# Coherent optical control of quantum Hall edge states

Ashutosh Singh<EMAIL_ADDRESS>Department of Physics and Astronomy,
Texas A&M University, College Station, TX, 77843 USA  Maria Sebastian
Department of Physics and Astronomy, Texas A&M University, College Station,
TX, 77843 USA  Mikhail Tokman  Department of Electrical and Electronic
Engineering and Schlesinger Knowledge Center for Compact Accelerators and
Radiation Sources, Ariel University, 40700 Ariel, Israel  Alexey Belyanin
<EMAIL_ADDRESS>Department of Physics and Astronomy, Texas A&M University,
College Station, TX, 77843 USA

###### Abstract

Current carrying chiral edge states in quantum Hall systems have fascinating
properties that are usually studied by electron spectroscopy and
interferometry. Here we demonstrate that electron occupation, current, and
electron coherence in chiral edge states can be selectively probed and
controlled by low-energy electromagnetic radiation in the microwave to
infrared range without affecting electron states in the bulk or destroying
quantum Hall effect conditions in the bulk of the sample. Both linear and
nonlinear optical control is possible due to inevitable violation of
adiabaticity and inversion symmetry breaking for electron states near the
edge. This opens up new pathways for frequency- and polarization-selective
spectroscopy and control of individual edge states.

##  I Introduction

The Quantum Hall (QH) effect is one of the most studied phenomena in condensed
matter physics Prange and Girvin (1987), with far-reaching applications in
many other areas. One of the most important insights in the QH effect physics
was the intricate relation between the electron states in the insulating bulk
of the sample and the current carrying chiral edge states Halperin (1982);
MacDonald and Středa (1984) which have been extensively studied with real
space imaging and momentum resolved electron spectroscopy Pascher  _et al._
(2014); Marguerite  _et al._ (2019); Johnsen  _et al._ (2023); Li  _et al._
(2013); Patlatiuk  _et al._ (2018); Kim  _et al._ (2021). Moreover, coherence
of unidirectional electron transport in QH edge states stimulated massive
research in QH edge state interferometry. Various types of electron
interferometers have been implemented, both in conventional semiconductor
quantum wells and in graphene samples, and for both integer and fractional
statistics of carriers van Wees  _et al._ (1989); de C. Chamon  _et al._
(1997); Ji  _et al._ (2003); Chung and Stone (2006); Stern and Halperin
(2006); Feldman and Kitaev (2006); Zhang  _et al._ (2009); Stern  _et al._
(2010); Déprez  _et al._ (2021); Feldman and Halperin (2022); Nakamura  _et
al._ (2020); Ronen  _et al._ (2021); Nakamura  _et al._ (2023).

Terahertz optical spectroscopy of Landau-quantized electron states in two-
dimensional (2D) electron gas is of course yet another massive field of
research. However, its utility in probing or manipulating the electron states
under the conditions of QH effect is highly problematic, because resonant
optical transitions between bulk Landau levels will lead to nonequilibrium
carrier population, which therefore enables nonzero bulk DC conductivity
across the sample. As was argued in Ref. Appugliese  _et al._ (2022), even
vacuum cavity fields under the ultra-strong coupling conditions could break
the topological protection of the integer QH effect and destroy some of the
high quantum number plateaux.

In this paper we show that the optical spectroscopy and even coherent optical
control of the QH edge states are still possible and can be in fact very
effective without destroying the QH effect conditions in the bulk. The key
physical reason for this is that the optical transitions between electron
states near the sample boundary (within a few magnetic lengths from the edge)
have significantly different transition energies and polarization selection
rules as compared to the bulk of the sample. This permits highly selective
excitation of a given 1D edge channel with single quasiparticle sensitivity
without disturbing the rest of the sample. Furthermore, inversion symmetry
breaking near the sample boundary (see highly asymmetric wave functions in the
supplemental video) enables strong second-order optical nonlinearity in
electric dipole approximation, resulting in efficient optical rectification of
incident radiation and direct optical driving of a quasi-DC current in edge
states. These qualitative features are illustrated in Figs. 1 and 2, with more
quantitative discussion in the sections below.

Figure 1: (a) Photons interacting with electrons moving near one of the edges
of a sample which is subjected to a quantizing magnetic field. (b) Near the
edge of the sample, energies and wave-functions of the electrons are modified
in such a way that new channels open up for the absorption of photons or
nonlinear optical rectification. Dipole forbidden (solid green) and Pauli
blocked (solid blue) channels deep in the bulk become active for electrons
moving near the edge (marked as dashed green and solid red, respectively).

High spatiotemporal and energy selectivity of the optical excitations of
chiral edge states not only makes it a sensitive spectroscopy tool
complementary to electron transport measurements but also enables coherent
control of individual edge channels in QH interferometers endowing them with
new optoelectronic functionality. Further enhancement in selectivity could be
possible with near field tip-enhanced optical microscopy as opposed to far-
field illumination sketched in Fig. 1. Therefore we hope that our paper will
stimulate further collaboration between optical and QH effect communities.

In this paper we focus at the integer QH effect in semiconductor quantum well
samples for parabolic electron dispersion. The graphene edge states offer a
greater variety of the optical transitions due to two kinds of edge
terminations and will be considered elsewhere, as well as edge states with
fractional statistics of carriers. The structure of the paper is as follows.
In section II we provide both asymptotic and exact numerical solutions for
electron eigenstates and eigenenergies in the presence of an edge treated as a
hard-wall boundary, which of course provides maximum nonadiabaticity. We also
calculate dipole matrix elements of the optical transitions between both bulk
and edge states. In section III we compute a single-photon absorption
probability for a quantized optical field and obtain spectra of 2D absorbance
which show a series of sharp characteristic peaks at high frequencies that are
entirely due to nonadiabatic edge states. In section IV, we demonstrate
different optical mechanisms that give rise to direct current (DC) generation
by nonlinear rectification of the incident radiation. We calculate the second-
order nonlinear DC current which exist already in electric dipole
approximation due to inversion symmetry breaking near the edge. We also
evaluate the DC current due to optical rectification beyond electric dipole
approximation, which exists due to the optical field gradient on the sample.
Appendix summarizes the results for electron eigenstates in the adiabatic
approximation for comparison, derives the electron density flux for an
arbitrary electric potential, and gives the list of second-order nonlinear
density matrix elements contributing to the rectification current.

Figure 2: (a) The plots of eigenenergies as a function of momentum $k$.
Electron states in shaded regions contribute to photon absorption for a given
Fermi level shown with red horizontal line. (b) The absorbance spectrum.
Absorption due to the edge states appears at frequencies higher than the bulk
cyclotron resonance frequency $\omega_{c}$. The shaded colored regions under
the black curve show the contributions of individual transitions between the
neighboring LLs, with matching colors between (a) and (b).

##  II Chiral edge states close to the boundary

Consider a QH system with electrons constrained to the $x,y$ plane in a
quantizing magnetic field $B$ along $z$-axis described by the vector potential
${\vec{A}}=-yB\hat{x}$. Near the boundary in $y$ direction located at $y=0$,
the electron states experience the edge potential $\Phi(y)$, so that the
Schrödinger equation for a wave-function of the form $\psi_{k}(x,y)\sim
e^{ikx}\chi(y)$ becomes

$\displaystyle\frac{\partial^{2}\chi}{\partial
y^{2}}+\frac{2m_{e}}{\hbar^{2}}\left[E+e\Phi(y)-\frac{1}{2}m_{e}\omega_{c}^{2}\left(y-y_{k}\right)^{2}\right]\chi(y)=0~{}.$
(1)

where $m$ is the electron mass, $E$ is the energy eigenvalue,
$\omega_{c}=eB/(m_{e}c)$ is the cyclotron frequency, $y_{k}=k\ell_{c}^{2}$ is
the position of the center of the cyclotron oscillator, and
$\ell_{c}^{2}=c\hbar/(eB)$ the magnetic length.

In the adiabatic approximation which is usually invoked when discussing edge
states (see Appendix A), $\Phi(y)$ is considered such a slow function of $y$
that its change over the magnetic length scale is neglected. In this case,
while the energies of the LLs increase towards the edge, the energy separation
between them does not change and the dipole matrix elements of all transitions
between them do not change either. Furthermore, the drift velocities of
electrons in the channels do not depend on the LL index.

The adiabatic approximation becomes increasingly inadequate when the distance
to the edge becomes of the order of a few magnetic lengths. In order to
proceed, we need to specify the shape of the edge potential. It obviously
depends on the details of the interface and is affected by the presence of the
surface states, any space charge accumulation which tends to further sharpen
the potential profile Chklovskii  _et al._ (1992), etc. In order to obtain
quantitative results we take the simplest nonadiabatic potential: the hard
wall condition Mei and Lee (1983); Patlatiuk  _et al._ (2018), or a step in
$\Phi(y)$ at $y=0$, which for a high enough potential barrier means that for a
given LL index $n$ and x-component of momentum $k$, $\chi_{nk}(0)=0$ and
$\chi_{nk}(y\to\infty)=0$. Although obviously an idealization of any realistic
boundary, the hard wall condition captures the main physical effects of the
nonadiabatic interface, namely asymmetry of the wave functions near the
boundary (see the supplemental video), nonequidistant eigenenergies $E_{nk}$
of the edge states as shown in Fig. 3, modified polarization selection rules
which increasingly favor $y$-polarization as the states are pressed to the
edge, and inversion symmetry breaking which leads to nonzero permanent dipole
moments and modified LL number $n$ selection rules: the transitions with $n$
changing by 2 become allowed in the electric dipole approximation. Finally,
the electron drift velocity in the state $|nk\rangle$ i.e., the diagonal
matrix element of the velocity operator, becomes $n$ and $k$ dependent and has
a standard form (see Appendix B for derivation):

$\displaystyle\langle\hat{v}_{x}\rangle_{nk}=\frac{1}{\hbar}\frac{\partial
E_{nk}}{\partial k}~{}.$ (2)

Note that this expression does not depend on the specific form of the
eigenstates $\psi_{nk}(x,y)$; moreover, Eq.(2) does not depend on any
assumptions about the nonuniformity scale of the potential $\Phi(y)$ in
comparison with the magnetic length $\ell_{c}$.

We will consider the observational consequences of these effects one by one
below, keeping the derivation details and lengthy formulas in the Appendix.
Before using a numerical solution of Eq. (1) with a hard wall potential to
calculate the optical response, we point out two approximate analytical
solutions of Eq. (1) for a hard-wall boundary condition $\Phi(y>0)=0$ and
$\chi_{nk}(0)=0$.

###  II.1 Asymptotic solution for $n\gg 1$

In this case, analytical solutions can be obtained within the quasi-classical
approximation Landau and Lifshitz (1981). For
$y_{k}>\sqrt{2E/(m\omega_{c}^{2})}$, the quasi-classical solution of Eq.(1)
corresponds to two turning points in the region $y>0$ (one at $y>y_{k}>0$ and
another at $y_{k}>y>0$), for which
$E=m\omega_{c}^{2}\left(y-y_{k}\right)^{2}/2$. In this case the solution obeys
Bohr-Sommerfeld quantization rule, which for Eq. (1) corresponds to standard
eigenenergies $E_{n}=\hbar\omega_{c}(n+1/2)$. For
$y_{k}<\sqrt{2E/(m\omega_{c}^{2})}$ there is one turning point at $y>y_{k}>0$
and another one due to reflection from the wall at $y=0$. Denoting the turning
point at $y\neq 0$ as $y^{*}$, we write the quasi-classical solution of Eq.
(1) as

$\displaystyle\chi_{nk}(y)=\frac{1}{N_{nk}\sqrt{q_{nk}}}\cos\left(\int_{y}^{y^{*}}dy~{}q_{nk}(y)-\frac{\pi}{4}\right),$
(3)

where $\hbar
q_{nk}(y)=\sqrt{2m\left[E_{nk}-m\omega_{c}^{2}(y-y_{k})^{2}/2\right]}=0$ and
$q_{nk}(y^{*})=0$. Here $N_{nk}$ is the normalization factor. Taking into
account the boundary condition on the ideally reflecting wall, i.e.,
$\chi_{nk}(0)=0$, we obtain the transcendental equation for the eigenenergies
as

$\displaystyle
E_{nk}=\frac{2\pi(n+3/4)\hbar\omega_{c}}{2\beta_{nk}-\sin(2\beta_{nk})}~{},$
(4)

with $\cos(\beta_{nk})=-\hbar k/\sqrt{2mE_{nk}}$. For $y_{k}\propto k>0$ and
$1-\hbar k/\sqrt{2mE_{nk}}\ll 1$, we have $\beta_{nk}\to\pi$, i.e.
$E_{nk}=(n+3/4)\hbar\omega_{c}$. As we see, the energy is increased by
$\hbar\omega_{c}/4$ as compared to the LL in the bulk. This is the result of
changing boundary conditions from a smooth effective potential to a hard wall.
For $y_{k}=0$, $\beta_{nk}=\pi/2$, which gives
$E_{nk}=(2n+3/2)\hbar\omega_{c}$. Here the energy is raised by more than a
factor of 2 as compared to the LLs in the bulk. For $y_{k}<0$, one can find
the asymptotic solution for $E_{nk}\gg n\hbar\omega_{c}$, which corresponds to
the large wave-number limit, $|k|\ell_{c}\gg\sqrt{n}$. In this case we need to
have $\beta_{nk}-\sin(2\beta_{nk})/2\ll 1$, i.e., $\beta_{nk}\to 0$. Denoting
$\varepsilon_{k}=\hbar^{2}k^{2}/(2m)$, we obtain

$\displaystyle
E_{nk}\approx\varepsilon_{k}+\left[\frac{3\pi}{2}\hbar\omega_{c}\left(n+\frac{3}{4}\right)\right]^{2/3}\varepsilon_{k}^{1/3}~{},$
(5)

implying that the energies increase and the distance between LLs grows non-
equidistantly. This already suggests that for a vertical wall or for any non-
adiabatic potential one can realize resonant optical transitions between edge
states without causing resonant absorption between bulk LLs.

