††thanks: M.M.K. and S.S. contributed equally to this manuscript.††thanks:
M.M.K. and S.S. contributed equally to this manuscript.

# Band geometry induced electro-optic effect and polarization rotation

M. Maneesh Kumar  Department of Physics, Indian Institute of Technology
Kanpur, Kanpur 208016  Sanjay Sarkar  Department of Physics, Indian Institute
of Technology Kanpur, Kanpur 208016  Amit Agarwal<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016

###### Abstract

Electric field-induced modulation of the optical properties is crucial for
amplitude and phase modulators used in photonic devices. Here, we present a
comprehensive study of the band geometry-induced electro-optic effect,
specifically focusing on the Fermi surface and disorder-induced contributions.
These contributions are crucial for metallic and semimetallic systems and for
optical frequencies comparable to or smaller than the scattering rates. We
highlight the importance of the quantum metric and metric connection in
generating the electro-optic effect in parity-time reversal ($\mathcal{PT}$)
symmetric systems such as CuMnAs. Our findings establish the electro-optic
effect as a novel tool to probe band geometric effects and open new avenues to
design electrically controlled efficient amplitude and phase modulators for
photonic applications.

##  I Introduction

Electro-optical effects showcase the remarkable ability of quantum materials
to transform their optical properties under the influence of an applied dc
electric field. Specifically, for linear electro-optical (EO) or Pockels
effect Shih and Yariv (1982); Boyd (2003); Chen  _et al._ (2014); Li  _et al._
(2020); Wang  _et al._ (2018), the change in the refractive index (RI) is
proportional to the strength of the applied dc electric field. This phenomenon
underpins the functionality of EO materials as amplitude and phase modulators,
which enable electrical tunability and manipulation of the optical parameters
such as amplitude, phase, and polarization Goldstein (1986); Meintjes and Raab
(1999); Cui  _et al._ (2017), for a variety of applications in photonic
devices. There is immense interest in the exciting phenomena of EO effects and
polarization rotation in optical experiments Gibson  _et al._ (2018); Liu  _et
al._ (2003); Luo  _et al._ (2020); Ye  _et al._ (2023). Earlier studies have
focused on material systems such as organic materials Benea-Chelmus  _et al._
(2021); Sun  _et al._ (2021), liquid crystals Blinov (1975), chiral
topological semimetals Ma and Pesin (2015); Sonowal  _et al._ (2019); Singh
_et al._ (2018), zinc blends Namba (1961), certain trigonal and piezoelectric
crystals Cheng  _et al._ (2019); Fu and Cohen (2000), Ferroelectric oxide
materials Jiang  _et al._ (2020), Si3N4 platforms Alexander  _et al._ (2018),
silicone waveguides Berciano  _et al._ (2018), and on certain stoichiometric
samples Fujiwara (1999); Peterson  _et al._ (1964); Kaminow and Johnston
(1967); Wang  _et al._ (2022). The EO modulators based on epitaxial BaTiO3
integrated on silicon Li and Tang (2019) play a pivotal role in optical
communication.

Despite extensive material-based studies of EO effects, a comprehensive
theoretical exploration of the phenomena of EO effect induced by nonlinear
optical susceptibilities is still lacking. Specifically, the contributions to
the nonlinear optical susceptibilities induced by Fermi surface effects and
scattering with impurities have not been explored earlier. These contributions
are essential for metallic and semimetallic systems and for exploring low-
frequency EO regimes. Recently, this effect has been studied in chiral
topological semimetals Li  _et al._ (2022), revealing that the second-order
response includes contributions from the shift current, injection current, and
nonlinear anomalous current induced by the Berry curvature. Here, we develop a
generalized theory of the linear EO effect, which works for all systems
without any symmetry constraints. Using the density-matrix based quantum
kinetic theory framework, we calculate the second-order optical response at
finite optical frequencies. We retain all Fermi surface and Fermi sea
contributions and disorder effects to identify six unique contributions to the
linear EO effect. We denote these contributions as shift, injection, Drude,
anomalous, higher-order pole, and double resonant contributions Bhalla  _et
al._ (2022). As an example to demonstrate our work, we focus on an
antiferromagnetic material CuMnAs, which breaks parity ($\cal P$) and time-
reversal symmetry ($\cal T$) but is ${\cal P}{\cal T}$ symmetric. In a
$\mathcal{PT}$ symmetric system, the Berry curvature and the symplectic
connection–the imaginary parts of the quantum geometric tensor (QGT), and the
quantum geometric connection (QGC) vanish. We show that the dominant
contribution in the linear EO effect in $\mathcal{PT}$ symmetric systems
emerges from the quantum metric and the metric connection, the real parts of
QGT and QGC, respectively. More interestingly, we show that the dominant EO
effect arises for optical energies lesser than the chemical potential. This
highlights that the Fermi surface effects play a significant role in EO
effects.

Our paper is structured as follows: In Sec. II, we develop the framework for
understanding the EO effect starting from Maxwell’s equations. In Sec. III, we
explore the linear EO effect, focusing on the relationship between
birefringence and the applied static electric field. In Sec. IV, we introduce
the distinct band geometric contributions to optical susceptibilities,
capturing the Fermi sea, Fermi surface, and disorder effects. We demonstrate
our results in Sec. V, for a model system of CuMnAs, a $\mathcal{PT}$ \-
symmetric material, with broken $\mathcal{P}$ and $\mathcal{T}$\- symmetry.
Finally, in Sec. VI, we summarise our work.

##  II The electro-optic effect

In this section, we employ Maxwell’s equations to calculate the modulation of
frequency-dependent refractive indices in a medium subjected to a dc electric
field. The spatio-temporal evolution of the electric field propagating in a
medium is given by Griffiths (2017); Jackson (2012),

${}{\nabla}^{2}{\bm{E}(\textbf{r},t)}=\frac{\partial}{\partial
t}\left[\mu{\bm{j}}_{f}+\epsilon\mu\frac{\partial{\bm{E}(\textbf{r},t)}}{\partial
t}\right]\leavevmode\nobreak\ .$ (1)

Here, $\epsilon$ ($\mu$) is the static permittivity (permeability) of the
medium, and ${\bf j}_{f}$ is the free charge current density. We consider a
light beam propagating with wave vector $\bf{q}$ along the $z$-axis
($\hat{z}$) with its polarization confined to the $x$-$y$ plane. An external
bias voltage (static or slowly varying) is applied along the $y$-axis,
resulting in a dc electric field $\textbf{E}^{d}=E_{y}^{d}\leavevmode\nobreak\
\hat{y}$. Thus, for a monochromatic light of frequency $\omega$, the dominant
electron current oscillating with a frequency $\omega$ and linear in the
optical field strength $E(\omega$), is given by Boyd (2003); Ashcroft  _et
al._ (1976),

