{}g_{0;\alpha}^{\omega_{l}}g_{mp;\alpha}^{\omega_{\delta}}=[g_{0;\alpha}^{\omega_{l}}g_{mp;\alpha}^{\omega_{\delta}}]^{*}$
(64)

Here, the Berry curvature vanishes, and the quantum metric remains non-zero.
Thus, the only contribution from the quantum metric $G_{mp}$ generates a
finite nonlinear current.

###  III.2 Spatial Geometrical analysis

We analyze the different components of the conductivity tensor due to the
application of the optical field in distinct directions.
Scheme - I: When the optical field is applied in the $\hat{x}$-direction such
as ${\bm{E}}(t)=\sum_{j}E_{x}^{\omega_{j}}e^{-i\omega_{j}t}$, the longitudinal
current $j_{x}^{(2)}$ for the time-reversal symmetric system vanishes due to
the vanishing Berry curvature $\Omega_{bc}=\Omega_{xx}$. However, it remains
finite for the parity-time reversal symmetric system due to non-zero quantum
metric $G_{xx}$. Conversely, the Berry curvature contributes to the transverse
current $j_{y}^{(2)}$ due to the non-zero response components $\sigma_{od}$
and $\sigma_{oo}$. Note that the injection current due to $\Omega$ here is
zero.
Scheme - II: For the optical field as the superposition of the beams in
$\hat{x}$ and $\hat{y}$-directions,
${\bm{E}}(t)=E_{x}^{\omega_{j}}e^{-i\omega_{j}t}+E_{y}^{\omega_{l}}e^{-i\omega_{l}t}$,
the current along $\hat{x}$-direction is mainly contributed by the two tensor
components $\sigma_{xxy}$ and $\sigma_{xyx}$. In the spatial geometry $xxy$,
the resonant part of the interband-intraband conductivity or the anomalous
conductivity vanishes, and the non-resonant part gives a finite value via the
finite quantum metric. In addition, the other parts also contribute to the
current through $G$. For $xyx$ geometry, the double resonant part
$\sigma_{xyx}^{oo,II}$ and the higher order pole part $\sigma_{xyx}^{oo,III}$
are zero for the time reversal symmetric system due to vanishing antisymmetric
Berry curvature $\Omega_{xx}$. Similarly for the nonlinear current
$j_{y}^{(2)}$, we have $\sigma_{yxy}$, and $\sigma_{yyx}$ components.

##  IV Applications

In this section, we discuss the application of the general kinetic approach
for the optical currents developed in the present study to known models. Our
focus in this section is to establish the connection of the geometric
quantities with the nonlinear response components using a few examples.
However, the framework is formulated in this paper in a more general way that
is appropriate for all systems.

Figure 2: Schematic picture of the energy dispersion in the two-dimensional
momentum plane for (a) Topological antiferromagnet CuMnAs and (b) Gapped
tilted Weyl semimetal. Here, the momentum coordinates are labeled as $k_{x}$
and $k_{y}$, and the third dimension corresponds to the energy. We consider
the parameter values in the units of eV as $\tilde{t}$=1.0, $t$ = 0.08,
$\alpha_{R}$=0.8, $\alpha_{D}$=0.0, $h_{AFx}$=0.85, $h_{AFy}$=0, $h_{AFz}$=0
for CuMnAs system. For tilted Weyl semimetal, $t_{x}=0.5$ eVÅ-1, $\Delta=0.05$
eV and $v$=1.0 eVÅ-1.

