# Foldy-Wouthuysen transformation for gapped Dirac fermions in two-dimensional
semiconducting materials and valley excitons under external fields

Yao-Wen Chang<EMAIL_ADDRESS>Yia-Chung Chang
<EMAIL_ADDRESS>Research Center for Applied Sciences, Academia
Sinica, Taipei 11529, Taiwan

###### Abstract

In this work, we provide a detailed derivation of Foldy-Wouthuysen (FW)
transformation for two-dimensional (2D) gapped Dirac fermions under external
fields and apply the formalism to study valley excitons in 2D semiconducting
materials. Similar to relativistic quantum few-body problem, the gapped Dirac
equation can be transformed into a Schrödinger equation with ”relativistic”
correction terms. In this 2D materials system, the correction terms can be
interpreted as the Berry-curvature effect. The Hamiltonian for a valley
exciton in external fields can be written based on the FW transformed Dirac
Hamiltonian. Various valley-dependent effects on excitons, such as fine-
structure splittings of exciton energy levels, valley-selected exciton
transitions, and exciton valley Zeeman effect are discussed within this
framework.

##  I Introduction

Two-dimensional (2D) semiconducting materials such as transition metal
dichalcogenides (TMDCs) are atomic thin semiconductors known for their
potential for future applications in electronic and photonic
deviceswang2012electronics ; xia2014two ; mak2016photonics . The valley-
dependent electronic structure of 2D materials provides a new degree of
freedom to manipulate, and the unique topology of band structure introduces an
intriguing Berry-curvature effect on their physical
propertiesschaibley2016valleytronics ; vitale2018valleytronics ; mak2018light
; zhao2021valley . The lower dimensionality also induces strong electron
correlations and thus excitonic effect becomes important. A valley exciton is
a Wannier-like exciton whose properties are decided by the additional valley
degree of freedom and affected by the band-structure geometrycai2013magnetic ;
berkelbach2015bright ; yu2015valley ; berkelbach2017optical ;
durnev2018excitons ; li2020fine . Various physical phenomena related to valley
excitons, such as Berry-curvature induced exciton energy-level
splittingsrivastava2015signatures ; zhou2015berry ; trushin2018model ;
van2019spectrum ; yong2019valley , valley-selected optical
transitionberkelbach2015bright ; yu2015valley ; gong2017optical ;
zhang2018optical , exciton valley Hall effectonga2017exciton ; glazov2020skew
, and exciton valley Zeeman effectvan2018strong ; bragancca2019magnetic ;
catarina2019optical ; koperski2018orbital ; YCPRL  have been observed
experimentally and discussed theoretically. While there are a lot of
theoretical works using different methods to study different issues of valley
excitons, the connection among different theories and interpretations is not
manifest.

One of theoretical issues to study valley excitons is to include external-
field interactions to the exciton model. For a Wannier exciton, which consists
of an electron and a hole bound by the Coulomb interaction, the system is
similar to a hydrogen atom in external fields. The electric-field interaction
can be included by the dipole-field interaction and the magnetic-field
interaction can be included as the vector fields in the kinetic momentums of
the electron and hole. For the valley excitons in 2D materials, the exciton
Hamiltonian is constructed by the Bloch-electron wavefunctions solved from
diagonalizing a Dirac Hamiltonian or a few-band tight-binding
Hamiltonianberkelbach2015bright ; berkelbach2017optical . However, the Dirac
Hamiltonian or tight-binding Hamiltonian with including external-field
interactions is much more difficult to solve, such that the exciton
Hamiltonian under external field also becomes more difficult to derive. To
simplify the problem, approximations are necessary. In fact, a similar problem
had been intensively studied in the fields of relativistic few-body
physicsbethe2012quantum ; reiher2014relativistic . A relativistic particle
which is described by a three-dimensional Dirac fermion under external fields
can be reduced to a nonrelativistic particle with relativistic corrections in
low-energy limit by the Foldy-Wouthuysen (FW)
transformationreiher2014relativistic ; foldy1950dirac ; eriksen1958foldy ;
silenko2008foldy ; silenko2016general . A relativistic few-body problem, such
as a real hydrogen atom in external fields, can be studied by solving the
nonrelativistic Hamiltonian with including the relativistic
correctionsclose1970relativistic ; krajcik1974relativistic ;
anthony1994relativistic . The approximation scheme can achieve extreme
accuracy in prediction of energy spectra and fine structures. Following the
same strategy, the 2D Dirac fermions can also be reduced to nonrelativistic
form by the FW transformation, and the valley-exciton Hamiltonian can be
approximated as the Wannier-exciton Hamiltonian with band-geometry
corrections. The valley excitons in external fields can then be studied by the
corrected exciton Hamiltonian.

Semiclassical methods have been used to study the band-structure correction to
the properties of excitonsxiao2005berry ; yao2008valley ; yao2008berry ;
chang2008berry ; gradhand2012first . The Berry curvature effect on excitons in
2D materials has been studied theoreticallycai2013magnetic ;
srivastava2015signatures ; zhou2015berry ; gong2017optical ; trushin2018model
; zhang2018optical ; van2019spectrum  and observed
experimentallyyong2019valley . The Berry-curvature effect causes energy-level
splittingsrivastava2015signatures ; zhou2015berry ; trushin2018model ;
van2019spectrum ; yong2019valley  and anomalous selection rulegong2017optical
; zhang2018optical  for valley excitons with nonzero angular momentum. It has
been shown that the exciton Hamiltonian can be derived from a FW
transformation of 2D gapped Dirac fermionszhou2015berry ; trushin2018model .
However, the joint effect of the Berry-curvature effect and external fields
has yet to be discussed. We intend to fill this gap by extending the
derivation of FW transformation to incorporate both effects for 2D gapped
Dirac fermions in external fields.

In the present work, the Bloch electron in 2D materials under external fields
is described by a 2D gapped Dirac model, and the FW transformation is used to
derive the electron-hole representation of the Dirac fermion. The FW
transformed single-particle Hamiltonian, two-particle interaction, and
interband transition are derived. Hamiltonians for valley excitons in external
fields, including an in-plane electric field, an in-plane electromagnetic
field, and an out-of-plane magnetic field can be written. The formalism is
applied to the study of physical properties of valley excitons. Within this
theoretical framework, several known valley-dependent excitonic effects
including exciton energy-level splittings, valley-selected exciton
transitions, and exciton valley Zeeman effect are studied and discussed.

The remainder of the article is organized as follows. In Sec. II, we review
the 2D gapped Dirac fermion model for the band structure of Bloch electrons
with a screened Coulomb interaction. The electromagnetic interaction,
electron-hole representation, and exciton states based on the Dirac fermion
model are also defined here. In Sec. III, the FW transformation for the 2D
gapped Dirac fermion model is introduced. The FW transformed single-particle
Hamiltonian, two-particle interaction, and interband transition are derived.
In Sec. IV, the formula derived from the FW transformation are applied to
valley excitons in 2D materials. The Hamiltonians for valley excitons in an
in-plane electric field, an in-plane electromagnetic field, and an out-of-
plane magnetic field are derived, and valley-dependent phenomena are studied.
Finally, the summary and some perspectives of this theoretical framework are
given in Sec. V. In Appendix A, the variational method used to solve exciton
eigenenergies and exciton wavefunctions is presented in detail.

##  II 2D gapped Dirac fermion

Before introducing the FW transformation, we first review the 2D gapped Dirac
model and derive the exciton model based on the formulation of the Dirac
model. In Sec. II.1, the many-body Dirac Hamiltonian is introduced in the
second quantization formalism. In Sec. II.2, the electromagnetic-field
interaction in the Dirac model and the formula for calculating optical spectra
are given. In Sec. II.3, the electron-hole representation of the many-body
Dirac Hamiltonian is introduced and the exciton state is written in this
representation.

###  II.1 Many-body electronic Hamiltonian

The many-body electronic Hamiltonian for 2D gapped Dirac fermions can be
written as berkelbach2015bright ; berkelbach2017optical

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\sum_{\tau}\int\hat{\psi}^{\dagger}_{\tau}(\mathbf{r})\mathtt{h}_{\tau}(\mathbf{r})\hat{\psi}_{\tau}(\mathbf{r})\text{d}^{2}r$
(1)
$\displaystyle+\frac{1}{2}\int{V}({r}_{12})\hat{\rho}(\mathbf{r}_{1})\hat{\rho}(\mathbf{r}_{2})\text{d}^{2}r_{1}\text{d}^{2}r_{2},$

where $\hat{\psi}^{\dagger}_{\tau}(\mathbf{r})$,
$\hat{\psi}_{\tau}(\mathbf{r})$ are the fermion field operators with $\tau$
being the valley index,
$\hat{\rho}(\mathbf{r})=\sum_{\tau}\hat{\psi}^{\dagger}_{\tau}(\mathbf{r})\hat{\psi}_{\tau}(\mathbf{r})$
is the density operator, $\mathtt{h}_{\tau}(\mathbf{r})$ is the single-
particle Hamiltonian, and ${V}({r}_{12})$ is a 2D screened Coulomb potential
with ${r}_{12}=|\mathbf{r}_{1}-\mathbf{r}_{2}|$. The single-particle
Hamiltonian is given by

$\displaystyle\mathtt{h}_{\tau}(\mathbf{r})$ $\displaystyle=$ $\displaystyle
v_{\text{F}}\bm{\pi}\cdot{\bm{\alpha}_{\tau}}+mv^{2}_{\text{F}}{\beta}-q\mathbf{F}\cdot\mathbf{r}$
(2) $\displaystyle=$
$\displaystyle\begin{pmatrix}mv^{2}_{\text{F}}-q\mathbf{F}\cdot\mathbf{r}&v_{\text{F}}\left(\tau\pi^{x}-\mathtt{i}\pi^{y}\right)\\\
v_{\text{F}}\left(\tau\pi^{x}+\mathtt{i}\pi^{y}\right)&-mv^{2}_{\text{F}}-q\mathbf{F}\cdot\mathbf{r}\end{pmatrix}$

with
$\bm{\alpha}_{\tau}=\tau\sigma_{x}\mathbf{e}_{x}+\sigma_{y}\mathbf{e}_{y}$ and
$\beta=\sigma_{z}$, and $\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$ being Pauli
matrices, $\mathbf{F}$ an in-plane electric field, $\bm{\pi}$ the kinetic
momentum, $q$ the charge, $v_{\text{F}}$ the Fermi velocity, and $m$ the
effective mass of the particle. The effective mass can be connected to a band-
gap energy by $\Delta_{0}=2mv^{2}_{\text{F}}$. Note that the band gap
$\Delta_{0}$ here is not the same with the observed transport band gap, which
also contains the contribution from electron correlations. The kinetic
momentum of a Dirac particle in an out-of-plane magnetic field is given by

$\displaystyle\bm{\pi}=\mathbf{p}-\frac{qB}{2}\mathbf{e}_{\perp}\times\mathbf{r},$
(3)

where $\mathbf{p}=-\mathtt{i}\bm{\nabla}$ is the momentum,
$\mathbf{e}_{\perp}$ is a unit vector perpendicular to the surface of the 2D
system, and $B$ is the amplitude of the out-of-plane magnetic field.