$\displaystyle
j_{a}(\omega)=\sigma_{ab}(\omega)E_{b}(\omega)+\sigma^{(2)}_{abc}(\omega;\omega,0)E_{b}(\omega)E^{d}_{c}(0)\leavevmode\nobreak\
.$ (2)

Here, the first term denotes the linear conductivity, and the second term
represents the NL current contribution responsible for the linear electro-
optic effect. In Eq. (2), the summation over repeated indices is implied. We
will focus on the propagating plane wave solutions of Eq. (1), which have the
form $\textbf{E}(\omega)=\textbf{E}_{0}e^{i(\textbf{q}\cdot\textbf{r}-\omega
t)}$. Using $\textbf{E}(\omega)$, and Eq. (2) in Eq. (1), we obtain,

$q^{2}E_{a}=\frac{\omega^{2}}{c^{2}}\left[\frac{i\sigma_{ab}E_{b}}{\omega\epsilon_{0}}+\frac{i\sigma^{(2)}_{abc}E_{b}E_{c}}{\omega\epsilon_{0}}+E_{a}\right]\leavevmode\nobreak\
.$ (3)

Here, we have used $E_{a(b)}=E_{a(b)}(\omega)$ for brevity. Defining the
finite frequency dielectric tensor Cohen and Louie (2016),
$\epsilon_{aa}(\omega)=1+\frac{i}{\omega\epsilon_{0}}\sigma_{aa}(\omega)$, we
can express the above equation as,

$\left[\left(\frac{cq}{\omega}\right)^{2}-\epsilon_{aa}(\omega)\right]E_{a}=\frac{i}{\omega\epsilon_{0}}[\sigma_{ab}E_{b}(1-\delta_{ab})+\sigma^{(2)}_{abc}E_{b}E_{c}]\leavevmode\nobreak\
.$ (4)

We define the complex RI, $\tilde{N}=(cq/\omega)$ and
$N^{2}_{aa}=\epsilon_{aa}(\omega)$. The corresponding frequency-dependendent
susceptibility tensors are given by
$\chi^{(2)}_{abc}(\omega;\omega,0)=\frac{i}{\omega\epsilon_{0}}\sigma^{(2)}_{abc}(\omega;\omega,0)$
and $\chi_{ab}(\omega)=\frac{i}{\omega\epsilon_{0}}\sigma_{ab}(\omega)$.
Expressing Eq. (4) in terms of these, we obtain,

$(\tilde{N}^{2}-N^{2}_{aa})E_{a}=\chi_{ab}(\omega)E_{b}(1-\delta_{ab})+\chi^{(2)}_{abc}(\omega;\omega,0)E_{b}E_{c}\leavevmode\nobreak\
.$ (5)

Explicitly writing Eq. (5) for $E_{x}$ and $E_{y}$ in a matrix form, we have,

$\begin{pmatrix}\tilde{N}^{2}-N_{xx}^{2}-\chi^{(2)}_{xxy}E^{d}_{y}&&-\chi_{xy}-\chi^{(2)}_{xyy}E^{d}_{y}\\\
-\chi_{yx}-\chi^{(2)\leavevmode\nobreak\
}_{yxy}E^{d}_{y}&&\tilde{N}^{2}-N_{yy}^{2}-\chi^{(2)}_{yyy}E^{d}_{y}\end{pmatrix}\begin{pmatrix}E_{x}\\\
E_{y}\end{pmatrix}=0\leavevmode\nobreak\ .$ (6)

For simplicity of notation and without loss of generality, we express
$N_{xx}=N$ and $N_{yy}=N\sqrt{1+\eta^{2}}$, where $\eta$ is a parameterizing
factor.

To have a non-trivial solution for $E_{x}\leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ E_{y}$, the determinant of the matrix must vanish.
This yields the following condition,

$(\tilde{N}^{2}-N^{2})^{2}-(\tilde{N}^{2}-N^{2})[\eta^{2}N^{2}+(\chi^{(2)}_{yyy}+\chi^{(2)}_{xxy})E^{d}_{y}]+\chi^{(2)}_{xxy}E^{d}_{y}(\eta^{2}N^{2}+\chi_{yyy}E^{d}_{y})-(\chi_{xy}+\chi^{(2)}_{xyy}E^{d}_{y})(\chi_{yx}+\chi^{(2)}_{yxy}E^{d}_{y})=0\leavevmode\nobreak\
.$ (7)

Solving the equation for roots of $(\tilde{N}^{2}-N^{2})$, we obtain the
complex RI $\tilde{N}$ to be

$\tilde{N}_{\pm}=\sqrt{N^{2}+\Theta_{\pm}},\leavevmode\nobreak\
\text{where}\hskip
2.84526pt\Theta_{\pm}\equiv\frac{1}{2}[\zeta\pm\sqrt{\zeta^{2}-4\xi}]\leavevmode\nobreak\
,$ (8)

is a dc electric field and frequency-dependent function. Here, we have
defined,

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle[\eta^{2}N^{2}+(\chi^{(2)}_{yyy}+\chi^{(2)}_{xxy})E^{d}_{y}]\leavevmode\nobreak\
,$ (9)

and,

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\chi^{(2)}_{xxy}E^{d}_{y}(\eta^{2}N^{2}+\chi^{(2)}_{yyy}E^{d}_{y})$
(10) $\displaystyle-$
$\displaystyle(\chi_{xy}+\chi^{(2)}_{xyy}E^{d}_{y})(\chi_{yx}+\chi^{(2)}_{yxy}E^{d}_{y})\leavevmode\nobreak\
.$

Note that for $E^{d}_{y}\to 0$, we have $\zeta\to\eta^{2}N^{2}$, and
$\xi\to|\chi_{xy}|^{2}$. The complex RI can be tuned and controlled by varying
$E^{d}_{y}$. $\tilde{N}_{\pm}$ corresponds to two solutions for the effective
propagating wavevectors,
$\bm{q}_{\pm}=(\omega\tilde{N}_{\pm}/c)\leavevmode\nobreak\ \hat{q}_{\pm},$ of
the optical field. The complex RI can be expressed in terms of the
refractivities ($n_{\pm}$) and the extinction (or absorption) coefficients
($\kappa_{\pm}$), $N_{\pm}=n_{\pm}+i\kappa_{\pm}$ Visnovsky (2018). The
eigenvectors corresponding to the propagating optical fields of Eq. (6) are
given by,

$\begin{pmatrix}E_{x}\\\ E_{y}\end{pmatrix}_{+}=|{\bf
E}(\omega)|A_{+}\begin{pmatrix}\chi_{xy}+\chi^{(2)}_{xyy}E^{d}_{y}\\\
{\Theta_{+}-\chi^{(2)}_{xxy}E^{d}_{y}}\end{pmatrix}\leavevmode\nobreak\ ,$
(11)

and,

$\begin{pmatrix}E_{x}\\\ E_{y}\end{pmatrix}_{-}=|{\bf
E}(\omega)|A_{-}\begin{pmatrix}{\Theta_{-}-\chi^{(2)}_{yyy}E^{d}_{y}}-\eta^{2}N^{2}\\\
\chi_{yx}+\chi^{(2)}_{yyy}E^{d}_{y}\end{pmatrix}\leavevmode\nobreak\ .$ (12)

Here, $|{\bf E}(\omega)|$ is the strength of the incident field, and
$A_{\pm}=A_{\pm}(\omega)$, are multiplicative factors normalizing the
eigenvectors. These eigenvectors are tunable and can be electrically
controlled by varying $E^{d}_{y}$.

