# Nonlinear Hall effect in Rashba systems with hexagonal warping

Soumadeep Saha  Undergraduate Programme, Indian Institute of Science,
Bangalore 560012, India  Awadhesh Narayan<EMAIL_ADDRESS>Solid State and
Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012,
India

###### Abstract

Rashba spin-orbit coupled systems are an important class of materials noted
for diverse fundamental and applied phenomena. Recently, the emergence of non-
linear Hall effect under conditions of time-reversal symmetry has been
discovered in materials with broken inversion symmetry. In this work, we study
the second- and third-order Hall response in Rashba systems with hexagonal
warping. Starting with a low-energy model, we obtain the analytic expressions
and discover the unique dipole profile in Rashba systems with hexagonal
warping. Furthermore, we extend the analysis using a realistic tight-binding
model. Next, we predict the existence of a third-order Hall effect in these
systems, and calculate the Berry connection polarizability tensor
analytically. We also show how the model parameters affect the third-order
conductivity. Our predictions can help in the experimental realization of
Berry curvature multipole physics in Rashba materials with hexagonal warping,
and provide a new platform for engineering the non-linear Hall effects.

##  I INTRODUCTION

The celebrated Hall effect is a consequence of breaking of the time-reversal
symmetry (TRS) in the linear response regime Cage  _et al._ (2012); Thouless
_et al._ (1982). Magnetic fields or magnetic dopants break TRS in materials
leading to the appearance of the Hall effect Luttinger (1958); Chien (2013).
Berry curvature, which is derived from Berry flux, plays the role of a
magnetic field in momentum space, and can be understood to lead to Hall
responses Vanderbilt (2018). Recently, pioneering work of Sodemann and Fu
explored the role of the dipole moment of the Berry curvature, called the
Berry curvature dipole (BCD) in transport properties of quantum systems
Sodemann and Fu (2015). BCD leads to non-linear Hall effects even in the
presence of TRS, provided that the inversion symmetry of the crystal is broken
Du  _et al._ (2021); Ortix (2021). The order of moment of the Berry curvature
can be connected to the order of the Hall effect. The zeroth order moment is
connected to the conventional Hall effect, whereas the second harmonics in
Hall signals are caused by the first-order moment of Berry curvature, i.e.,
the BCD. Continuing further, higher order moments of Berry curvature are
proposed to be connected to the appearance of higher harmonics of Hall signal.
As an instance, the third-order Hall effect is described by another intrinsic
geometric property of bands known as Berry connection polarizability (BCP)
tensor Liu  _et al._ (2022).

Non-linear Hall effect and its connection to BCD have been predicted for a
variety of materials. These include, among others, two-dimensional materials
such as transition metal dichalcogenides (TMDs), tilted massive Dirac and Weyl
cone systems Du  _et al._ (2018); Xiao  _et al._ (2010); Singh  _et al._
(2020); Joseph and Narayan (2021), strained monolayer and bilayer graphene
Battilomo  _et al._ (2019), Weyl semimetals Yan and Felser (2017), buckled
honeycomb lattices of silicene, germanene and stanene Bandyopadhyay  _et al._
(2022), strong Rashba systems Facio  _et al._ (2018), and Janus TMDs Joseph
_et al._ (2021). Complementary developments have also been achieved in the
experimental direction. The first observation of second-order Hall effect was
made in few layer TMD, WTe2 Kang  _et al._ (2019); Ma  _et al._ (2019).
Following this discovery, non-linear Hall signals have also been reported in
materials such as Dirac semimetals Shvetsov  _et al._ (2019), Kondo materials
Dzsaber  _et al._ (2021), WSe2 Qin  _et al._ (2021); Huang  _et al._ (2020),
topological insulator surfaces Lv  _et al._ (2021), and oxide interfaces of
LaAlO3/SrTiO3 Lebedev  _et al._ (2021); Lesne  _et al._ (2022), among others.
Very recently, third-order Hall signals have been theoretically predicted in
multi-weyl semimetals Roy and Narayan (2022), and detected experimentally in
MoTe2 Lai  _et al._ (2021).

An important group of materials for exploring fundamental spin-orbit physics
and spintronic applications are materials with Rashba spin-orbit coupling
Manchon  _et al._ (2015). Recently, pressure driven phase transition have been
discovered in three-dimensional bismuth tellurium iodine Bahramy  _et al._
(2012); Facio  _et al._ (2018), a material with a strong Rashba effect. Here
the usual quadratic Hamiltonian is augmented by a linear-in-momentum term
arising from the Rashba interaction. On the other hand, surface states of
topological insulators such as Bi2Te3, Bi2Se3, and Sb2Te3 are described by a
massless Dirac equation in the low energy regime Zhang  _et al._ (2009). These
materials belong to a larger class of systems having single Dirac cone. Angle-
resolved photoemission spectroscopy has revealed that the Fermi surface of
these systems have a snowflake structure Fu (2009). This uncommon structure is
explained by the addition of a hexagonal warping term in the Hamiltonian,
which manifests as a cubic-in-momentum term. These exciting developments
suggest the need of exploring non-linear Hall response theoretically in a
generalized framework, governed by a common Hamiltonian incorporating these
different terms.

In this work, we study the second- and third-order Hall response in Rashba
systems with hexagonal warping using a low energy model, as well as a tight-
binding model. Through analytical calculations, we obtain the expressions for
Berry curvature, and numerically calculate the BCD profile. We show that the
unusual BCD profile obtained can be explained as an interplay of various terms
in the Hamiltonian and the band structures arising from it. We also perform
tight-binding calculations and find complementary results. We find that the
particle-hole asymmetry plays an important role in the formation of an unusual
BCD profile in these systems. Further, we calculate the BCP tensor components
for the low energy model and predict the existence of a third-order Hall
effect. We also study the effect of warping strength on the BCP tensor
components. Furthermore, we explore third-order signals after introducing a
band gap in the system, and find interesting connections between the warping
strength and the gap parameter. Our work can help in experimental studies of
Rashba systems with hexagonal warping, and potentially enable characterization
of new materials belonging to this class.

The rest of the paper is organized as follows. In Section II, we give an
overview of the relevant theory on Berry curvature, BCD, BCP and Rashba
physics. Next we describe the results from the low-energy model in Section
IIIA. In Section IIIB, we outline the tight-binding model and related
calculations. Section IV discusses the third-order Hall response and the
conductivity tensor. Finally, Section V provides experimental consideration of
the effects investigated in the manuscript.