###  II.2 Electromagnetic-field interaction

The electromagnetic-field interaction with the Dirac fermion can be included
by extending the kinetic momentum

$\displaystyle\bm{\pi}^{\star}(t)=\bm{\pi}-q\int\tilde{\bm{\mathcal{A}}}(\mathbf{k},t)e^{\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\frac{\text{d}^{2}k}{(2\pi)^{2}},$
(4)

where
$\tilde{\bm{\mathcal{A}}}(\mathbf{k},t)e^{\mathtt{i}\mathbf{k}\cdot\mathbf{r}}$
is the vector potential in an electromagnetic mode with wavevector
$\mathbf{k}$. In the long-wavelength limit, the vector potential can be
simplified as

$\displaystyle\tilde{\bm{\mathcal{A}}}(\mathbf{k},t)e^{\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\simeq\tilde{\bm{\mathcal{A}}}(\mathbf{k},t)\left(1+\mathtt{i}\mathbf{k}\cdot\mathbf{r}+\cdots\right)\simeq\tilde{\bm{\mathcal{A}}}(\mathbf{k},t).$
(5)

Including the electromagnetic-field interaction, the electronic Hamiltonian
can be rewritten as

$\displaystyle\hat{\mathcal{H}}^{\star}(t)=\hat{\mathcal{H}}-\hat{\bm{j}}\cdot\bm{\mathcal{A}}(t),$
(6)

where
$\bm{\mathcal{A}}(t)=\int\tilde{\bm{\mathcal{A}}}(\mathbf{k},t)\text{d}^{2}k/(2\pi)^{2}$
is the electromagnetic vector potential and

$\displaystyle\hat{\bm{j}}=q\sum_{\tau}\int\hat{\psi}^{\dagger}_{\tau}(\mathbf{r})\bm{u}_{\tau}\hat{\psi}_{\tau}(\mathbf{r})\text{d}^{2}r$
(7)

is the current operator with $\bm{u}_{\tau}=v_{\text{F}}\bm{\alpha}_{\tau}$
the velocity matrix. We define
$\text{d}\hat{\bm{\xi}}(t)=\int_{0}^{t}\hat{\bm{j}}(t^{\prime})dt^{\prime}$ as
the polarization operator. It can be shown that the polarization operator can
be related to the current operator by

$\displaystyle\frac{\text{d}\hat{\bm{\xi}}}{\text{d}t}=\mathtt{i}\left[\hat{\mathcal{H}},\;\hat{\bm{\xi}}\right]=\hat{\bm{j}},$
(8)

where the interaction picture is used for the time-dependent fermion field
operators. By using a length-gauge transformation, the electromagnetic-field
interaction in the electronic Hamiltonian can be rewritten as a dipole-field
interaction, and we have

$\displaystyle\hat{\mathcal{H}}^{\star}_{\text{L}}(t)$ $\displaystyle=$
$\displaystyle
e^{-\mathtt{i}\hat{\bm{\xi}}\cdot\bm{\mathcal{A}}(t)}\left[\hat{\mathcal{H}}^{\star}(t)-\mathtt{i}\frac{\partial}{\partial{t}}\right]e^{\mathtt{i}\hat{\bm{\xi}}\cdot\bm{\mathcal{A}}(t)}+\mathtt{i}\frac{\partial}{\partial{t}}$
(9) $\displaystyle=$
$\displaystyle\hat{\mathcal{H}}-\hat{\bm{\xi}}\cdot\bm{\mathcal{F}}(t),$

with $\bm{\mathcal{F}}=-{\partial{\bm{\mathcal{A}}}}/{\partial{t}}$ being the
electromagnetic field. The formula of electromagnetic-field interaction in Eq.
(9) is known as the electric-dipole approximation. On the other hand, if the
polarization matrix element is difficult to solve, the current matrix element
can also be used to calculate the transition probabilities. The current matrix
element can be related to the polarization matrix element by using Eq. (8), as

$\displaystyle\langle N|\hat{\bm{j}}|M\rangle$ $\displaystyle=$
$\displaystyle\langle
N|\frac{\text{d}\hat{\bm{\xi}}}{\text{d}t}|M\rangle=\mathtt{i}(E_{N}-E_{M})\langle
N|\hat{\bm{\xi}}|M\rangle,$ (10)

with $|N\langle$ the $N$-th state wavefunction and ${E}_{N}$ the $N$-th state
eigenenergy. Therefore, the transition probabilities can be related to the
current matrix element via the relation
${\xi}^{\mu}_{NM}=-\mathtt{i}{j}^{\mu}_{NM}/({E_{N}-E_{M}})$, where
$\xi^{\mu}_{NM}=\langle{N}|\hat{\xi}^{\mu}|{M}\rangle$ is the polarization
matrix element and $j^{\mu}_{NM}=\langle{N}|\hat{j}^{\mu}|{M}\rangle$ is the
current matrix element. It is what we will do to solve one-exciton transition
probability in Sec. IV, because it is more convenient to find the current
matrix element by the FW-transformed formulations.

Based on Fermi’s golden rule, the one-photon transition probability is given
byberkelbach2015bright

$\displaystyle\Gamma^{\mu}_{\text{1P}}(\omega)$ $\displaystyle=$
$\displaystyle{2\pi}\sum_{NM}\varrho_{M}|\xi^{\mu}_{NM}|^{2}\delta(\omega-
E_{N}+E_{M}),$ (11)

where
$\varrho_{M}=\exp\left({-\beta_{\vartheta}{E}_{M}}\right)/\left[\sum_{M^{\prime}}\exp\left({-\beta_{\vartheta}\mathcal{E}_{M^{\prime}}}\right)\right]$
is the distribution function of the initial state with $\beta_{\vartheta}$ the
inverse temperature. The two-photon transition probability is given by the
Kramers-Heisenberg formulaberkelbach2015bright

$\displaystyle\Gamma^{\mu}_{\text{2P}}(2\omega)$ $\displaystyle=$
$\displaystyle
2\pi\sum_{NM}\varrho_{M}|\Upsilon^{\mu}_{NM}(\omega)|^{2}\delta(2\omega-
E_{N}+E_{M}),$

where
$\Upsilon^{\mu}_{NM}(\omega)=\sum_{Q}{\xi^{\mu}_{NQ}\xi^{\mu}_{QM}}/({E_{Q}-E_{M}-\omega-\mathtt{i}\eta})$
is the two-photon transition amplitude with $\eta$ a line-broadening factor.
By using these formulations, the one-photon and two-photon absorption spectra
can be studied.

###  II.3 Electron-hole representation

The Fourier transformed single-particle Hamiltonian and fermion field operator
are defined as
$\tilde{\mathtt{h}}_{\tau}(\mathbf{k})=\int\mathtt{h}_{\tau}(\mathbf{r})e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\text{d}^{2}r$,
$\hat{\tilde{\psi}}_{\tau}(\mathbf{k})=\int\hat{\psi}_{\tau}(\mathbf{r})e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\text{d}^{2}r$.
It can be found that there is an unitary transformation matrix such that

$\displaystyle{\mathcal{U}}_{\tau}(\mathbf{k})\tilde{\mathtt{h}}_{\tau}(\mathbf{k}){\mathcal{U}}^{\dagger}_{\tau}(\mathbf{k})=\tilde{\varepsilon}_{\tau}(\mathbf{k})\beta,$
(13) $\displaystyle\hat{\tilde{\psi}}_{\tau}(\mathbf{k})$ $\displaystyle=$
$\displaystyle{\mathcal{U}}^{\dagger}_{\tau}(\mathbf{k})\begin{pmatrix}\hat{c}_{\tau}(\mathbf{k})\\\
\hat{d}^{\dagger}_{\tau}(-\mathbf{k})\end{pmatrix},$ (14)
$\displaystyle\hat{\tilde{\psi}}^{\dagger}_{\tau}(\mathbf{k})$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\hat{c}^{\dagger}_{\tau}(\mathbf{k})&\hat{d}_{\tau}(-\mathbf{k})\end{pmatrix}{\mathcal{U}}_{\tau}(\mathbf{k}),$
(15)

where $\tilde{\varepsilon}_{\tau}(\mathbf{k})$ is the single-particle energy,
$\hat{c}^{\dagger}_{\tau}(\mathbf{k})$, $\hat{c}_{\tau}(\mathbf{k})$ are the
electron creation, annihilation operators and
$\hat{d}^{\dagger}_{\tau}(\mathbf{k})$, $\hat{d}_{\tau}(\mathbf{k})$ are the
hole creation, annihilation operators. The kinetic part of the many-body
Hamiltonian can be transformed as

$\displaystyle\hat{\mathcal{H}}_{\text{kinetic}}$ $\displaystyle=$
$\displaystyle\sum_{\tau}\int\hat{\psi}^{\dagger}_{\tau}(\mathbf{r})\mathtt{h}_{\tau}(\mathbf{r})\hat{\psi}_{\tau}(\mathbf{r})\text{d}^{2}r$
(16) $\displaystyle=$ $\displaystyle
E_{0}+\sum_{\tau}\int\tilde{\varepsilon}_{\tau}(\mathbf{k})\big{[}\hat{c}^{\dagger}_{\tau}(\mathbf{k})\hat{c}_{\tau}(\mathbf{k})$
$\displaystyle+\hat{d}^{\dagger}_{\tau}(-\mathbf{k})\hat{d}_{\tau}(-\mathbf{k})\big{]}\frac{\text{d}^{2}k}{(2\pi)^{2}},$

where $E_{0}$ is the ground-state energy.

The polarization and current operators can be rewritten by integrations in
quasi-momentum space as

$\displaystyle\hat{\bm{\xi}}=\mathtt{i}q\sum_{\tau}\int\hat{\tilde{\psi}}^{\dagger}_{\tau}(\mathbf{k})\bm{\nabla}_{\mathbf{k}}\hat{\tilde{\psi}}_{\tau}(\mathbf{k})\frac{\text{d}^{2}k}{(2\pi)^{2}},$
(17)
$\displaystyle\hat{\bm{j}}=q\sum_{\tau}\int\hat{\tilde{\psi}}^{\dagger}_{\tau}(\mathbf{k})\bm{u}_{\tau}\hat{\tilde{\psi}}_{\tau}(\mathbf{k})\frac{\text{d}^{2}k}{(2\pi)^{2}}.$
(18)

Because the hole can be seen as the electron with negative energy in the
present formulation, the electron with charge $q=-e$ is chosen for the
polarization and current operators. The momentum matrix elements are defined
by

$\displaystyle\begin{pmatrix}\tilde{\bm{\mathcal{P}}}_{\text{ee},\tau}(\mathbf{k})&\tilde{\bm{\mathcal{P}}}_{\text{eh},\tau}(\mathbf{k})\\\
\tilde{\bm{\mathcal{P}}}_{\text{he},\tau}(\mathbf{k})&\tilde{\bm{\mathcal{P}}}_{\text{hh},\tau}(\mathbf{k})\end{pmatrix}={\mathcal{U}}_{\tau}(\mathbf{k})\bm{u}_{\tau}{\mathcal{U}}^{\dagger}_{\tau}(\mathbf{k}),$
(19)

and then the current operator can be rewritten as

$\displaystyle\hat{\bm{j}}$ $\displaystyle=$
$\displaystyle-e\sum_{\tau}\int\Big{[}\hat{c}^{\dagger}_{\tau}(\mathbf{k})\tilde{\bm{\mathcal{P}}}_{\text{ee},\tau}(\mathbf{k})\hat{c}_{\tau}(\mathbf{k})$
(20)
$\displaystyle+\hat{d}_{\tau}(-\mathbf{k})\tilde{\bm{\mathcal{P}}}_{\text{he},\tau}(\mathbf{k})\hat{c}_{\tau}(\mathbf{k})+\hat{c}^{\dagger}_{\tau}(\mathbf{k})\tilde{\bm{\mathcal{P}}}_{\text{eh},\tau}(\mathbf{k})\hat{d}^{\dagger}_{\tau}(-\mathbf{k})$
$\displaystyle-\hat{d}^{\dagger}_{\tau}(-\mathbf{k})\tilde{\bm{\mathcal{P}}}_{\text{hh},\tau}(\mathbf{k})\hat{d}_{\tau}(-\mathbf{k})\Big{]}\frac{\text{d}^{2}k}{(2\pi)^{2}}.$

The dipole-moment matrix elements are defined by

$\displaystyle\begin{pmatrix}\tilde{\bm{\mathcal{D}}}_{\text{ee},\tau}(\mathbf{k})&\tilde{\bm{\mathcal{D}}}_{\text{eh},\tau}(\mathbf{k})\\\
\tilde{\bm{\mathcal{D}}}_{\text{he},\tau}(\mathbf{k})&\tilde{\bm{\mathcal{D}}}_{\text{hh},\tau}(\mathbf{k})\end{pmatrix}=\mathtt{i}\bm{\nabla}_{\mathbf{k}}+{\mathcal{U}}_{\tau}(\mathbf{k})\mathtt{i}\bm{\nabla}_{\mathbf{k}}{\mathcal{U}}^{\dagger}_{\tau}(\mathbf{k}),$
(21)

and the polarization operator becomes

$\displaystyle\hat{\bm{\xi}}$ $\displaystyle=$
$\displaystyle-e\sum_{\tau}\int\Big{[}\hat{c}^{\dagger}_{\tau}(\mathbf{k})\tilde{\bm{\mathcal{D}}}_{\text{ee},\tau}(\mathbf{k})\hat{c}_{\tau}(\mathbf{k})$
(22)
$\displaystyle+\hat{d}_{\tau}(-\mathbf{k})\tilde{\bm{\mathcal{D}}}_{\text{he},\tau}(\mathbf{k})\hat{c}_{\tau}(\mathbf{k})+\hat{c}^{\dagger}_{\tau}(\mathbf{k})\tilde{\bm{\mathcal{D}}}_{\text{eh},\tau}(\mathbf{k})\hat{d}^{\dagger}_{\tau}(-\mathbf{k})$
$\displaystyle-\hat{d}^{\dagger}_{\tau}(-\mathbf{k})\tilde{\bm{\mathcal{D}}}_{\text{hh},\tau}(\mathbf{k})\hat{d}_{\tau}(-\mathbf{k})\Big{]}\frac{\text{d}^{2}k}{(2\pi)^{2}}.$

By using the electron-hole representation, it is convenient to write the
excitonic excited states, and the optical transition amplitudes can also be
formulated.