An interesting consequence of the unequal refractive indices is the induced
polarization rotation, $\Delta\tilde{\phi},$ for a linearly polarized light
incident on the medium, and traverses through it. It can be quantified in
terms of the finite birefringence given by $\Delta\tilde{n}={\rm
Re}[\tilde{N}_{+}-\tilde{N}_{-}]$. The linearly polarised light splits into
two circularly polarised (CP) or elliptically polarised (EP) lights of
opposite chirality upon entering the medium.

Consequently, upon exiting the medium, these superimpose into a linearly
polarised light, with a rotated plane of polarisation w.r.t. the incident
light. This frequency-dependent polarization rotation is given by Li  _et al._
(2022); Cui  _et al._ (2014),

$\Delta\tilde{\phi}(\omega)=\Delta\tilde{n}(\omega)\frac{\omega
L}{c}\leavevmode\nobreak\ .$ (13)

Here, $c$ is the speed of light in free space, and $L$ is the length of the
light path through the sample. This phenomenon is similar to the polarization
rotation caused by the optical activity of chiral molecules Landau  _et al._
(2013). However, in this case, the optical activity is controlled by a dc
electric field.

In addition to the change in the refractive index, a static electric field can
also induce a change in the absorption coefficient. This is instrumental for
modulating the intensity of the propagating optical beams. Additionally, the
two different absorption coefficients give rise to a frequency-dependent
ellipticity of the propagating light and are also related to the dichroism.
Analogous to the polarization rotation, the ellipticity ($\tan\varepsilon$) is
given by Visnovsky (2018),

$\tan\varepsilon=-\tanh\left[\frac{\omega
L}{2c}\Delta\tilde{\kappa}\right]\leavevmode\nobreak\ ,$ (14)

where we have defined
$\Delta\tilde{\kappa}=(\tilde{\kappa}_{+}-\tilde{\kappa}_{-})=\text{Im}[\tilde{N}_{+}-\tilde{N}_{-}]$.

In this section, we have developed a general theory on birefringence arising
from the impact of a static dc field captured via the nonlinear conductivity.
In the next section, we focus on the linear EO effect.

##  III Linear Electro-optic effect

The static electric-field dependent refractive indices obtained in Eq. (8), is
a nonlinear function of $E_{y}^{d}$. Here, we focus on Pockels effect, which
captures the linear $E_{y}^{d}$ contribution to the polarization rotation. For
this, we expand the altered complex RI in Eq. (8) in powers of $E_{y}^{d}$, to
obtain

$\displaystyle\zeta=\zeta^{(0)}+\zeta^{(1)}E^{d}_{y}\leavevmode\nobreak\ .$
(15)

Here, $\zeta^{(0)}=\eta^{2}N^{2}$ and $\zeta^{(1)}=(\chi_{yyy}+\chi_{xxy})$.
Similarly, we have,

$\displaystyle\xi=\xi^{(0)}+\xi^{(1)}E_{y}^{d}+\xi^{(2)}(E_{y}^{d})^{2}\leavevmode\nobreak\
,$ (16)

where $\xi^{(0)}=-\chi_{xy}\chi_{yx}$,
$\xi^{(1)}=(\eta^{2}N^{2}\chi_{xxy}-\chi_{xy}\chi_{yxy}-\chi_{yx}\chi_{xyy})$,
and $\xi^{(2)}=(\chi_{xxy}\chi_{yyy}+\chi_{xyy}\chi_{yxy})$. These expansions
allow us to express $\tilde{N}_{\pm}$ up to the first order of $E_{y}^{d}$ as,

$\tilde{N}_{\pm}=\frac{\sqrt{2N^{2}+\zeta^{(0)}\pm\rho}}{\sqrt{2}}+\frac{\zeta^{(1)}\left(\rho\pm\zeta^{(0)}\right)\mp
2\xi^{(1)}}{2\sqrt{2}\rho\sqrt{2N^{2}+\zeta^{(0)}\pm\rho}}E_{y}^{d}\leavevmode\nobreak\
,$ (17)

with, $\rho=\sqrt{[\zeta^{(0)}]^{2}-4\xi^{(0)}}$. The difference in complex RI
arising from the Pockels effect is given by,

$\displaystyle\Delta\tilde{N}=\tilde{N}_{+}-\tilde{N}_{-}$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{\zeta^{(1)}\left(\rho+\zeta^{(0)}\right)-2\xi^{(1)}}{2\sqrt{2}\rho\sqrt{2N^{2}+\zeta^{(0)}+\rho}}-\frac{\zeta^{(1)}\left(\rho-\zeta^{(0)}\right)+2\xi^{(1)}}{2\sqrt{2}\rho\sqrt{2N^{2}+\zeta^{(0)}-\rho}}\bigg{]}E^{d}_{y}\leavevmode\nobreak\
.$ (18)

The real and imaginary parts of the above expression capture the birefringence
($\Delta\tilde{n}$) and dichroism ($\Delta\tilde{\kappa}$) induced by the
applied static electric field to first order. To evaluate these effects, we
compute the second-order susceptibilities $\chi^{(2)}_{abc}(\omega;\omega,0)$.
The following section will detail this calculation using the density matrix-
based quantum kinetic theory.