###  II.2 Solution for any $n$ but large wavenumbers $k$

Let us denote $V(y)=m\omega_{c}^{2}(y-y_{k})^{2}/2$. Under the condition
$V^{\prime}(y)|_{y=y^{*}}-V^{\prime}(y)|_{y=0}\ll V^{\prime}(y)|_{y=0}$, where
$E=V(y^{*})$, and for
$\xi=\left(2|k|/\ell_{c}^{2}\right)^{1/3}\left(y-(E-\varepsilon_{k})/(\hbar\omega_{c}|k|)\right)$,
Eq. (1) can be transformed to

$\frac{\partial^{2}\chi}{\partial\xi^{2}}-\xi\chi=0.$

We can choose our solution as the Airy function ${\rm Ai}(\xi)$, which goes to
zero for $\xi\to\infty$ and satisfies the boundary condition $\chi(y=0)=0$,
with the corresponding eigenenergies

$\displaystyle~{}~{}~{}~{}E_{nk}=\varepsilon_{k}+\left(\hbar\omega_{c}\right)^{2/3}\varepsilon_{k}^{1/3}|\xi_{n}|~{},$
(6)

where $\xi_{n}$ are zeros of the Airy function, i.e., ${\rm
Ai}(\xi_{n})=0,n=0,1,2...$. Here $\xi_{0}$ is the zero with value closest to
$\xi=0$, and their values $|\xi_{n}|$ increase with increasing $n$. One can
verify that Eq. (6) is valid as long as $|k|\ell_{c}\gg\sqrt{n}$. The
eigenfunctions $\chi_{nk}(y)$ corresponding to eigenenergies $E_{nk}$ are
defined in the interval $0\leq y<\infty$ as

$\displaystyle~{}~{}~{}~{}\chi_{nk}(y)=\frac{1}{{\tilde{N}}_{nk}}{\rm
Ai}\left(y\left(\frac{2|k|}{\ell_{c}^{2}}\right)^{1/3}+\xi_{n}\right)~{},$ (7)

with ${\tilde{N}}_{nk}$ being the normalization factor which is given as,

$\displaystyle{\tilde{N}}_{nk}^{2}=\left(\frac{\ell_{c}^{2}}{2|k|}\right)^{1/3}\int_{\xi_{n}}^{\infty}{\rm
Ai}^{2}(\xi)d\xi~{}.$ (8)

The dispersion of eigenenergies (6) has the similar structure to those in the
previous subsection, even though the former does not rely on $n$ being large.

Figure 3: (a) Numerically obtained energy eigenvalues (normalized with
$\hbar\omega_{c}$) from Eq. (1) in the presence of hard wall at $y=0$ for the
first eleven eigenvalues corresponding to $n=0,1...10$ respectively, as a
function of the center of cyclotron rotation $y_{k}/\ell_{c}=k\ell_{c}$. (b)
Difference between consecutive energy eigenvalues as a function of
$k\ell_{c}$. Here $\tilde{E}_{nk}\equiv E_{nk}/\hbar\omega_{c}$.

###  II.3 Exact numerical solution for eigenstates and dipole matrix elements

Finally, we compute the eigenvalues and the eigenfunctions for Eq. (1)
numerically using Numerov’s algorithm. Fig. 3 shows the eigenenergies for the
first 11 LLs as a function of $y_{k}/\ell_{c}=k\ell_{c}$, and the energy
differences between neighboring eigenvalues. It demonstrates not only the
bending of flat LLs of the bulk states near the edge of the sample, but also
the fact that the optical transitions become increasingly nonequidistant and
move to higher energies as compared to the transitions between bulk LLs. It is
also worth pointing out strong inhomogeneous broadening of the inter-LL
optical transitions.

Figure 4: Magnitude of the normalized dipole matrix elements as a function of
the center of the cyclotron rotation $y_{k}/\ell_{c}=k\ell_{c}$ for (a). $n\to
n+1$, (b) $n\to n+2$ and (c) the difference of intra-LL dipole elements. Here
$\tilde{y}_{ij}\equiv y_{ik;jk}/\ell_{c}$.

The dipole matrix elements and selection rules are modified drastically in the
presence of the boundary, mostly due to inversion symmetry breaking. In the
electric dipole approximation, the general expression for carrier velocity
along $\hat{x}$ direction which includes both diagonal ($n=m$) and off-
diagonal ($n\neq m$) elements is

$\displaystyle\langle
nk|\hat{v}_{x}|mk^{\prime}\rangle=\delta_{kk^{\prime}}\omega_{c}\left(y_{k}\delta_{nm}-y_{nk;mk}\right)~{},$
(9)

with $y_{nk;mk}=\langle nk|y|mk\rangle$. In the bulk, vertical transitions
between $m$th and $n$th LLs obeys the following selection rule,

$\displaystyle\frac{y_{nk;mk}}{\ell_{c}}=\delta_{n,m+1}\sqrt{\frac{n}{2}}~{}.$
(10)

This simple relation, however, does not hold for edge states. In Fig. 4, we
plot the dipole elements as a function of the center of the cyclotron rotation
$y_{k}/\ell_{c}=k\ell_{c}$ for transitions involving the first several LLs.
The panel (a) shows that the dipole elements deviate from Eq. (10) as $y_{k}$
approaches the edge. At the same time, previously forbidden transitions $n\to
n+2$ are activated, as shown in panel (b). Furthermore, the permanent, intra-
LL dipole matrix elements differ from their bulk values as well (panel (c)),
such that their differences between consecutive LLs change from zero to finite
values. Both effects are a consequence of inversion symmetry breaking and lead
to large second-order nonlinearity in electric-dipole approximation, as we
will see below.

##  III Quantum theory of the photon absorption

Since one of the most interesting possibilities offered by the optical field
is to excite a single electron into a given edge state, we need a fully
quantized theory of fermions in a QH sample interacting with a quantized
electromagnetic field. We will derive both the probability of a single photon
absorption and the absorbance of a classical field.

It is convenient to describe the quantum state of electrons in terms of
occupation numbers of $|nk\rangle=\chi_{nk}(y)e^{ikx}$ states, for example
$|\cdots 1_{nk}\cdots 0_{n^{\prime}k^{\prime}}\cdots\rangle$, where $1_{nk}$
and $0_{n^{\prime}k^{\prime}}$ are occupied and unoccupied states. Fermionic
annihilation and creation operators are acting on these states in the usual
way: $\hat{a}_{nk}|\cdots 1_{nk}\cdots\rangle=|\cdots 0_{nk}\cdots\rangle$ and
$\hat{a}_{nk}^{\dagger}|\cdots 0_{nk}\cdots\rangle=|\cdots
1_{nk}\cdots\rangle$. The electron Hamiltonian is then
$\hat{H}_{e}=\sum_{n,k}E_{nk}\hat{a}_{nk}^{\dagger}\hat{a}_{nk}$.

The Hamiltonian and the operator of the vector potential of the EM field
incident on the sample are

$\hat{H}_{\rm ph}=\sum_{\nu,{\bf q}}\hbar\omega_{\nu{\bf
q}}\left(\hat{b}_{\nu{\bf q}}^{\dagger}\hat{b}_{\nu{\bf
q}}+\frac{1}{2}\right),$ (11) $\displaystyle\hat{\bf
A}=\sum_{\nu,\mathbf{q}}\sqrt{\frac{2\pi c^{2}\hbar}{V\omega_{\bf
q}}}\left({\bf e}_{\nu{\bf q}}\hat{b}_{\nu{\bf q}}e^{i{\bf q}{\bf r}}+{\bf
e}^{*}_{\nu{\bf q}}\hat{b}^{\dagger}_{\nu{\bf q}}e^{-i{\bf q}{\bf
r}}\right)~{},$ (12)

where $\hat{b}_{\nu\mathbf{q}}$ and $\hat{b}^{\dagger}_{\nu\mathbf{q}}$ are
the photon annihilation and creation operators acting on the photon Fock
states $|n_{\nu\mathbf{q}}\rangle$, $\omega_{q}=c|{\bf q}|$, $V$ is the
quantization volume with periodic boundary conditions, and ${\bf e}_{\nu{\bf
q}}$ is the polarization vector such that ${\bf q}\cdot{\bf e}_{\nu{\bf
q}}=0.$ The total Hamiltonian is

$\displaystyle\hat{H}=\hat{H}_{e}+\hat{H}_{\rm ph}+\hat{H}_{int}~{},$ (13)

where the interaction Hamiltonian $\hat{H}_{int}=-\frac{1}{c}\hat{\bf
j}\cdot\hat{\bf A}$ and the current operator is $\hat{\bf j}=-e\hat{\bf v}$.
From the general relation ${\bf
v}_{nk;n^{\prime}k^{\prime}}=-i\omega_{nk;n^{\prime}k^{\prime}}{\bf
r}_{nk;n^{\prime}k^{\prime}}$, where
$\hbar\omega_{nk;n^{\prime}k^{\prime}}=E_{nk}-E_{n^{\prime}k^{\prime}}$, we
can express the current matrix elements through the dipole matrix elements
discussed in the previous section:

$\displaystyle{\bf j}_{nk;n^{\prime}k^{\prime}}$ $\displaystyle=$
$\displaystyle{\bf j}_{nk;n^{\prime}k}\delta_{kk^{\prime}},$ (14)
$\displaystyle(j_{x})_{nk;n^{\prime}k}$ $\displaystyle=$ $\displaystyle
e\omega_{c}y_{nk;n^{\prime}k},$ (15) $\displaystyle(j_{y})_{nk;n^{\prime}k}$
$\displaystyle=$ $\displaystyle ie\omega_{nk;n^{\prime}k}y_{nk;n^{\prime}k},$
(16)

which results in the modified polarization selection rules:

$(j_{x})_{nk;n^{\prime}k}+i\frac{\omega_{c}}{\omega_{nk;n^{\prime}k}}(j_{y})_{nk;n^{\prime}k}=0.$
(17)

As follows from Eq. (17) and Fig. 3(b), for electron states near the edge the
transition frequency $\omega_{nk;n^{\prime}k}$ becomes several times larger
than $\omega_{c}$ and therefore the y-component of the current matrix element
becomes significantly larger than the x-component.

Now consider normal incidence of the radiation on the Hall sample, when ${\bf
q}=q{\bf z}_{0}$, and take the field quantization volume as a ray bundle of
volume $V=l_{x}l_{y}l_{z}$. For our purpose it is sufficient to consider only
one spatial mode at frequency $\omega_{q}=cq$ and polarization ${\bf e}_{q}$.
It is straightforward to generalize it to a multimode wave-packet. As a
result, in the rotating-wave approximation (RWA) the interaction Hamiltonian
becomes

$\displaystyle\hat{H}_{int}=-\sum_{k,n>n^{\prime}}\sqrt{\frac{2\pi\hbar}{V\omega_{q}}}\left({\bf
e}_{q}\cdot{\bf
j}_{nk;n^{\prime}k}{\hat{b}}_{q}{\hat{a}}^{\dagger}_{kn}{\hat{a}}_{kn^{\prime}}+H.c.\right)~{}.$

As an initial state of electrons we take

$\Psi_{e}(0)=\prod_{\\{{nk}\\}_{occ}}{\hat{a}}^{\dagger}_{kn}|0_{e}\rangle$

where $|0_{e}\rangle$ is the vacuum state of the electron system,
$\\{{nk}\\}_{occ}$ denotes occupied states. The values of $n$ change from
$n=0$ to $n=n_{F}$ where $n_{F}$ is the index of the highest occupied LL in
the bulk, whereas the values of $k$ change up to $k_{nF}$, for which
$E_{nk}=E_{F}$, where $E_{F}$ is the Fermi energy. An initial single-photon
state of the field is

$\Psi_{\rm ph}(0)={\hat{b}}^{\dagger}_{q}|0_{p}\rangle,$

where $|0_{p}\rangle$ is the vacuum state of the field. We seek the solution
of the Schrödinger equation as

$\displaystyle\Psi(t)$ $\displaystyle=$ $\displaystyle C_{q}(t)\Psi_{\rm
ph}(0)\Psi_{e}(0)$ (18) $\displaystyle+$
$\displaystyle\sum_{\\{{n^{\prime}k}\\}_{\rm
occ},n}C_{nn^{\prime}k}(t)|0_{p}\rangle\hat{a}_{nk}^{\dagger}\hat{a}_{n^{\prime}k}\Psi_{e}(0)~{}.$

In Eq. (18) the summation is performed only over the indices $n,n^{\prime}$
and $k$ which at the initial moment of time correspond to occupied states
$|n^{\prime}k\rangle$ and empty states $|nk\rangle$. At $t=0$ we have
$C_{q}(0)=1$ and $C_{nn^{\prime}k}(0)=0.$ The Schrödinger equation leads to
linear equations for the complex amplitudes $C_{q}(t)$ and
$C_{nn^{\prime}k}(t)$, that are similar to those for a quantum field
interacting with an inhomogeneously broadened ensemble of two-level quantum
emitters Tokman  _et al._ (2023a). Solving them, one can obtain a complete
quantum dynamics of light-matter interaction. In particular, in the
perturbative linear regime we obtain the absorption probability per unit time
and per given transition $n^{\prime}\to n$ as

$\displaystyle{\mathcal{A}}_{n^{\prime}\to n}=\sum_{k}\frac{4\pi|{\bf
j}_{nk;n^{\prime}k}\cdot{\bf
e}^{*}_{q}|^{2}}{\hbar\omega(l_{x}l_{y}l_{z})}\frac{\Gamma\left(\rho^{(0)}_{n^{\prime}k;n^{\prime}k}-\rho^{(0)}_{nk;nk}\right)}{\Gamma^{2}+(\omega_{nk;n^{\prime}k}-\omega)^{2}}~{},$
(19)

where $\Gamma$ is the homogeneous broadening of inter-LL transitions
determined by disorder-induced scattering and
$\rho^{(0)}_{jk;jk}=\left(1+e^{(E_{jk}-E_{F})/(k_{B}T)}\right)^{-1}$ is the
Fermi distribution function at temperature $T$. For small $\Gamma$ as compared
to the inhomogeneous broadening of transitions between edge states, the
Lorentzian in Eq. (19) becomes the delta function
$\pi\delta(\omega_{nk;n^{\prime}k}-\omega)$ and one recovers the Fermi’s
golden rule expression. In the case of a finite $\Gamma$ stochastic noise
terms appear in the equations for complex amplitudes according to the
stochastic Schroedinger equation approach Tokman  _et al._ (2023a), but this
will not affect the linear absorption probability.