Based on the electron-hole representation, an one-exciton excited state, also
known as an exciton, can be written as

$\displaystyle|{\text{X}_{I\tau}}\rangle=\int\tilde{\Psi}_{\text{X},I\tau}(\mathbf{k})\hat{c}^{\dagger}_{\tau}(\mathbf{k})\hat{d}^{\dagger}_{\tau}(-\mathbf{k})\frac{\text{d}^{2}k}{(2\pi)^{2}}|{0}\rangle,$
(23)

where the exciton wavefunction ${\Psi}_{\text{X},I\tau}(\mathbf{k})$ can be
solved from the eigenvalue equation

$\displaystyle\int\tilde{\mathcal{H}}_{\text{X},\tau}(\mathbf{k},\mathbf{k}^{\prime})\tilde{\Psi}_{\text{X},I\tau}(\mathbf{k}^{\prime})\frac{\text{d}^{2}k^{\prime}}{(2\pi)^{2}}=E_{\text{X},I\tau}\tilde{\Psi}_{\text{X},I\tau}(\mathbf{k}),$
(24)

and the exciton Hamiltonian can be derived from

$\displaystyle\tilde{\mathcal{H}}_{\text{X},\tau}(\mathbf{k},\mathbf{k}^{\prime})$
$\displaystyle=$
$\displaystyle|{0}\rangle{{0}}|\hat{d}_{\tau}(-\mathbf{k})\hat{c}_{\tau}(\mathbf{k})\hat{\mathcal{H}}\hat{c}^{\dagger}_{\tau}(\mathbf{k}^{\prime})\hat{d}^{\dagger}_{\tau}(-\mathbf{k}^{\prime})|{0}\rangle.$
(25)

The transition between an one-exciton excited state and the ground state is
known as the “one-exciton transition”, and the transition amplitude can be
calculated in terms of either $\xi^{\mu}_{N0}$ or $j^{\mu}_{N0}$, where

$\displaystyle\xi^{\mu}_{N0}=-e\int\tilde{\Psi}^{*}_{\text{X},I_{N}\tau_{N}}(\mathbf{k})\tilde{\mathcal{D}}^{\mu}_{\text{eh},\tau_{N}}(\mathbf{k})\frac{\text{d}^{2}k}{(2\pi)^{2}}$
(26)

and

$\displaystyle
j^{\mu}_{N0}=-e\int\tilde{\Psi}^{*}_{\text{X},I_{N}\tau_{N}}(\mathbf{k})\tilde{\mathcal{P}}^{\mu}_{\text{eh},\tau_{N}}(\mathbf{k})\frac{\text{d}^{2}k}{(2\pi)^{2}}.$
(27)

The transition between different one-exciton excited states is known as the
“intra-exciton transition” and the transition amplitudes can be calculated in
terms of either $\xi^{\mu}_{NM}$ or $j^{\mu}_{NM}$, where

$\displaystyle\xi^{\mu}_{NM}$ $\displaystyle=$
$\displaystyle-e\int\tilde{\Psi}^{*}_{\text{X},I_{N}\tau_{N}}(\mathbf{k})\tilde{\mathcal{D}}^{\mu}_{\text{X},\tau_{N}}(\mathbf{k})\tilde{\Psi}_{\text{X},I_{M}\tau_{N}}(\mathbf{k})\frac{\text{d}^{2}k}{(2\pi)^{2}}$

and

$\displaystyle j^{\mu}_{NM}$ $\displaystyle=$
$\displaystyle-e\int\tilde{\Psi}^{*}_{\text{X},I_{N}\tau_{N}}(\mathbf{k})\tilde{\mathcal{P}}^{\mu}_{\text{X},\tau_{N}}(\mathbf{k})\tilde{\Psi}_{\text{X},I_{M}\tau_{N}}(\mathbf{k})\frac{\text{d}^{2}k}{(2\pi)^{2}}$

with
$\tilde{\mathcal{D}}^{\mu}_{\text{X},\tau}(\mathbf{k})=\tilde{\mathcal{D}}^{\mu}_{\text{ee},\tau}(\mathbf{k})-\tilde{\mathcal{D}}^{\mu}_{\text{hh},\tau}(\mathbf{k})$
and
$\tilde{\mathcal{P}}^{\mu}_{\text{X},\tau}(\mathbf{k})=\tilde{\mathcal{P}}^{\mu}_{\text{ee},\tau}(\mathbf{k})-\tilde{\mathcal{P}}^{\mu}_{\text{hh},\tau}(\mathbf{k})$.
The ground-state energy can be assigned as $E_{0}=0$ and the excited-state
energy for a one-exciton excited state is given by
$E_{N}=E_{\text{X},I_{N}\tau_{N}}$. With the relations given above optical
spectra involving one-exciton transitions and intra-exciton transitions can be
calculated.

##  III FW transformation

Based on the derivation in Sec. II.3, we find that the key to connect the
Dirac equation and the exciton Hamiltonian is the unitary transformation
matrix ${\mathcal{U}}(\mathbf{k})$. We need to solve the diagonalization
problem in Eq. (13) first to find the electron and hole energies. However, it
will be difficult if external fields are included in the Dirac Hamiltonian
matrix. One way to solve this problem is to replace the wavevector
$\mathbf{k}$ by $-\mathtt{i}\bm{\nabla}$ in Eq. (13). Namely,
$\tilde{\mathtt{h}}_{\tau}(-\mathtt{i}\bm{\nabla})\equiv\mathtt{h}_{\tau}(\mathbf{r})$
and
$\tilde{\varepsilon}_{\tau}(-\mathtt{i}\bm{\nabla})\equiv{\varepsilon}_{\tau}(\mathbf{r})$.
We get

$\displaystyle{\mathcal{U}}_{\tau}(-\mathtt{i}\bm{\nabla})\mathtt{h}_{\tau}(\mathbf{r}){\mathcal{U}}^{\dagger}_{\tau}(-\mathtt{i}\bm{\nabla})={\varepsilon}_{\tau}(\mathbf{r})\beta.$
(30)

Eq. (30) is the basic idea of FW transformation. By a proper choice of the
unitary transformation, the analytic formulation of the single-particle
Hamiltonian $\varepsilon_{\tau}(\mathbf{r})$ can be derived. Since
$\varepsilon_{\tau}(\mathbf{r})\beta$ is still a functional of
$-\mathtt{i}\bm{\nabla}$, $\varepsilon_{\tau}(\mathbf{r})$ is actually the
electron Hamiltonian and $-\varepsilon_{\tau}(\mathbf{r})$ is the hole
Hamiltonian.

###  III.1 Eriksen method

The FW transformation connecting the initial Hamiltonian $\mathcal{H}$ and the
transformed Hamiltonian $\mathcal{H}_{\text{FW}}$ can be written
asfoldy1950dirac ; eriksen1958foldy

$\displaystyle\mathcal{H}_{\text{FW}}=\mathcal{U}_{\text{FW}}\left(\mathcal{H}-\mathtt{i}\frac{\partial}{\partial{t}}\right)\mathcal{U}^{\dagger}_{\text{FW}}+\mathtt{i}\frac{\partial}{\partial{t}}.$
(31)

where $\mathcal{U}_{\text{FW}}$ is an unitary operator named as the FW
transformation operator. Based on Eriksen’s methoderiksen1958foldy ;
silenko2008foldy ; silenko2016general , we assume that the initial Hamiltonian
can be divided into

$\displaystyle\mathcal{H}={\beta}mv^{2}_{\text{F}}+\mathcal{E}+\mathcal{O}+\mathcal{O}^{\prime},$
(32)

with ${\beta}\mathcal{E}=\mathcal{E}{\beta}$,
${\beta}\mathcal{O}=-\mathcal{O}{\beta}$,
${\beta}\mathcal{O}^{\prime}=-\mathcal{O}^{\prime}{\beta}$, where
$\mathcal{E}$ is an even operator and $\mathcal{O}$ is an odd operator in the
Hamiltonian, $\mathcal{O}^{\prime}$ is an odd operator of the external
interaction of the Hamiltonian. Accordingly, the FW transformation operator up
to the order of $1/m^{2}$ has the expressionsilenko2008foldy ;
silenko2016general

$\displaystyle\mathcal{U}_{\text{FW}}$ $\displaystyle=$
$\displaystyle\exp\left(\frac{1}{2}\left[\mathcal{S},\;\mathcal{S}^{\prime}\right]\right)\exp\left(\mathtt{i}\mathcal{S}^{\prime}\right)\exp\left(\mathtt{i}\mathcal{S}\right),$
(33)

where $\mathcal{S}=-\mathtt{i}\beta\mathcal{O}/(2mv^{2}_{\text{F}})$,
$\mathcal{S}^{\prime}=-\mathtt{i}\left[\mathcal{O},\mathcal{G}\right]/(4m^{2}v^{4}_{\text{F}})$,
with

$\displaystyle\mathcal{G}=\mathcal{E}+\mathcal{O}^{\prime}-\mathtt{i}\frac{\partial}{\partial{t}}.$
(34)

The FW Hamiltonian up to the order of $1/m^{2}$ can be written as

$\displaystyle\mathcal{H}_{\text{FW}}$ $\displaystyle=$
$\displaystyle{\beta}mv^{2}_{\text{F}}+\mathcal{E}+\mathcal{O}^{\prime}+\frac{{\beta}\mathcal{O}^{2}}{2mv^{2}_{\text{F}}}-\frac{\left[\mathcal{O},\left[\mathcal{O},\mathcal{G}\right]\right]}{8m^{2}v^{4}_{\text{F}}}.$
(35)

Note that the odd operator $\mathcal{O}^{\prime}$ is not discussed by cited
literatureseriksen1958foldy ; silenko2008foldy ; silenko2016general . As we
will show lately in Sec. III.4, the operator contributes mainly to the
interband transition, which is not considered in atomic systems.

###  III.2 Single-particle Hamiltonian

In the present case, the correspondences between the single-particle
Hamiltonian and the Eriksen’s Hamiltonian are given by

$\displaystyle\mathcal{H}={\mathtt{h}}-q\mathbf{r}\cdot\bm{\mathcal{F}},\hskip
8.61108pt\mathcal{E}=-q\mathbf{r}\cdot\left(\mathbf{F}+\bm{\mathcal{F}}\right),$
(36)
$\displaystyle\mathcal{O}=v_{\text{F}}\bm{\pi}\cdot{\bm{\alpha}_{\tau}},\hskip
8.61108pt\mathcal{O}^{\prime}=0.$ (37)

By using $\bm{\pi}\times\bm{\pi}=\mathtt{i}qB\mathbf{e}_{\perp}$ and
$\left(\mathbf{a}\cdot\bm{\alpha}_{\tau}\right)\left(\mathbf{b}\cdot\bm{\alpha}_{\tau}\right)=\mathbf{a}\cdot\mathbf{b}+\mathtt{i}\tau\mathbf{e}_{\perp}\cdot\mathbf{a}\times\mathbf{b}\beta$,
we find $\mathcal{O}^{2}={v^{2}_{\text{F}}}\left(|\bm{\pi}|^{2}-\beta\tau
qB\right)$,
$\left[\mathcal{O},\mathcal{G}\right]=\mathtt{i}qv_{\text{F}}\left(\mathbf{F}+\bm{\mathcal{F}}\right)\cdot{\bm{\alpha}_{\tau}}$,
and $\left[\mathcal{O},\left[\mathcal{O},\mathcal{G}\right]\right]=2\tau
qv^{2}_{\text{F}}\beta\mathbf{e}_{\perp}\cdot\left(\mathbf{F}+\bm{\mathcal{F}}\right)\times\bm{\pi}$.
The FW Hamiltonian is given by

$\displaystyle\mathcal{H}_{\text{FW}}$ $\displaystyle=$
$\displaystyle\beta\left[mv^{2}_{\text{F}}+\frac{|\bm{\pi}|^{2}}{2m}-\tau\frac{q\mathbf{e}_{\perp}\cdot\left(\mathbf{F}+\bm{\mathcal{F}}\right)\times\bm{\pi}}{4m^{2}v^{2}_{\text{F}}}\right]$
(38)
$\displaystyle-q\left(\mathbf{F}+\bm{\mathcal{F}}\right)\cdot\mathbf{r}-\tau\frac{qB}{2m}.$

Based on the FW single-particle Hamiltonian, the effective single-particle
energy is given by

$\displaystyle\varepsilon_{\text{FW},i}$ $\displaystyle=$ $\displaystyle
mv^{2}_{\text{F}}+\frac{|\bm{\pi}_{i}|^{2}}{2m}-\tau_{i}\frac{q_{i}\mathbf{e}_{\perp}\cdot\left(\mathbf{F}+\bm{\mathcal{F}}\right)\times\bm{\pi}_{i}}{4m^{2}v^{2}_{\text{F}}}$
(39) $\displaystyle-
q_{i}\left(\mathbf{F}+\bm{\mathcal{F}}\right)\cdot\mathbf{r}_{i}-\tau_{i}\frac{q_{i}B}{2m},$

where $i\in\\{\text{e},\text{h}\\}$, $q=-e$ for electrons and $q=e$ for holes,
$\tau_{\text{e}}$ and $\tau_{\text{h}}$ are valley indices of the effective
electron and hole.