##  IV second order Optical responses

In this section, we calculate the components of the second-order
susceptibility $\chi_{abc}^{(2)}(\omega;\omega,0)$ using the density-matrix
based quantum kinetic theory formalism. We will include the impact of a finite
Fermi surface and disorder in the second-order responses, which are usually
neglected Sipe and Shkrebtii (2000); Li  _et al._ (2022). We use the quantum
Liouville equation (QLE) to calculate the non-equilibrium density matrix. The
time evolution of the density matrix is given by,

$\dfrac{\partial\rho(\bm{k},t)}{\partial t}+\frac{i}{\hbar}[{\cal
H},\rho(\bm{k},t)]=0\leavevmode\nobreak\ .$ (19)

Here, $\rho=\rho(\bm{k},t)$ is the non-equilibrium density matrix, and ${\cal
H}=\hat{H}_{0}+\hat{H}_{\rm E}$, with $\hat{H}_{0}$ describing the unperturbed
Bloch Hamiltonian and $\hat{H}_{\rm E}$ is the perturbation Hamiltonian
capturing the effect of the electric field. Using the dipole approximation in
the length gauge Aversa and Sipe (1995); Taghizadeh  _et al._ (2017)
framework, the light-matter interaction term can be expressed as,

$\hat{H}_{\rm E}=e{\bm{E}}(t)\cdot\hat{\bm{r}}\leavevmode\nobreak\ .$ (20)

Here, $\hat{\bm{r}}$ is the position operator, and ${\bm{E}}(t)$ is the time-
dependent electric field. We perturbatively solve for the density matrix up to
second order in the electric field strength. To solve Eq. (19), we use the
adiabatic switching on of the perturbing fields,
${\bm{E}}(t)={\bm{E}}e^{-i(\omega+i/\tau)t}$. This approach, combined with the
interaction picture, has been extensively used in literature for the
perturbative solution of the density matrix Boyd (2003); Das  _et al._ (2023).
Incorporating such an adiabatic switching on approximation gives rise to a
relaxation term in the kinetic equation. For the $N$th order density matrix
($\rho^{(N)}\propto E^{N}$), Eq. (19) yields Bhalla  _et al._ (2022); Mandal
_et al._ (2024); Varshney  _et al._ (2023),

$\dfrac{\partial\rho^{(N)}}{\partial
t}+\dfrac{i}{\hbar}[\hat{H}_{0},\rho^{(N)}]+\dfrac{\rho^{(N)}}{\tau/N}=\dfrac{-ie{\bm{E}}}{\hbar}\cdot[\bm{\hat{r}},\rho^{(N-1)}]\leavevmode\nobreak\
.$ (21)

In the above expression, we consider the relaxation time $\tau$ to be a
constant. It accounts for the scattering of electrons with static and dynamic
(phonons) impurities. We use this equation to obtain the density matrix of
various orders in the electric field, as shown in Appendix A. Equipped with
the expressions of the density matrices, we calculate the NL electric current
using the expression,

${\bm{j}}=-e{\rm Tr}[{\rho}\hat{\bm{v}}]\leavevmode\nobreak\ .$ (22)

Here, the velocity operator matrix, expressed in the eigenbasis of
$\hat{H}_{0}$, is given by
$v_{pm}^{a}=v_{m}^{a}\delta_{pm}+(1-\delta_{pm})i\omega_{pm}\mathcal{R}_{pm}^{a}\mathcal{R}_{pm}^{a}$,
where $\mathcal{R}_{pm}^{a}=\langle
u_{\bm{k},p}|\partial_{k_{a}}u_{\bm{k},m}\rangle$ with $|u_{\bm{k},m}\rangle$
being the $m$th band Bloch electron’s wavefunction, is the interband Berry
connection. From the current density, the second-order conductivities can be
extracted via the relation,

$j_{a}^{(2)}(\omega)=\sigma_{abc}^{(2)}(\omega;\omega,0)E_{b}(\omega)E_{c}\leavevmode\nobreak\
.$ (23)

We obtain the second-order susceptibility from the second-order conductivity
using the relation,

$\chi^{(2)}_{abc}(\omega;\omega,0)=\frac{i}{\omega\epsilon_{0}}\sigma^{(2)}_{abc}(\omega;\omega,0)\leavevmode\nobreak\
.$ (24)

Figure 1: (a) Band dispersion for two dimensional $\mathcal{PT}$-symmetric
CuMnAs. We set the hopping parameters as t = 0.08 eV and $\tilde{t}=1$ eV. The
other parameters are $\alpha_{R}=0.8$, $\alpha_{D}=0$, and
${\bm{f}}_{AF}=(0.85,0,0)$ eV. The momentum space distribution of the quantum
metric components: (b) $\mathcal{G}^{xx}_{cv}$, (c) $\mathcal{G}^{yy}_{cv}$,
and (d) $\mathcal{G}^{xy}_{cv}$. The momentum space distribution of the metric
connection (e) $\Gamma^{xxy}_{cv}$ and (f) $\Gamma^{yyy}_{cv}$. We have set
the temperature to be $T=12$K, and the scattering time $\tau=10^{-14}$ s for
all our calculations.

Following Ref. Bhalla  _et al._ (2022), we express the second-order
susceptibility into six distinct forms based on their origin. We denote these
contributions as the shift ($\chi_{abc}^{\rm Sh}$), injection
($\chi_{abc}^{\rm In}$), Drude ($\chi_{abc}^{\rm D}$), anomalous
($\chi_{abc}^{\rm An}$), higher-order pole ($\chi_{abc}^{\rm HOP}$), and
double resonant ($\chi_{abc}^{\rm DR}$) susceptibility, respectively. The
shift and the injection currents arise from electron position and electron
velocity transition (along current direction) during an optical interband
excitation Ahn  _et al._ (2020); Han  _et al._ (2024). These are both Fermi
sea effects. The NL Drude contribution arises from the intraband transition
and is a Fermi surface contribution Bhalla  _et al._ (2023). The anomalous and
the double resonant currents originate from the Fermi surface effects. The
higher-order pole contribution occurs due to the velocity injection along the
applied field direction, and it is a Fermi sea phenomenon. The NL
susceptibilities in Eq. (8) is an aggregate of all these contributions. In
addition to the contributions from the shift and injection currents Sonowal
_et al._ (2019), we highlight additional contributions to the NL conductivity
Bhalla  _et al._ (2022). The pairwise field-symmetrized NL susceptibilities
are given by,