The summation over $k$ can be replaced by integration, $\sum_{k}\to
g_{s}\frac{L_{x}}{2\pi}\int dk$, where we also added the spin degeneracy
factor $g_{s}$. Here $L_{x}$ is the sample length (quantization length of
electron states) in x-direction along the edge. The total absorption
probability is obtained by adding the contributions from all LL transitions,
${\mathcal{A}}_{tot}=\frac{l_{z}}{c}\sum_{n,n^{\prime}}{\mathcal{A}}_{n^{\prime}\to
n}$. Here we also converted the probability per unit time into the total
dimensionless probability of absorbing a photon by multiplying the former by
the photon pulse duration $\Delta t\approx l_{z}/c$. Defined this way, the
photon absorption probability will also describe the dimensionless absorbance
of the classical monochromatic wave by a 2D system. Here we obviously assumed
a rectangular temporal profile of the pulse for simplicity. Propagation of
single-photon pulses of an arbitrary shape involves a bit more algebra and can
be found, e.g., in Ref. Tokman  _et al._ (2023b).

Figure 5: Normalized absorbance spectra given by Eq. (20) (solid black line)
of $\hat{y}$-polarized photons for (a) Fermi energy above the fifth LL and (c)
Fermi energy above the tenth LL. Panels (b) and (d) show the difference
between the absorbance of $\hat{y}$ and $\hat{x}$ polarization for the same
Fermi levels ($x$-polarized absorbance subtracted from $y$ polarized one). The
shaded colored regions under the black curves show the contributions of
individual transitions between the neighboring LLs. The insets in (a) and (c)
show the regions of phase space which contribute to the absorption of photons
for each pair of neighboring LLs. The colors of shades in the inset match
those in the main figure for each panel. Other parameters are
$\Gamma=0.02\,\omega_{c}$ and temperature $T=0$.

The resulting expression depends on the geometric parameter
$\frac{L_{x}}{l_{x}l_{y}}$, which can be written as a dimensionless factor

$f=\frac{L_{x}}{l_{x}}\frac{\ell_{c}}{l_{y}}.$

This factor measures the overlap of an incident radiation beam with electron
states in the (x,y) plane and is expected to be much smaller than 1,
especially due to the ratio $\ell_{c}/l_{y}$. Indeed, for a GaAs/AlGaAs
quantum well with the conduction band effective mass of $0.067m_{0}$ Lo  _et
al._ (1991) and in a magnetic field $B=1$ T, we have $\hbar\omega_{c}\approx
1.7$ meV and $\ell_{c}\approx 26$ nm. If we want to focus incident THz
radiation tightly on one edge of the sample in order to bring the overlap
closer to 1, one should illuminate a metal tip or fabricate a metallic
nanoantenna along the sample edge.

In order to show universal, geometry-independent absorbance spectra, we
normalize the absorbance by this geometric overlap factor, namely

${\mathscr{A}}(\omega)=\frac{{\mathcal{A}}_{tot}}{f}=\frac{1}{f}\frac{l_{z}}{c}\sum_{n,n^{\prime}}{\mathcal{A}}_{n^{\prime}\to
n}.$ (20)

The resulting spectra ${\mathscr{A}}(\omega)$ of absorbance (or single-photon
absorption probability) are shown in Fig. 5(a) and (c) for the case of
$y$-polarized radiation. The shaded regions under the black curve show the
contributions of individual transitions between neighboring LLs; the range of
these transitions in momentum space is shown in the insets, with colors
matching the shaded regions in the main figure. The salient feature of the
total absorbance spectra is the existence of the sharp peaks at frequencies
where the contributions from two different transitions overlap. The overlaps
exist in the region of momenta $k>0$ and frequencies
$\omega_{c}<\omega<2\omega_{c}$, as one can also see from the spread of
transition energies in Fig. 3(b). For higher frequencies and for the states
closer to the edge ($k<0$) the overlaps disappear and the peaks in Fig. 5 turn
into dips, as is also clear from Fig. 3(b) where the order of transition
energies between neighboring LLs gets inverted upon crossing $k=0$. These
peaks and dips provide a clear spectroscopic signature of individual edge
states. Moreover, they also provide an opportunity of exciting nonequilibrium
carriers into a given edge state and then observe a corresponding change in
the edge-state interferometry. Due to a large spectral detuning from the bulk
transitions at $\omega=\omega_{c}$ this is possible without destroying the QH
effect conditions in the bulk of the sample.

One detrimental factor that could possibly affect the edge state spectroscopy,
especially with broadband pulses, is the presence of a strong peak at
$\omega=\omega_{c}$ due to the bulk state transition $n_{F}\to n_{F}+1$.
However, this peak can be subtracted out in the measurements of difference
between absorbance in x- and y-polarizations, as shown in Fig. 5 (b) and (d).
Here we make use of the fact that near the edge the x- and y-components of
transition matrix elements become increasingly different, with a much stronger
y-component, as follows from Eq. (17). Thus, difference measurements will get
rid of the bulk peak and may result in easier observable peaks and dips for
edge states.

As one can see from Fig. 5, the magnitude of absorbance peaks normalized by
the geometric factor $f$ is of the order of 1-2. Therefore, the fraction of
light absorbed in individual peaks is essentially the above factor $f$
measuring the overlap of the incident radiation with the sample area. While
there is no doubt that $f\ll 1$, one needs to keep it large enough to ensure
that the absorption is detectable.

One can also estimate the magnitude of absorbance analytically by taking the
limit $\Gamma\to 0$ which converts Eq. (19) into the one given the Fermi’s
Golden Rule, with the delta-function in the integrand describing energy
conservation. Performing integration in $k$, we obtain

$\displaystyle\frac{l_{z}}{c}{\mathcal{A}}_{n^{\prime}\to n}=f\frac{2\pi|{\bf
j}_{nk;n^{\prime}k}\cdot{\bf e}^{*}_{q}|^{2}}{\hbar\omega
c\ell_{c}\partial_{k}\omega_{nk;n^{\prime}k}}\Bigg{|}_{\omega_{nk;n^{\prime}k}=\omega},$
(21)

where we also took the population difference equal to 1. Using the expression
for energy dispersion provided in the Eq. (5) (or Eq. (6)), we find
$\partial_{k}\omega_{nk;n^{\prime}k}=2\omega_{nk;n^{\prime}k}/(3k)$. For a
$y$-polarized field we have $|{\bf j}_{nk;n^{\prime}k}\cdot{\bf
e}^{*}_{q}|^{2}=e^{2}\omega_{nk;n^{\prime}k}^{2}|y_{nk;n^{\prime}k}|^{2}$.
This gives

$\displaystyle\frac{l_{z}}{c}{\mathcal{A}}_{n^{\prime}\to
n}=f\frac{3\pi\alpha}{\ell_{c}}|y_{nk;n^{\prime}k}|^{2}k\big{|}_{\omega_{nk;n^{\prime}k}=\omega}~{},$
(22)

where $\alpha$ is the fine structure constant, $k$ scales as
$\omega_{nk;n^{\prime}k}^{3/2}$, and the momentum dependence of
$|y_{nk;n^{\prime}k}|^{2}$ can be extracted from Fig. 4. For example, if we
take the wavenumber of the initial state at the Fermi level, $k\approx
k_{F}\approx\frac{1}{\ell_{c}}\sqrt{2n_{F}+1}$, the absorbance due to the
transition between these LLs is

$\displaystyle\frac{l_{z}}{c}{\mathcal{A}}_{n^{\prime}\to n}\approx
f3\pi\alpha\sqrt{2n_{F}+1}\frac{|y_{nk_{F};n^{\prime}k_{F}}|^{2}}{\ell_{c}^{2}}.$
(23)

##  IV Generation of DC current by optical rectification

The DC current carried by chiral edge states in QH effect experiments is
proportional to the diagonal elements of the density matrix,

$\displaystyle\mathcal{J}_{0}=-\frac{e}{L_{x}}\sum_{n,k}\left(v_{x}\right)_{nk;nk}\rho^{(0)}_{nk;nk}~{}.$
(24)

The net current from both edges is zero, but it becomes nonzero if a DC
voltage $\Delta V$ is applied. Indeed, using Eq. (24) together with Eq. (2),
one can find out that in equilibrium, the difference between the currents
along opposite edges $\Delta\mathcal{J}_{0}=\frac{e^{2}n_{F}}{2\pi\hbar}\Delta
V$, where $n_{F}$ is the number of filled LLs in the bulk.

However, inversion symmetry breaking for edge states gives rise to a
possibility of generating the DC or quasi-DC current by an optical field
through the second-order process of the optical rectification, which becomes
allowed in the electric dipole approximation and moreover quite efficient. In
order to support a net current only one edge needs to be illuminated;
otherwise the contributions of two opposite edges will still cancel each
other.

This optically driven DC current is due to the nondiagonal elements of the
density matrix,

$\displaystyle\mathcal{J}_{\rm dc}=\frac{e^{2}B}{L_{x}m_{e}c}\sum_{n\neq
m}\sum_{k}y_{nk;mk}\rho_{mk;nk}~{}.$ (25)

The density matrix elements are found by solving the master equation,

$\displaystyle\frac{\partial\rho_{nm}}{\partial
t}+\frac{i}{\hbar}\sum_{\nu}\left(H_{n\nu}\rho_{\nu m}-\rho_{n\nu}H_{\nu
m}\right)=0~{}.$ (26)

to which we will add phenomenological relaxation terms. Here
$H_{\alpha\beta}=\delta_{\alpha\beta}E_{\alpha}+V_{\alpha\beta}(t)$ and
$\alpha,\beta$ represent the quantum state comprising the Landau level index
and the momentum. The light-matter interaction Hamiltonian in electric-dipole
approximation is $\hat{V}(t)=ey({\mathcal{E}}e^{-i\omega
t}+{\mathcal{E}}^{*}e^{i\omega t})$, where we assume the field to be classical
in this section. Since all optical transitions are vertical in the electric-
dipole approximation, we suppressed the momentum index for simplicity, such
that $y_{ik;jk}\to y_{ij}$ and $\rho_{ik;jk}\to\rho_{ij}$. Of course the
momentum index has to be restored when integrating over $k$ in Eq. (25).

We proceed by expanding the elements of the density matrix in perturbative
series with respect to the interaction Hamiltonian,

$\displaystyle\frac{\partial\rho^{(j+1)}_{nm}}{\partial
t}=-i\omega_{nm}{\rho}^{(j+1)}_{nm}-\frac{i}{\hbar}\sum_{\nu}\left(V_{n\nu}\rho^{(j)}_{\nu
m}-\rho^{(j)}_{n\nu}V_{\nu m}\right),$ (27)

with $n$ and $m$ denoting the quantum states corresponding to Landau level
indices $n$ and $m$ respectively. We are only interested in the terms up to
the second order in the optical field. The linear in ${\mathcal{E}}$ term
yields

$\displaystyle{\rho}^{(1)}_{\beta\alpha}=-\frac{e}{\hbar}\frac{y_{\beta\alpha}\left(\rho^{(0)}_{\alpha\alpha}-\rho^{(0)}_{\beta\beta}\right)}{\omega_{\beta\alpha}-\omega-i\gamma}~{}\mathcal{E}e^{-i\omega
t}~{},$ (28)

where
${\rho}^{(1)}_{\alpha\beta}=\left({\rho}^{(1)}_{\beta\alpha}\right)^{*}$.

From Eq. (25) and taking into account the structure of the perturbative
solution, it is clear that there are two types of contributions to the DC
current at zero frequency, i.e., $\propto|{\mathcal{E}}|^{2}$. One type of
terms in Eq. (25) scale as $y_{n;(n+2)}\rho^{(2)}_{(n+2);n}$. They are enabled
by nonzero dipole matrix elements $y_{n;(n+2)}$ of the transitions which
change the LL index by 2, i.e., $n\to n+2$. The second group of terms scale as
$y_{n;(n+1)}\rho^{(2)}_{(n+1);n}$ where
$\rho^{(2)}_{(n+1);n}\propto(y_{(n+1);(n+1)}-y_{n;n})$. As one can see from
Fig. 4, both $y_{n;(n+2)}$ and $(y_{(n+1);(n+1)}-y_{n;n})$ are only nonzero
for electron states close to the edge, when the center of the cyclotron
rotation $y_{k}$ is within several magnetic lengths $\ell_{c}$ from the edge.

The second-order nonlinear density matrix elements contributing to the DC
current satisfy the equations of motion,

$\displaystyle\frac{\partial\rho^{(2)}_{n;(n-1)}}{\partial
t}+i\omega_{n;(n-1)}\rho^{(2)}_{n;(n-1)}=-\frac{ie}{\hbar}{\mathscr{F}}(t)\times$
$\displaystyle\left[y_{n;(n-2)}\rho^{(1)}_{(n-2);(n-1)}-y_{(n-2);(n-1)}\rho^{(1)}_{n;(n-2)}\right.$
$\displaystyle+y_{n;(n+1)}\rho^{(1)}_{(n+1);(n-1)}-y_{(n+1);(n-1)}\rho^{(1)}_{n;(n+1)}$
$\displaystyle\left.+(y_{n;n}-y_{(n-1);(n-1)})\rho^{(1)}_{n;(n-1)}\right],$
(29) $\displaystyle\frac{\partial\rho^{(2)}_{(n+1);(n-1)}}{\partial
t}+i\omega_{(n+1);(n-1)}\rho^{(2)}_{(n+1);(n-1)}=$
$\displaystyle-\frac{ie}{\hbar}{\mathscr{F}}(t)\left[y_{(n+1);n}\rho^{(1)}_{n;(n-1)}-y_{n;(n-1)}\rho^{(1)}_{(n+1);n}\right.$
$\displaystyle\left.+(y_{(n+1);(n+1)}-y_{(n-1);(n-1)})\rho^{(1)}_{(n+1);(n-1)}\right],$
(30)

where we have defined ${\mathscr{F}}(t)={\mathcal{E}}e^{-i\omega
t}+{\mathcal{E}}^{*}e^{i\omega t}$. Time-independent solution to these
equations under illumination with a monochromatic field gives rise to a large
number of terms contributing to the DC current, which grows rapidly with
increasing doping. For convenience, in Appendix C below we list the density
matrix elements that contribute to the rectification current for the Fermi
level $E_{F}=3\hbar\omega_{c}$ between $n=2$ and $n=3$ bulk LLs, chosen to
make plots in Fig. 6.