###  III.3 Two-particle interaction

Now we consider a two-particle Hamiltonian

$\displaystyle\mathcal{H}=\mathtt{h}_{1}+\mathtt{h}_{2}+\mathcal{V}_{12},$
(40)

where $\mathcal{V}_{12}$ is a two-particle interaction, and
${\mathtt{h}}_{i}=v_{\text{F}}\bm{\pi}_{i}\cdot{\bm{\alpha}_{\tau}}_{i}+mv^{2}_{\text{F}}\beta_{i}+q_{i}\Phi_{i}$
is the single-particle Hamiltonian matrix. The Hamiltonian can be divided as

$\displaystyle\mathcal{H}=\beta_{1}mv^{2}_{\text{F}}+\beta_{2}mv^{2}_{\text{F}}+\mathcal{E}+\mathcal{O}_{1}+\mathcal{O}_{2},$
(41) $\displaystyle\mathcal{E}=q_{1}\Phi_{1}+q_{2}\Phi_{2}+\mathcal{V}_{12},$
(42)
$\displaystyle\mathcal{O}_{1}=v_{\text{F}}\bm{\pi}_{1}\cdot{\bm{\alpha}_{\tau_{1}}},\hskip
8.61108pt\mathcal{O}_{2}=v_{\text{F}}\bm{\pi}_{2}\cdot{\bm{\alpha}_{\tau_{2}}}.$
(43)

The FW transformed Hamiltonian is then given by

$\displaystyle\mathcal{H}_{\text{FW}}$ $\displaystyle=$
$\displaystyle\mathtt{h}_{1}+\mathtt{h}_{2}-\frac{\left[\mathcal{O}_{1},\left[\mathcal{O}_{1},\mathcal{G}\right]\right]}{8m^{2}v^{4}_{\text{F}}}-\frac{\left[\mathcal{O}_{2},\left[\mathcal{O}_{2},\mathcal{G}\right]\right]}{8m^{2}v^{4}_{\text{F}}}$
(44)
$\displaystyle-\frac{\left[\mathcal{O}_{1},\left[\mathcal{O}_{2},\mathcal{G}\right]\right]+\left[\mathcal{O}_{2},\left[\mathcal{O}_{1},\mathcal{G}\right]\right]}{8m^{2}v^{4}_{\text{F}}},$

with $\mathcal{G}=\mathcal{V}_{12}-\mathtt{i}\partial/\partial{t}$. It can
shown
$\left[\mathcal{O}_{i},\mathcal{G}\right]=-\mathtt{i}v_{\text{F}}\bm{\alpha}_{\tau_{i}}\cdot\bm{\nabla}_{i}\mathcal{V}_{12}$,
$\left[\mathcal{O}_{i},\left[\mathcal{O}_{i},\mathcal{G}\right]\right]=-v^{2}_{\text{F}}\nabla^{2}_{i}\mathcal{V}_{12}-2v^{2}_{\text{F}}\beta_{i}\tau_{i}\mathbf{e}_{\perp}\cdot\bm{\nabla}_{i}\mathcal{V}_{12}\times\bm{\pi}_{i}$,
and
$\left[\mathcal{O}_{i},\left[\mathcal{O}_{j},\mathcal{G}\right]\right]=-v^{2}_{\text{F}}(\bm{\alpha}_{\tau_{i}}\cdot\bm{\nabla}_{i})(\bm{\alpha}_{\tau_{j}}\cdot\bm{\nabla}_{j})\mathcal{V}_{12}$
for $i\neq{j}$. The FW transformed two-particle interaction is given by

$\displaystyle\mathcal{V}_{\text{FW},12}$ $\displaystyle=$
$\displaystyle\mathcal{V}_{12}+\beta_{1}\tau_{1}\frac{\mathbf{e}_{\perp}\cdot\bm{\nabla}_{1}\mathcal{V}_{12}\times\bm{\pi}_{1}}{4m^{2}v^{2}_{\text{F}}}+\frac{\nabla^{2}_{1}\mathcal{V}_{12}}{8m^{2}v^{2}_{\text{F}}}$
(45)
$\displaystyle+\beta_{2}\tau_{2}\frac{\mathbf{e}_{\perp}\cdot\bm{\nabla}_{2}\mathcal{V}_{12}\times\bm{\pi}_{2}}{4m^{2}v^{2}_{\text{F}}}+\frac{\nabla^{2}_{2}\mathcal{V}_{12}}{8m^{2}v^{2}_{\text{F}}}$
$\displaystyle+\frac{\left(\bm{\alpha}_{\tau_{1}}\cdot\bm{\nabla}_{1}\right)\left(\bm{\alpha}_{\tau_{2}}\cdot\bm{\nabla}_{2}\right)\mathcal{V}_{12}}{4m^{2}v^{2}_{\text{F}}}.$

For 2D materials, the two-particle interaction is assumed to be given by the
screened Coulomb interaction

$\displaystyle\mathcal{V}_{ij}=\frac{q_{i}q_{j}}{e^{2}}V({r}_{ij}).$ (46)

The FW transformed screened Coulomb potential is then given by

$\displaystyle V_{\text{FW},ij}$ $\displaystyle=$ $\displaystyle
V({r}_{ij})+\tau_{i}\frac{\mathbf{e}_{\perp}\cdot\bm{\nabla}_{i}V({r}_{ij})\times\bm{\pi}_{i}}{4m^{2}v^{2}_{\text{F}}}+\frac{\nabla^{2}_{i}V({r}_{ij})}{8m^{2}v^{2}_{\text{F}}}$
(47)
$\displaystyle+\tau_{j}\frac{\mathbf{e}_{\perp}\cdot\bm{\nabla}_{j}V({r}_{ij})\times\bm{\pi}_{j}}{4m^{2}v^{2}_{\text{F}}}+\frac{\nabla^{2}_{j}V({r}_{ij})}{8m^{2}v^{2}_{\text{F}}}$

for $i,j\in\\{\text{e},\text{h}\\}$. The FW transformed screened Coulomb
potential can be used to describe the electron-electron interaction, the
electron-hole interaction, and the hole-hole interaction in 2D materials.

###  III.4 Interband transition

Assuming that the contributions from the static electric and magnetic fields
to the interband transition can be neglected, the correspondences between
electromagnetic-field interaction and Eriksen’s Hamiltonian are given by

$\displaystyle\mathcal{H}=\mathtt{h}^{(0)}-qv_{\text{F}}\bm{\alpha}_{\tau}\cdot\bm{\mathcal{A}},\hskip
8.61108pt\mathcal{E}=0,\hskip
8.61108pt\mathcal{O}=v_{\text{F}}\mathbf{p}\cdot{\bm{\alpha}_{\tau}},$ (48)
$\displaystyle\mathcal{O}^{\prime}=-qv_{\text{F}}\bm{\alpha}_{\tau}\cdot\bm{\mathcal{A}}.$
(49)

We find
$\left[\mathcal{O},\mathcal{G}\right]=2\mathtt{i}qv^{2}_{\text{F}}\tau\mathbf{e}_{\perp}\cdot\bm{\mathcal{A}}\times\mathbf{p}\beta$,
$\left[\mathcal{O},\left[\mathcal{O},\mathcal{G}\right]\right]=-4qv^{3}_{\text{F}}\left(\mathbf{e}_{\perp}\cdot\bm{\mathcal{A}}\times\mathbf{p}\right)(\mathbf{e}_{\perp}\cdot\bm{\alpha}_{\tau}\times\mathbf{p})$.
The FW transformed Hamitonian with electromagnetic-field interaction is given
by

$\displaystyle\mathcal{H}_{\text{FW}}$ $\displaystyle=$
$\displaystyle\mathtt{h}^{(0)}_{\text{FW}}-q\bm{u}_{\text{FW}}\cdot\bm{\mathcal{A}},$
(50)

where the FW effective velocity matrix is

$\displaystyle\bm{u}_{\text{FW}}\cdot\bm{\mathcal{A}}$ $\displaystyle=$
$\displaystyle
v_{\text{F}}\bm{\alpha}_{\tau}\cdot\bm{\mathcal{A}}-\frac{v_{\text{F}}\left(\mathbf{e}_{\perp}\cdot\bm{\mathcal{A}}\times\mathbf{p}\right)(\mathbf{e}_{\perp}\cdot\bm{\alpha}_{\tau}\times\mathbf{p})}{2m^{2}v^{2}_{\text{F}}}.$

By defining the circular-polarized components of variables
$\mathcal{A}^{\pm}=(\mathcal{A}^{x}\pm\mathtt{i}\mathcal{A}^{y})/\sqrt{2}$,
$p^{\pm}=(p^{x}\pm\mathtt{i}p^{y})/\sqrt{2}$, and

$\displaystyle\alpha^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\alpha^{x}\pm\mathtt{i}\alpha^{y}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}0&{\tau\pm{1}}\\\
{\tau\mp{1}}&0\end{pmatrix},$ (52)

we can find
$\bm{\alpha}_{\tau}\cdot\bm{\mathcal{A}}=\alpha^{+}\mathcal{A}^{-}+\alpha^{-}\mathcal{A}^{+}$
and

$\displaystyle\left(\mathbf{e}_{\perp}\cdot\bm{\mathcal{A}}\times\mathbf{p}\right)(\mathbf{e}_{\perp}\cdot\bm{\alpha}_{\tau}\times\mathbf{p})$
$\displaystyle=$
$\displaystyle\left[\alpha^{-}p^{2}/2-\alpha^{+}(p^{-})^{2}\right]\mathcal{A}^{+}$
$\displaystyle+\left[\alpha^{+}p^{2}/2-\alpha^{-}(p^{+})^{2}\right]\mathcal{A}^{-}.$

The FW velocity matrix,
${u}^{\pm}_{\text{FW}}=({u}^{x}_{\text{FW}}\pm\mathtt{i}{u}^{y}_{\text{FW}})/\sqrt{2}$,
is given by

$\displaystyle{u}^{\pm}_{\text{FW}}=\left(1-\frac{p^{2}}{4m^{2}v^{2}_{\text{F}}}\right)v_{\text{F}}\alpha^{\pm}+\frac{(p^{\pm})^{2}}{2m^{2}v^{2}_{\text{F}}}v_{\text{F}}\alpha^{\mp}.$
(54)

The velocity matrix will be used to derive the selection rule and transition
amplitude in Sec. IV.3.

##  IV Applications to valley excitons

In this section, we will use the formulation derived from the FW
transformation of a gapped Dirac Hamiltonian to study valley excitons in 2D
materials. Based on the formulations, the Hamiltonian for a valley exciton in
external fields can be written. The valley-exciton Hamiltonian is similar to
the Wannier-exciton Hamiltonian, which is written as an electron and a hole
bound by an attractive Coulomb interaction, except that the band-geometry
correction terms are added. The effects of external-field interactions,
including electric-field interaction, electromagnetic-field interaction, and
magnetic-field interaction, on valley excitons can then be studied.