$\displaystyle\chi^{{\text{Sh}}}_{abc}$ $\displaystyle=$
$\displaystyle-\frac{ie^{3}}{2\omega\epsilon_{o}\hbar^{2}}\sum_{m,p}\int_{\rm
BZ}[d\bm{k}]\leavevmode\nobreak\
\omega_{mp}\tilde{g}^{\omega}_{mp}\left[g^{\omega}_{mp}\left(\Gamma^{abc}_{mp}-i\tilde{\Gamma}^{abc}_{mp}\right)+g^{0}_{mp}\left(\Gamma^{acb}_{mp}-i\tilde{\Gamma}^{acb}_{mp}\right)\right]F_{mp}\leavevmode\nobreak\
,$ (25a) $\displaystyle\chi^{{\text{Inj}}}_{abc}$ $\displaystyle=$
$\displaystyle\frac{ie^{3}}{2\epsilon_{0}\omega\hbar^{2}}\sum_{m,p}\int_{\rm
BZ}[d\bm{k}]\leavevmode\nobreak\
\partial_{a}(\omega_{mp})g^{\omega}_{0}\left[g^{\omega}_{mp}\left(\mathcal{G}^{bc}_{mp}-i\frac{\Omega^{bc}_{mp}}{2}\right)+g^{0}_{mp}\left(\mathcal{G}^{cb}_{mp}-i\frac{\Omega^{cb}_{mp}}{2}\right)\right]F_{mp}\leavevmode\nobreak\
,$ (25b) $\displaystyle\chi^{{\text{D}}}_{abc}$ $\displaystyle=$
$\displaystyle-\frac{ie^{3}}{2\epsilon_{0}\omega\hbar^{2}}\sum_{m}\int_{\rm
BZ}[d\bm{k}]\leavevmode\nobreak\
\left[\frac{1}{\hbar}\partial_{a}\epsilon_{\textbf{k},m}(\tilde{g}^{\omega}_{0}{g}^{\omega}_{0}+\tilde{g}^{0}_{0}{g}^{\omega}_{0})\partial_{b}\partial_{c}f_{m}\right]\leavevmode\nobreak\
,$ (25c) $\displaystyle\chi^{{\text{An}}}_{abc}$ $\displaystyle=$
$\displaystyle-\frac{ie^{3}}{2\epsilon_{0}\omega\hbar^{2}}\sum_{m,p}\int_{\rm
BZ}[d\bm{k}]\leavevmode\nobreak\
\omega_{mp}g^{\omega}_{mp}\left[g^{\omega}_{0}\left(\mathcal{G}^{ab}_{mp}-i\frac{\Omega^{ab}_{mp}}{2}\right)\partial_{c}F_{mp}+g^{0}_{0}\left(\mathcal{G}^{ac}_{mp}-i\frac{\Omega^{ac}_{mp}}{2}\right)\partial_{b}F_{mp}\right]\leavevmode\nobreak\
,$ (25d) $\displaystyle\chi^{{\text{HOP}}}_{abc}$ $\displaystyle=$
$\displaystyle-\frac{ie^{3}}{2\epsilon_{0}\omega\hbar^{2}}\sum_{m,p}\int_{\rm
BZ}[d\bm{k}]\leavevmode\nobreak\
\omega_{mp}g^{\omega}_{mp}\left[\left(\mathcal{G}^{ac}_{mp}-i\frac{\Omega^{ac}_{mp}}{2}\right)\partial_{b}g^{\omega}_{mp}+\left(\mathcal{G}^{ab}_{mp}-i\frac{\Omega^{ab}_{mp}}{2}\right)\partial_{c}g^{0}_{mp}\right]F_{mp}\leavevmode\nobreak\
,$ (25e) $\displaystyle\chi^{{\text{DR}}}_{abc}$ $\displaystyle=$
$\displaystyle-\frac{ie^{3}}{2\epsilon_{0}\omega\hbar^{2}}\sum_{m,p}\int_{\rm
BZ}[d\bm{k}]\leavevmode\nobreak\
\omega_{mp}g^{\omega}_{mp}\left[g^{\omega}_{mp}\left(\mathcal{G}^{ac}_{mp}-i\frac{\Omega^{ac}_{mp}}{2}\right)\partial_{b}F_{mp}+g^{0}_{mp}\left(\mathcal{G}^{ab}_{mp}-i\frac{\Omega^{ab}_{mp}}{2}\right)\partial_{c}F_{mp}\right]\leavevmode\nobreak\
.$ (25f)

In the above expressions, $[d\textbf{k}]={d^{d}k}/{(2\pi)^{d}}$ in $d$ spatial
dimensions. The NL susceptibilities are primarily determined by four band
geometric quantities: the Berry curvature ($\Omega$), the quantum metric
($\mathcal{G}$), the metric connection ($\Gamma$), and the symplectic
connection ($\tilde{\Gamma}$) Xiao  _et al._ (2010); Kozii  _et al._ (2021);
PURICE (1996); Mandal  _et al._ (2024). The former two are the real and
imaginary parts of the QGT ($\mathcal{Q}=\mathcal{G}-i\Omega/2$), while the
latter two are the real and imaginary parts of the QGC
($\mathcal{C}=\Gamma-i\tilde{\Gamma}$), respectively. The functions,
$g^{\omega}_{mp}=[1/\tau-i(\omega-\omega_{mp})]^{-1}$ and
$\tilde{g}^{\omega}_{mp}=[2/\tau-i(\omega-\omega_{mp})]^{-1}$ with
$\hbar\omega_{mp}=(\epsilon_{m,\textbf{k}}-\epsilon_{p,\textbf{k}})$ are
related to the joint density of states broadened by disorder.
$F_{mp}=f^{(0)}_{m}-f^{(0)}_{p}$ is the difference in the occupation of the
$m$ and $p$ bands in equilibrium. The Fermi-Dirac distribution function gives
the normalized occupation density in the $m$th band,
$f^{(0)}_{m}=[1+e^{\beta(\epsilon_{m,\textbf{k}}-\mu)}]^{-1}$ with
$\beta(=1/k_{B}T)$. Here, $T$, $\mu$, and $k_{B}$ are the thermodynamic
temperature, chemical potential, and the Boltzmann constant, respectively.

These NL responses at frequency $\omega$, combined with Eq. (8) complete the
theoretical framework of electric field-induced modulation of the RI of a
material or the electro-optic effect. The results obtained so far are general
and not constrained by any symmetry restrictions. However, different band
geometric quantities can either dominate or vanish in systems with specific
symmetries, making the induced EO effect an indirect probe of the band
geometry of the system. An interesting system is where both $\mathcal{P}$ and
$\mathcal{T}$ symmetries are broken, while their combined action,
$\mathcal{PT}$, remains intact Xia  _et al._ (2021); Klauck  _et al._ (2019);
Kremer  _et al._ (2019); Tiwari and Bzdušek (2020). In ${\cal P}{\cal T}$
symmetric systems, the Berry curvature and the symplectic connection vanish
over the entire Brillouin zone Bhalla  _et al._ (2022). As a consequence, the
electro-optic effect is induced entirely by the quantum metric and the metric
connection. The linear order anomalous Hall response also vanishes as the
Berry curvature is zero over the entire Brillouin zone in ${\cal P}{\cal T}$
symmetric systems.

In the next section, we apply our theory to a $\mathcal{PT}$-symmetric model.
This will allow us to illustrate the contributions of the quantum metric and
the metric connection.

##  V Linear Electro-Optic Effect in $\mathcal{PT}$ \- symmetric systems

Figure 2: The frequency dependence of the refractivities (a) $\tilde{n}_{+}$
and c) $\tilde{n}_{-}$, respectively. (b) and (d) depict the frequency
dependence of the absorption coefficients, $\tilde{\kappa}_{+}$ and
$\tilde{\kappa}_{-}$, respectively. The variation of (e) $\Delta\tilde{n}$ and
(f) $\Delta\tilde{\kappa}$ with frequency. The response for different values
of the dc electric field, $\bm{E}^{d}=E^{d}_{y}\hat{y}$, in (a)-(f) are
indicated by the blue dots ($E^{d}_{y}=0$), green line ($E^{d}_{y}=5\times
10^{7}$ V/m), and red the line ($E^{d}_{y}=10^{8}$ V/m) respectively. The
dotted vertical line represents $\hbar\omega=\mu$, where $\mu$ is the chemical
potential. For all our calculations, we have set the temperature to be $T=12$
K and the scattering time $\tau$ to be $10^{-14}$ s.