Figure 6: Nonlinear DC current resulting from second-order optical
rectification process, as a function of the normalized optical field
frequency. The calculations included the first five Landau levels for chemical
potential $E_{F}=3\hbar\omega_{c}$. Red curve: the electric dipole
contribution to the nonlinear current from Eq. (32),
$\frac{{\mathcal{J}}^{(I)}_{\rm dc}+{\mathcal{J}}^{(II)}_{\rm
dc}}{\zeta\mathcal{J}_{0}}$, normalized by the “diagonal” DC current
$\mathcal{J}_{0}$ from Eq. (24) and factor $\zeta$ from Eq. (31). Blue dashed
curve: beyond electric-dipole contribution to the nonlinear current,
$\frac{{\mathcal{J}}^{(III)}_{\rm dc}+{\mathcal{J}}^{(IV)}_{\rm
dc}}{\zeta_{\rm grad}\mathcal{J}_{0}}$, normalized by $\mathcal{J}_{0}$ and
factor $\zeta_{\rm grad}$ from Eq. (35).

In Fig. 6 the red curve shows the rectification current normalized by the
conventional “diagonal” DC current (24) and by the factor

$\zeta=\frac{e^{2}|{\mathcal{E}}|^{2}\ell_{c}^{2}}{\hbar^{2}\omega_{c}^{2}}$
(31)

proportional to the optical field intensity, as a function of the optical
field frequency. The current is a sum of the above two contributions,
${\mathcal{J}}^{(I)}_{\rm dc}$ and ${\mathcal{J}}^{(II)}_{\rm dc}$ given by

$\displaystyle{\mathcal{J}}^{(I/II)}_{\rm
dc}=\frac{e^{2}B}{m_{e}c}\int\frac{dk}{2\pi}~{}J^{(I/II)}~{},$ (32)

where

$\displaystyle J^{(I)}$ $\displaystyle=$ $\displaystyle
y_{01}{\rho}^{(2)}_{10}+y_{12}{\rho}^{(2)}_{21}+y_{23}{\rho}^{(2)}_{32}+c.c.~{},$
(33) $\displaystyle J^{(II)}$ $\displaystyle=$ $\displaystyle
y_{02}{\rho}^{(2)}_{20}+y_{13}{\rho}^{(2)}_{31}+y_{24}{\rho}^{(2)}_{42}+c.c~{}.$
(34)

The expressions for the second-order density matrix elements are given in
Appendix C.

The main peak at the cyclotron frequency in Fig. 6 is due to the $n=2\to n=3$
transition near the edge. With increasing optical frequency, the term
$(E_{3k}-E_{2k})/\hbar-\omega$ flips sign and consequently we observe a
negative peak in the plot. Additional kinks are due to activation/deactivation
of various transitions. They are going to be smeared by finite temperature
effects and/or increased scattering.

Since the normalized nonlinear current in Fig. 6 is of the order of 1, the
dimensional current magnitude is fraction $\zeta$ of the conventional edge
current (24). The factor $\zeta$ is essentially the Rabi frequency squared of
incident light normalized to the cyclotron frequency squared. When $\zeta$
becomes of the order of 1, population transfer effects between the LLs in the
bulk become important even despite the detuning from the bulk inter-LL
resonance. This could create bulk conductance across the sample and destroy
the QH effect. Therefore, to maintain the QH effect conditions in the bulk, it
is desirable to keep $\zeta\ll 1$. The optimal situation to observe or utilize
the nonlinear rectification current would be to illuminate one edge of the
sample under zero DC bias conditions, when the conventional current is zero.

###  IV.1 Beyond electric-dipole approximation

Yet another second-order nonlinear optical contribution to the rectified DC
current originates from the spatial nonuniformity of the electric field of the
incident radiation, i.e., beyond the electric-dipole approximation. In the
lowest order, the light-mater interaction Hamiltonian becomes

$\hat{V}(t)=e\left[\left(y{\mathcal{E}}+\frac{y^{2}}{2}\partial_{y}{\mathcal{E}}\right)e^{-i\omega
t}+\left(y{\mathcal{E}}+\frac{y^{2}}{2}\partial_{y}{\mathcal{E}}\right)^{*}e^{i\omega
t}\right].$

This gives rise to the following additional matrix elements contributing to
the rectification current,

$V_{(n+2);n}=\frac{e}{2}(y^{2})_{(n+2);n}\left(\partial_{y}{\mathcal{E}}e^{-i\omega
t}+\partial_{y}{\mathcal{E}}^{*}e^{i\omega t}\right),$
$V_{(n+1);(n-1)}=\frac{e}{2}(y^{2})_{(n+1);(n-1)}\left(\partial_{y}{\mathcal{E}}e^{-i\omega
t}+\partial_{y}{\mathcal{E}}^{*}e^{i\omega t}\right).$

Solving the density matrix equations in the second order of the field as in
the previous subsection, we obtain additional contributions to the density
matrix elements. For example the factor
$y_{12}y_{20}{\mathcal{E}}^{*}{\mathcal{E}}$ in the first term in
${\rho}^{(2)}_{10}$ in Appendix C is modified as
$y_{12}y_{20}{\mathcal{E}}^{*}{\mathcal{E}}+y_{12}y^{2}_{20}{\mathcal{E}}^{*}\partial_{y}{\mathcal{E}}/2+y^{2}_{12}y_{20}{\mathcal{E}}\partial_{y}{\mathcal{E}}^{*}/2$,
and similarly for other terms.

We denote the resulting contributions as $J^{III}$ and $J^{IV}$ in the spirit
of Eq. (33) and Eq. (34), respectively, and calculate the resulting
rectification current ${\mathcal{J}}^{(III)}_{\rm
dc}+{\mathcal{J}}^{(IV)}_{\rm dc}$ similarly to Eq. (32). It is shown as a
blue curve in Fig. 6. The current is normalized by the conventional “diagonal”
DC current (24) and by the factor

$\zeta_{\rm
grad}=\frac{e^{2}\ell_{c}^{3}\partial_{y}|{\mathcal{E}}|^{2}}{\hbar^{2}\omega_{c}^{2}}$
(35)

proportional to the gradient of the optical field intensity. With this
normalization, the optical rectification current is close in magnitude to the
one obtained in electric dipole approximation and shown as a red curve. Note,
however, that the normalization factor in the denominator contains the optical
field intensity gradient $\ell_{c}\partial_{y}|{\mathcal{E}}|^{2}$ which is
much smaller than $|{\mathcal{E}}|^{2}$, unless the optical field is focused
to the size of the order of $\ell_{c}$, i.e., tens of nm, by using a nanotip
or nanoantenna at the sample edge as sketched in the inset to Fig. 6. Without
nanofocusing, the electric-dipole current is greater in magnitude by the
factor
$\sim\frac{|{\mathcal{E}}|^{2}}{\ell_{c}\partial_{y}|{\mathcal{E}}|^{2}}$.

##  V Conclusions

In conclusion, we demonstrated the feasibility of the optical spectroscopy and
selective coherent optical control of chiral edge state populations and
current under the conditions of the integer QH effect. The physical mechanism
enabling the selective control is inversion symmetry breaking and violation of
adiabaticity for electron states near the edge. As a result, optical
transitions between the edge states have significantly different transition
energies and polarization selection rules as compared to the bulk of the
sample. This enables selective excitation of a given edge channel with single
quasiparticle sensitivity without disturbing the rest of the sample. The 2D
absorbance shows characteristic peaks at different edge state resonances at
frequencies significantly higher than the bulk cyclotron transition. A large
fraction of incident light in the illuminated area of the sample can be
absorbed. The overall absorption probability of single photons depends on
their overlap with the edge state area and will benefit from nanofocusing with
a tip or nanoantenna. Furthermore, inversion symmetry breaking near the sample
boundary enables strong second-order optical nonlinearity already in electric
dipole approximation, resulting in efficient optical rectification of incident
radiation and direct optical driving of a quasi-DC current in edge states. The
predicted optical effects can be used to to study or control edge currents by
optical means and to control the interference pattern in QH interference
experiments.

While the calculations in this paper were performed for nonrelativistic
electron dispersion in semiconductor quantum wells, one should expect
qualitatively similar results in graphene samples. Another natural extension
of this work is to investigate the optical control of edge current under
fractional QH effect conditions. We hope that this study will attract
attention of both QH and optical communities and lead to interesting
collaborative experiments.

##  VI Acknowledgements

This work has been supported in part by the Air Force Office for Scientific
Research Grant No. FA9550-21-1-0272 and National Science Foundation Award No.
1936276.

##  Appendix A Electron states in adiabatic approximation

For the reader’s convenience, here we summarize the results for electron
states in the adiabatic approximation, when the electric potential $\Phi(y)$
varies much slower than the magnetic length. For a magnetic field,
${\vec{B}}=B\hat{z}$ we choose the vector potential of the form
${\vec{A}}=-yB\hat{x}$, which yields the single-electron Hamiltonian

$\displaystyle
H=\frac{1}{2m}\left(p_{x}-\frac{eBy}{c}\right)^{2}+\frac{p_{y}^{2}}{2m}-e\Phi~{},$
(36)

where $m$ is the electron mass. Since there is no explicit $x$ dependence, we
can seek the wavefunction as

$\displaystyle\psi(x,y)=e^{ikx}\chi(y)~{},$ (37)

for which the Schroedinger equation yields

$\displaystyle\frac{\partial^{2}\chi}{\partial
y^{2}}+\frac{2m}{\hbar^{2}}\left[E+e\Phi(y)-\frac{1}{2}m\omega_{c}^{2}\left(y-y_{k}\right)^{2}\right]\chi(y)=0~{}.$
(38)

where we have defined the cyclotron frequency, $\omega_{c}=eB/(mc)$, the
position of the center of the cyclotron rotation $y_{k}=k\ell_{c}^{2}$, and
the magnetic length $\ell_{c}=\sqrt{c\hbar/(eB)}$.

In the adiabatic approximation, the potential $\Phi(y)$ is varying slowly near
the edge,
$\Phi(y)\approx\Phi(y_{k})+\left(y-y_{k}\right)\left(\partial_{y}\Phi(y)|_{y=y_{k}}\right)$.
The particle energies are

$\displaystyle E_{nk}$ $\displaystyle=$
$\displaystyle\left(n+\frac{1}{2}\right)\hbar\omega_{c}-e\Phi\left(y_{k}+\frac{c\partial_{y}\Phi(y)|_{y=y_{k}}}{\omega_{c}B}\right)$
(39) $\displaystyle+$
$\displaystyle\frac{m}{2}\left(\frac{c\partial_{y}\Phi(y)|_{y=y_{k}}}{B}\right)^{2}.$

The term $\frac{c\partial_{y}\Phi(y)|_{y=y_{k}}}{\omega_{c}B}$ in the right-
hand side of Eq. (39) is the shift of the center of the cyclotron rotation
with respect to the initial point $y_{k}$ under the action of the electric
field $E_{y}\approx-\partial_{y}\Phi(y)|_{y=y_{k}}$. The last term on the
right-hand side of Eq. (39) is the kinetic energy of the drift motion in
crossed $E$ and $B$ fields.

The eigenfunctions become

$\displaystyle\chi_{nk}(y)$ $\displaystyle=$
$\displaystyle\frac{e^{-\left(y-y_{k}-c\partial_{y}\Phi(y)|_{y=y_{k}}/(\omega_{c}B)\right)^{2}/(\sqrt{2}\ell_{c})^{2}}}{\sqrt{2^{n}n!\sqrt{\pi}\ell_{c}}}$
(40) $\displaystyle\times$
$\displaystyle{\mathcal{H}}_{n}\left(\frac{y}{\ell_{c}}-\frac{y_{k}}{\ell_{c}}-\frac{c}{\ell_{c}}\frac{\partial_{y}\Phi(y)|_{y=y_{k}}}{\omega_{c}B}\right),$

where $n$ is the LL index and ${\mathcal{H}}_{n}$ is a Hermite polynomial. In
this case, while the energies of the LLs increase towards the edge, the energy
separation between them does not change as compared to the bulk states, and
the dipole matrix elements of all transitions between them do not change
either. Furthermore, the drift velocity of electrons along the edge does not
depend on the LL index:

$\displaystyle\langle\hat{v}_{x}\rangle_{k}=-\frac{c}{B}\partial_{y}\Phi(y)|_{y=y_{k}}~{}.$
(41)

##  Appendix B Electron density flux for an arbitrary potential

It is interesting that the electron density flux and drift velocity can be
found for an arbitrary potential $\Phi(y)$, provided Eq. (1) allows a finite
solution corresponding to a certain discrete spectrum of $E_{nk}$ and a set of
eigenfunctions of the form $\psi_{nk}(x,y)=e^{ikx}\chi_{nk}(y)$, where
$\chi_{nk}(y)|_{y\to\pm\infty}\to 0$, $\int
dxdy~{}\psi_{nk}^{*}(x,y)\psi_{mk^{\prime}}(x,y)=\delta_{nm}\delta_{kk^{\prime}}$.
To prove this, we introduce the operator of two-dimensional electron density
for the $n$th LL,
${\hat{N}}_{n}(x,y)=\hat{\Psi}^{\dagger}(x,y)\hat{\Psi}(x,y)$ where the field
operators are given in the second-quantized form:
$\hat{\Psi}(x,y)=\sum_{k}\psi_{nk}(x,y){\hat{a}}_{nk},\hat{\Psi}^{\dagger}=\sum_{k}\psi^{*}_{nk}(x,y){\hat{a}}^{\dagger}_{nk}$
with the annihilation and creation operators as $\hat{a}_{nk}$ and
$\hat{a}^{\dagger}_{nk}$ respectively. The operator of two-dimensional spatial
electron density is then

$\displaystyle{\hat{N}}_{n}(x,y)=\sum_{k,k^{\prime}}\chi^{*}_{nk^{\prime}}(y)\chi_{nk}(y)e^{i(k-k^{\prime})x}\hat{\rho}_{nk;nk^{\prime}}~{},$
(42)

where
$\hat{\rho}_{nk;nk^{\prime}}={\hat{a}}^{\dagger}_{nk}{\hat{a}}_{nk^{\prime}}$.
The same result can be obtained by using the transition operator
$\hat{\rho}_{nk;nk^{\prime}}=|nk\rangle\langle nk^{\prime}|$. The operator
$\hat{\rho}_{nk;nk^{\prime}}$ obeys the Heisenberg equation,

$\displaystyle\frac{\partial\hat{\rho}_{nk;nk^{\prime}}}{\partial
t}=-\frac{i}{\hbar}\left(E_{nk}-E_{nk^{\prime}}\right)\hat{\rho}_{nk;nk^{\prime}}~{}.$
(43)