###  IV.1 Valley excitons in electric fields

An exciton in an electric field ($|\mathbf{F}|\neq{0}$) and an electromagnetic
field ($|\bm{\mathcal{A}}|\neq{0}$) without magnetic field (${B}={0}$) is
considered in this section. The effective exciton Hamiltonian derived from the
FW transformed Dirac Hamiltonian is given by

$\displaystyle\mathcal{H}_{\text{X}}$ $\displaystyle=$
$\displaystyle\delta\Delta+\left(\varepsilon_{\text{FW},\text{e}}+\delta\varepsilon_{\text{e}}\right)+\left(\varepsilon_{\text{FW},\text{h}}+\delta\varepsilon_{\text{h}}\right)-V_{\text{FW},\text{eh}}$
$\displaystyle=$
$\displaystyle\Delta+\frac{|\mathbf{p}_{\text{e}}|^{2}}{2m_{\text{e}}}+\frac{|\mathbf{p}_{\text{h}}|^{2}}{2m_{\text{h}}}-V({r}_{\text{eh}})$
$\displaystyle+\tau_{\text{e}}\frac{\mathbf{e}_{\perp}\cdot\left[e\left(\mathbf{F}+\bm{\mathcal{F}}\right)-\bm{\nabla}_{\text{e}}V({r}_{\text{eh}})\right]\times\mathbf{p}_{\text{e}}}{4{m}^{2}v^{2}_{\text{F}}}$
$\displaystyle-\tau_{\text{h}}\frac{\mathbf{e}_{\perp}\cdot\left[e\left(\mathbf{F}+\bm{\mathcal{F}}\right)+\bm{\nabla}_{\text{h}}V({r}_{\text{eh}})\right]\times\mathbf{p}_{\text{h}}}{4{m}^{2}v^{2}_{\text{F}}}$
$\displaystyle+e(\mathbf{r}_{\text{e}}-\mathbf{r}_{\text{h}})\cdot\left(\mathbf{F}+\bm{\mathcal{F}}\right)-\frac{\nabla^{2}_{\text{e}}V({r}_{\text{eh}})}{8{m}^{2}v^{2}_{\text{F}}}-\frac{\nabla^{2}_{\text{h}}V({r}_{\text{eh}})}{8{m}^{2}v^{2}_{\text{F}}},$

where $\delta\Delta$ is the electron-correlation modification to the band-gap
energy, $\Delta=\delta\Delta+2mv^{2}_{\text{F}}$ is the band-gap energy,
$\delta\varepsilon_{\text{e}}$ and $\delta\varepsilon_{\text{h}}$ are band-
asymmetry modifications to the electron and hole energies, $m_{\text{e}}$ and
$m_{\text{h}}$ are effective electron mass and effective hole mass, which are
defined by

$\displaystyle\frac{|\mathbf{p}|^{2}}{2m_{\text{e}}}=\frac{|\mathbf{p}|^{2}}{2m}+\delta\varepsilon_{\text{e}},\hskip
8.61108pt\frac{|\mathbf{p}|^{2}}{2m_{\text{h}}}=\frac{|\mathbf{p}|^{2}}{2m}+\delta\varepsilon_{\text{h}}.$
(56)

Assuming $\delta\varepsilon_{\text{e}}=-\delta\varepsilon_{\text{h}}$, we can
find $m=2\mu$, with $\mu=m_{\text{e}}m_{\text{h}}/(m_{\text{e}}+m_{\text{h}})$
the reduced mass. A coordinate transformation to the relative coordinate
system is applied

$\displaystyle\mathbf{R}$ $\displaystyle=$
$\displaystyle\frac{m_{\text{e}}\mathbf{r}_{\text{e}}+m_{\text{h}}\mathbf{r}_{\text{h}}}{m_{\text{X}}},\hskip
8.61108pt\mathbf{r}=\mathbf{r}_{\text{e}}-\mathbf{r}_{\text{h}},$ (57)
$\displaystyle\mathbf{p}_{\text{e}}=\frac{m_{\text{e}}}{m_{\text{X}}}\mathbf{P}+\mathbf{p},\hskip
4.30554pt\mathbf{p}_{\text{h}}=\frac{m_{\text{h}}}{m_{\text{X}}}\mathbf{P}-\mathbf{p},$
(58)

where $m_{\text{X}}=m_{\text{e}}+m_{\text{h}}$ is the exciton mass,
$\mathbf{R}$ and $\mathbf{P}=-\mathtt{i}\bm{\nabla}_{\mathbf{R}}$ are the
center-of-mass position and momentum, $\mathbf{r}$ and
$\mathbf{p}=-\mathtt{i}\bm{\nabla}$ are internal-coordinate position and
momentum. The exciton Hamiltonian becomes

$\displaystyle\mathcal{H}_{\text{X}}=\mathcal{H}^{\prime}_{\text{X}}+\mathcal{H}^{\prime\prime}_{\text{X}}+\mathcal{J}_{\text{X}},$
(59)

where

$\displaystyle\mathcal{H}^{\prime}_{\text{X}}$ $\displaystyle=$
$\displaystyle\frac{P^{2}}{2m_{\text{X}}}+\frac{e\Omega(m_{\text{e}}\tau_{\text{e}}-m_{\text{h}}\tau_{\text{h}})}{4m_{\text{X}}}\mathbf{e}_{\perp}\cdot\left(\mathbf{F}+\bm{\mathcal{F}}\right)\times\mathbf{P}$
(60)

is the exciton translational Hamiltonian,

$\displaystyle\mathcal{H}^{\prime\prime}_{\text{X}}$ $\displaystyle=$
$\displaystyle\Delta+\frac{p^{2}}{2\mu}-V({r})+e\mathbf{r}\cdot\left(\mathbf{F}+\bm{\mathcal{F}}\right)-\frac{\Omega}{4}\nabla^{2}V({r})$
(61)
$\displaystyle+\frac{\Omega(\tau_{\text{e}}+\tau_{\text{h}})}{4}\mathbf{e}_{\perp}\cdot\left[e\left(\mathbf{F}+\bm{\mathcal{F}}\right)-\bm{\nabla}V({r})\right]\times\mathbf{p},$

is the exciton internal Hamiltonian with

$\displaystyle\Omega=\frac{1}{m^{2}v^{2}_{\text{F}}}=\frac{1}{4\mu^{2}v^{2}_{\text{F}}}$
(62)

the Berry curvature, and

$\displaystyle\mathcal{J}_{\text{X}}$ $\displaystyle=$
$\displaystyle-\frac{e\Omega(m_{\text{e}}\tau_{\text{e}}-m_{\text{h}}\tau_{\text{h}})}{4m_{\text{X}}}\mathbf{e}_{\perp}\cdot\bm{\nabla}V({r})\times\mathbf{P}$
(63)

is the exciton translational-internal coupling. For the exciton internal
Hamiltonian, the first four terms describe a Wannier exciton in an electric
field, the fifth term is the Darwin interaction, and the last term is the
exciton valley-orbit coupling, which resembles 2D version of spin-orbit
coupling for atomistic systems. Both the exciton valley-orbit coupling and the
Darwin interaction can be considered as the contributions from the Berry-
curvature effect to the exciton energy levels of intravalley excitons
($\tau_{\text{e}}=\tau_{\text{h}}$).

Based on the translational Hamiltonian in Eq. (60), it is also shown that the
Berry-curvature effect contributes to the valley-dependent exciton transport
and the exciton translational-internal coupling in the electric field for
intervalley excitons ($\tau_{\text{e}}=-\tau_{\text{h}}$). By using the
Heisenberg equation of motion and omitting the translational-internal
coupling, the exciton mobility in an electric field can be obtained by

$\displaystyle\Big{\langle}\frac{\text{d}\mathbf{R}}{\text{d}t}\Big{\rangle}$
$\displaystyle=$
$\displaystyle-\mathtt{i}\langle\left[\mathbf{R},\mathcal{H}^{\prime}_{\text{X}}\right]\rangle$
(64) $\displaystyle=$
$\displaystyle\frac{\langle\mathbf{P}\rangle}{m_{\text{X}}}+\frac{e\Omega(m_{\text{e}}\tau_{\text{e}}-m_{\text{h}}\tau_{\text{h}})}{4m_{\text{X}}}\mathbf{e}_{\perp}\times\mathbf{F},$

where $\langle\mathbf{P}\rangle=0$ for the ensemble of excitons. The second
term in Eq. (64) can be seen as a contribution to the exciton valley Hall
effect, which has been observed experimentallyonga2017exciton . This is effect
is believed to be driven by the side-jump together with the skew-scattering
mechanismglazov2020skew . While the contribution in Eq. (64) to the exciton
valley Hall effect is infinitesimal for intravalley excitons, it may be
measurable if intervalley excitons also contribute to the exciton mobility.
Due to the simplicity of the present model, the contribution of intervalley
excitons to the exciton Hall conductivity will be studied by an improved model
in future research.

###  IV.2 Exciton energy-level splitting

Materials |  $\Delta$(eV) |  $a$(Å) |  $t$(eV) | $m_{\text{e}}/m_{0}$ | $m_{\text{h}}/m_{0}$ |  $r_{0}$(Å) | $\kappa$ | $1s$ | $2s$ | $3s$ | $2p_{+}/2p_{-}$ | $3p_{+}/3p_{-}$ | $3d_{+}/3d_{-}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\text{MoS}_{2}$ | $1.918$ | $3.193$ | $1.0799$ | $0.47$ | $0.54$ | $44.68$ | $1.0$ | $506.2$ | $243.3$ | $151.3$ | $292.4/303.0$ | $174.6/178.8$ | $197.7/202.5$
$2.0$ | $333.6$ | $124.4$ | $66.1$ | $156.7/163.4$ | $78.6/80.8$ | $89.8/92.0$
$\text{MoSe}_{2}$ | $1.516$ | $3.313$ | $0.8984$ | $0.55$ | $0.59$ | $53.16$ | $1.0$ | $458.7$ | $230.5$ | $147.6$ | $274.0/284.1$ | $169.0/173.2$ | $190.5/195.3$
$2.0$ | $309.2$ | $122.8$ | $67.6$ | $152.9/159.7$ | $79.9/82.2$ | $91.4/93.8$
$\text{WS}_{2}$ | $2.042$ | $3.197$ | $1.3315$ | $0.32$ | $0.35$ | $40.17$ | $1.0$ | $488.2$ | $215.8$ | $127.0$ | $263.8/274.2$ | $148.3/152.3$ | $169.0/173.4$
$2.0$ | $308.2$ | $102.4$ | $51.1$ | $130.8/136.6$ | $61.1/62.9$ | $69.4/71.1$
$\text{WSe}_{2}$ | $1.761$ | $3.310$ | $1.1395$ | $0.34$ | $0.36$ | $47.57$ | $1.0$ | $437.2$ | $200.6$ | $120.9$ | $242.9/253.2$ | $140.3/144.3$ | $159.5/164.0$
$2.0$ | $281.2$ | $98.3$ | $50.4$ | $124.5/130.6$ | $60.0/61.9$ | $68.4/70.2$
$0.38$ | $0.43$ | $46.80$ | $4.0$ | $172.0$ | $43.7$ | $19.4$ | $57.9/60.1$ | $23.5/24.0$ | $25.5/25.9$
Table 1: Parameters of TMDCs ($\Delta$ the band-gap energy; $a$ the lattice
constant; $t$ the hopping coupling; $r_{0}$ the screening length; $\kappa$ the
dielectric constant) and calculated exciton binding energies (in meV) of
intravalley excitons with $\tau=1$. The Fermi velocity is given by
$v_{\text{F}}=at$. The parameters are obtained from Ref. wu2019exciton,
except the last row. The parameters ($m_{\text{e}}/m_{0}$,
$m_{\text{h}}/m_{0}$, $r_{0}$(Å), $\kappa$) in the last row are chosen to fit
the experimentally observed photoluminescence spectra reported in Ref.
liu2019magnetophotoluminescence, . The electron-hole exchange effect has been
included in the last row according to Ref. mypaper0 .