##  II BERRY CURVATURE, BERRY CURVATURE DIPOLE, BERRY CONNECTION
POLARIZABILITY TENSOR AND RASHBA EFFECT

In this section, we briefly review the central theoretical concepts of Berry
curvature, BCD and BCP tensor, along with their role in transport properties.
When an oscillating electric field, $\mathbf{E}(t)=\textrm{Re}[\epsilon
e^{i\omega t}]$ is applied, a non-linear current flows through the material,
given by $J_{a}=\textrm{Re}[J_{a}^{(0)}+J_{a}^{(2)}e^{2i\omega t}]$, with
$J_{a}^{(0)}=\chi_{abc}^{(0)}\epsilon_{b}\epsilon_{c}^{*}$ and
$J_{a}^{(2)}=\chi_{abc}^{(2)}\epsilon_{b}\epsilon_{c}$. The coefficients
$\chi_{abc}^{(0)}$ and $\chi_{abc}^{(2)}$ are found to be equal for systems
with time-reversal symmetry, and are given by Sodemann and Fu (2015)

$\chi_{abc}^{(0)}=\chi_{abc}^{(2)}=\frac{\epsilon^{acd}D_{bd}e^{3}\tau}{2\hbar^{2}(1+i\omega
t)},$ (1)

where $\epsilon^{acd}$ is the Levi-Civita antisymmetric tensor, $\omega$ is
the frequency, $-e$ is the charge of the electron, and $\tau$ is the
relaxation time. $D_{bd}$ is the BCD, which is given by

$D_{bd}=-\sum_{i}\int[d\mathbf{k}]\frac{\partial\epsilon^{i}_{\mathbf{k}}}{\partial
k_{b}}\Omega^{d}_{i\mathbf{k}}\frac{\partial
f_{\mathbf{k}}}{\partial\epsilon^{i}_{\mathbf{k}}}.$ (2)

Here $f_{\mathbf{k}}$ denotes the equilibrium Fermi-Dirac distribution and
$a,b,c,d\in\\{x,y,z\\}$. Further, $\epsilon^{i}_{\mathbf{k}}$ is the energy of
the $i^{th}$ band with wavevector $\mathbf{k}$, and
$[d\mathbf{k}]=d^{3}\mathbf{k}/(2\pi)^{3}$. The kernel of the integral of
Equation 2 is the BCD density Du  _et al._ (2018),

$d_{bd}(\mathbf{k})=(\partial^{b}_{\mathbf{k}}\epsilon_{\mathbf{k}})\Omega^{d}_{\mathbf{k}}.$
(3)

The fundamental quantity of interest, appearing in Equation 2 and Equation 3
is the Berry curvature, $\Omega^{d}_{i\mathbf{k}}$, which stands for the Berry
curvature component in the direction $d$ for the $i$-th energy band with
wavevector $\mathbf{k}$. Berry curvature, in turn, can be found as Xiao  _et
al._ (2010)

$\Omega^{a}_{i\mathbf{k}}=-\epsilon^{abc}\textrm{Im}\sum_{i\neq
j}\frac{\langle i|\frac{\partial H}{\partial k_{b}}|j\rangle\langle
j|\frac{\partial H}{\partial k_{c}}|i\rangle-\langle i|\frac{\partial
H}{\partial k_{c}}|j\rangle\langle j|\frac{\partial H}{\partial
k_{b}}|i\rangle}{(\epsilon^{i}_{\mathbf{k}}-\epsilon^{j}_{\mathbf{k}})^{2}},$
(4)

where $|i\rangle$ is eigenstate of the Hamiltonian $H$, corresponding to the
band $i$, having energy $\epsilon^{i}_{\mathbf{k}}$ and wavevector
$\mathbf{k}$. The integral in Equation 2 survives even in the case of time-
reversal symmetric systems, provided inversion symmetry is broken. This
breaking of symmetry allows for a non-zero value of BCD for cases with finite
Berry curvature.

The third-order Hall effect can be connected to the BCP tensor, an intrinsic
geometric property of bands. The BCP tensor components are given by Lai  _et
al._ (2021)

$G_{ab}=2\textrm{R}e\sum_{i\neq
j}\frac{(\mathcal{A}_{a})_{ij}(\mathcal{A}_{b})_{ji}}{\epsilon_{i}-\epsilon_{j}},$
(5)

with $\mathcal{A}$ being the Berry connection. When an electric field is
applied, the Berry curvature gets perturbed, and needs to be corrected to
first order as Liu  _et al._ (2022)

$\tilde{\Omega}(\mathbf{k})=\Omega(\mathbf{k})+\Omega^{(1)}(\mathbf{k}).$ (6)

The first order correction is given by

$\Omega^{(1)}=\nabla_{\mathbf{k}}\times\mathcal{A}^{(1)},$ (7)

and the first order correction of the Berry connection, $\mathcal{A}^{(1)}$
can be related to the BCP tensor through

$\mathcal{A}^{(1)}_{a}(\mathbf{k})=G_{ab}(\mathbf{k})E_{b}.$ (8)

Berry connection polarizability is then defined as

$P_{ab}=\frac{\partial\tilde{\Omega}_{a}}{\partial
E_{b}}=\epsilon_{acd}\partial_{c}G_{db}.$ (9)

The BCP components can be used to evaluate the third-order conductivity tensor
$\chi$. Continuing from the second-order response, the third-order response is
related to the electric field components as
$j_{a}^{(3)}=\chi_{abcd}E_{b}E_{c}E_{d}$, assuming Einstein summation
convention. In the case of third-order response, $\chi$ can be separated into
two parts, $\chi^{I}$ being linear in $\tau$, and $\chi^{II}$ being cubic in
$\tau$. The conductivity components are given by

$\chi_{abcd}^{I}=\tau\left(2\int\left[d\mathbf{k}\right](\partial_{a}\partial_{b}G_{cd})f_{\mathbf{k}}-\int\left[d\mathbf{k}\right](\partial_{c}\partial_{d}G_{ab})f_{\mathbf{k}}-\frac{1}{2}\int\left[d\mathbf{k}\right](v_{a}v_{b}G_{cd})f^{{}^{\prime\prime}}_{\mathbf{k}}\right),$
(10a)
$\chi_{abcd}^{II}=-\tau^{3}\int\left[d\mathbf{k}\right]v_{a}\partial_{b}\partial_{c}\partial_{d}f_{\mathbf{k}},$
(10b)

where $v_{i}=\frac{\partial E}{\partial k_{i}}$ is the group velocity. We will
use this formalism to investigate the non-linear Hall properties in our Rashba
systems with hexagonal warping.