If the fermions are interacting with classical electromagnetic fields, the
Heisenberg operator $\hat{\rho}_{nk;nk^{\prime}}$, after averaging over the
initial state of the system, becomes the matrix element of a standard density
matrix $\rho_{nk;nk^{\prime}}$. The general expression for the probability
flux of the electron density can be expressed through the Wigner function,
defined as

$\displaystyle{\hat{\mathcal{W}}}_{n}(x,y,K)=$
$\displaystyle\sum_{\kappa}\psi^{*}_{n(K-\kappa/2)}(x,y)\psi_{n(K+\kappa/2)}(x,y)\hat{\rho}_{n(K+\kappa/2);n(K-\kappa/2)},$

where $k-k^{\prime}=\kappa$, $(k+k^{\prime})/2=K$. The Wigner function
${\hat{\mathcal{W}}}_{n}(x,y,K)$ determines both the operator of spatial
density,

$\displaystyle\sum_{K}{\hat{\mathcal{W}}}_{n}(x,y,K)=\hat{N}_{n}(x,y),$ (44)

and the distribution over momenta,

$\displaystyle\int
dxdy~{}{\hat{\mathcal{W}}}_{n}(x,y,K)|_{K=k}=\hat{\rho}_{nk;nk}.$ (45)

It follows from Eq. (43) that

$\displaystyle\frac{\partial\hat{\rho}_{n(K+\kappa/2);n(K-\kappa/2)}}{\partial
t}=$
$\displaystyle-\frac{i}{\hbar}\left(E_{n(K+\kappa/2)}-E_{n(K-\kappa/2)}\right)\hat{\rho}_{n(K+\kappa/2);n(K-\kappa/2)}~{}.$
(46)

Suppose that we have a well-localized state in $k$ so that $\delta k\ll k$ (a
narrow spectrum in $k$ means narrow localization along $y$ near the edge). In
this case we have
$E_{n(K+\kappa/2)}-E_{n(K-\kappa/2)}\approx\kappa\partial_{k}E_{nk}|_{k=K}$.
Then from Eq. (B) and taking into account Eqs. (44) and (45) one can obtain

$\displaystyle\frac{\partial{\hat{\mathcal{W}}}_{n}(x,y,K)}{\partial
t}+\frac{1}{\hbar}\frac{\partial E_{nk}}{\partial
k}\bigg{|}_{k=K}\frac{\partial{\hat{\mathcal{W}}}_{n}(x,y,K)}{\partial
x}=0~{}.$ (47)

For a quadratic spectrum of the type $E_{nk}=E_{n0}+\alpha_{n}k^{2}$ Eq. (47)
is exact and does not require a narrow spectrum.

Using Eqs. (47) and (44), one can obtain

$\displaystyle\frac{\partial{\hat{N}}_{n}(x,y)}{\partial
t}+\frac{1}{\hbar}\frac{\partial E_{nk}}{\partial
k}\frac{\partial{\hat{N}}_{n}(x,y)}{\partial x}=0$ (48)

where $k$ is the central value of the given narrow spectrum. From Eq.(48) one
obtains the expression for the observed particle velocity in the state
$|nk\rangle$ i.e., the diagonal matrix element of the velocity operator, which
has a standard form:

$\displaystyle\langle\hat{v}_{x}\rangle_{nk}=\frac{1}{\hbar}\frac{\partial
E_{nk}}{\partial k}~{}.$ (49)

Note that this expression coincides with the one obtained from the solution
(40) for a uniform electric field but it does not depend on the specific form
of the eigenstates $\psi_{nk}(x,y)$; furthermore, Eq. (49) does not depend on
any assumptions about the nonuniformity scale of the potential $\Phi(y)$ in
comparison with the magnetic length $\ell_{c}$.

##  Appendix C Second-order nonlinear density matrix elements

For reader’s convenience, we provide below the the list of second-order
nonlinear density matrix elements that contribute to the optical rectification
current in the electric-dipole approximation for our chosen value of the Fermi
level, $E_{F}=3\hbar\omega_{c}$:

$\displaystyle\frac{{\rho}^{(2)}_{10}}{e^{2}|\mathcal{E}|^{2}/\hbar^{2}}$
$\displaystyle=$
$\displaystyle\frac{y_{12}y_{20}\Delta\rho^{(0)}_{02}}{\omega_{10}\left(\Delta_{20}-i\gamma\right)}-\frac{y_{20}y_{12}\Delta\rho^{(0)}_{12}}{\omega_{10}\left(\Delta_{21}+i\gamma\right)}-\frac{\left(y_{00}-y_{11}\right)y_{10}\Delta\rho^{(0)}_{01}}{\omega_{10}\left(\Delta_{10}-i\gamma\right)},$
$\displaystyle\frac{{\rho}^{(2)}_{21}}{e^{2}|\mathcal{E}|^{2}/\hbar^{2}}$
$\displaystyle=$
$\displaystyle\frac{y_{20}y_{01}\Delta\rho^{(0)}_{01}}{\omega_{21}\left(\Delta_{10}+i\gamma\right)}-\frac{y_{01}y_{20}\Delta\rho^{(0)}_{02}}{\omega_{21}\left(\Delta_{20}-i\gamma\right)}+\frac{y_{23}y_{31}\Delta\rho^{(0)}_{13}}{\omega_{21}\left(\Delta_{31}-i\gamma\right)}-\frac{y_{31}y_{23}\Delta\rho^{(0)}_{23}}{\omega_{21}\left(\Delta_{32}+i\gamma\right)}-\frac{\left(y_{11}-y_{22}\right)y_{21}\Delta\rho^{(0)}_{12}}{\omega_{21}\left(\Delta_{21}-i\gamma\right)},$
$\displaystyle\frac{{\rho}^{(2)}_{32}}{e^{2}|\mathcal{E}|^{2}/\hbar^{2}}$
$\displaystyle=$
$\displaystyle\frac{y_{31}y_{12}\Delta\rho^{(0)}_{12}}{\omega_{32}\left(\Delta_{21}+i\gamma\right)}-\frac{y_{12}y_{31}\Delta\rho^{(0)}_{13}}{\omega_{32}\left(\Delta_{31}-i\gamma\right)}+\frac{y_{34}y_{42}\Delta\rho^{(0)}_{24}}{\omega_{32}\left(\Delta_{42}-i\gamma\right)}-\frac{y_{42}y_{34}\Delta\rho^{(0)}_{34}}{\omega_{32}\left(\Delta_{43}+i\gamma\right)}-\frac{\left(y_{22}-y_{33}\right)y_{32}\Delta\rho^{(0)}_{23}}{\omega_{32}\left(\Delta_{32}-i\gamma\right)},$
$\displaystyle\frac{{\rho}^{(2)}_{43}}{e^{2}|\mathcal{E}|^{2}/\hbar^{2}}$
$\displaystyle=$
$\displaystyle\frac{y_{42}y_{23}\Delta\rho^{(0)}_{23}}{\omega_{43}\left(\Delta_{32}+i\gamma\right)}-\frac{y_{23}y_{42}\Delta\rho^{(0)}_{24}}{\omega_{43}\left(\Delta_{42}-i\gamma\right)},$
$\displaystyle\frac{{\rho}^{(2)}_{20}}{e^{2}|\mathcal{E}|^{2}/\hbar^{2}}$
$\displaystyle=$
$\displaystyle\frac{y_{21}y_{10}\Delta\rho^{(0)}_{01}}{\omega_{20}\left(\Delta_{10}-i\gamma\right)}-\frac{y_{10}y_{21}\Delta\rho^{(0)}_{12}}{\omega_{20}\left(\Delta_{21}-i\gamma\right)}-\frac{\left(y_{00}-y_{22}\right)y_{20}\Delta\rho^{(0)}_{02}}{\omega_{20}\left(\Delta_{20}-i\gamma\right)},$
$\displaystyle\frac{{\rho}^{(2)}_{31}}{e^{2}|\mathcal{E}|^{2}/\hbar^{2}}$
$\displaystyle=$
$\displaystyle\frac{y_{32}y_{21}\Delta\rho^{(0)}_{12}}{\omega_{31}\left(\Delta_{21}-i\gamma\right)}-\frac{y_{21}y_{32}\Delta\rho^{(0)}_{23}}{\omega_{31}\left(\Delta_{32}-i\gamma\right)}-\frac{\left(y_{11}-y_{33}\right)y_{31}\Delta\rho^{(0)}_{13}}{\omega_{31}\left(\Delta_{31}-i\gamma\right)},$
$\displaystyle\frac{{\rho}^{(2)}_{42}}{e^{2}|\mathcal{E}|^{2}/\hbar^{2}}$
$\displaystyle=$
$\displaystyle\frac{y_{43}y_{32}\Delta\rho^{(0)}_{23}}{\omega_{42}\left(\Delta_{32}-i\gamma\right)}-\frac{\left(y_{22}-y_{44}\right)y_{42}\Delta\rho^{(0)}_{24}}{\omega_{42}\left(\Delta_{42}-i\gamma\right)}.$

Here
$\Delta\rho^{(0)}_{\alpha\beta}=\left(\rho^{(0)}_{\alpha\alpha}-\rho^{(0)}_{\beta\beta}\right)$
is the equilibrium population difference and
$\Delta_{\alpha\beta}=\omega_{\alpha\beta}-\omega$.

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# Projective dimension and regularity of edge ideals of some vertex-weighted
oriented unicyclic graphs

Guangjun Zhu∗ and Hong Wang  School of Mathematical Sciences, Soochow
University, Suzhou, Jiangsu, 215006, P. R. China
zhuguangjun@suda.edu.cn(Corresponding author:Guangjun Zhu),
651634806@qq.com(Hong wang).

###### Abstract.

In this paper we provide some exact formulas for the projective dimension and
the regularity of edge ideals of vertex-weighted oriented unicyclic graphs.
These formulas are in function of the weight of the vertices, the numbers of
edges. We also give some examples to show that these formulas are related to
direction selection and the assumptions that $w(x)\geq 2$ for any vertex $x$
cannot be dropped.

2020 Mathematics Subject Classification. Primary: 13F20; Secondary 05C20,
05C22, 05E40.

projective dimension, regularity, edge ideal, vertex-weighted oriented
unicyclic graph, vertex-weighted rooted forest

##  1\. Introduction

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring in $n$ variables over a
field $k$ and let $I\subset S$ be a homogeneous ideal. There are two central
invariants associated to $I$, the regularity
$\mbox{reg}\,(I):=\mbox{max}\\{j-i\ |\ \beta_{i,j}(I)\neq 0\\}$ and the
projective dimension $\mbox{pd}\,(I):=\mbox{max}\\{i\ |\ \beta_{i,j}(I)\neq 0\
\text{for some}\ j\\}$, that in a sense, they measure the complexity of
computing the graded Betti numbers $\beta_{i,j}(I)$ of $I$. In particular, if
$I$ is a monomial ideal, its polarization $I^{\mathcal{P}}$ has the same
projective dimension and regularity as $I$ and is squarefree. Thus one can
associate $I^{\mathcal{P}}$ to a graph or a hypergraph or a simplicial
complex. Many authors have studied the regularity and Betti numbers of edge
ideals of graphs, e.g. [AF2, BHTN, BHO, HT3, J, KM, MV, Z1, Z2, Z3].

A directed graph or digraph $D$ consists of a finite set $V(D)$ of vertices,
together with a collection $E(D)$ of ordered pairs of distinct points called
edges or arrows. A vertex-weighted directed graph is a triplet
$D=(V(D),E(D),w)$, where $w$ is a weight function
$w:V(D)\rightarrow\mathbb{N}^{+}$, where $N^{+}=\\{1,2,\ldots\\}$. Some times
for short we denote the vertex set $V(D)$ and the edge set $E(D)$ by $V$ and
$E$ respectively. The weight of $x_{i}\in V$ is $w(x_{i})$, denoted by $w_{i}$
or $w_{x_{i}}$.

The edge ideal of a vertex-weighted digraph was first introduced by Gimenez et
al [GBSVV]. Let $D=(V,E,w)$ be a vertex-weighted digraph with the vertex set
$V=\\{x_{1},\ldots,x_{n}\\}$. We consider the polynomial ring
$S=k[x_{1},\dots,x_{n}]$ in $n$ variables over a field $k$. The edge ideal of
$D$, denoted by $I(D)$, is the ideal of $S$ given by

$I(D)=(x_{i}x_{j}^{w_{j}}\mid x_{i}x_{j}\in E).$

Edge ideals of weighted digraphs arose in the theory of Reed-Muller codes as
initial ideals of vanishing ideals of projective spaces over finite fields
[MPV, PS]. If a vertex $x_{i}$ of $D$ is a source (i.e., has only arrows
leaving $x_{i}$) we shall always assume $w_{i}=1$ because in this case the
definition of $I(D)$ does not depend on the weight of $x_{i}$. If $w_{j}=1$
for all $j$, then $I(D)$ is the edge ideal of underlying graph $G$ of $D$. It
has been extensively studied in the literature [HT3, MV, SVV]. Especially the
study of algebraic invariants corresponding to their minimal free resolutions
has become popular(see [AF2, BHTN, BHO, J, KM, Z1, Z2, Z3]). In [Z3], the
first three authors derive some exact formulas for the projective dimension
and regularity of the edge ideals associated to some vertex-weighted digraphs
such as rooted forests, oriented cycles. To the best of our knowledge, little
is known about the projective dimension and the regularity of $I(D)$ for some
vertex-weighted digraphs.

In this article, we are interested in algebraic properties corresponding to
the projective dimension and the regularity of the edge ideals of vertex-
weighted oriented unicyclic graphs. By using the approaches of Betti splitting
and polarization, we derive some exact formulas for the projective dimension
and the regularity of these edge ideals. The results are as follows:

######  Theorem 1.1.

Let $T_{j}=(V(T_{j}),E(T_{j}),w_{j})$ be a vertex-weighted rooted forest such
that $w(x)\geq 2$ if $d(x)\neq 1$ for $1\leq j\leq s$ and $C_{n}$ a vertex-
weighted oriented cycle with vertex set $\\{x_{1},x_{2},\ldots,x_{n}\\}$ and
$w_{x_{i}}\geq 2$ for $1\leq i\leq n$. Let
$D=C_{n}\cup(\bigcup\limits_{j=1}^{s}T_{j})$ be a vertex-weighted oriented
graph obtained by attaching the root of $T_{j}$ to vertex $x_{i_{j}}$ of the
cycle $C_{n}$ for $1\leq j\leq s$. Then

  1. 1

$\mbox{reg}\,(I(D))=\sum\limits_{x\in V(D)}w(x)-|E(D)|+1$,

  2. 2

$\mbox{pd}\,(I(D))=|E(D)|-1$.