In this section, exciton internal Hamiltonians without external fields are
considered, and the eigenspectrum are used to study the Berry-curvature effect
on exciton energy levels. By assuming that
$\tau_{\text{e}}=\tau_{\text{h}}=\tau$, the exciton internal Hamiltonian for
the intravalley exciton can be written as

$\displaystyle\mathcal{H}^{\prime\prime}_{\text{intravalley-X},\tau}(\mathbf{r})$
$\displaystyle=$
$\displaystyle\Delta+\frac{p^{2}}{2\mu}-V({r})-\frac{\Omega}{4}\nabla^{2}V({r})$
(65)
$\displaystyle-\frac{\tau\Omega}{2}\mathbf{e}_{\perp}\cdot\bm{\nabla}V({r})\times\mathbf{p}.$

On the other hand, by assuming that $\tau_{\text{e}}=-\tau_{\text{h}}=\tau$,
the exciton internal Hamiltonian for the intervalley exciton can be written as

$\displaystyle\mathcal{H}^{\prime\prime}_{\text{intervalley-X},\tau}(\mathbf{r})$
$\displaystyle=$
$\displaystyle\Delta+\frac{p^{2}}{2\mu}-V({r})-\frac{\Omega}{4}\nabla^{2}V({r}).$
(66)

The Darwin interaction is found in both Hamiltonians, but only the
intravalley-exciton Hamiltonian contains the exciton valley-orbit coupling.
The exciton eigenenergy and wavefunction can be calculated by the eigenvalue
equation

$\displaystyle\mathcal{H}^{\prime\prime}_{\text{X},\tau}(\mathbf{r})\Psi_{\text{X},I\tau}(\mathbf{r})=E_{\text{X},I\tau}\Psi_{\text{X},I\tau}(\mathbf{r}),$
(67)

for both intravalley excitons and intervalley excitons. Since the exciton
internal Hamiltonian and the angular momentum operator commutes,
$\left[\mathcal{H}^{\prime\prime}_{\text{X},\tau}(\mathbf{r}),\;\mathcal{L}(\mathbf{r})\right]=0$,
with the angular momentum operator being given by

$\displaystyle\mathcal{L}(\mathbf{r})=-\mathtt{i}\mathbf{e}_{\perp}\cdot\mathbf{r}\times\bm{\nabla}=-\mathtt{i}\frac{\partial}{\partial\varphi},$
(68)

the exciton wavefunction is also an eigenfunction of the angular momentum
operator,
$\mathcal{L}(\mathbf{r})\Psi_{\text{X},I\tau}(\mathbf{r})=l\Psi_{\text{X},I\tau}(\mathbf{r})$,
with $l$ the angular momentum of the exciton. The exciton wavefunction can be
written as
$\Psi_{\text{X},I\tau}(\mathbf{r})=e^{\mathtt{i}l\varphi}\mathcal{R}_{n}(r)$,
where $\mathcal{R}_{n}(r)$ is the radial wavefunction of the exciton, with $n$
the principal quantum number. By using $\mathbf{p}=-\mathtt{i}\bm{\nabla}$ and

$\displaystyle\mathbf{e}_{\perp}\cdot\bm{\nabla}V(r)\times\bm{\nabla}=\frac{1}{r}\frac{\partial
V(r)}{\partial{r}}\mathbf{e}_{\perp}\cdot\mathbf{r}\times\bm{\nabla},$ (69)

it can be shown that the exciton vally-orbit coupling
($\mathcal{V}_{\text{X,VOC}}$), the fifth term in Eq. (65) and Eq. (66) is
proportional to the angular momentum operator

$\displaystyle\mathcal{V}_{\text{X,VOC}}(\mathbf{r})\propto-\frac{\Omega}{2}\frac{1}{r}\frac{\partial
V(r)}{\partial{r}}\mathcal{L}(\mathbf{r}),$ (70)

such that $\langle\mathcal{V}_{\text{X,VOC}}\rangle_{I\tau}\propto{l}$ with

$\displaystyle\langle\mathcal{O}\rangle_{I\tau}\equiv\int\Psi^{*}_{\text{X},I\tau}(\mathbf{r})\mathcal{O}_{\tau}(\mathbf{r})\Psi_{\text{X},I\tau}(\mathbf{r})\text{d}^{2}r$
(71)

being the expectational value. It is shown that the exciton valley-orbit
coupling only causes energy shifts of degenerate exciton states with angular
momentums other than zero. Since the only difference between the intravalley
and intervalley excitons is the exciton valley-orbit couplings, the binding
energies and the wavefunctions of intravalley and intervalley exctions with
zero angular momentum ($1s$ excitons) are the same.

Based on the Hamiltonian given in Eq. (65), we can solve the exciton energy
levels by the variational methodzhang2019two ; wu2019exciton ; mypaper0 ;
henriques2021calculation  (see Appendix A). In Table. 1, the calculated
exciton binding energies of different intravalley excitons in TMDCs and the
parameters used are listed. The exciton Hamiltonian, screened Coulomb
potential, and the calculation method are given in Appendix A. It is found
that the energy-level splittings for $2p_{+}/2p_{-}$ excitons are about $10.6$
meV for $\text{MoS}_{2}$ with $\kappa=1$ and $10.1$ meV for $\text{MoSe}_{2}$
with $\kappa=1$. The former value is consistent with the calculated result in
Ref. srivastava2015signatures, . However, the latter value is lower than the
experimentally observed value ${14}$ meV in Ref. yong2019valley, . It suggests
that the Berry curvature calculated by the present model might be
underestimated in comparison to the full band-structure model.

###  IV.3 Exciton transitions

Figure 1: Calculated one-photon (up) and two-photon (down) absorption spectra
of $\text{MoS}_{2}$ with the parameters given in Table 1., the valley index
$\tau=1$, and the screening constant $\kappa=2$. The line-broadening factor
for the two-photon absorption is $\eta=0.1$ eV.

In this section, the one-exciton and intra-exciton transitions for intravalley
excitons are studied, and their transition amplitudes are derived. The exciton
wavefunction in momentum space is given by the Fourier transform

$\displaystyle\tilde{\Psi}_{\text{X},I\tau}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\int
e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\Psi_{\text{X},I\tau}(\mathbf{r})\text{d}^{2}r=e^{\mathtt{i}l\varphi_{\mathbf{k}}}\tilde{\mathcal{R}}_{nl}(k),$
(72)

where $\tilde{\mathcal{R}}_{nl}(k)$ is the radial part of the exciton
wavefunction in momentum space, with $n$ the principal quantum number and $l$
the angular momentum. By using Eq. (19) and the FW transformed velocity matrix
in Eq. (54), the momentum matrix element is given by

$\displaystyle\tilde{\mathcal{P}}^{\pm}_{\text{eh},\tau}(\mathbf{k})$
$\displaystyle=$
$\displaystyle\left[\tilde{\mathcal{P}}^{\mp}_{\text{he},\tau}(\mathbf{k})\right]^{*}=\left[\tilde{\mathcal{P}}^{x}_{\text{eh},\tau}(\mathbf{k})\pm\mathtt{i}\tilde{\mathcal{P}}^{y}_{\text{eh},\tau}(\mathbf{k})\right]/\sqrt{2}$
$\displaystyle\simeq$ $\displaystyle
v_{\text{F}}\left[\delta_{\tau,\pm{1}}\left(1-\frac{\Omega
k^{2}}{4}\right)+\delta_{\tau,\mp{1}}\frac{\Omega
k^{2}}{4}e^{\pm\mathtt{i}2\varphi_{\mathbf{k}}}\right].$

The same result has been derived in Ref. gong2017optical, . By inserting the
exciton wavefunction in Eq. (72) and the momentum matrix element in Eq.
(LABEL:momentum_matrix_element) into the transition amplitude formula in Eq.
(27), the transition amplitudes for one-exciton transition can be solved, and
the selection rule for linear optical absorption can be derived. It is found
that the absorption of $\mathbf{e}^{+}$ polarized photon generates an exciton
at $\tau=1$ valley with ${l}=0$ or at $\tau=-1$ valley with ${l}=2$, and the
absorption of $\mathbf{e}^{-}$ polarized photon generates an exciton at
$\tau=-1$ valley with ${l}=0$ or at $\tau=1$ with ${l}=-2$, with
$\mathbf{e}^{\pm}=\mathbf{e}_{x}\pm\mathtt{i}\mathbf{e}_{y}$. The selection
rule for one-exciton transition can be summarized as

$\displaystyle\begin{cases}\Delta{l}+\tau=1&\text{for $\mathbf{e}^{+}$
polarized photon}\\\ \Delta{l}+\tau=-1&\text{for $\mathbf{e}^{-}$ polarized
photon}\end{cases}$ (74)

with $\Delta{l}=0,\pm{2}$ and $\tau=\pm{1}$. While the valley index $\tau$
stands for the angular momentum difference between the ground state and the
one-exciton state, the selection rule can be seen as the consequence of
angular momentum conservation in the photoabsorption process.

To derive the intra-exciton transition, which is defined as the transition
between exciton stateshenriques2021calculation , we consider the exciton
internal Hamiltonian with an in-plane electromagnetic-field interaction. The
internal Hamiltonian for intravalley exctions is written as

$\displaystyle\mathcal{H}^{\prime\prime}_{\text{X},\tau}(\mathbf{r})$
$\displaystyle=$
$\displaystyle\Delta+\frac{p^{2}}{2\mu}-V(r)-\frac{\Omega}{4}\nabla^{2}V({r})+e\bm{\mathcal{D}}_{\text{X},\tau}(\mathbf{r})\cdot\bm{\mathcal{F}}$
(75)
$\displaystyle-\frac{\tau\Omega}{2}\mathbf{e}_{\perp}\cdot\bm{\nabla}V({r})\times\mathbf{p},$

where the exciton dipole-moment operator is defined as

$\displaystyle\bm{\mathcal{D}}_{\text{X},\tau}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\mathbf{r}-\frac{\tau\Omega}{2}\mathbf{e}_{\perp}\times\mathbf{p}.$
(76)

The electromagnetic-field interaction can be rewritten as

$\displaystyle\bm{\mathcal{D}}_{\text{X},\tau}(\mathbf{r})\cdot\bm{\mathcal{F}}=\mathcal{D}^{+}_{\text{X},\tau}(\mathbf{r})\mathcal{F}^{-}+\mathcal{D}^{-}_{\text{X},\tau}(\mathbf{r})\mathcal{F}^{+},$
(77)

where
$\mathcal{F}^{\pm}=(\mathcal{F}^{x}\pm\mathtt{i}\mathcal{F}^{y})/\sqrt{2}$ and

$\displaystyle\mathcal{D}^{\pm}_{\text{X},\tau}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\mathcal{D}^{x}_{\text{X},\tau}(\mathbf{r})\pm\mathtt{i}\mathcal{D}^{y}_{\text{X},\tau}(\mathbf{r})\right]=r^{\pm}\mp\mathtt{i}\frac{\tau\Omega}{2}p^{\pm},$

with $r^{\pm}=(x\pm\mathtt{i}y)/\sqrt{2}=re^{\pm\mathtt{i}\varphi}/\sqrt{2}$
and

$\displaystyle
p^{\pm}=-\frac{\mathtt{i}}{\sqrt{2}}\left(\frac{\partial}{\partial{x}}\pm\mathtt{i}\frac{\partial}{\partial{y}}\right)=-\frac{\mathtt{i}}{\sqrt{2}}e^{\pm\mathtt{i}\varphi}\left(\frac{\partial}{\partial{r}}\pm\frac{\mathtt{i}}{r}\frac{\partial}{\partial{\varphi}}\right).$

Since the exciton dipole-moment operator is related to the dipole-momentum
matrix element by
$\mathcal{D}^{\mu}_{\text{X},\tau}(\mathbf{r})=\int\tilde{\mathcal{D}}^{\mu}_{\text{X},\tau}(\mathbf{k})e^{\mathtt{i}\mathbf{k}\cdot\mathbf{r}}{\text{d}^{2}k}/{(2\pi)^{2}}$,
and the intra-exciton transition amplitude can be calculated from Eq. (28), we
find

$\displaystyle\xi^{\mu}_{NM}=-e\int\Psi^{*}_{\text{X},I_{N}\tau_{N}}(\mathbf{r})\mathcal{D}^{\mu}_{\text{X},\tau_{N}}(\mathbf{r})\Psi_{\text{X},J_{M}\tau_{M}}(\mathbf{r})\text{d}^{2}r$
(80)

and $|\xi^{\pm}_{NM}|\propto\delta_{l_{N},l_{M}\pm{1}}$. The selection rule
for the intra-exciton transition can be found as

$\displaystyle\begin{cases}\Delta{l}=1&\text{for $\mathbf{e}^{+}$ polarized
photon}\\\ \Delta{l}=-1&\text{for $\mathbf{e}^{-}$ polarized
photon}\end{cases}$ (81)

Therefore, given that an $1s$ exciton is generated by the one-exciton
transition, an intra-exciton transition from the $1s$ exciton to $2p_{\pm}$
excitons can be induced by a two-photon process.