###  IV.1 Topological antiferromagnetic CuMnAs

Here we consider the case of Dirac semimetal, which has attracted attention as
a host of massless Dirac quasiparticles with two doubly degenerate bands in
the momentum space Liu  _et al._ (2014a, b); Yang and Nagaosa (2014). These
doubly degenerate bands having band crossing between them generate four-fold
degenerate Dirac points. However, such Dirac points are unstable and require
symmetry protection Yang and Nagaosa (2014). Specifically, upon breaking
either symmetry, such as time-reversal and parity, the double degeneracy of
the bands is lifted, and the massless Dirac quasiparticles break down. This
raises a natural question about the existence of such fermions in the absence
of individual symmetry and in the presence of the combination of both the
time-reversal and parity symmetry. Such a picture has been addressed by
considering the example of a 2D material, CuMnY where Y is As or P, having the
spin-orbit coupling that protects the band crossings in the Dirac semimetal
Tang  _et al._ (2016). In the paramagnetic phase, this material preserves the
time-reversal and parity symmetries. This results in the formation of Kramers
pairs by each band. On the other hand, in the antiferromagnetic phase material
breaks both $\mathcal{T}$ and $\mathcal{P}$ symmetry. However, it preserves
the degeneracy due to the $\mathcal{P}\mathcal{T}$ symmetry which relates to
the spin degrees of freedom. This exciting feature makes the antiferromagnetic
phase of the CnMnAs an excellent choice to study transport effects such as the
spin-orbit torque, spin Hall effect, and anomalous Hall effect Wadley  _et
al._ (2016); Šmejkal  _et al._ (2017a); Máca  _et al._ (2012); Smejkal  _et
al._ (2022); Wang  _et al._ (2021); Liu  _et al._ (2021).

The $\mathcal{P}\mathcal{T}$ symmetric topological antiferromagnetic CuMnAs
material is described by the low energy 2D model Hamiltonian in the momentum
space as Šmejkal  _et al._ (2017b)

$\begin{array}[]{rl}\mathcal{H}({\bm{k}})&\displaystyle=\begin{pmatrix}\varepsilon_{0}({\bm{k}})+{\bm{h}}_{\rm
A}({\bm{k}})\cdot{\bm{\sigma}}&V_{\rm AB}({\bm{k}})\\\ V_{\rm
AB}({\bm{k}})&\varepsilon_{0}({\bm{k}})+{\bm{h}}_{\rm
B}({\bm{k}})\cdot{\bm{\sigma}}\end{pmatrix},\end{array}$ (65)

where $V_{\rm AB}({\bm{k}})=-2\tilde{t}\cos(k_{x}/2)\cos(k_{y}/2)$ is the
inter-sublattice hopping term having $\tilde{t}$ the first nearest neighbour
hopping parameter,
$\varepsilon_{0}({\bm{k}})=-t\left[\cos(k_{x})+\cos(k_{y})\right]$ is the
intra-sublattice hopping term with parameter $t$ as the second-nearest
neighbour and $\sigma_{i}$ represent the Pauli matrices for spin. Further, the
quantity ${\bm{h}}_{\rm A}({\bm{k}})$ for the sublattice $A$ that includes the
antiferromagnetic (AF) magnetization field and the spin-orbit coupling (SOC)
term ${\bm{h}}_{\rm A}({\bm{k}})={\bm{h}}_{\rm AF}+{\bm{h}}_{\rm
SOC}({\bm{k}})$ is defined like

$\begin{array}[]{rl}{\bm{h}}_{\rm A}({\bm{k}})=\begin{pmatrix}h_{\rm
AF}^{x}-\alpha_{\rm R}\sin(k_{y})+\alpha_{\rm D}\sin(k_{y})\\\ h_{\rm
AF}^{y}+\alpha_{\rm R}\sin(k_{x})+\alpha_{\rm D}\sin(k_{x})\\\ h_{\rm
AF}^{z}\end{pmatrix},\end{array}$ (66)

having $\alpha_{R}$ and $\alpha_{D}$ as the spin-orbit coupling coefficients
and for sublattice $B$, ${\bm{h}}_{\rm B}({\bm{k}})=-{\bm{h}}_{\rm
A}({\bm{k}})$. The energy eigenvalues corresponding to the Hamiltonian Eq.
(65) are

$\varepsilon({\bm{k}})=\varepsilon_{0}\pm\sqrt{V_{\rm AB}^{2}+h_{{\rm
A}x}^{2}+h_{{\rm A}y}^{2}+h_{{\rm A}z}^{2}}~{}.$ (67)