In this section, we study the linear EO effect in CuMnAs, a $\mathcal{PT}$
symmetric antiferromagnet with broken $\mathcal{P}$ and $\mathcal{T}$ symmetry
Tang  _et al._ (2016); Cao  _et al._ (2023); Rui  _et al._ (2019); Wadley  _et
al._ (2016). CuMnAs hosts oppositely magnetized sublattices (or Mn atoms -
designated as $A$ and $B$), which break the $\mathcal{P}$ and $\mathcal{T}$
symmetry. However, the simultaneous interchange of the $A$ and $B$ sublattices
with a magnetization flip keeps the system unchanged, reflecting its
${\mathcal{P}}{\mathcal{T}}$ symmetry. The opposite magnetization also leads
to sublattice-dependent anti-symmetric spin-orbit coupling. The effective
tight-binding Hamiltonian of this system Watanabe and Yanase (2021); Das  _et
al._ (2023) is given by

$\mathcal{H}(\textbf{k})=\begin{pmatrix}\epsilon_{0}(\textbf{k})+\textbf{f}_{A}(\textbf{k})\cdot{\bm{\sigma}}&&V_{AB}(\textbf{k})\\\
V_{AB}(\textbf{k})&&\epsilon_{0}(\textbf{k})+\textbf{f}_{B}(\textbf{k})\cdot{\bm{\sigma}}\end{pmatrix}\leavevmode\nobreak\
.$ (26)

Here, we have $\epsilon_{0}(\textbf{k})=-t(\cos k_{x}+\cos k_{y})$,
$V_{AB}=-2\tilde{t}\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}$, and
${\bm{\sigma}}$ represents the Pauli spin matrices. The hopping between
orbitals of the same sublattice is quantified by $t$, while $\tilde{t}$
denotes the hopping between orbitals on different sublattices. The sublattice-
dependent spin-orbit coupling and the magnetization field are included in
${\bm{f}}_{A}({\bm{k}})=-{\bm{f}}_{B}({\bm{k}})$ and we have,

$\displaystyle\bm{f}_{A}(\bm{k})=\begin{pmatrix}f^{x}_{AF}-\alpha_{r}\sin(k_{y})+\alpha_{d}\sin(k_{y})\\\
f^{y}_{AF}+\alpha_{r}\sin(k_{x})+\alpha_{d}\sin(k_{x})\\\ f^{z}_{AF}\\\
\end{pmatrix}\leavevmode\nobreak\ .$ (27)

The electronic band dispersion of the Hamiltonian in Eq. (26) are given by
$\epsilon(\textbf{k})=\epsilon_{0}\pm\sqrt{V^{2}_{ab}+f^{2}_{Ax}+f^{2}_{Ay}+f^{2}_{Az}}$,
and shown in Fig. 1(a). Here, $+(-)$ denotes the conduction (valence) band. We
present the momentum space distribution of different components of the quantum
metric and the metric connection in Fig. 1(b)-(f). These components are
pivotal for inducing polarization rotation within $\mathcal{PT}$-symmetric
systems. Fig. 3 in Appendix A shows the real and imaginary parts of
$\chi_{xxy}$ and $\chi_{yyy}$ for CuMnAs. As the Berry curvature and
symplectic connection vanish for a $\mathcal{PT}$ symmetric system, the shift
current is induced by the metric connection, while the other contributions
arise from the quantum metric. For CuMnAs, all the contributions of
$\chi_{xyy}$ and $\chi_{yxy}$ components are zero.

In Fig. 2(a) and 2(b), we show the variation of $\tilde{n}_{+}$ and
$\tilde{\kappa}_{+}$ with optical frequency. Figure 2(c) and 2(d) show the
variation of $\tilde{n}_{-}$ and $\tilde{\kappa}_{-}$ with frequency. Since
the NL susceptibilities are approximately seven orders smaller than the linear
susceptibility, the effect of the dc electric field becomes significant around
$E^{d}_{y}=10^{7}$ V/m or so. Experimentally, external field
($E^{d}_{y}=10^{6}$ V/m) induced linear EO effect (Pockels effect) has been
observed in ferroelectric materials like C8H8N2 and LiBNO3 with birefringence
being of the order of $\sim 10^{-5}$ and $\sim 10^{-4}$ respectively Uemura
_et al._ (2020). We present the altered birefringence,
$\Delta\tilde{n}=\Delta\tilde{n}_{+}-\Delta\tilde{n}_{-},$ as a function of
frequency in Fig. 2(e). In Fig. 2(f), we present the dichroism,
$\Delta\tilde{\kappa},$ as a function of frequency. We note that in Fig. 3,
all the non-monotonic variation of the optical susceptibilities occur for
frequencies $\hbar\omega\ll 2\mu$. Beyond the frequencies $\hbar\omega\ll
2\mu$, the nonlinear susceptibilities vanish as $1/\omega$. A similar trend
can also be seen in all panels of Fig. 2.

There are two distinct differences between the electro-optic effect in
conventional ferroelectric materials and ${\cal P}{\cal T}$ symmetric systems.
In contrast to ferroelectric materials, the EO effect in CuMnAs is induced by
the band geometric quantities, such as the quantum metric and the metric
connection. Additionally, CuMnAs supports the EO effect for optical energies
below the chemical potential [see the region to the left of the dashed
vertical line for $\mu=0.2$ eV in Fig. 2]. This is a distinct signature of
Fermi surface effects playing a significant role in the total optical
susceptibility in metallic systems.

##  VI Conclusion

To summarize, our study highlights the intricate phenomena of polarization
rotation induced by linear electro-optic effect-induced alteration in their
refractive index. Using quantum kinetic theory and density matrix formalism,
we have identified six distinct contributions to the linear EO effect: shift,
injection, Drude, anomalous, higher-order pole, and double resonant currents.
Our work emphasizes the significant roles of the quantum metric and metric
connection in $\mathcal{PT}$-symmetric systems, where Berry curvature and
symplectic connection are nullified. We derived expressions for the change in
refractive index (RI) and calculated the resulting polarization rotation of
emergent light in the presence of a transverse static electric field.