In Fig. 1, the calculated one-photon (up) and two-photon (down) absorption
spectra of $\text{MoS}_{2}$ based on the one-exciton transition amplitude and
the intra-exciton transition amplitude are shown. The formula for the one-
photon absorption spectrum is given by Eq. (11) and the formula for the two-
photon absorption spectrum is given by Eq. (LABEL:two_photon). The resonance
peaks in the one-photon absorption spectrum can be assigned as the transitions
of $ns$ excitons, and the peaks in the one-photon absorption spectrum can be
assigned as the transitions of $np_{-}/np_{+}$ excitons. It is found that the
Berry-curvature corrections to the transition amplitudes are about two order-
of-magnitude smaller than the uncorrected transition amplitudes. Therefore,
the contributions of Berry-curvature corrections to the one-exciton transition
probability and the intra-exciton transition probability are not shown in the
optical spectra of TMDCs. The finding is consistent with the calculations in
Ref. gong2017optical, .

###  IV.4 Exciton valley Zeeman effect

In this section, the case of an exciton in a out-of-plane magnetic field
(${B}\neq{0}$, $|\mathbf{F}|=|\bm{\mathcal{F}}|=0$) is considered and the
exciton valley Zeeman effect is studied. Based on the FW transformed single-
particle Hamiltonian and the two-particle potential, the exciton Hamiltonian
can be written as

$\displaystyle\mathcal{H}_{\text{X}}$ $\displaystyle=$
$\displaystyle\Delta+\frac{|\bm{\pi}_{\text{e}}|^{2}}{2m_{\text{e}}}+\frac{|\bm{\pi}_{\text{h}}|^{2}}{2m_{\text{h}}}-V({r}_{\text{eh}})+\frac{\tau_{\text{e}}eB}{2{m}}-\frac{\tau_{\text{h}}eB}{2{m}}$
(82)
$\displaystyle-\frac{\tau_{\text{e}}\mathbf{e}_{\perp}\cdot\bm{\nabla}_{\text{e}}V({r}_{\text{eh}})\times\bm{\pi}_{\text{e}}}{4{m}^{2}v^{2}_{\text{F}}}-\frac{\nabla^{2}_{\text{e}}V({r}_{\text{eh}})}{8{m}^{2}v^{2}_{\text{F}}}$
$\displaystyle-\frac{\tau_{\text{h}}\mathbf{e}_{\perp}\cdot\bm{\nabla}_{\text{h}}V({r}_{\text{eh}})\times\bm{\pi}_{\text{h}}}{4{m}^{2}v^{2}_{\text{F}}}-\frac{\nabla^{2}_{\text{h}}V({r}_{\text{eh}})}{8{m}^{2}v^{2}_{\text{F}}}.$

Again, we use the coordinate transformation in Eq. (57) and Eq. (58) to
rewrite the Hamiltonian by the relative coordinate system. Additionally, we
apply the unitary transformation to the exciton Hamiltonian,
$\tilde{\mathcal{H}}_{\text{X}}=\mathcal{U}_{\text{X}}\mathcal{H}_{\text{X}}\mathcal{U}^{\dagger}_{\text{X}}$,
with the unitary transformation operator

$\displaystyle\mathcal{U}_{\text{X}}$ $\displaystyle=$
$\displaystyle\exp\left[\frac{\mathtt{i}eB}{2}\mathbf{e}_{\perp}\cdot\left(\mathbf{R}\times\mathbf{r}\right)\right],$
(83)

which causes the momentum operators (note that
$\mathbf{p}=-\mathtt{i}\bm{\nabla}$ and
$\mathbf{P}=-\mathtt{i}\bm{\nabla}_{\mathbf{R}}$) being transformed as
$\mathcal{U}_{\text{X}}\mathbf{p}\mathcal{U}^{\dagger}_{\text{X}}=\mathbf{p}-(eB/2)\mathbf{e}_{\perp}\times\mathbf{R}$,
$\mathcal{U}_{\text{X}}\mathbf{P}\mathcal{U}^{\dagger}_{\text{X}}=\mathbf{P}+(eB/2)\mathbf{e}_{\perp}\times\mathbf{r}$.
The exciton Hamiltonian is rewritten as

$\displaystyle\tilde{\mathcal{H}}_{\text{X}}=\mathcal{H}^{\prime}_{\text{X}}+\mathcal{H}^{\prime\prime}_{\text{X}}+\mathcal{J}_{\text{X}},$
(84)

where $\mathcal{H}^{\prime}_{\text{X}}={P^{2}}/({2m_{\text{X}}})$ is the
exciton translational Hamiltonian,

$\displaystyle\mathcal{H}^{\prime\prime}_{\text{X}}$ $\displaystyle=$
$\displaystyle\Delta+\frac{p^{2}}{2\mu}-{V}({r})-\frac{\Omega}{4}\nabla^{2}{V}({r})$
(85)
$\displaystyle-\frac{\Omega(\tau_{\text{e}}+\tau_{\text{h}})}{4}\mathbf{e}_{\perp}\cdot\bm{\nabla}{V}({r})\times\mathbf{p}$
$\displaystyle+\frac{eB}{2}\Bigg{[}\left(\frac{1}{m_{\text{e}}}-\frac{1}{m_{\text{h}}}\right)\mathcal{L}(\mathbf{r})+\frac{\tau_{\text{e}}-\tau_{\text{h}}}{2\mu}$
$\displaystyle-\frac{\Omega(\tau_{\text{e}}-\tau_{\text{h}})}{4}\mathbf{r}\cdot\bm{\nabla}V({r})\Bigg{]}+\frac{e^{2}B^{2}}{8\mu}r^{2}$

is the exciton internal Hamiltonian, and

$\displaystyle\mathcal{J}_{\text{X}}$ $\displaystyle=$
$\displaystyle\mathbf{e}_{\perp}\cdot\left[\frac{eB}{m_{\text{X}}}\mathbf{r}-\frac{e\Omega(m_{\text{e}}\tau_{\text{e}}-m_{\text{h}}\tau_{\text{h}})}{4m_{\text{X}}}\bm{\nabla}V({r})\right]\times\mathbf{P}$
(86)

is the exciton translational-internal coupling. Note that we have used the
identity
$\mathbf{e}_{\perp}\cdot\left[\bm{\nabla}{V}({r})\times(\mathbf{e}_{\perp}\times\mathbf{r})\right]=\mathbf{r}\cdot\bm{\nabla}V({r})$
in Eq. (85). The first three terms of the exciton internal Hamiltonian are the
Wannier exciton Hamiltonian, the forth term is the Darwin interaction, the
fifth term is the exciton valley-orbit coupling, the sixth term is the valley
Zeeman interaction and the last term is the diamagnetic interaction.

To study the valley Zeeman effect of excitons, it is important to note that
the Zeeman splittings are also contributed from the spin and the atomic
orbital of the electron or hole. The contributions from the Zeeman splittings
of spins and atomic orbitals to the electron energy and the hole energy are
given bybragancca2019magnetic

$\displaystyle\tilde{\varepsilon}_{\text{e}}$ $\displaystyle=$
$\displaystyle\varepsilon^{(0)}_{\text{FW,e}}+\frac{s_{\text{e}}g_{\text{spin}}+\tau_{\text{e}}g_{\text{e}}}{2}\mu_{\text{B}}B,$
(87) $\displaystyle\tilde{\varepsilon}_{\text{h}}$ $\displaystyle=$
$\displaystyle\varepsilon^{(0)}_{\text{FW,h}}-\frac{s_{\text{h}}g_{\text{spin}}+\tau_{\text{h}}g_{\text{h}}}{2}\mu_{\text{B}}B,$
(88)

where $s_{\text{e}}$ and $s_{\text{h}}$ are electron-spin and hole-spin
indices, $g_{\text{spin}}$ is the spin Lande g-factor, $g_{\text{e}}$ and
$g_{\text{h}}$ are the atomic orbital Lande g-factors for electrons and holes,
and $\mu_{\text{B}}={e}/({2m_{0}})$ is the Bohr magneton with $m_{0}$ the free
electron mass. The band-gap energy is given by

$\displaystyle\tilde{\Delta}$ $\displaystyle=$
$\displaystyle\Delta+\frac{1}{2}\left[\left(s_{\text{e}}-s_{\text{h}}\right)g_{\text{spin}}+\tau_{\text{e}}g_{\text{e}}-\tau_{\text{h}}g_{\text{h}}\right]\mu_{\text{B}}B.$
(89)

We assume that the spin and valley indices of the hole are given by
$s_{\text{e}}=s_{\text{h}}=\tau_{\text{h}}=\tau$ for both intravalley and
intervalley excitons, and the valley index of the electron is given by
$\tau_{\text{e}}=\tau$ for intravalley bright excitons and
$\tau_{\text{e}}=-\tau$ for intervalley excitons. The exciton valley Zeeman
shift is defined by

$\displaystyle
E_{\text{X},I}=E^{(0)}_{\text{X},I}+\frac{1}{2}g_{\text{X},I}\mu_{\text{B}}B+\cdots,$
(90)

where $E^{(0)}_{\text{X},I}$ is the exciton energy level without magnetic-
field interaction, $g_{\text{X},I}$ is the exciton valley g-factor for the
$I$-th exciton state. We can find

$\displaystyle g_{\text{X},I}$ $\displaystyle=$
$\displaystyle\left(s_{\text{e}}-s_{\text{h}}\right)g_{\text{spin}}+\tau_{\text{e}}g_{\text{e}}-\tau_{\text{h}}g_{\text{h}}$
(91)
$\displaystyle+\frac{el_{N}}{\mu_{\text{B}}}\left(\frac{1}{m_{\text{e}}}-\frac{1}{m_{\text{h}}}\right)+\frac{e(\tau_{\text{e}}-\tau_{\text{h}})}{\mu_{\text{B}}(2\mu)}$
$\displaystyle-\frac{e\Omega(\tau_{\text{e}}-\tau_{\text{h}})}{4\mu_{\text{B}}}\langle{\mathbf{r}\cdot\bm{\nabla}V({r})}\rangle_{I},$

where the last term in Eq. (91) is an interaction-induced Zeeman splitting.
Assuming $g_{\text{spin}}=2$, $g_{\text{e}}=0$, $g_{\text{h}}=4$ and using the
parameters in the last line of Table. 1
($m_{\text{e}}=m_{\text{h}}=0.44m_{0}$, $r_{0}=50.00$, $\kappa=4.0$), the
g-factors with the exciton state $I$ assigned as the $1s$ state are given by
$|g_{\text{X},1s}|=4$ for the intravalley exciton and $|g_{\text{X},1s}|=14.5$
for the intervalley exciton, while the interaction-induced Zeeman splitting
contributes about $1.4$ for the g-factor of the intervalley exciton. These
values are consistent with experimental measurements of excitonic states of
$\text{WSe}_{2}$liu2019magnetophotoluminescence ; YCPRL , where the measured
g-factors are $|g_{\text{X},1s}|\simeq{4}$ for the intravalley exciton and
$|g_{\text{X},1s}|\simeq{13}$ for the intervalley exciton.

##  V Summary and perspectives

In the present work, the FW transformation is applied to 2D gapped Dirac
fermions under external fields. The single-particle Hamiltonian, two-particle
interaction, and interband transition obtained from the FW transformation are
used to study the valley-dependent physical properties of 2D materials.
Exciton Hamiltonians for intravalley and intervalley excitons in an in-plane
electromagnetic field and in an out-of-plane magnetic field are derived.
Exciton energy-level splittings, valley-selected exciton transitions, and
exciton valley Zeeman effect are formulated analytically and studied. The
variational method is used to solve the exciton energy levels and optical
spectra. The calculated results are quantitatively coincident with
literatures. Even though these effects have been discussed by different
theoretical methods in literatures, we believe that the present theoretical
framework still provides a new viewpoint on these topics and a straightforward
derivation procedure to study related problems.

While the Berry-curvature effects on valley excitons have been studied in
literatures, many related topics still demand investigation. One topics of
particular importance is the Berry-curvature effect on valley trions and other
valley-dependent exciton complexes. For problems involving more than two
particles, a semiclassical derivation of Berry-curvature effect becomes much
more difficult. Therefore, the present FW transformation method becomes quite
useful for deriving effective Hamiltonians for valley-dependent exciton
complexes. For trions, some theoretical studies addressing the Berry-curvature
effect have been publishedhichri2019charged ; hichri2020trion , but joint
effects of valley degree of freedom and external fields on trions have yet to
be studied. Another possible extension of the present framework is to apply
the FW transformation to 2D gapped Dirac fermions with additional interactions
or band-structure modifications. For example, the trigonal warping in the band
structures of 2D materials can be included by the FW transformation as a
nonlocal kinetic correction to the exciton Hamiltonian and interband
transition. Such extensibility shows the potential and versatility of the
present theoretical framework. These topics will be considered in future
studies.