Here, (+) sign is for the conduction band and (-) for the valence band.
Further, the dispersion $\varepsilon({\bm{k}})\neq\varepsilon(-{\bm{k}})$ due
to the broken particle-hole symmetry by $\varepsilon_{0}$. The schematic
picture of the dispersion is shown in Fig. 2 (a), and the corresponding
geometric quantities are shown in Fig. 3 (a)-(c). Here, the band crossing at
the Dirac points is protected due to the glide planer symmetry Šmejkal  _et
al._ (2017a). Further, the quantum geometric quantities Berry curvature and
symplectic connection are zero in this system due to symmetry arguments.
However, the other quantities quantum metric $G_{ab}$ and symplectic
connection $\Gamma_{abc}$ are non-zero where $(a,b,c)\in(x,y)$. To demonstrate
the nature of these quantities, we have shown a few components in the top
panel of Fig. 3 where we have considered the antiferromagnetic magnetization
field along $\hat{x}$-direction and zero in other directions.

Figure 3: Distribution of the geometric quantities in the momentum space. Top
panel: for Topological antiferromagnetic CuMnAs where (a) corresponds to the
quantum metric, (b) and (c) to metric connection. Bottom panel: for thin film
tilted Weyl semimetal where (d) refers to the Berry curvature, (e) and (f) to
the symplectic connection.

In the parity time-reversal symmetric CuMnAs system, eight components of the
nonlinear conductivities contribute to the second-order current, in general.
However with the finite magnetization field along $\hat{x}$-direction, the
components with odd number of spatial $x$ indices such as $\sigma_{xxx}$,
$\sigma_{xyy}$, $\sigma_{yyx}$ and $\sigma_{yxy}$ vanish. Thus, we left with
$\sigma_{yxx}$, $\sigma_{xxy}=\sigma_{xyx}$ and $\sigma_{yyy}$. The behavior
of these components with the incident beam frequency at the low temperature is
shown in Fig. 4(a) and (b). Here we fix the frequency of one incident beam
$\omega_{2}$ and tune the frequency of another beam $\omega_{1}$, while the
chemical potential is kept at $\mu=0.2$ eV and the scattering time scale
$\tau=1$ ps. We observe that the total response, a sum of different components
such as dd, do, od, and oo is mainly contributed by the geometric quantities
$G_{ab}$ and $\Gamma_{abc}$. In CuMnAs, we find the following features. (i)
The absorption peaks are generated at energies around $\mu$ and $2\mu$. The
observed behavior represents interference between the Fermi surface (i.e., the
momentum derivative of the Fermi distribution function) and the Fermi sea
effects. Note that the deviation in peaks is due to the absence of particle-
hole symmetry in the considered system. (ii) The nonlinear conductivity
$\sigma_{xxy}$ is opposite in sign to the other components. This arises due to
the opposite sign of the $\sigma^{do}$, $\sigma^{od}$ and $\sigma^{oo,I}$
stemming from the nature of the factors $G_{bc}v_{a}$, $G_{ab}v_{c}$ and
$\mathcal{C}_{abc}$ respectively.

Figure 4: Different components of the second-order response due to the two
beams having frequency $\omega_{1}$ and $\omega_{2}$. Column 1: (a) and (c)
depict the sum frequency generation phenomenon, and Column 2: (b) and (d) show
the difference in frequency generation effect. The output is obtained at fixed
frequency $\omega_{2}=0.1$ eV, and the chemical potential $\mu=0.2$ eV,
$\tau=1$ ps and temperature $T=10$ K, but varies $\omega_{1}$. Figures (a) and
(b) correspond to the CuMnAs where we set the hopping $t=0.08$ eV and
$\tilde{t}=1$ eV. The other parameters are $\alpha_{\rm R}=0.8,\alpha_{\rm
D}=0$ and ${\bm{h}}_{\rm AF}=(0.85,0,0)$ eV. Figures (c) and (d) refer to the
thin film tilted Weyl semimetal where we consider the gap $\Delta=0.05$ eV and
the tilt $t_{x}=0.1$ eVÅand temperature $T=1$K.