##  III RESULTS AND DISCUSSION

###  III.1 Low-energy model

First, let us present a low-energy model to describe our Rashba system with
hexagonal warping. For the rest of the discussion, we set $\hbar=e=1$. The
Hamiltonian reads Lesne  _et al._ (2022)

$\mathscr{H}=\frac{\textbf{k}^{2}}{2m}-\alpha_{R}(\sigma_{x}k_{y}-\sigma_{y}k_{x})+\frac{\lambda}{2}(k^{3}_{+}+k^{3}_{-})\sigma_{z},$
(11)

where $\alpha_{R}$ is the Rashba coefficient, $\lambda$ is the strength of
warping, $m$ is the effective mass of electron, and $k_{\pm}=k_{x}\pm ik_{y}$.
Let us recall the symmetries of this Hamiltonian, which will also be important
for subsequent discussions. Under threefold rotation operator $C_{3}$ and
mirror symmetry operator $M_{x}$, the momentum $k$ and Pauli spin matrices
$\sigma_{x},\sigma_{y}$ and $\sigma_{z}$ transform as Lesne  _et al._ (2022);
Zhou  _et al._ (2020); Fu (2009)

$\displaystyle C_{3}:k_{\pm}\rightarrow e^{\pm 2\pi
i/3}k_{\pm},\quad\sigma_{\pm}\rightarrow e^{\pm 2\pi
i/3}\sigma_{\pm},\quad\sigma_{z}\rightarrow\sigma_{z},$ (12) $\displaystyle
M_{x}:k_{+}\rightarrow-
k_{-},\quad\sigma_{x}\rightarrow\sigma_{x},\quad\sigma_{y,z}\rightarrow-\sigma_{y,z}.$

Along with the spatial symmetry constraints, there are constraints from time-
reversal symmetry. The time-reversal symmetry operator is represented as
$\mathcal{T}=i\sigma_{y}\mathcal{K}$, where $\mathcal{K}$ is the complex
conjugation operator. The Hamiltonian should be invariant under time reversal
operator, which translates to the constraint
$\mathcal{H}(\mathbf{k})=\mathcal{T}\mathcal{H}(-\mathbf{k})\mathcal{T}^{-1}=\sigma_{y}\mathcal{H}^{*}(-\mathbf{k})\sigma_{y}$.

At the leading order, the breaking of threefold rotation symmetry for the
system is modeled by assuming unequal coefficients for spin-orbit coupling
terms which are linear in momentum. The coefficients of $k_{x}$ and $k_{y}$
are then taken to be $v_{x}$ and $v_{y}$, respectively. Thus, our modified
Hamiltonian now becomes

$\mathscr{H}=\frac{k^{2}_{x}+k^{2}_{y}+k^{2}_{z}}{2m}-(v_{y}\sigma_{x}k_{y}-v_{x}\sigma_{y}k_{x})+\frac{\lambda}{2}(k^{3}_{+}+k^{3}_{-})\sigma_{z}.$
(13)

The corresponding dispersion relation is given by

$E_{\pm}(\textbf{k})=\frac{k^{2}_{x}+k^{2}_{y}+k^{2}_{z}}{2m}\pm\sqrt{v^{2}_{x}k_{x}^{2}+v^{2}_{y}k_{y}^{2}+k^{6}_{x}\lambda^{2}-6k_{x}^{4}k_{y}^{2}\lambda^{2}+9k_{x}^{2}k_{y}^{4}\lambda^{2}}.$
(14)

We note that the bands touch each other only at $k_{x}=k_{y}=k_{z}=0$ in the
momentum space. Energy is measured in units of $k_{F}^{2}$, and hence value of
$m$ is chosen to be 0.5. For this dispersion relation, $k_{z}$ only provides a
constant shift in the energy value, and does not affect the band structure for
the system. Thus, later we will choose to set $k_{z}=0$, which allows us to
describe our model as an effective two-dimensional one.

We begin our analysis by deriving analytical expressions for Berry curvature.
The Berry curvature components along the three cartesian directions read

$\Omega^{x}_{\pm}=0,$ (15a) $\Omega^{y}_{\pm}=0,$ (15b)
$\Omega^{z}_{\pm}=\frac{\pm
v_{x}v_{y}k_{x}(-3k_{y}^{2}+k_{x}^{2})\lambda}{(v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}+k_{x}^{2}(k_{x}^{2}-3k_{y}^{2})^{2}\lambda^{2})^{3/2}}.$
(15c)

We note that $\Omega^{z}_{\pm}$ is independent of $k_{z}$. This dependence of
$\Omega^{z}_{\pm}$ on only $k_{x}$ and $k_{y}$ motivates us to write
$\Omega^{z}_{\pm}$ in polar form as

$\Omega^{z}_{\pm}(k,\theta)=\pm\frac{2\sqrt{2}\lambda
v_{x}v_{y}k^{3}\cos{3\theta}}{[2k^{2}(v_{x}^{2}\cos^{2}{\theta}+v_{y}^{2}\sin^{2}{\theta})+\lambda^{2}k^{6}\cos{6\theta}+\lambda^{2}k^{6}]^{3/2}},$
(16)

where $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$, and $\theta$ is taken with reference
from the $k_{x}$ direction.

Symmetry properties possessed by $\Omega^{z}$ are: (a) $\Omega^{z}$ is odd in
$k_{x}$ , i.e., $\Omega^{z}(-k_{x},k_{y})=-\Omega^{z}(k_{x},k_{y})$. (b)
$\Omega^{z}$ is even in $k_{y}$ ,i.e.,
$\Omega^{z}(k_{x},-k_{y})=\Omega^{z}(k_{x},k_{y})$. (c) It vanishes along the
high symmetry directions of $k_{x}=0,k_{x}=\pm\sqrt{3}k_{y}$. These symmetry
directions correspond to the directions where the cubic term in the
Hamiltonian, namely $\frac{\lambda}{2}(k_{+}^{3}+k_{-}^{3})$ vanishes. As
noted, this term is the hexagonal warping term, and is essential for
occurrence of Berry curvature in the systems under consideration. The effect
of this term is thus directly manifested in the expression for Berry
curvature, and provides a good connection between underlying physics and
mathematical symmetry.

Figure 1:  Band structure and Berry curvature for low energy Hamiltonian. The
band structure superimposed with the Berry curvature along $k_{x}$ (setting
$k_{z}=0$), for (a) $k_{y}=0$, (b) $k_{y}=0.2$, and (c) $k_{y}=0.4$. The
warping effect of band structure can be clearly observed from the plots,
predominant in the valence band. There is a slight asymmetry in the
distribution of Berry curvature along $k_{x}$, prominently seen in (c). We
observe that Berry curvature changes rapidly near the band edges, which is
crucial for the dipole profile. Other parameters are chosen to be
$\lambda=0.1$, $v_{y}=v_{x}=0.4$, and $m=1/2$.

The band structures along $k_{x}$ (for $k_{z}=0$), are plotted in Fig. 1 for
varying $k_{y}$. The value of $z$ component of Berry curvature is superimposed
on the band structure. As noted earlier, the bands are degenerate at the
origin, and rapid changes in Berry curvature can be observed around $k_{x}=0$.
The band structure is not symmetric about the zero energy, because of the
particle-hole asymmetry, which is reflected by the $\frac{k^{2}}{2m}$ term in
the Hamiltonian. A slight asymmetry of Berry curvature distribution on the
band structure can also be observed along $k_{x}$, especially as $k_{y}$
becomes non-zero. This is a combined effect of the Rashba coefficients as well
as the warping term $\lambda$. This deviation is necessary for the occurrence
of the dipole profile, as we will discuss later.