######  Theorem 1.2.

Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented unicyclic graph such that
$w(x)\geq 2$ if $d(x)\neq 1$. Then

  1. 1

$\mbox{reg}\,(I(D))=\sum\limits_{x\in V(D)}w(x)-|E(D)|+1$,

  2. 2

$\mbox{pd}\,(I(D))=|E(D)|-1$.

Our paper is organized as follows. In section $2$, we recall some definitions
and basic facts used in the following sections. In section $3$, we provide
some exact formulas for the projective dimension and the regularity of the
edge ideals of vertex-weighted oriented unicyclic graphs. Meanwhile, we give
some examples to show the projective dimension and the regularity of these
edge ideals are related to direction selection and the assumption that
$w(x)\geq 2$ for any vertex $x$ cannot be dropped.

For all unexplained terminology and additional information, we refer to [JG]
(for the theory of digraphs), [BM] (for graph theory), and [BH, HH] (for the
theory of edge ideals of graphs and monomial ideals). We greatfully
acknowledge the use of the computer algebra system CoCoA ([C]) for our
experiments.

##  2\. Preliminaries

In this section, we gather together the needed definitions and basic facts,
which will be used throughout this paper. However, for more details, we refer
the reader to [AF2, BM, FHT, HT1, HH, J, JG, MPV, PRT, Z3].

A directed graph or digraph $D$ consists of a finite set $V(D)$ of vertices,
together with a collection $E(D)$ of ordered pairs of distinct points called
edges or arrows. If $\\{u,v\\}\in E(D)$ is an edge, we write $uv$ for
$\\{u,v\\}$, which is denoted to be the directed edge where the direction is
from $u$ to $v$ and $u$ (resp. $v$) is called the starting point (resp. the
ending point). An oriented graph is a directed graph having no bidirected
edges (i.e. each pair of vertices is joined by a single edge having a unique
direction). In other words an oriented graph $D$ is a simple graph $G$
together with an orientation of its edges. We call $G$ the underlying graph of
$D$.

Every concept that is valid for graphs automatically applies to digraphs too.
For example, the degree of a vertex $x$ in a digraph $D$, denoted by $d(x)$,
is simply the degree of $x$ in $G$. Likewise, a digraph is said to be
connected if its underlying graph is connected. A digraph $H$ is called an
induced subgraph of digraph $D$ if $V(H)\subseteq V(D)$, and for any $x,y\in
V(H)$, $xy$ is an edge in $H$ if and only if $xy$ is an edge in $D$. For
$P\subset V(D)$, we denote $D\setminus P$ the induced subgraph of $D$ obtained
by removing the vertices in $P$ and the edges incident to these vertices. If
$P=\\{x\\}$ consists of a single element, then we write $D\setminus x$ for
$D\setminus\\{x\\}$. For $W\subseteq E(D)$, we define $D\setminus W$ to be the
subgraph of $D$ with all edges in $W$ deleted (but its vertices remained).
When $W=\\{e\\}$ consists of a single edge, we write $D\setminus e$ instead of
$D\setminus\\{e\\}$. An oriented path or oriented cycle is an orientation of a
path or cycle in which each vertex dominates its successor in the sequence. An
oriented acyclic graph is a simple digraph without oriented cycles. An
oriented tree or polytree is an oriented acyclic graph formed by orienting the
edges of undirected acyclic graphs. A rooted tree is an oriented tree in which
all edges are oriented either away from or towards the root. Unless
specifically stated, a rooted tree in this article is an oriented tree in
which all edges are oriented away from the root. An oriented forest is a
disjoint union of oriented trees. A rooted forest is a disjoint union of
rooted trees.

A vertex-weighted oriented graph is a triplet $D=(V(D),E(D),w)$, where $V(D)$
is the vertex set, $E(D)$ is the edge set and $w$ is a weight function
$w:V(D)\rightarrow\mathbb{N}^{+}$, where $N^{+}=\\{1,2,\ldots\\}$. Some times
for short we denote the vertex set $V(D)$ and edge set $E(D)$ by $V$ and $E$
respectively. The weight of $x_{i}\in V$ is $w(x_{i})$, denoted by $w_{i}$ or
$w_{x_{i}}$. Given a vertex-weighted oriented graph $D=(V,E,w)$ with the
vertex set $V=\\{x_{1},\ldots,x_{n}\\}$, we consider the polynomial ring
$S=k[x_{1},\dots,x_{n}]$ in $n$ variables over a field $k$. The edge ideal of
$D$, denoted by $I(D)$, is the ideal of $S$ given by

$I(D)=(x_{i}x_{j}^{w_{j}}\mid x_{i}x_{j}\in E).$

If a vertex $x_{i}$ of $D$ is a source (i.e., has only arrows leaving $x_{i}$)
we shall always assume $w_{i}=1$ because in this case the definition of $I(D)$
does not depend on the weight of $x_{i}$.

For any homogeneous ideal $I$ of the polynomial ring $S=k[x_{1},\dots,x_{n}]$,
there exists a graded minimal finite free resolution

$0\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p,j}(I)}\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p-1,j}(I)}\rightarrow\cdots\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{0,j}(I)}\rightarrow
I\rightarrow 0,$

where the maps are exact, $p\leq n$, and $S(-j)$ is the $S$-module obtained by
shifting the degrees of $S$ by $j$. The number $\beta_{i,j}(I)$, the
$(i,j)$-th graded Betti number of $I$, is an invariant of $I$ that equals the
number of minimal generators of degree $j$ in the $i$th syzygy module of $I$.
Of particular interests are the following invariants which measure the “size”
of the minimal graded free resolution of $I$. The projective dimension of $I$,
denoted pd $(I)$, is defined to be

$\mbox{pd}\,(I):=\mbox{max}\,\\{i\ |\ \beta_{i,j}(I)\neq 0\\}.$

The regularity of $I$, denoted $\mbox{reg}\,(I)$, is defined by

$\mbox{reg}\,(I):=\mbox{max}\,\\{j-i\ |\ \beta_{i,j}(I)\neq 0\\}.$

We now derive some formulas for $\mbox{pd}\,(I)$ and $\mbox{reg}\,(I)$ in some
special cases by using some tools developed in [FHT].

######  Definition 2.1.

Let $I$ be a monomial ideal, and suppose that there exist monomial ideals $J$
and $K$ such that $\mathcal{G}(I)$ is the disjoint union of $\mathcal{G}(J)$
and $\mathcal{G}(K)$, where $\mathcal{G}(I)$ denotes the unique minimal set of
monomial generators of $I$. Then $I=J+K$ is a Betti splitting if

$\beta_{i,j}(I)=\beta_{i,j}(J)+\beta_{i,j}(K)+\beta_{i-1,j}(J\cap K)\hskip
5.69054pt\mbox{for all}\hskip 5.69054pti,j\geq 0,$

where $\beta_{i-1,j}(J\cap K)=0\hskip 5.69054pt\mbox{if}\hskip 5.69054pti=0$.

In [FHT], the authors describe some sufficient conditions for an ideal $I$ to
have a Betti splitting. We need the following lemma.

######  Lemma 2.2.

([FHT, Corollary 2.7]) Suppose that $I=J+K$ where $\mathcal{G}(J)$ contains
all the generators of $I$ divisible by some variable $x_{i}$ and
$\mathcal{G}(K)$ is a nonempty set containing the remaining generators of $I$.
If $J$ has a linear resolution, then $I=J+K$ is a Betti splitting.

When $I$ is a Betti splitting ideal, Definition 2.1 implies the following
results:

######  Corollary 2.3.

If $I=J+K$ is a Betti splitting ideal, then

  1. 1

$\mbox{reg}\,(I)=\mbox{max}\\{\mbox{reg}\,(J),\mbox{reg}\,(K),\mbox{reg}\,(J\cap
K)-1\\}$,

  2. 2

$\mbox{pd}\,(I)=\mbox{max}\\{\mbox{pd}\,(J),\mbox{pd}\,(K),\mbox{pd}\,(J\cap
K)+1\\}$.

The following lemmas is often used in this article.

######  Lemma 2.4.

([HTT, Lemma 1.3])  Let $R$ be a polynomial ring over a field and let $I$ be a
proper non-zero homogeneous ideal in $R$. Then

  1. 1

$\mbox{pd}\,(I)=\mbox{pd}\,(S/I)-1$,

  2. 2

$\mbox{reg}\,(I)=\mbox{reg}\,(S/I)+1$.

######  Lemma 2.5.

([HT1, Lemma 2.2 and Lemma 3.2 ]) Let $S_{1}=k[x_{1},\dots,x_{m}]$,
$S_{2}=k[x_{m+1},\dots,x_{n}]$ and $S=k[x_{1},\dots,x_{n}]$ be three
polynomial rings, $I\subseteq S_{1}$ and $J\subseteq S_{2}$ be two proper non-
zero homogeneous ideals. Then

  1. 1

$\mbox{pd}\,(S/(I+J))=\mbox{pd}\,(S_{1}/I)+\mbox{pd}\,(S_{2}/J)$,

  2. 2

$\mbox{reg}\,(S/(I+J))=\mbox{reg}\,(S_{1}/I)+\mbox{reg}\,(S_{2}/J)$.

From Lemma 2.4 and Lemma 2.5, we have

######  Lemma 2.6.

([Z1, Lemma 3.1]) Let $S_{1}\\!\\!=\\!k[x_{1},\dots,x_{m}]$ and
$S_{2}\\!=\\!k[x_{m+1},\dots,x_{n}]$ be two polynomial rings, $I\subseteq
S_{1}$ and $J\subseteq S_{2}$ be two non-zero homogeneous ideals. Then

  1. 1

$\mbox{pd}\,(I+J)=\mbox{pd}\,(I)+\mbox{pd}\,(J)+1$,

  2. 2

$\mbox{reg}\,(I+J)=\mbox{reg}\,(I)+\mbox{reg}\,(J)-1$.

Let $\mathcal{G}(I)$ denote the minimal set of generators of a monomial ideal
$I\subset S$ and let $u\in S$ be a monomial, we set
$\mbox{supp}(u)=\\{x_{i}:x_{i}|u\\}$. If
$\mathcal{G}(I)=\\{u_{1},\ldots,u_{m}\\}$, we set
$\mbox{supp}(I)=\bigcup\limits_{i=1}^{m}\mbox{supp}(u_{i})$. The following
lemma is well known.

######  Lemma 2.7.

Let $I,J\\!=(u)$ be two monomial ideals such that
$\mbox{supp}\,(u)\cap\mbox{supp}\,(I)\\!\\!=\emptyset$. If the degree of
monomial $u$ is $d$. Then

  1. 1

$\mbox{reg}\,(J)=d$,

  2. 2

$\mbox{reg}\,(JI)=\mbox{reg}\,(I)+d$,

  3. 3

$\mbox{pd}\,(JI)=\mbox{pd}\,(I)$.

######  Definition 2.8.

Suppose that $u=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ is a monomial in $S$. We
define the polarization of $u$ to be the squarefree monomial

$\mathcal{P}(u)=x_{11}x_{12}\cdots x_{1a_{1}}x_{21}\cdots x_{2a_{2}}\cdots
x_{n1}\cdots x_{na_{n}}$

in the polynomial ring $S^{\mathcal{P}}=k[x_{ij}\mid 1\leq i\leq n,1\leq j\leq
a_{i}]$. If $I\subset S$ is a monomial ideal with
$\mathcal{G}(I)=\\{u_{1},\ldots,u_{m}\\}$, the polarization of $I$, denoted by
$I^{\mathcal{P}}$, is defined as:

$I^{\mathcal{P}}=(\mathcal{P}(u_{1}),\ldots,\mathcal{P}(u_{m})),$

which is a squarefree monomial ideal in the polynomial ring $S^{\mathcal{P}}$.

Here is an example of how polarization works.

######  Example 2.9.

Let $I(D)=(x_{1}x_{2}^{3},x_{2}x_{3}^{2},x_{3}x_{4}^{4},x_{4}x_{1}^{5})$ be
the edge ideal of a vertex-weighted oriented cycle $D$, then the polarization
of $I(D)$ is the ideal
$I(D)^{\mathcal{P}}=(x_{11}x_{21}x_{22}x_{23},x_{21}x_{31}x_{32},\\\
x_{31}x_{41}x_{42}x_{43}x_{44},x_{41}x_{11}x_{12}x_{13}x_{14}x_{15})$.

A monomial ideal $I$ and its polarization $I^{\mathcal{P}}$ share many
homological and algebraic properties. The following is a very useful property
of polarization.

######  Lemma 2.10.

([HH, Corollary 1.6.3]) Let $I\subset S$ be a monomial ideal and
$I^{\mathcal{P}}\subset S^{\mathcal{P}}$ its polarization. Then

  1. 1

$\beta_{ij}(I)=\beta_{ij}(I^{\mathcal{P}})$ for all $i$ and $j$,

  2. 2

$\mbox{reg}\,(I)=\mbox{reg}\,(I^{\mathcal{P}})$,

  3. 3

$\mbox{pd}\,(I)=\mbox{pd}\,(I^{\mathcal{P}})$.

The following lemma can be used for computing the projective dimension and the
regularity of an ideal.

######  Lemma 2.11.

([HTT, Lemma 1.1 and Lemma 1.2]) Let $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ be a short exact sequence of finitely generated graded
$S$-modules. Then

  1. 1

$\mbox{reg}\,(B)\leq\mbox{max}\,\\{\mbox{reg}\,(A),\mbox{reg}\,(C)\\}$,

  2. 2

$\mbox{pd}\,(B)\leq\mbox{max}\,\\{\mbox{pd}\,(A),\mbox{pd}\,(C)\\}$.

##  3\. Projective dimension and regularity of edge ideals of vertex-weighted
oriented unicyclic graphs

In this section, we will provide some exact formulas for the projective
dimension and the regularity of the edge ideals of vertex-weighted unicyclic
graphs. Meanwhile, we give some examples to show the projective dimension and
the regularity of the edge ideals of vertex-weighted oriented unicyclic graphs
are related to direction selection and the assumption that $w(x)\geq 2$ if
$d(x)\neq 1$ cannot be dropped. We shall start from the definition of the
union of some digraphs.