## acknowledgment

This work was supported in part by the Ministry of Science and Technology
(MOST), Taiwan under Contract No. 109-2112-M-001-046 and 110-2112-M-001-042.
Y.-W.C. thanks the financial support from the Postdoctoral Scholar Program at
Academia Sinica, Taiwan, ROC.

##  Appendix A Variational method

In the appendix, the variational method to solve the exciton wavefunction and
eigenenergy is introduced. By a rescaling of the length unit to the effective
Bohr radius ”$a_{0}=({m_{0}}/{m_{\text{e}}})a_{\text{B}}$” and the energy unit
to the effective Hartree
”$\varepsilon_{0}=({m_{\text{e}}}/{m_{0}})(2\text{Ry})$”, with $m_{0}$ the
free electron mass, $a_{\text{B}}\simeq 0.5291772$ Å the Bohr radius and
$\text{Ry}\simeq 13.60569$ eV the Rydberg constant, the exciton Hamiltonian
can be rewritten as

$\displaystyle\mathcal{H}^{\prime\prime}_{\text{X}}(\mathbf{r})$
$\displaystyle=$
$\displaystyle-\frac{\sigma+1}{2}\nabla^{2}-{V}({r})-\frac{\Omega}{4}\nabla^{2}{V}({r})$
(92)
$\displaystyle-\frac{\Omega(\tau_{\text{e}}+\tau_{\text{h}})}{4}\frac{1}{r}\frac{\partial{V}(r)}{\partial{r}}\mathcal{L}_{\text{X}}(\mathbf{r}),$

where $\sigma=m_{\text{e}}/m_{\text{h}}$ is the mass ratio, $V(r)$ is a
screened Coulomb potential,
$\mathcal{L}_{\text{X}}(\mathbf{r})=-\mathtt{i}\mathbf{e}_{\perp}\cdot\mathbf{r}\times\bm{\nabla}=-\mathtt{i}{\partial}/{\partial\varphi}$
is the angular momentum operator,

$\displaystyle\Omega=\frac{(\sigma+1)^{2}a^{2}_{0}\varepsilon^{2}_{0}}{4a^{2}t^{2}}$
(93)

is the Berry curvature with $a$ the lattice constant and $t$ the hopping
coupling. The 2D screened Coulomb potential is assumed to be given by the
Rytova-Keldysh potentialrytova ; Keldysh

$\displaystyle V(r)$ $\displaystyle=$
$\displaystyle\frac{\pi}{\kappa\rho_{0}}\left[H_{0}\left(\frac{r}{\rho_{0}}\right)-Y_{0}\left(\frac{r}{\rho_{0}}\right)\right]$
(94) $\displaystyle=$
$\displaystyle\int\;e^{\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\left[\frac{2\pi}{\kappa|\mathbf{k}|\left(1+|\mathbf{k}|\rho_{0}\right)}\right]\frac{\text{d}^{2}k}{(2\pi)^{2}},$

where $\rho_{0}=r_{0}/(\kappa a_{0})$ with $r_{0}$ the screening length in
unit of Å, $\kappa$ is the dielectric constant, $H_{0}(r)$ and $Y_{0}(r)$ are
the Struve function and the Bessel function of the second kind.

We use 2D Slater-type-orbital (STO) as the basis function to expand the
exciton wavefunctionzhang2019two ; wu2019exciton ; mypaper0 . The exciton
wavefunction can be written as

$\displaystyle\Psi_{\text{X},I}=\sum_{a}c_{a,I}\phi_{a}(\mathbf{r}),$ (95)

where $c_{a,I}$ is the linear variational parameter and

$\displaystyle\phi_{a}(\mathbf{r})=\frac{e^{\mathtt{i}l_{a}\varphi}}{\sqrt{2\pi}}r^{n_{a}-1}e^{-\zeta_{a}r}$
(96)

is the STO with $\zeta_{a}$ the screening constant and also a variational
parameter, $n_{a}$ the principal quantum number and $l_{a}$ angular momentum.
By using linear variational method, the exciton coefficient can be solved by
the eigenvalue equation

$\displaystyle\sum_{b}h_{ab}c_{b,I}=\varepsilon_{\text{X},I}\sum_{b}o_{ab}c_{b,I},$
(97)

where $h_{ab}$ is exciton Hamiltonian matrix, $o_{ab}$ is the overlap matrix,
and the eigenvalue of the equation $\varepsilon_{\text{X},M}$ is the exciton
energy. The exciton Hamiltonian matrix is given by
$h_{ab}=t_{ab}+v_{ab}+d_{ab}$, with $t_{ab}$ the kinetic integral, $v_{ab}$
the potential integral, and $d_{ab}$ the band-geometry integral. The
formulations of the integrals $t_{ab}$, $v_{ab}$, $o_{ab}$ can be found in
Ref. mypaper0, . The band-geometry integral is given by

$\displaystyle d_{ab}$ $\displaystyle=$
$\displaystyle-\frac{\Omega(\tau_{\text{e}}+\tau_{\text{h}})}{4}\int\phi^{*}_{a}(\mathbf{r})\frac{1}{r}\frac{\partial{V}(r)}{\partial{r}}\mathcal{L}_{\text{X}}(\mathbf{r})\phi_{b}(\mathbf{r})\text{d}^{2}r$
(98)
$\displaystyle-\frac{\Omega}{4}\int\phi^{*}_{a}(\mathbf{r})\nabla^{2}{V}({r})\phi_{b}(\mathbf{r})\text{d}^{2}r.$

To calculate these orbital integrals containing the screened Coulomb
potential, we use the following formula

$\displaystyle\frac{\partial}{\partial{r}}=\cos\varphi\frac{\partial}{\partial{x}}+\sin\varphi\frac{\partial}{\partial{y}}=e^{\mathtt{i}\varphi}\frac{\partial}{\partial{z}}+e^{-\mathtt{i}\varphi}\frac{\partial}{\partial{z^{*}}},$
(99)

with $z=x+\mathtt{i}y$ and $z^{*}=x-\mathtt{i}y$, and

$\displaystyle\mathbf{k}\cdot\mathbf{r}=k_{x}x+k_{y}y=\frac{kz}{2}e^{-\mathtt{i}\varphi_{\mathbf{k}}}+\frac{kz^{*}}{2}e^{\mathtt{i}\varphi_{\mathbf{k}}},$
(100) $\displaystyle\frac{\partial{V}}{\partial{z}}=\int
e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\left[-\frac{\mathtt{i}k}{2}e^{-\mathtt{i}\varphi_{\mathbf{k}}}\tilde{V}(k)\right]\frac{\text{d}^{2}k}{(2\pi)^{2}},$
(101) $\displaystyle\nabla^{2}{V}=\int
e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\left[-k^{2}\tilde{V}(k)\right]\frac{\text{d}^{2}k}{(2\pi)^{2}},$
(102)

with
$\tilde{V}(k)=\int{V}(r)e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\text{d}^{2}r$
the Fourier transform of the screened potential. Given the screened potential
being the Rytova-Keldysh potential, the Laplacian of the potential function
can be rewritten by

$\displaystyle\nabla^{2}{V}(\mathbf{r})$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{\kappa\rho_{0}}\int
e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\frac{k^{2}}{k\left(1/\rho_{0}+k\right)}\frac{\text{d}^{2}k}{(2\pi)^{2}}$
(103) $\displaystyle=$ $\displaystyle-\frac{2\pi}{\kappa\rho_{0}}\int
e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\frac{\text{d}^{2}k}{(2\pi)^{2}}+\frac{2\pi}{\kappa\rho_{0}}\int\frac{e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}}{1+k\rho_{0}}\frac{\text{d}^{2}k}{(2\pi)^{2}}$
$\displaystyle=$
$\displaystyle-\frac{2\pi}{\kappa\rho_{0}}\delta(\mathbf{r})+\int
e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\frac{k}{\rho_{0}}\tilde{V}(k)\frac{\text{d}^{2}k}{(2\pi)^{2}}.$

In order to derive the orbital integrals, we can take advantage of the
analytic formula for the Fourier transform of the 2D STO

$\displaystyle\tilde{\phi}_{a}(\mathbf{k})$ $\displaystyle\equiv$
$\displaystyle\int\phi_{a}(\mathbf{r})e^{-\mathtt{i}\mathbf{k}\cdot\mathbf{r}}\text{d}^{2}r=\frac{e^{\mathtt{i}l_{a}\varphi_{\mathbf{k}}}}{\sqrt{2\pi}}\mathcal{R}_{n_{a},l_{a}}(\zeta_{a},k),$

where the radial function is generated bymypaper0

$\displaystyle\mathcal{R}_{n,l}(\zeta,k)$ $\displaystyle=$
$\displaystyle\frac{2\pi(-\mathtt{i})^{n}}{k^{n+1}}\left[\frac{\text{d}^{n}}{\text{d}z^{n}}\frac{\left(z-\mathtt{i}\eta\sqrt{1-z^{2}}\right)^{|l|}}{\sqrt{1-z^{2}}}\right]_{z=\mathtt{i}{\zeta}/{k}},$

with $\eta=l/|l|$ being the sign of $l$. The band-geometry integral is given
by

$\displaystyle d_{ab}$ $\displaystyle=$
$\displaystyle\delta_{l_{a},l_{b}}\frac{\mathtt{i}l_{b}\Omega(\tau_{\text{e}}+\tau_{\text{h}})}{8}\int\Big{[}\mathcal{R}_{n_{a}+n_{b}-2,1}(\zeta_{a}+\zeta_{b},k)$
(106)
$\displaystyle+\mathcal{R}_{n_{a}+n_{b}-2,-1}(\zeta_{a}+\zeta_{b},k)\Big{]}\tilde{V}(k)\frac{k^{2}\text{d}k}{(2\pi)^{2}}$
$\displaystyle-\delta_{l_{a},l_{b}}\frac{\Omega}{4\rho_{0}}\int
k\tilde{V}(k)\mathcal{R}_{n_{a}+n_{b}-1,0}(\zeta_{a}+\zeta_{b},k)\frac{k\text{d}k}{(2\pi)^{2}}$
$\displaystyle+\delta_{l_{a},l_{b}}\delta_{n_{a},1}\delta_{n_{b},1}\frac{\Omega}{4\kappa\rho_{0}}.$

The integrals of the transition amplitudes can also be calculated by using the
STOs and the analytical formulation. The one-exciton transition amplitude is
given by

$\displaystyle j^{\pm}_{N0}$ $\displaystyle=$
$\displaystyle-\frac{ev_{\text{F}}}{(2\pi)^{3/2}}\sum_{\tau,a}c^{*}_{a,N}$
(107)
$\displaystyle\times\Bigg{[}\delta_{\tau,\pm{1}}\delta_{l_{a},0}\int^{k_{\Lambda}}_{0}\mathcal{R}^{*}_{n_{a},0}(\zeta_{a},k)\left(1-\frac{\Omega
k^{2}}{4}\right)kdk$
$\displaystyle+\delta_{\tau,\mp{1}}\delta_{l_{a},\pm{2}}\int^{k_{\Lambda}}_{0}\mathcal{R}^{*}_{n_{a},\pm{2}}(\zeta_{a},k)\frac{\Omega
k^{2}}{4}kdk\Bigg{]}$

and $j^{\pm}_{0N}=(j^{\mp}_{N0})^{*}$. Note that the integration in Eq. (107)
could diverge as the cut-off momentum $k_{\Lambda}\rightarrow\infty$.
Therefore, the cut-off momentum $k_{\Lambda}=2\pi/a$ with $a$ the lattice
constant is chosen to ensure convergence. The intra-exciton transition
amplitude is given by

$\displaystyle\xi^{\pm}_{NM}$ $\displaystyle=$
$\displaystyle-\frac{e}{\sqrt{2}}\sum_{\tau,ab}c^{*}_{a,N}c_{b,M}\delta_{l_{a},l_{b}\pm{1}}\frac{(n_{a}+n_{b}-1)!}{(\zeta_{a}+\zeta_{b})^{n_{a}+n_{b}}}\Bigg{\\{}\frac{n_{a}+n_{b}}{\zeta_{a}+\zeta_{b}}$
(108) $\displaystyle\mp\frac{\tau\Omega}{2}\Bigg{[}\frac{n_{b}-1\mp
l_{b}}{n_{a}+n_{b}-1}(\zeta_{a}+\zeta_{b})-\zeta_{b}\Bigg{]}\Bigg{\\}}.$

## References

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