###  IV.2 Thin film tilted Weyl semimetal

Secondly, we consider the tilted Weyl semimetal, which is a three-dimensional
topological semimetal in general Potter  _et al._ (2014); Burkov (2016). Here
the conduction and valence bands touch each other at the Weyl nodes having
opposite chirality. Further, this material shows the phase transition from
topological to trivial by tuning the gap controlled by an out-of-plane
component of the momentum Culcer  _et al._ (2017). In the case of an ultrathin
film of the Weyl semimetal, the out-of-plane component of the momentum is
quantized, then the system becomes a two-dimensional fermion system Lu  _et
al._ (2010); Abanin and Pesin (2011). Further, this quantized component
results in the mass or the gap between the bands, leading to intriguing
quantum transport effects such as anomalous Hall effect, planar Hall effect,
and weak localization and anti-localization effects Burkov (2014); Trescher
_et al._ (2015); Ferreiros  _et al._ (2017); Liu  _et al._ (2017); Ma  _et
al._ (2019).

The effective Hamiltonian for the time-reversal symmetry broken tilted Weyl
semimetal around a Weyl point Ma  _et al._ (2019) is

$\mathcal{H}({\bm{k}})=v{\bm{k}}\cdot{\bm{\sigma}}+{\bm{t}}\cdot{\bm{k}}\sigma_{0}+\Delta\sigma_{z}.$
(68)

where the first term represents the spin-orbit coupling term having $v$ as the
effective velocity in units of eV m s-1, $t$ is the tilt vector and $\Delta$
refers to the gap which distinguish thin film Weyl semimetal from the
topological insulator. The energy eigenvalues are

$\varepsilon({\bm{k}})=t_{x}k_{x}+t_{y}k_{y}\pm\sqrt{v^{2}k^{2}+\Delta^{2}}.$
(69)

Here, the tilt term breaks the time-reversal and inversion symmetry due to the
linear momentum factor. Such tilt term does not affect the eigenvectors, hence
the topology of the system. However, it affects the response of the system. In
addition, the type of Weyl semimetal is defined by $|t|<v$ (Type-I) and
$|t|>v$ (Type-II). The corresponding dispersion for the Type-I Weyl semimetal
is shown in Fig. 2 (b), and the geometric quantities are shown in Fig. 3
(d)-(f). Here, the gap between conduction and valence band is $2\Delta$.
Without tilt, the band dispersion becomes identical to the topological
insulator, with the mass term smaller than the spin-orbit coupling term. In
the opposite case, i.e., at the large mass, it behaves as a massive fermion
system. In addition, all the quantum geometric quantities are non-zero, and a
few are shown in Fig. 3. The Berry curvature is finite only due to the
presence of the gap around the Dirac point and vanishes at $\Delta=0$.

As distinct from the antiferromagnetic CuMnAs system, all eight nonlinear
response components contribute to the dynamical current for the thin film
tilted Weyl semimetal. However, these reduce to six due to symmetrical
properties for the tensor components such as $\sigma_{xyx}=\sigma_{xxy}$, and
$\sigma_{yxy}=\sigma_{yyx}$. These are mainly dictated by the quantities
$G_{ab}$, $\Omega_{ab}$, $\Gamma_{abc}$, and $\tilde{\Gamma}_{abc}$. The
behavior of the nonlinear response components is shown in Fig. 4(c) and (d).
Here, we observe the following features. (i) The occurrence of the two
absorption peaks, one at $\hbar\omega_{\Sigma}=2\mu$ and the other at energy
$\hbar\omega_{\Sigma}=2\mu\pm\hbar\omega_{2}$. Note that the peak
corresponding to the red and green curves is shifted as it is influenced by
the contribution stemming from both finite Berry curvature and quantum metric.
(ii) The generation of the resonant behavior of the response happens due to
the finite Fermi surface effect. (iii) The $\sigma_{yxy}$ yields a larger
magnitude than other nonlinear tensor components due to the stronger
$\sigma^{oo,II}$ contribution stemming from the shifted Fermi surface in the
momentum space along $\hat{x}$-direction because of the tilt $t_{x}$ and the
quantum metric.