From an application perspective, the electro-optic effect is the cornerstone
for electro-optic modulation, which is crucial for precise control of optical
parameters. Electro-optic modulators are indispensable optical components for
modulating light’s intensity (intensity modulators) and phase or polarization
(phase modulators) Bull  _et al._ (2004); Forouzmand and Mosallaei (2019);
Zhang  _et al._ (2019); Chen  _et al._ (2014); Li  _et al._ (2020); Wang  _et
al._ (2018). In addition to serving as a potential probe for band geometric
quantities, efficient EO modulators drive innovation across the diverse fields
of optical communications Wooten  _et al._ (2000); Benea-Chelmus  _et al._
(2022a), computing Portner  _et al._ (2021), microwave Marpaung  _et al._
(2019) and quantum photonics Sun  _et al._ (2015); Carlson  _et al._ (2018);
Karpiński  _et al._ (2017), sensing Benea-Chelmus  _et al._ (2022b); Prasad
_et al._ (2017), and telecommunications Dalton (2019); Kumar  _et al._ (2015);
Krasavin and Zayats (2012).

##  VII Acknowledgement

We acknowledge Debasis Dutta, Kamal Das, and Debottam Mandal for the useful
discussions. M.M.K. acknowledges the Department of Science and Technology,
Government of India, for financial support via Project No.
DST/NM/TUE/QM-6/2019(G)-IIT Kanpur. S. S. thanks the MHRD, India, for funding
through the Prime Minister’s Research Fellowship (PMRF).

##  Appendix A Optical susceptibility calculations

In this section, we provide a detailed calculation of the optical conductivity
using the density matrix up to the second order in the electric field.
Perturbatively treating the density matrix in the quantum Liouville equation,
the $N$th order density matrix can be shown to be related to the $(N-1)$th
order density matrix via the recursive relation,

$\dfrac{\partial\rho^{(N)}_{mp}}{\partial
t}+\dfrac{i}{\hbar}[\hat{H}_{0},\rho^{(N)}]_{mp}+\dfrac{\rho^{(N)}_{mp}}{\tau/N}=\frac{-ie\bm{E}}{\hbar}\cdot[{\bm{r}},\rho^{(N-1)}]_{mp}\leavevmode\nobreak\
.$ (28)

In the steady state, this simplifies to,

$\left(-i\omega_{\Sigma}+i\omega_{mp}+\dfrac{N}{\tau}\right)\rho^{(N)}_{mp}=\dfrac{e{\bm{E}}}{\hbar}\cdot[\mathcal{D}_{\bm{k}}\rho^{(N-1)}]_{mp}\leavevmode\nobreak\
,$ (29)

which leads us to the recursive equation for $\rho^{(N)}$,

$\rho^{(N)}_{mp}=\frac{e}{\hbar}g_{mp}^{\omega_{\Sigma}}{\bm{E}}\cdot[\mathcal{D}_{\bm{k}}\rho^{(N-1)}]_{mp}\leavevmode\nobreak\
.$ (30)

The function
$g_{mp}^{\omega_{\Sigma}}=[N/\tau-i(\omega_{\Sigma}-\omega_{mp})]^{-1}$
characterizes the density of states broadened by scattering corresponding to
incident frequency $\omega_{\Sigma}$. Here, $\omega_{\Sigma}$ and
$\omega_{mp}=\hbar^{-1}(\epsilon_{m,\bm{k}}-\epsilon_{n,\bm{k}})$ are the sum
of the harmonic frequencies incident on the system, and difference of the
frequencies corresponding to bands indexed $m$ and $n,$ respectively. The
first-order density matrix is given by,

$\rho^{(1)}_{mp}=\frac{e}{\hbar}g_{mp}^{\omega}{\bm{E}}\cdot\left(\partial_{\bm{k}}\rho_{mp}^{(0)}-i[\mathcal{R}_{\bm{k}},\rho^{(0)}]_{mp}\right)\leavevmode\nobreak\
.$ (31)

Here, $\rho^{(0)}_{mp}=f_{m}\leavevmode\nobreak\ \delta_{mp}$ represents the
equilibrium distribution of electrons, with
$f_{m}=[1+e^{\beta(\epsilon_{m{\bm{k}}}-\mu)}]^{-1}$ being the Fermi-Dirac
distribution for the $m$th band, and $\beta=1/(k_{B}T)$. With some
simplifications, the first-order density matrix can be expressed as,

$\rho^{(1)}_{mp}=\frac{e}{\hbar}g^{\omega}_{mp}(\partial_{c}f_{m}+i\mathcal{R}^{c}_{mp}F_{mp})E_{c}e^{-i\omega
t}\leavevmode\nobreak\ .$ (32)

We note that for $m=p$, the second term of the above equation vanishes, and
for $m\neq p$ the first term vanishes. Calculating the second-order density
matrix, we obtain,

$\rho^{(2)}_{mp}=\frac{e}{\hbar}g_{mp}^{\omega_{\Sigma}}{\bm{E}}\cdot\left[\partial_{\bm{k}}\rho_{mp}^{(1)}-i[\mathcal{R}_{\bm{k}},\rho^{(1)}]_{mp}\right]\leavevmode\nobreak\
\\\
=\frac{e}{\hbar}g_{mp}^{\omega_{\Sigma}}E_{c}\left[\partial_{c}\rho_{mp}^{(1)}-i\sum_{n}\left(\mathcal{R}^{c}_{mn}\rho_{np}^{(1)}-\rho_{mn}^{(1)}\mathcal{R}^{c}_{np}\right)\right]\leavevmode\nobreak\
.$ (33)

Figure 3: Frequency dependence of various components - shift (Sh), Drude (D),
injection (Inj), anomalous (An), higher-order pole (HOP), and double-resonant
(DR), of nonlinear susceptibility. Panels (a) and (b) show the real and
imaginary parts of the susceptibility $\chi_{xxy}$, respectively. Panels (c)
and (d) show the real and imaginary parts of the susceptibility $\chi_{yyy}$,
respectively. The susceptibilities without the Drude contribution are shown in
the inset of panels (b) and (d). The model parameters are the same as in fig.
1.

For a two-band model, the second-order density matrix can be separated into
four parts: two diagonal and two off-diagonal terms. We define them as,
$\rho_{mm}^{(2),I}$, $\rho_{mm}^{(2),II}$, $\rho_{mp}^{(2),I}$, and
$\rho_{mp}^{(2),II}$, respectively. These four parts after pairwise field
symmetrization can be expressed as,