Figure 2:  Distribution of Berry curvature. Density plots of Berry curvature
with hexagonal warping for (a) $v_{y}/v_{x}=1.2$, (b) $v_{y}/v_{x}=1$, and (c)
$v_{x}/v_{y}=1.2$. The effect of warping is clear from the plots, where the
Berry curvature is mostly concentrated in six regions. Berry curvature is zero
along the dashed straight lines, as expected from the symmetries of the
Hamiltonian. The relative concentration of Berry curvature in different
regions is determined by the ratio of Rashba coefficients. Other parameters
common to the plots are $m=0.5$ and $\lambda=0.1$.

The distribution of the $z$ component of Berry curvature in $k_{x}-k_{y}$
plane is shown in Fig. 2. Except at the origin, where the bands are
degenerate, Berry curvature is well-defined in the $k_{x}-k_{y}$ plane.
Hexagonal warping forces Berry curvature to get distributed into six regions,
as can be observed from the plots. Berry curvature vanishes along the mirror
lines, which is expected from symmetry considerations. However, the relative
strengths of Berry curvature in each region is determined by the ratio of
Rashba coefficients $v_{x}$ and $v_{y}$. Thus, there are two independent, but
essential factors which determine the distribution of Berry curvature for our
system.

The rapidly changing Berry curvature near the band edges and the asymmetry of
the band structure suggest the presence of BCD for the material systems
described by our Hamiltonian. In these systems, because of the Rashba spin-
orbit coupling, there is only one mirror symmetry, $M_{x}$. The presence of
$M_{x}$ symmetry leads to $k_{x}$ and $k_{y}$ transforming as odd and even
parameters, respectively. Thus, considering two-dimensional system
$(k_{z}=0)$, only non-zero component of Berry curvature is the $z$ component,
$\Omega^{z}$. Noting that $\Omega^{z}$ is odd in momentum, we find that
$D_{xz}$ is the only non-zero component of BCD, owing to the symmetry
considerations.

We address the second order nonlinear conductivity using Eq. 1. In typical
materials, the relaxation time $\tau$ is of the order of picoseconds, and
typically the frequency of alternating electric field is in the range 10-1000
Hz. In this regime, $\omega\tau<<1$ . Thus, in effect from Eq. 1, we get that
second order nonlinear coefficients ($\chi$) are directly proportional to the
Berry curvature dipole. Specifically, we have

$\chi_{xxy}\propto D_{xz},\quad\chi_{yxx}\propto-D_{xz}.$ (17)

The quantitative details of conductivity depend on the properties of the
material system under consideration. However, from qualitative point of view,
the dipole profiles (Fig. 3) provides a good picture of the behavior of $\chi$
as a function of Fermi energy.

Next, we calculate the BCD component $D_{xz}$ using Equation 2, as a function
of energy. The plots are shown in Fig. 3, and present several interesting
features that can be understood from the interplay of the band structure and
the Berry curvature. We notice that for negative values of $E-E_{F}$, $D_{xz}$
is vanishing away from the Fermi level, starting from approximately
$E-E_{F}=-0.25$. Observing the band structures (Fig. 1) for warping strength
$\lambda=0.1$, we find that there are very small values of Berry curvature
present at these energies. This explains the smooth vanishing of BCD for large
negative values of energy.

For positive values of energy, there is a gradual linear increase in $D_{xz}$.
This finite value of $D_{xz}$ arises because of the presence of substantial
Berry curvature in the conduction band. The concentration of Berry curvature
decreases away from the origin, but as pointed out before, there is a slight
asymmetry in its distribution along $k_{x}$. This asymmetry is the cause of
the linear increase in an otherwise stable value of BCD for positive values of
$E-E_{F}$.

Figure 3: Berry curvature dipole for the low energy model. The non-zero
component $D_{xz}$ as a function of energy, for varying warping strengths (a)
$\lambda=0.1$, (b) $\lambda=0.2$, and (c) $\lambda=0.3$. $D_{xz}$ increases
nearly linearly at positive values of $E-E_{F}$. For negative values of
$E-E_{F}$, the value of $D_{xz}$ is vanishing away from the Fermi level. Near
the Fermi level, there are multiple peaks and dips arising from rapid changes
in the value of the Berry curvature. We notice the difference between maximum
positive and negative values of BCD, caused by warping and the influence of
the group velocity.

Near the Fermi level, there are multiple peaks and dips occurring for BCD,
which can be understood from the rapid changes of the Berry curvature in this
energy region. In addition to Berry curvature, the group velocity proportional
to $\frac{\partial E}{\partial k_{x}}$ plays an important role in determining
the dipole profile. At the band edges, the group velocity vanishes, forcing
the dipole to vanish. This results in multiple sign changes in BCD at negative
values of $E-E_{F}$ very close to the Fermi level. Near the band edges, group
velocity changes sign rapidly, which can be qualitatively visualized through
the change in the curvature of the band structure in this energy window. When
integrated over the Brillouin zone, these small sign changes lead to large
fluctuations in BCD values near the Fermi level, as seen in Fig. 3.

From Fig. 2, we note that the warping strength $\lambda$ plays a very
sensitive role in determining the BCD. The reason for this is two-fold.
Firstly, the distribution of Berry curvature caused by hexagonal warping (as
shown in Fig. 2), and the warping of band structure itself, leads to rapid
changes in the group velocity. Here we find a generic feature of $D_{xz}$ for
these systems – the BCD is neither symmetric, nor anti-symmetric with respect
to $E-E_{F}$. This is an unusual behavior of BCD amongst the materials well-
studied in literature Du  _et al._ (2018); Zhang  _et al._ (2018); Xu  _et
al._ (2018); Battilomo  _et al._ (2019).

Figure 4: Berry curvature dipole density. BCD density plotted for the
conduction band (panel (a)) and valence band (panel (b)), plotted in the
$k_{x}-k_{y}$ plane ($k_{z}=0$), for different values of warping strength
$\lambda$. The distribution of dipole density is symmetric along $k_{x}$ and
$k_{y}$. An increase in warping causes the features to get more concentrated
near the origin, because of increase in changes in group velocity. It is worth
noting that the positive and negative values of density are not symmetric for
conduction band, whereas they are almost symmetric for valence band. This
distribution of the BCD density, along with group velocity effects results in
a finite dipole. Here $m$ is taken to be 0.5.

To gain further insights into BCD of hexagonally warped Rashba systems, we
next look into the BCD density. Analytically, we find that $d_{xz}\sim\pm
k_{x}^{2}(k_{x}^{2}-3k_{y}^{2})$, thus being symmetric in both $k_{x}$ and
$k_{y}$. The dipole density, $d_{xz}$, for conduction band and valence band is
plotted in Fig. 4. It is worth noting that as $\lambda$ increases, the density
gets more concentrated near the origin. This results in a higher BCD occurring
around low negative values of $E-E_{F}$, as shown in Fig. 3. However, since
the density is distributed symmetrically in the $k_{x}-k_{y}$ plane, the
unusual dipole profile is essentially caused by the warping of band structure
itself, with the magnitude being largely dictated by the warping strength.