Experimentally, the presented results for the SFG and DFG are significant in
terms of measurement geometry and doping. First, these nonlinear signals can
be measured by invoking the measurement geometry of the nonlinear response in
the distinct direction of the applied field, i.e., along and perpendicular to
the field. In CuMnAs, one can have nonlinear current for the particular
geometry, i.e., $j_{y}^{(2)}=\sigma_{yxx}E_{0}^{2}\cos^{2}\gamma$, where
$\gamma$ is a polarization angle made by applied field along $x$-axis. This
results in the maximum current at $\gamma=0$. However, the current
$j_{x}^{(2)}=\sigma_{xxy}/2E_{0}^{2}\sin 2\gamma$ yields more value at $\pi/4$
polarization angle. On the other hand, in thin film tilted Weyl semimetal, the
nonlinear current can be obtained irrespective of the polarization angle as it
contributes to all response components. Along $\hat{x}$-direction, the current
follows
$j_{x}^{(2)}=[\sigma_{xxx}\cos^{2}\gamma+\sigma_{xyy}\sin^{2}\gamma+\sigma_{xxy}\sin
2\gamma]E_{0}^{2}$ and along $\hat{y}$-direction,
$j_{y}^{(2)}=[\sigma_{yyy}\sin^{2}\gamma+\sigma_{yxx}\cos^{2}\gamma+\sigma_{yxy}\sin
2\gamma]E_{0}^{2}$. Second, the strength of the nonlinear current can be tuned
with the chemical potential or by doping. By taking into account the Fermi
level inside the band, the Fermi surface terms such as the resonant (a subpart
of interband-intraband) and double resonant (a subpart of interband-interband)
strengthen the peak value of the second-order response, hence the nonlinear
current.

##  V Summary

We have systematically developed a general platform for evaluating the
nonlinear response of a crystal to an oscillating electric field or laser
field by taking into account the interband and intraband counterparts of the
density matrix. In the linear regime, the intraband part of the response is
captured by the band-diagonal component of the density matrix and the
interband part by off-diagonal component which are responsible for the linear
longitudinal conductivity, current-induced spin polarizations in spin-orbit
coupled systems, anomalous Hall, and spin-Hall effects Nagaosa  _et al._
(2010); Inoue  _et al._ (2004); Culcer and Winkler (2007); Culcer and Das
Sarma (2011). However, on going beyond the linear regime by expanding the
density matrix in terms of the external stimuli, it is not trivial to express
the interband and intraband components directly. These are interconnected to
each other and lead to distinct contributions to the nonlinear currents, such
as intraband-intraband, intraband-interband, interband-intraband, and
interband-interband. Further, the corresponding nonlinear current gives a
significant contribution on account of the finite Fermi surface, which was not
discussed earlier to the best of our knowledge.

We employ our theory to describe the phenomena of sum frequency and difference
frequency summation, which lead to the second-harmonic and rectification
effect as a special case, respectively. Similarly, we identified the
fundamental connection between the geometric quantum quantities and the
nonlinear response. Based on the connection, we showed how the fundamental
symmetries play a significant role in examining the physical origin of the
different components of the nonlinear currents. An interesting and important
fact is that the nonlinear optical currents are dominated due to the interband
coherence contribution. We highlight the contribution of the individual part
of the nonlinear optical current that provides insightful information. First,
the intraband-interband current, known as the injection current, is calculated
earlier in specific regimes of frequency and scattering time scale within the
Fermi Golden rule, which lacks the detailed behavior of the particular
current. Here, we calculated the explicit expression of the injection current
without considering assumptions applicable to all regimes of interest. Second,
we naively express the interband-intraband response in two parts. The non-
resonant part corresponds to the well-known anomalous nonlinear current and is
non-zero only if the Berry curvature is finite. On the other hand, the
resonant part yields finite value in all systems and participates to have an
absorption peak in the nonlinear response. Third, the interband-interband
current leads to the shift, double resonant and higher-order pole sum
frequency, and difference frequency summation nonlinear currents. We
demonstrated the whole analysis for CuMnAs and thin film Weyl semimetal
systems.

Further, our theory considers the transport and optical responses on an equal
footing, reflecting their interplay in the second-order response. The method
developed in this work serves as a key tool to examine the intrinsic and
extrinsic contributions of nonlinear currents obtained on general and
fundamental grounds, and yield physical insight into the behavior of fermions.

###### Acknowledgements.

P.B. thanks SRM-AP for providing a high-performance computing facility. DC is
supported by the Australian Research Council Centre of Excellence in Future
Low-Energy Electronics Technologies, project number CE170100039.

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