$\displaystyle\rho_{mm}^{(2),I}(\omega;\omega,0)$ $\displaystyle=$
$\displaystyle\dfrac{e^{2}}{\hbar^{2}}(\tilde{g}^{\omega}_{0}{g}^{\omega}_{0}+\tilde{g}^{0}_{0}{g}^{\omega}_{0})\partial_{c}\partial_{d}f_{m}E_{c}E^{d}_{y}\leavevmode\nobreak\
,$ (34) $\displaystyle\rho_{mm}^{(2),II}(\omega;\omega,0)$ $\displaystyle=$
$\displaystyle\dfrac{e^{2}}{\hbar^{2}}\tilde{g}^{\omega}_{0}\sum_{n}[({g}^{\omega}_{nm}\mathcal{R}^{b}_{mn}\mathcal{R}^{c}_{nm}F_{nm}-{g}^{\omega}_{mn}\mathcal{R}^{c}_{mn}\mathcal{R}^{b}_{nm}F_{mn})+({g}^{0}_{nm}\mathcal{R}^{c}_{mn}\mathcal{R}^{b}_{nm}F_{nm}-{g}^{0}_{mn}\mathcal{R}^{b}_{mn}\mathcal{R}^{c}_{nm}F_{mn})]\leavevmode\nobreak\
,$ $\displaystyle\rho_{mp}^{(2),I}(\omega;\omega,0)$ $\displaystyle=$
$\displaystyle
i\dfrac{e^{2}}{\hbar^{2}}\tilde{g}^{\omega}_{mp}\left[\left({g}^{\omega}_{mp}F_{mp}\mathcal{D}^{b}_{mp}\mathcal{R}^{c}_{mp}+\mathcal{R}^{c}_{mp}\partial_{b}({g}^{\omega}_{mp}F_{mp})\right)+\left({g}^{0}_{mp}F_{mp}\mathcal{D}^{c}_{mp}\mathcal{R}^{b}_{mp}+\mathcal{R}^{b}_{mp}\partial_{c}({g}^{0}_{mp}F_{mp})\right)\right]\leavevmode\nobreak\
,$ (36) $\displaystyle\rho_{mp}^{(2),II}(\omega;\omega,0)$ $\displaystyle=$
$\displaystyle
i\dfrac{e^{2}}{\hbar^{2}}\tilde{g}^{\omega}_{mp}\left[{g}^{\omega}_{0}\mathcal{R}^{b}_{mp}\partial_{c}F_{mp}+{g}^{0}_{0}\mathcal{R}^{c}_{mp}\partial_{b}F_{mp}\right]\leavevmode\nobreak\
.$ (37)

In the above equations, we have considered the frequency combination of
$(\omega,0)$. Further, we calculate the current densities using the
definition,

${\bm{j}}(t)=-e{\rm Tr}[{\rho}\hat{\bm{v}}]\leavevmode\nobreak\ .$ (38)

So, the $N$th order current can be calculated as,

$j^{(N)}_{a}(t)=-e\sum_{m,p}v_{pm}^{a}\rho_{mp}^{(N)}(t)\leavevmode\nobreak\
.$ (39)

Here, $v^{a}_{pm}=v^{0a}_{pm}\delta_{pm}+i\omega_{pm}\mathcal{R}^{a}_{pm}$ is
the general velocity operator with
$v^{0a}_{mm}=\hbar^{-1}\partial_{a}\epsilon_{m},$ and $\mathcal{R}^{q}_{pm}$
is the $q$th component of the interband Berry connection. Using Eq. (39), for
$N=1$, we can write the linear current as,

$\displaystyle j^{(1)}_{a}(t)={}$
$\displaystyle-\frac{e^{2}}{\hbar}\sum_{m,p}\sum_{c}g^{\omega}_{mp}(v^{0a}_{pm}\delta_{pm}+i\omega_{mp}\mathcal{R}^{a}_{pm})(\partial_{c}f_{m}+i\mathcal{R}^{c}_{mp}F_{mp})E^{c}e^{-i\omega
t}\leavevmode\nobreak\ ,$
$\displaystyle=-\frac{e^{2}}{\hbar}\sum_{c}\bigg{[}\sum_{m}\frac{1}{\hbar}(\partial_{a}\epsilon_{m})(\partial_{c}f_{m})g^{\omega}_{0}-\sum_{m,p}\omega_{mp}g^{\omega}_{mp}\mathcal{R}^{a}_{pm}\mathcal{R}^{c}_{mp}F_{mp}\bigg{]}E^{c}e^{-i\omega
t}\leavevmode\nobreak\ ,$
$\displaystyle=-\frac{e^{2}}{\hbar}\sum_{c}\bigg{[}\sum_{m}\frac{1}{\hbar}(\partial_{a}\epsilon_{m})(\partial_{c}f_{m})g^{\omega}_{0}-\sum_{m,p}\omega_{mp}g^{\omega}_{mp}\mathcal{Q}^{ac}_{mp}F_{mp}\bigg{]}E^{c}e^{-i\omega
t}\leavevmode\nobreak\ .$ (40)

The product of two Berry connections has been expressed in terms of the QGT.
It can be expressed in terms of the band-resolved Berry curvature and the
quantum metric as,

$\mathcal{Q}^{bc}_{mp}=\mathcal{R}^{b}_{pm}\mathcal{R}^{c}_{mp}=\mathcal{G}^{bc}_{mp}-(i/2)\Omega^{bc}_{mp}\leavevmode\nobreak\
.$ (41)

We have two contributions to the current density. Under $\mathcal{PT}$
symmetry, as the Berry curvature vanishes, the linear conductivity
corresponding to each of the contributions can therefore be given as,

$\displaystyle\sigma^{\text{intra}}_{ac}(\omega)$
$\displaystyle=-\frac{e^{2}}{\hbar}\sum_{m}\int[d\textbf{k}]\frac{1}{\hbar}(\partial_{a}\epsilon_{m})g^{\omega}_{0}(\partial_{c}f^{(0)}_{m})\leavevmode\nobreak\
,$ (42) $\displaystyle\sigma^{\text{inter}}_{ac}(\omega)$
$\displaystyle=\frac{e^{2}}{\hbar}\sum_{m,p}\int[d\textbf{k}]\omega_{pm}g^{\omega}_{mp}\mathcal{G}^{ac}_{mp}F_{mp}\leavevmode\nobreak\
.$ (43)

The linear conductivity $\sigma_{ac}(\omega)$ is given as,

$\sigma_{ac}(\omega)=\sigma^{\text{intra}}_{ac}(\omega)+\sigma^{\text{inter}}_{ac}(\omega)\leavevmode\nobreak\
.$ (44)

The corresponding susceptibility tensor is given from
$\sigma_{ac}(\omega)=-i\omega\epsilon_{0}\chi_{ac}(\omega)$, where
$\epsilon_{0}$ is the permittivity of vacuum. It can be shown that the
intraband conductivity is the infamous Drude conductivity Ashcroft  _et al._
(1976) for frequency-dependent electric fields and that the interband
contribution we see for $\sigma^{\text{inter}}_{ac}(\omega)$, is the linear
anomalous Hall conductivity Nagaosa  _et al._ (2010). As the Berry curvature
vanishes in a $\mathcal{PT}$-symmetric system, the total linear contribution
to the interband conductivity would be from the quantum metric. At $N=2$ in
Eq. (39), we get the second-order current. The expressions of the pairwise
field-symmetrized second-order susceptibilities are shown in the main text.

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