The density distribution is seen to be largely unaffected by the ratio of
$v_{x}$ and $v_{y}$. However, the positive and negative values of BCD density
are not equal in magnitude, which results in the net dipole being finite. To
get a better understanding of the BCD, we employ next tight-binding models,
which provide a more realistic description of material systems.

###  III.2 Tight-binding model

After finding several interesting behaviors of BCD and Berry curvature in the
low-energy model for our system, we now employ a tight-binding model Lima  _et
al._ (2022); Moca and Marinescu (2005); Ast and Gierz (2012); Van Miert  _et
al._ (2014) derived from the low-energy Hamiltonian. The tight-binding
Hamiltonian reads

$\mathscr{H}=\frac{1}{2m}(6-2\cos{k_{x}}-2\cos{k_{y}}-2\cos{k_{z}})-(v_{y}\sin{k_{y}}\sigma_{x}-v_{x}\sin{k_{x}}\sigma_{y})+\lambda\sin{k_{x}}(6\cos{k_{y}}-2\cos{k_{x}}-4)\sigma_{z},$
(18)

where $v_{x}$ and $v_{y}$ are the Fermi velocities in the $x$ and $y$
directions, respectively, $\lambda$ is the strength of warping, and $m$ is the
effective mass of the electron.

For numerical calculations, we use PythTB Yusufaly  _et al._ (2013) package
for the generation of the lattice system. The output from PythTB is then fed
into the Wannier-Berri package Tsirkin (2021) for calculating band structures,
Berry curvature and BCD. For ease of calculation, we set $v_{x}=v_{y}=1$, and
the on-site energy to zero. In this setting, $\lambda$ becomes proportional to
the hopping parameter $t$ ($t=1$), in units of which energy is measured. For
calculation of BCD, the convergence was checked on a $100\times 100\times 100$
$k$-grid.

Figure 5: Band structure, Berry curvature and Berry curvature dipole for
tight-binding model. Panel (a) corresponds to the case with particle-hole
symmetry. The bands are plotted along the path $k_{y}=k_{x}(\tan\theta)$, with
$\theta=30^{\circ}$ for (a1), $\theta=45^{\circ}$ for (a2),
$\theta=60^{\circ}$ for (a3). Panel (b) corresponds to the case with particle-
hole asymmetry. The bands are plotted along the path
$k_{y}=k_{x}(\tan\theta)$, with $\theta=30^{\circ}$ for (b1),
$\theta=45^{\circ}$ for (b2), $\theta=60^{\circ}$ for (b3). Figures (a4) and
(b4) show the plot of dipole component, $D_{xz}$, as a function of the Fermi
Energy. (a4) is symmetric about the energy axis as expected. Other parameters
common to the plots are $\lambda=0.2$ and $v_{y}=v_{x}=1$.

Our tight-binding model provides a good description of the effect of particle-
hole asymmetry on the BCD of these systems. Fig. 5 shows the band structure
with superimposed Berry curvature, along with the BCD for the tight-binding
model, both in the presence and absence of the particle-hole asymmetry. The
band structures show the Berry curvature being concentrated near the band
edges, with maximum values for region around $k_{x}=k_{y}$, as can be observed
from band structures in Fig. 5(a2) and Fig. 5(b2), consistent with the low-
energy model. Fig. 5(a4) shows the dipole profile when particle-hole symmetry
is present. As expected, the profile is symmetric with respect to $E-E_{F}=0$.
Moreover, $D_{xz}$ goes to zero when $E=E_{F}$. This is because of the exact
cancellation of the dipole density at that energy.

It is worth noting that as particle-hole asymmetry is introduced, the band
structures also become asymmetric with respect to Fermi energy, as observed in
Fig. 5(b1)-(b4). This affects the dipole profile, causing a larger magnitude
of dipole for negative values of Fermi energies [see Fig. 5(b4)]. We also note
a near constant dipole value followed by a linear increase of BCD for positive
values of $E-E_{F}$ in Fig. 5(b4). This can be understood by the decrease of
Berry curvature in the conduction band when asymmetry is present. Our analysis
suggests that particle-hole asymmetry, a property often overlooked, plays a
crucial role in determining the BCD in these systems, in addition to the
important role of hexagonal warping.

##  IV Third-order Hall response

A natural extension of second-order non-linear Hall response is to next higher
order, i.e., to investigate the third-order Hall response. So far, only a
handful of systems have been identified to exhibit the third-order response.
We next explore whether our Rashba systems with hexagonal warping can show
such effects. Third-order Hall response is understood in the formalism of the
BCP tensor Lai  _et al._ (2021); Liu  _et al._ (2022). For our low-energy
Hamiltonian (Equation 13), using Equation 5, we venture to calculate the
different BCP components for our Rashba system. We obtain them to be

$G_{xx}=-\frac{4v_{x}^{2}k_{x}^{6}\lambda^{2}+v_{y}^{2}k_{y}^{2}(v_{x}^{2}+9(k_{x}^{2}-k_{y}^{2})^{2}\lambda^{2})}{4(v_{x}^{2}k_{x}^{2}+v_{y}k_{y}^{2}+k_{x}^{2}(k_{x}^{2}-3k_{y}^{2})^{2}\lambda^{2})^{5/2}},$
(19a)

$G_{yy}=-\frac{k_{x}^{2}(36v_{x}^{2}k_{x}^{2}k_{y}^{2}\lambda^{2}+v_{y}^{2}(v_{x}^{2}+(k_{x}^{2}+3k_{y}^{2})^{2}\lambda^{2}))}{4(v_{x}^{2}k_{x}^{2}+v_{y}k_{y}^{2}+k_{x}^{2}(k_{x}^{2}-3k_{y}^{2})^{2}\lambda^{2})^{5/2}},$
(19b)

$G_{xy}=G_{yx}=\frac{12v_{x}^{2}k_{x}^{5}k_{y}\lambda^{2}+v_{y}^{2}k_{x}k_{y}(v_{x}^{2}+3(k_{x}^{4}+2k_{x}^{2}k_{y}^{2}-3k_{y}^{4})\lambda^{2})}{4(v_{x}^{2}k_{x}^{2}+v_{y}k_{y}^{2}+k_{x}^{2}(k_{x}^{2}-3k_{y}^{2})^{2}\lambda^{2})^{5/2}},$
(19c)

$G_{xz}=G_{yz}=G_{zx}=G_{zy}=G_{zz}=0.$ (19d)

We find that out of the nine components of BCP tensor, only four components
are non-zero. Further, the off-diagonal components are symmetric. From the
expressions of BCP components, we notice that the denominator goes to zero as
$k_{x},k_{y}\rightarrow 0$. This forces $G_{xx}$, $G_{yy}$ and $G_{xy}$ to
diverge near the origin, hindering a complete analysis and understanding of
the physical aspects of our hexagonally-warped Rashba systems.