######  Definition 3.1.

Let $D_{i}=(V(D_{i}),E(D_{i}))$ be a digraph with the underlying graph $G_{i}$
for $1\leq i\leq k$. If $e$ is an edge of $G_{j}$ for $i_{1}\leq j\leq i_{s}$
with $1\leq i_{1}\leq\cdots\leq i_{s}\leq k$, then the direction of edge $e$
is the same in $D_{j}$ for $i_{1}\leq j\leq i_{s}$. The union of digraphs
$D_{1},D_{2},\dots,D_{k}$, written as $\bigcup\limits_{i=1}^{k}D_{i}$, is the
digraph with vertex set $\bigcup\limits_{i=1}^{k}V(D_{i})$ and edge set
$\bigcup\limits_{i=1}^{k}E(D_{i})$.

The following two lemmas is needed to facilitate calculating the projective
dimension and the regularity of the edge ideal of some graphs.

######  Lemma 3.2.

([Z3, Theorem 3.5, Theorem 3.3 ]) Let $D=(V(D),E(D),w)$ be a vertex-weighted
rooted forest such that $w(x)\geq 2$ if $d(x)\neq 1$. Then

  1. 1

$\mbox{reg}\,(I(D))=\sum\limits_{x\in V(D)}w(x)-|E(D)|+1$,

  2. 2

$\mbox{pd}\,(I(D))=|E(D)|-1$.

######  Lemma 3.3.

([Z3, Theorem 4.1]) Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented cycle
such that $w(x)\geq 2$ for any $x\in V(D)$. Then

  1. 1

$\mbox{reg}\,(I(D))=\sum\limits_{x\in V(D)}w(x)-|E(D)|+1$,

  2. 2

$\mbox{pd}\,(I(D))=|E(D)|-1$.

Let $D=(V,E)$ be a digraph and $x\in V$, then we call $N_{D}^{+}(x)=\\{y:xy\in
E\\}$ and $N_{D}^{-}(x)=\\{y:yx\in E\\}$ to be the out-neighbourhood and in-
neighbourhood of $x$, respectively. The neighbourhood of $x$ is the set
$N_{D}(x)=N_{D}^{+}(x)\cup N_{D}^{-}(x)$.

Now we are ready to present the main result of this section.

######  Theorem 3.4.

Let $T_{j}=(V(T_{j}),E(T_{j}),w_{j})$ be a vertex-weighted rooted forest such
that $w(x)\geq 2$ if $d(x)\neq 1$ for $1\leq j\leq s$ and $C_{n}$ a vertex-
weighted oriented cycle with vertex set $\\{x_{1},x_{2},\ldots,x_{n}\\}$ and
$w_{x_{i}}\geq 2$ for $1\leq i\leq n$. Let
$D=C_{n}\cup(\bigcup\limits_{j=1}^{s}T_{j})$ be a vertex-weighted oriented
graph obtained by attaching the root of $T_{j}$ to vertex $x_{i_{j}}$ of the
cycle $C_{n}$ for $1\leq j\leq s$. Then

  1. 1

$\mbox{reg}\,(I(D))=\sum\limits_{x\in V(D)}w(x)-|E(D)|+1$,

  2. 2

$\mbox{pd}\,(I(D))=|E(D)|-1$.

###### Proof.

Let $N_{T_{j}}^{+}(x_{i_{j}})\\!=\\!\\{y_{j1},\ldots,y_{j,t_{j}}\\}$ for
$1\leq j\leq s$. Assume that $i_{1}\\!<\\!i_{2}\\!<\\!\cdots\\!<\\!i_{s}$. We
consider two cases:

Case (1): If $T_{j}$ is a rooted star graph for any $1\leq j\leq s$, then we
have

$\displaystyle I(D)=$
$\displaystyle(x_{1}x_{2}^{w_{2}},\ldots,x_{n-1}x_{n}^{w_{n}},x_{n}x_{1}^{w_{1}},x_{i_{1}}y_{11}^{w_{y_{11}}},\ldots,x_{i_{1}}y_{1,t_{1}}^{w_{y_{1,t_{1}}}},x_{i_{2}}y_{21}^{w_{y_{21,}}},\ldots,x_{i_{2}}y_{2,t_{2}}^{w_{y_{2,t_{2}}}},\ldots,$
$\displaystyle
x_{i_{s}}y_{s1}^{w_{y_{s1}}},\ldots,x_{i_{s}}y_{s,t_{s}}^{w_{y_{s,t_{s}}}}).$

Case (2): If there exists some $1\leq j\leq s$ such that $T_{j}$ is a rooted
tree but not a star digraph. We may assume that $T_{j}$ is a rooted tree but
not a star digraph for any $1\leq j\leq s$. Because other cases follow the
same line of arguments. In this case, we have

$\displaystyle I(D)=$
$\displaystyle(x_{1}x_{2}^{w_{2}}\\!,\ldots,x_{n-1}x_{n}^{w_{n}}\\!,x_{n}x_{1}^{w_{1}}\\!,x_{i_{1}}y_{11}^{w_{y_{11}}}\\!\\!\\!,\ldots,x_{i_{1}}y_{1,t_{1}}^{w_{y_{1,t_{1}}}}\\!\\!\\!\\!\\!,x_{i_{2}}y_{21}^{w_{y_{21}}}\\!\\!\\!,\ldots,x_{i_{2}}y_{2,t_{2}}^{w_{y_{2,t_{2}}}}\\!\\!\\!\\!,$
$\displaystyle\ldots,x_{i_{s}}y_{s1}^{w_{y_{s1}}}\\!\\!\\!,\ldots,x_{i_{s}}y_{s,t_{s}}^{w_{y_{s,t_{s}}}})+I(T_{1}\setminus
x_{i_{1}})+I(T_{2}\setminus x_{i_{2}})+\cdots+I(T_{s}\setminus x_{i_{s}}),$

where $I(T_{j}\setminus x_{i_{j}})$ is the edge ideal of the induced subgraph
of $T_{j}$ obtained by removing vertex $x_{i_{j}}$ and the edges incident to
$x_{i_{j}}$ for $j=1,\ldots,s$.

Case (1) can be shown by similar arguments as Case (2), so we only prove the
statement holds in Case (2).

Let $I(D)^{\mathcal{P}}$ be the polarization of $I(D)$, then

$\displaystyle I(D)^{\mathcal{P}}=$
$\displaystyle(x_{11}\\!\prod\limits_{j=1}^{w_{2}}x_{2j},\ldots,x_{n-1,1}\prod\limits_{j=1}^{w_{n}}x_{nj},x_{n1}\prod\limits_{j=1}^{w_{1}}x_{1j},x_{i_{1}1}\prod\limits_{j=1}^{w_{y_{11}}}y_{11,\,j},\ldots,x_{i_{1}1}\\!\\!\\!\prod\limits_{j=1}^{w_{y_{1,t_{1}}}}\\!\\!\\!y_{1,t_{1},\,j},x_{i_{2}1}\\!\\!\prod\limits_{j=1}^{w_{y_{21}}}\\!\\!y_{21,\,j},$
$\displaystyle\ldots,x_{i_{2}1}\\!\\!\prod\limits_{j=1}^{w_{y_{2,t_{2}}}}\\!\\!y_{2,t_{2},\,j},\ldots,x_{i_{s}1}\prod\limits_{j=1}^{w_{y_{s1}}}y_{s1j},\ldots,x_{i_{s}1}\\!\prod\limits_{j=1}^{w_{y_{s,t_{s}}}}y_{s,t_{s},\,j})+I(T_{1}\setminus
x_{i_{1}})^{\mathcal{P}}+I(T_{2}\setminus x_{i_{2}})^{\mathcal{P}}$
$\displaystyle+$ $\displaystyle\cdots+I(T_{s}\setminus
x_{i_{s}})^{\mathcal{P}}.$

We set $i_{1}=1$, $K=(x_{n1}\\!\prod\limits_{j=1}^{w_{1}}x_{1j})$ and

$\displaystyle J=$
$\displaystyle(x_{11}\\!\\!\prod\limits_{j=1}^{w_{2}}x_{2j},\ldots,x_{n-1,1}\\!\\!\prod\limits_{j=1}^{w_{n}}x_{nj},\widehat{x_{n1}\\!\\!\prod\limits_{j=1}^{w_{1}}x_{1j}},x_{11}\\!\\!\prod\limits_{j=1}^{w_{y_{11}}}y_{11,\,j},\ldots,\linebreak
x_{11}\\!\\!\prod\limits_{j=1}^{w_{y_{1,t_{1}}}}y_{1,t_{1},\,j},x_{i_{2}1}\prod\limits_{j=1}^{w_{y_{21}}}y_{21,\,j},\ldots,$
$\displaystyle
x_{i_{2}1}\prod\limits_{j=1}^{w_{y_{2,t_{2}}}}\\!\\!y_{2,t_{2},\,j},\ldots,x_{i_{s}1}\prod\limits_{j=1}^{w_{y_{s1}}}y_{s1,\,j},\ldots,x_{i_{s}1}\prod\limits_{j=1}^{w_{y_{s,t_{s}}}}y_{s,t_{s},\,j})+I(T_{1}\setminus
x_{i_{1}})^{\mathcal{P}}+I(T_{2}\setminus x_{i_{2}})^{\mathcal{P}}$
$\displaystyle+$ $\displaystyle\cdots+I(T_{s}\setminus
x_{i_{s}})^{\mathcal{P}}$

where $\widehat{x_{n1}\\!\prod\limits_{j=1}^{w_{1}}x_{1j}}$ denotes the
element $x_{n1}\\!\prod\limits_{j=1}^{w_{1}}x_{1j}$ being omitted from the
ideal $J$.

Note that $J$ is actually the polarization of the edge ideal $I(D\setminus e)$
of the subgraph $D\setminus e$ where $e=x_{n}x_{1}$, and $D\setminus e$ is a
rooted tree, whose root is $x_{1}$. By Lemmas 2.10 and 3.2, we obtain

$\displaystyle\mbox{reg}\,(J)$
$\displaystyle=\mbox{reg}\,(J^{\mathcal{P}})=\sum\limits_{x\in V(D\setminus
e)}w(x)-|E(D\setminus e)|+1$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-(w_{1}-1)-(|E(D)|-1)+1$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+1+2-w_{1}$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+3-w_{1}$

where third equality holds because we have weighted one in vertex $x_{1}$ in
the expression $\sum\limits_{x\in V(D\setminus e)}w(x)$, and

$\mbox{pd}\,(J)=\mbox{pd}\,(J^{\mathcal{P}})=|E(D\setminus e)|-1=|E(D)|-2.$

Now, we will compute $\mbox{reg}\,(J\cap K)$ and $\mbox{pd}\,(J\cap K)$. We
distinguish into the following two cases:

(1) If $s=1$, or $s\geq 2$ and $i_{s}\neq n$, then $d(x_{n})=2$. In this case,
we set $J\cap K=KL$. We write $L$ as follows:

$L=L_{1}+L_{2}$

where
$L_{1}=(\prod\limits_{j=1}^{w_{2}}x_{2j},x_{n-1,1}\\!\prod\limits_{j=2}^{w_{n}}x_{nj},\prod\limits_{j=1}^{w_{y_{11}}}y_{11,j},\ldots,\prod\limits_{j=1}^{w_{y_{1k}}}y_{1k,j})+I(D\setminus\\{x_{1},x_{n}\\})^{\mathcal{P}}$
is the polarization of the edge ideal of a rooted forest $H$, here $H$ is the
union of the induced subgraph $D\setminus\\{x_{1},x_{n}\\}$ of $D$ and a
vertex-weighted oriented graph $H^{\prime}$ with the vertex set
$V(H^{\prime})=\\{x_{21},x_{22},x_{n-1,1},x_{n2},y_{11,1},y_{11,2},\ldots,y_{1,k,1},y_{1,k,2}\\}$,
the edge set $E(H^{\prime})=\\{x_{21}x_{22},x_{n-1,1}x_{n2},\\\
y_{11,1}y_{11,2},\ldots,y_{1,k,1}y_{1,k,2}\\}$ and a weight function
$w^{\prime}\\!:\\!V(H^{\prime})\rightarrow\mathbb{N}^{+}$ such that
$w^{\prime}(x_{21})=1$, $w^{\prime}(x_{22})=w_{2}-1$,
$w^{\prime}(x_{n-1,1})=1$, $w^{\prime}(x_{n2})=w_{n}-1$,
$w^{\prime}(y_{1,j,1})=1$, $w^{\prime}(y_{1,j,2})=w_{y_{1,j}}-1\geq 1$ for any
$1\leq j\leq k$, and
$L_{2}=(\prod\limits_{j=1}^{w_{y_{1,k+1}}}\\!\\!y_{1,k+1,\,j},\ldots,\\!\\!\\!\prod\limits_{j=1}^{w_{y_{1,t_{1}}}}\\!\\!y_{1,t_{1},\,j})$
for any $k+1\leq\ell\leq t_{1}$. Thus
$|E(H)|=|E(D)|-(t_{1}+3)+(k+2)=|E(D)|-(t_{1}-k+1)$. We consider the following
two cases:

$(i)$ If $k<t_{1}$, notice that the variables that appear in $K$ and $L$, and
in $L_{1}$ and $L_{2}$ are different, respectively. Then using Lemmas 2.6, 2.7
and 3.2, we obtain

$\displaystyle\mbox{reg}\,(J\cap K)$
$\displaystyle=(1+w_{1})+\mbox{reg}\,(L)=(1+w_{1})+\mbox{reg}\,(L_{1})+\mbox{reg}\,(L_{2})-1$
$\displaystyle=(1+w_{1})+(\sum\limits_{x\in V(H)}w(x)-|E(H)|+1)$
$\displaystyle+((\sum\limits_{j=k+1}^{t_{1}}w_{y_{1j}}-(t_{1}-k))+1)-1$
$\displaystyle=(1+w_{1}+\sum\limits_{x\in
V(H)}w(x)+\sum\limits_{j=k+1}^{t_{1}}w_{y_{1j}})$
$\displaystyle-(|E(D)|-(t_{1}-k+1))-(t_{1}-k)+1$
$\displaystyle=\sum\limits_{x\in V(D)}w(x)-|E(D)|+2$

where the last equality holds because of $x_{n2}\in V(H^{\prime})$ with
$w^{\prime}(x_{n2})=w_{n}-1$ and $x_{1}\notin V(H)$, and