$G_{xx}$ and $G_{yy}$ both are even in $k_{x}$ as well as $k_{y}$. This
symmetry is unaffected by hexagonal warping. This property also explains the
same sign of ‘legs’ occurring for BCP components with increasing warping, as
can be observed from columns 1 and 2 of Fig. 6. In contrast $G_{xy}$ is odd
with respect to $k_{x}$ and $k_{y}$.

Figure 6: Berry connection polarizability tensor components. Distribution of
Berry connection polarizability tensor components plotted in the $k_{x}-k_{y}$
plane (with $k_{z}=0$). $G_{xx}$ is plotted for (a1) $\lambda=0$, (b1)
$\lambda=0.05$, (c1) $\lambda=0.1$; $G_{yy}$ for (a2) $\lambda=0$, (b2)
$\lambda=0.05$, (c2) $\lambda=0.1$, and $G_{xy}$ for (a3) $\lambda=0$, (b3)
$\lambda=0.05$, (c3) $\lambda=0.1$. $G_{xx}$ and $G_{yy}$ have a double-lobed
structure, with ‘legs’ forming away from origin as warping increases. These
‘legs’ have the same sign of the BCP component. $G{xy}$ also forms ‘legs’, but
these have different signs. The distribution of BCP components near the origin
is not much affected by warping, as can be noticed from all the plots.

Nevertheless, Fig. 6 shows the BCP tensor components in the $k_{x}-k_{y}$
plane, with varying warping strengths. As noted, $G_{xx}$ and $G_{yy}$ take
arbitrarily large values near the origin, indicated by the white portion of
the plots. When warping is zero, we find that $G_{xx}$ and $G_{yy}$ have a
double-lobed structure joining near the origin. As warping is turned on and
increased, this double-lobed structure distorts and forms ‘leg’ like
structures, extending outwards to larger momenta. Increase in warping makes
these ‘legs’ more profound, which effectively means the components become less
concentrated near the origin and become more spread out. However, the effect
of warping is negligible near the origin, which is expected analytically from
Equation 6, by taking the limit $\lambda\rightarrow 0$. Interestingly,
$G_{xy}$ takes small values near the origin, a behavior opposite to that of
$G_{xx}$ and $G_{yy}$. However, in this case, we find the presence of four
‘legs’ extending from infinity to the origin. In general, we find that the
nature of BCP components is substantially different from that of Berry
curvature. This suggests that these two quantities manifest in different order
phenomena, which turns out to be second and third-order non-linear Hall
effect.

The difficulty imposed on obtaining physical insights into the third-order
Hall response due to the behavior of BCP tensor components near the origin can
be overcome, if a band gap parameter is introduced in our effective model.
This is achieved by introducing a parameter $\Delta$ as the coefficient of
$\sigma_{z}$ in the Hamiltonian. The new Hamiltonian for the gapped Rashba
system with hexagonal warping now reads

$\mathscr{H}=\frac{k^{2}_{x}+k^{2}_{y}+k^{2}_{z}}{2m}-(v_{y}\sigma_{x}k_{y}-v_{x}\sigma_{y}k_{x})+\frac{\lambda}{2}(k^{3}_{+}+k^{3}_{-})\sigma_{z}+\Delta\sigma_{z}.$
(20)

Figure 7:  Berry connection polarizability tensor components with gap
parameter. Distribution for Berry connection polarizability tensor components
plotted in $k_{x}-k_{y}$ plane (with $k_{z}=0$.) $G_{xx}$ is plotted for (a1)
$\lambda=0$, (b1) $\lambda=0.05$, (c1) $\lambda=0.1$; $G_{yy}$ for (a2)
$\lambda=0$, (b2) $\lambda=0.05$, (c2) $\lambda=0.1$, and $G_{xy}$ for (a3)
$\lambda=0$, (b3) $\lambda=0.05$, (c3) $\lambda=0.09$. For $\lambda=0$, we
recover the usual monopole and quadrupole nature of $G_{xx}$ and $G_{xy}$,
respectively. For $\lambda\neq 0$, their nature deviates and new features
start appearing. Magnitudes of the components peak where Berry curvature is
maximum. Here $\Delta$ is taken to be 0.3 for panels (a1)-(c1) and (a2)-(c2),
and $\Delta$=0.25 for panels (a3)-(c3).

Proceeding with the same analysis as before, we calculate the analytical
expressions for BCP tensor. As before, the non-zero components of BCP tensor
are $G_{xx},G_{yy},G_{xy},$ and $G_{yx}$. The expressions of the BCP tensor
components in this case is rather lengthy and cumbersome, therefore we do not
write them out explicitly here. However, the plots for the components as shown
in Fig. 7 reveal several interesting features. Without the warping, the BCP
components exhibit a monopole and quadrupole kind of behavior Liu  _et al._
(2022). With the introduction of warping, however, this monopole or quadrupole
like behavior strikingly breaks down. The warping effect is clear from panels
(b) and (c) of Fig. 7. As warping increases, the distribution of BCP
components become increasingly asymmetric, and gets elongated along the
$k_{x}$. For the $G_{xy}$ components, we find that for non-zero warping, BCP
gets concentrated more in the region with negative $k_{x}$.

An analytic calculation of the full conductivity tensor involves contribution
of several components of $G$ and $\chi$, leading to rather complicated
expressions. To gain better insights, we calculate the conductivity tensor
after setting $k_{z}=0$. The third-order Hall response vanishes when electric
field is applied along or perpendicular to the mirror line, which in this case
is $M_{x}$. Hence the components
$\chi_{yyyx},\chi_{xyyy},\chi_{yxyy},\chi_{yyxy}$ and
$\chi_{xxxy},\chi_{xxyx},\chi_{xyxx},\chi_{yxxx}$ will be zero, and will not
contribute to the conductivity tensor. So, an in-plane electric field, given
by $\mathbf{E}=E(\cos\theta,\sin\theta)$ results in a third-order response
$j_{H}^{(3)}=\mathbf{j^{(3)}\cdot\hat{n}}$, where $\mathbf{\hat{n}}$ is the
unit vector normal to ${\mathbf{E}}$, given by $(-\sin\theta,\cos\theta)$.

Figure 8: Third-order conductivity tensor. (a) The angular dependence of the
third-order conductivity tensor with varying warping strength $\lambda$. In
this case $\Delta=0.1$. (b) The angular dependence the third-order
conductivity tensor for varying $\Delta$, with $\lambda=0.1$. We observe that
as $\lambda$ and $\Delta$ take close values, the maximum magnitude of
conductivity increases.

The third-order transverse conductivity tensor is then given by

$\chi_{H}(\theta)=\frac{j_{H}^{(3)}}{E^{3}},$ (21)

which can be measured experimentally Lai  _et al._ (2021). For systems with
mirror symmetry $M_{x}$, we obtain

$\chi_{H}(\theta)=(-\chi_{11}+3\chi_{21})\sin\theta\cos^{3}\theta+(\chi_{22}-3\chi_{12})\sin^{3}\theta\cos\theta,$
(22)

with $\theta$ measured from the mirror line, and the shorthand notations being
used are $\chi_{11}=\chi_{xxxx}$, $\chi_{22}=\chi_{yyyy}$,
$\chi_{21}=(\chi_{yyxx}+\chi_{yxyx}+\chi_{yxxy})/3$, and
$\chi_{12}=(\chi_{xxyy}+\chi_{xyxy}+\chi_{xyyx})/3$. We evaluate the third-
order conductivity tensor numerically, and it is shown as a function of
$\theta$ in Fig. 8. The most remarkable feature of conductivity tensor, in
this case, is the interplay of the band gap $\Delta$ and warping term
$\lambda$. For all values of $\Delta$ and $\lambda$, we find that $\chi_{H}$
varies with $\theta$ with a period of $\pi$. Here $\chi_{H}$ vanishes when
$\theta$ is a multiple of $\pi/2$, consistent with our symmetry analysis.

It is worth noting that as the values of $\Delta$ and $\lambda$ become closer,
the magnitude of $\chi_{H}$ increases. Moreover, there is a substantial
deviation from the sinusoidal nature, however the periodicity is maintained.
This property originates from the structure of $\chi$, and can be used in
experimental estimation of warping strengths. To illustrate, suppose we choose
a material, in which the band gap can be altered with an external
perturbation, such as by the application of an electric field. In such a case,
the warping strength is given by the value of band gap where a ‘resonance’ is
observed for the maximum value of conductivity. Thus, our proposed hexagonally
warped Rashba systems are intriguing platforms for observing and tuning
higher-order Hall effects.

In a very recent independent study Nag  _et al._ (2023), results on third-
order nonlinear Hall effect are presented, which are complementary to ours.
The consistency of results between the two works provides a check about the
soundness of the approach used. The major similarities between the two works
are the study of BCP tensor components with varying warping strength, and
pattern of sign changes between consecutive legs of $G_{xy}$. However, there
are important differences between the works worth noting. While Ref. Nag  _et
al._ (2023) uses a tilt term along with the low-energy Hamiltonian, our work
introduces a gap parameter, $\Delta$, advantages of which are discussed in the
manuscript. Similarly, we study the third order conductivity with the
interplay between warping and the band gap. This approach provides intriguing
new insights about behaviour of third-order conductivity, and a possible
experimental use of the same.

##  V Experimental Consideration

Based on our calculations, we next estimate the second order nonlinear Hall
response that can be observed in experiments. For this, we use the
approximation $\omega\tau<<1$, as before. With typical relaxation time in
picoseconds, under this approximation, we have

$\chi_{xxy}=-\chi_{yxx}=0.185D_{xz}.$ (23)

Consider a typical sample size of $10\times 5\times 1\;\mu$m3, and isotropic
resistivity in all directions to be $50\;\mu\Omega$cm. Let $V_{x}=V_{y}=1$V.
Then we have,

$J^{2\omega}_{x}=\chi_{xxy}E_{x}E_{y}=\chi_{xxy}\frac{V_{x}V_{y}}{L_{x}L_{y}}=(3.7\times
10^{9})D_{xz}\;\textrm{A/m}^{2}.$ (24)

The corresponding voltage produced is then
$V_{x}^{2\omega}=0.0185D_{xz}\;\textrm{V}$. The maximum voltage depending on
the warping strength thus is evaluated to be $2.86\times 10^{-4}$ V for
$\lambda=0.1$, $3.77\times 10^{-4}$ V for $\lambda=0.2$, and $11.6\times
10^{-4}$ V for $\lambda=0.3$. These can be measured using standard
experimental setups.

The third order nonlinear Hall voltage can be estimated by the following
relation

$V^{(3)}_{H}\propto\chi_{H}(\theta)E^{3}.$ (25)

We consider a typical uniform electric field of 100 V/cm in the $x$ direction.
For $\theta=45^{o}$ and $\Delta=0.1$, keeping other parameters identical to
the previous case, the induced voltage magnitude can be estimated to be
$V^{(3)}_{H}\approx 81$ mV for $\lambda=0$, and $V^{(3)}_{H}\approx 1.22$ V
for $\lambda=0.05$ . The interplay between $\Delta$ and $\lambda$ in third-
order nonlinear conductivity response can be used in experiments to find the
strength of warping in suitable systems. To illustrate, suppose we choose a
material, in which the band gap can be altered with an external perturbation,
such as by the application of an electric field. In such a case, the warping
strength is given by the value of band gap where a ‘resonance’ is observed for
the maximum value of conductivity. Thus, our proposed hexagonally warped
Rashba systems are intriguing platforms for observing and tuning higher-order
Hall effects.

##  VI Summary

In this work, we systematically investigated the Berry curvature multipole
physics in Rashba systems with hexagonal warping. We calculated the general
expressions for Berry curvature, using a low-energy model, and studied how the
model parameters affect its distribution. We next investigated the behaviour
of Berry curvature dipole in these systems, and found an unconventional dipole
profile. This could be explained using the band structure and the Berry
curvature distribution. Additionally, we looked at the dipole density profiles
and concluded that the warping of the band structure itself is an important
factor in determining the dipole profile. Motivated by the low-energy model
results, we used a tight-binding model to study the band structure and the BCD
in a more realistic setting. Here, we found another important factor for the
unusual dipole profile, which is the particle-hole asymmetry. Further, we
performed analytic calculations for Berry connection polarizability tensor,
which is related to the third-order Hall effect. We found that for our Rashba
systems with hexagonal warping, four of the nine components are non-zero.
Further, by introducing a band gap in our model, we found interesting
connections of the band gap with the warping strength, and their combined
effect on the third-order Hall response. We hope that our work can help in the
experimental realization of Berry curvature multipole physics in Rashba
materials with hexagonal warping, and provide a new avenue for engineering
non-linear Hall effects.

In an independent work, the second-order nonlinear Hall effect was studied
using a low-energy model Yar  _et al._ (2022). Our work goes beyond this by
the use of a tight-binding model in addition to the low-energy Hamiltonian.
Tight binding model provides much more realistic insights into the material
properties, and brings out crucial physics that is often not seen in the use
of low-energy model Zeng  _et al._ (2021). In this independent work Yar  _et
al._ (2022), the BCD density is studied using a low-energy model similar to
ours. However, there are crucial differences between our work and Ref. Yar
_et al._ (2022). In the context of BCD density, our work studies the
relationship between BCD density and warping strength. This is in contrast to
Yar  _et al._ (2022), which focuses on a single warping strength. Considering
different warping strengths also provides intriguing connections to the way
BCD is distributed near the origin ($k_{x}\approx 0$ and $k_{y}\approx 0$).

## Acknowledgments

We acknowledge useful discussions with S. Roy, D. Varghese, A. Bandyopadhyay,
A. Bose, and A. Banerjee. S. S. thanks the Kishore Vaigyanik Protsahan Yojana
(KVPY) for a fellowship. A. N. acknowledges the startup grant of the Indian
Institute of Science (SG/MHRD-19-0001).