$\displaystyle\mbox{pd}\,(J\cap K)$
$\displaystyle=\mbox{pd}\,(L)=\mbox{pd}\,(L_{1}+L_{2})=\mbox{pd}\,(L_{1})+\mbox{pd}\,(L_{2})+1$
$\displaystyle=|E(H)|-1+(t_{1}-k-1)+1$
$\displaystyle=(|E(D)|-(t_{1}-k+1))-1+(t_{1}-k-1)+1$ $\displaystyle=|E(D)|-2.$

$(ii)$ If $k=t_{1}$, then $L_{2}=(0)$ and
$|E(H)|=|E(D)|-(t_{1}+3)+(t_{1}+2)=|E(D)|-1$. Again by Lemmas 2.6, 2.7 and
3.2, we obtain

$\displaystyle\mbox{reg}\,(J\cap K)$
$\displaystyle=\mbox{reg}\,(KL)=(1+w_{1})+\mbox{reg}\,(L)$
$\displaystyle=(1+w_{1})+\sum\limits_{x\in V(H)}w(x)-|E(H)|+1$
$\displaystyle=(1+w_{1}+\sum\limits_{x\in V(H)}w(x))-(|E(D)|-1)+1$
$\displaystyle=\sum\limits_{x\in V(D)}w(x)-|E(D)|+2$

where the last equality holds because of $x_{n2}\in V(H^{\prime})$ with
$w^{\prime}(x_{n2})=w_{n}-1$ and $x_{1}\notin V(H)$, and

$\mbox{pd}\,(J\cap K)=\mbox{pd}\,(KL)=\mbox{pd}\,(L)=|E(H)|-1=|E(D)|-2.$

In brief, no matter case $(i)$ or $(ii)$, we always have

$\displaystyle\mbox{reg}\,(J\cap K)$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+2;$ $\displaystyle\mbox{pd}\,(J\cap K)$
$\displaystyle=|E(D)|-2.$

Since $K$ has a linear resolution, it follows that $I(D)^{\mathcal{P}}=J+K$ is
a Betti splitting. Thus by Lemma 2.10 and Corollary 2.3, we get

$\displaystyle\mbox{reg}\,(I(D))$
$\displaystyle=\mbox{reg}\,(I(D)^{\mathcal{P}})=\mbox{max}\\{\mbox{reg}\,(J),\mbox{reg}\,(K),\mbox{reg}\,(J\cap
K)-1\\}$ $\displaystyle=\mbox{max}\\{\\!\\!\\!\\!\sum\limits_{x\in
V(D)}\\!\\!\\!\\!\\!w(x)\\!-|E(D)|+3\\!-w_{1},w_{1}+1,\\!\\!\\!\\!\\!\sum\limits_{x\in
V(D)}\\!\\!\\!\\!w(x)\\!-|E(D)|+1\\}$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+1,$

and

$\displaystyle\mbox{pd}\,(I(D))$
$\displaystyle=\mbox{pd}\,(I(D)^{\mathcal{P}})=\mbox{max}\\{\mbox{pd}\,(J),\mbox{pd}\,(K),\mbox{pd}\,(J\cap
K)+1\\}$ $\displaystyle=\mbox{max}\\{|E(D)|-2,0,|E(D)|-2+1\\}$
$\displaystyle=|E(D)|-1.$

(2) If $i_{s}=n$, then $d(x_{n})>2$. In this case, we still set $J\cap K=KL$
and write $L$ as follows:

$L=L_{1}+L_{2}$

where
$L_{1}=(\prod\limits_{j=1}^{w_{2}}x_{2j},x_{n-1,1}\\!\prod\limits_{j=2}^{w_{n}}x_{nj},\prod\limits_{j=1}^{w_{y_{11}}}y_{11,\,j},\ldots,\prod\limits_{j=1}^{w_{y_{1k}}}y_{1k,\,j},\prod\limits_{j=1}^{w_{y_{s1}}}y_{s1,\,j},\ldots,\prod\limits_{j=1}^{w_{y_{s,\ell}}}y_{s,\ell,\,j})+I(D\setminus\\{x_{1},x_{n}\\})^{\mathcal{P}}$
is the polarization of the edge ideal of the graph $H$, here $H$ is the union
of the induced subgraph $D\setminus\\{x_{1},x_{n}\\}$ of $D$ and the vertex-
weighted oriented graph $H^{\prime}$ with the vertex set
$V(H^{\prime})=\\{x_{21},x_{22},x_{n-1},x_{n2},y_{11,1},y_{11,2},\ldots,y_{1k,1},y_{1k,2},y_{s1,1},y_{s1,2},\ldots,y_{s,\ell,1},y_{s,\ell,2}\\}$,
the edge set
$E(H^{\prime})=\\{x_{21}x_{22},x_{n-1}x_{n2},y_{11,1}y_{11,2},\ldots,y_{1k,1}y_{1k,2},y_{s1,1}y_{s1,2},\ldots,y_{s,\ell,1}y_{s,\ell,2}\\}$
and a weight function $w^{\prime}:V(H^{\prime})\rightarrow\mathbb{N}^{+}$ such
that
$w^{\prime}(x_{21})=w^{\prime}(x_{n-1,1})=w^{\prime}(y_{11,1})=\cdots=w^{\prime}(y_{1k,1})=w^{\prime}(y_{s1,1})=\cdots=w^{\prime}(y_{s,\ell,1})=1$,
$w^{\prime}(x_{22})=w_{2}-1,w^{\prime}(x_{n2})=w_{n}-1,w^{\prime}(y_{1j,\,2})=w_{y_{1j}}-1\geq
1$ for any $1\leq j\leq k\leq t_{1}$ and
$w^{\prime}(y_{s,p,2})=w_{y_{s,p}}-1\geq 1$ for any $1\leq p\leq\ell\leq
t_{s}$, and
$L_{2}=(\prod\limits_{j=1}^{w_{y_{1,k+1}}}\\!\\!\\!y_{1,k+1,\,j},\ldots,\\!\prod\limits_{j=1}^{w_{y_{1,t_{1}}}}\\!\\!\\!y_{1,t_{1},\,j},\prod\limits_{j=1}^{w_{y_{s,\ell+1}}}y_{s,\ell+1,\,j},\ldots,\\!\prod\limits_{j=1}^{w_{y_{s,t_{s}}}}y_{s,t_{s},\,j})$.
In this case, $H$ is a forest and
$|E(H)|=|E(D)|-(t_{s}-\ell)-(t_{1}-k)-1=|E(D)|-(t_{s}-\ell+t_{1}-k+1)$. We
consider the following two cases:

$(i)$ If $k<t_{1}$ or $\ell<t_{s}$, notice again that the variables that
appear in $K$ and $L$, and in $L_{1}$ and $L_{2}$ are different, respectively.
Then using Lemmas 2.6, 2.7 and 3.2, we obtain

$\displaystyle\mbox{reg}\,(J\cap K)$
$\displaystyle=(1+w_{1})+\mbox{reg}\,(L)=(1+w_{1})+\mbox{reg}\,(L_{1})+\mbox{reg}\,(L_{2})-1$
$\displaystyle=(1+\\!w_{1})+(\sum\limits_{x\in V(H)}w(x)-|E(H)|+1)$
$\displaystyle+((\sum\limits_{j=k+1}^{t_{1}}w_{y_{1j}}-(t_{1}-k))+1)-1$
$\displaystyle=(1+w_{1}+\sum\limits_{x\in
V(H)}w(x)+\sum\limits_{j=k+1}^{t_{1}}w_{y_{1j}})-(|E(D)|$
$\displaystyle-(t_{1}-k+1))-(t_{1}-k)+1$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+2$

where the last equality holds because of $x_{n2}\in V(H^{\prime})$ with
$w^{\prime}(x_{n2})=w_{n}-1$ and $x_{1}\notin V(H)$, and

$\displaystyle\mbox{pd}\,(J\cap K)$
$\displaystyle=\mbox{pd}\,(L)=\mbox{pd}\,(L_{1}+L_{2})=\mbox{pd}\,(L_{1})+\mbox{pd}\,(L_{2})+1$
$\displaystyle=|E(H)|\\!-\\!1+\\![\sum\limits_{j=k+1}^{t_{1}}\\!\mbox{pd}\,((y_{1j}^{w_{y_{1j}}}))\\!+\\!\sum\limits_{j=\ell+1}^{t_{s}}\\!\mbox{pd}\,((y_{sj}^{w_{y_{sj}}}))\\!+\\!1]+\\!1$
$\displaystyle=[|E(D)|-(t_{s}-\ell+t_{1}-k+1)]-1+(t_{s}-\ell+t_{1}-k-1)+1$
$\displaystyle=|E(D)|-2.$

$(ii)$ If $k=t_{1}$ and $\ell=t_{s}$, then $L_{2}=(0)$ and $|E(H)|=|E(D)|-1$.
From (ii) of case (1), we also have

$\displaystyle\mbox{reg}\,(J\cap K)$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+2,$ $\displaystyle\mbox{pd}\,(J\cap K)$
$\displaystyle=|E(D)|-2.$

Similar arguments as case (1), we obtain

$\displaystyle\mbox{reg}\,(I(D))$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+1,$ $\displaystyle\mbox{pd}\,(I(D))$
$\displaystyle=|E(D)|-1.$

The proof is completed. ∎

######  Theorem 3.5.

Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented unicyclic graph such that
$w(x)\geq 2$ if $d(x)\neq 1$. Then

  1. 1

$\mbox{reg}\,(I(D))=\sum\limits_{x\in V(D)}w(x)-|E(D)|+1$,

  2. 2

$\mbox{pd}\,(I(D))=|E(D)|-1$.

###### Proof.

Let $D_{1},\ldots,D_{m}$ be all connected components of $D$. Thus by Lemma
2.6, we get

$\displaystyle\mbox{reg}\,(I(D))$
$\displaystyle=\mbox{reg}\,(\sum\limits_{i=1}^{m}I(D_{i}))=\sum\limits_{i=1}^{m}\mbox{reg}\,(I(D_{i}))-(m-1),$
$\displaystyle\mbox{pd}\,(I(D))$
$\displaystyle=\mbox{pd}\,(\sum\limits_{i=1}^{m}I(D_{i}))=\sum\limits_{i=1}^{m}\mbox{pd}\,(I(D_{i}))+(m-1).$

By Lemma 3.2, it is enough to show

$\displaystyle\mbox{reg}\,(I(D))$ $\displaystyle=\sum\limits_{x\in
V(D)}w(x)-|E(D)|+1,$ $\displaystyle\mbox{pd}\,(I(D))$ $\displaystyle=|E(D)|-1$

where $D$ is a connected vertex-weighted oriented unicyclic graph. The
conclusion follows from Theorem 3.4. ∎

An immediate consequence of the above theorem is the following corollary.

######  Corollary 3.6.

Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented unicyclic graph with $mm$
connected components. Let $w(x)\geq 2$ for any $d(x)\neq 1$. Then

$\mbox{depth}\,(I(D))=m.$

###### Proof.

It follows from Auslander-Buchsbaum formula. ∎

The following two examples show that the assumption that $w(x)\geq 2$ if
$d(x)\neq 1$ in Theorem 3.4 and Theorem 3.5 cannot be dropped.

######  Example 3.7.

Let
$I(D)=(x_{1}x_{2}^{2},x_{2}x_{3}^{2},x_{3}x_{4}^{2},x_{4}x_{5}^{2},x_{5}x_{1}^{2},x_{1}x_{6},x_{6}x_{7},x_{7}x_{8}^{2})$
be the edge ideal of a vertex-weighted oriented unicyclic graph
$D=(V(D),E(D),w)$ with $w_{1}=w_{2}=w_{3}=w_{4}=w_{5}=w_{8}=2$ and
$w_{6}=w_{7}=1$. By using CoCoA, we get $\mbox{reg}\,(I(D))=8$ and
$\mbox{pd}\,(I(D))=6$. But we have
$\mbox{reg}\,(I(D))=\sum\limits_{i=1}^{8}w_{i}-|E(D)|+1=7$ and
$\mbox{pd}\,(I(D))=|E(D)|-1=7$ by Theorem 3.4.

The following two examples show that the projective dimension and the
regularity of the edge ideals of vertex-weighted oriented unicyclic graphs are
related to direction selection in Theorem 3.4.

######  Example 3.8.

Let
$I(D)=(x_{1}x_{2}^{3},x_{2}x_{3}^{2},x_{4}x_{3}^{2},x_{4}x_{1}^{4},x_{1}x_{5}^{2})$
be the edge ideal of a vertex-weighted oriented unicyclic graph
$D=(V(D),E(D),w)$ with $w_{1}=4$, $w_{2}=3$, $w_{3}=w_{5}=2$ and $w_{4}=1$. By
using CoCoA, we get $\mbox{reg}\,(I(D))=9$ and $\mbox{pd}\,(I(D))=3$. But we
have $\mbox{reg}\,(I(D))=\sum\limits_{i=1}^{5}w_{i}-|E(D)|+1=12-5+1=8$ and
$\mbox{pd}\,(I(D))=|E(D)|-1=5-1=4$ by Theorem 3.4.

######  Example 3.9.

Let
$I(D)=(x_{1}x_{2}^{2},x_{2}x_{3}^{2},x_{3}x_{4}^{4},x_{1}x_{4}^{4},x_{2}x_{5}^{2},x_{5}x_{6}^{2},x_{3}x_{7}^{2},x_{7}x_{8}^{2})$
be the edge ideal of a vertex-weighted oriented unicyclic graph
$D=(V(D),E(D),w)$ with $w_{1}=1$, $w_{2}=w_{3}=w_{5}=w_{6}=w_{7}=w_{8}=2$ and
$w_{4}=4$. By using CoCoA, we get $\mbox{reg}\,(I(D))=11$ and
$\mbox{pd}\,(I(D))=6$. But we have
$\mbox{reg}\,(I(D))=\sum\limits_{i=1}^{8}w_{i}-|E(D)|+1=17-8+1=10$ and
$\mbox{pd}\,(I(D))=|E(D)|-1=7$ by Theorem 3.4.

######  Acknowledgment.

This research is supported by the Natural Science Foundation of Jiangsu
Province (No. BK20221353). We gratefully acknowledge the use of the computer
algebra system CoCoA ([C]) for our experiments.

## References