## Appendix

The eigenvectors corresponding to the eigenvalues of Eqn. (14) are given, upto
normalisation constant and a phase, by

$\psi_{\pm}=\begin{pmatrix}k_{x}^{3}\lambda-3k_{x}k_{y}^{2}\lambda\pm\sqrt{k_{x}^{2}v_{x}^{2}+k_{y}^{2}v_{y}^{2}+k_{x}^{6}\lambda^{2}-6k_{x}^{4}k_{y}^{2}\lambda^{2}+9k_{x}^{2}k_{y}^{4}\lambda^{2}}\\\
-k_{y}v_{y}+ik_{x}v_{x}\end{pmatrix}.$ (26)

The group velocity terms, $\frac{\partial H}{\partial k}$, are given by

$\frac{\partial H}{\partial
k_{x}}=\begin{pmatrix}k_{x}/m+3\lambda(k_{x}^{2}-k_{y}^{2})&-iv_{x}\\\
iv_{x}&k_{x}/m-3\lambda(k_{x}^{2}-k_{y}^{2})\end{pmatrix},$ (27a)

$\frac{\partial H}{\partial
k_{y}}=\begin{pmatrix}k_{y}/m-6k_{y}k_{x}\lambda&-v_{y}\\\
-v_{y}&k_{y}/m+6k_{y}k_{x}\lambda\end{pmatrix},$ (27b)

$\frac{\partial H}{\partial k_{z}}=\begin{pmatrix}k_{z}/m&0\\\
0&k_{z}/m\end{pmatrix}.$ (27c)

Plugging the eigenvectors and the group velocity terms in Eqn. 4, the
expressions for Berry curvature follow.

In order to derive the tight-binding Hamiltonian (Eqn. 18), we use the
approximations $\sin{k_{x}}\approx k_{x}$,
$\cos{k_{x}}=1-\frac{k_{x}^{2}}{2}$, and similar for $k_{y}$ and $k_{z}$.
Thus, the various terms of the low-energy Hamiltonian transform, as follows

$\frac{1}{2m}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})=\frac{1}{2m}(2-2\cos{k_{x}}+2-2\cos{k_{y}}+2-2\cos{k_{z}})=\frac{1}{2m}(6-2\cos{k_{x}}-2\cos{k_{y}}-2\cos{k_{z}}),$
(28a)

$-v_{y}k_{y}\sigma_{x}=-v_{y}\sin{k_{y}}\sigma_{x},$ (28b)

$v_{x}k_{x}\sigma_{y}=v_{x}\sin{k_{x}}\sigma_{y},$ (28c)

$\begin{split}\frac{\lambda}{2}(k_{+}^{3}+k_{-}^{3})\sigma_{z}&=\frac{\lambda}{2}\times
2k_{x}(k_{x}^{2}-3k_{y}^{2})\sigma_{z}=\lambda\sin{k_{x}}((2(1-\cos{k_{x}})-3(2(1-\cos{k_{y}}))))\sigma_{z}\\\
&=\lambda\sin{k_{x}}(6\cos{k_{y}}-2\cos{k_{x}}-4)\sigma_{z}.\end{split}$ (28d)

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# Behaviors of pairs of dimensions and depths of edge ideals

Akihiro Higashitani  ,  Akane Kanno  and  Ryota Ueji  Department of Pure and
Applied Mathematics, Graduate School of Information Science and Technology,
Osaka University, Suita, Osaka 565-0871, Japan<EMAIL_ADDRESS>Department of Pure and Applied Mathematics, Graduate School of Information
Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
<EMAIL_ADDRESS>Department of Pure and Applied Mathematics,
Graduate School of Information Science and Technology, Osaka University,
Suita, Osaka 565-0871, Japan<EMAIL_ADDRESS>
###### Abstract.

Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$
of the polynomial ring $S$ in $n$ variables generated by the quadratic
monomials associated with the edges of $G$. In this paper, we consider the
possible pairs of dimensions and depths of $S/I(G)$ for connected graphs with
a fixed number of vertices. We discuss such pairs in the case where dimension
is relatively large. As a corollary, we completely determine the pairs for
connected graphs with small number of vertices. We also study the possible
pairs for connected chordal graphs.

###### Key words and phrases:

Edge ideals, dimension, depth, projective dimension.

###### 2020 Mathematics Subject Classification:

Primary: 13D02, Secondary: 13F55, 05E40, 05C70.

##  1\. Introduction

Monomial ideals are one of the most well-studied objects in the area of
combinatorial commutative algebra. Especially, edge ideals of graphs are of
particular interest in this area. On the other hand, dimensions and depths are
fundamental invariants on graded (or local) rings. The goal of this paper is
the investigation of the behaviors of pairs of dimensions and depths for the
quotient rings by edge ideals.

Throughout the present paper, we only treat finite simple graphs, so we omit
“finite simple”. Let $\operatorname{Graph}(n)$ denote the set of all connected
graphs on the vertex set $[n]:=\\{1,2,\ldots,n\\}$. Let
$S=\Bbbk[x_{1},\ldots,x_{n}]$ be a polynomial ring in $n$ variables over a
field $\Bbbk$. The edge ideal $I(G)$ of $G\in\operatorname{Graph}(n)$ is
defined by $I(G)=(x_{i}x_{j}:\\{i,j\\}\in E(G))\subset S$. Invariants on edge
ideals have been well studied in combinatorial commutative algebra.
Especially, the graded Betti numbers of the edge ideals of graphs have been
considered in various viewpoints. Since the (Castelnuovo–Mumford) regularity
and the projective dimension of edge ideals can be obtained from the
information on Betti numbers, we can also obtain some characterizations or
bounds on them. For example, see [3, 4, 6, 7, 8, 9, 10]. Note that the depths
of edge ideals and the projective dimensions are equivalent invariants in some
sense (see Subsection 2.3). For more information on edge ideals, see, e.g.,
[5, Section 9].

In this paper, we focus on the relationship between dimensions and depths of
edge ideals for connected graphs. The study of their relationship is initiated
in [6, Section 2] as far as the authors know. Let us briefly recall what is
obtained there. Given a positive integer $n$, let

$\operatorname{Graph}_{\operatorname{depth},\dim}(n):=\left\\{(\operatorname{depth}S/I(G),\dim
S/I(G)):G\in\operatorname{Graph}(n)\right\\},$

where $\dim G$ (resp. $\operatorname{depth}G$) denotes the dimension (resp.
depth) of $S/I(G)$. On the other hand, let

$C^{*}(n):=\left\\{(a,b)\in\mathbb{Z}^{2}:1\leq a\leq b\leq n-1,\;a\leq
b+1-\left\lceil\frac{b}{n-b}\right\rceil\right\\}.$

Then the following is proved:

######  Theorem 1.1 ([6, Theorem 2.8]).

Let $n\geq 2$. Then we have
$C^{*}(n)\subset\operatorname{Graph}_{\operatorname{depth},\dim}(n)$.

By taking this theorem into account, the following question naturally arises:

######  Question 1.2.

Does the equality
$C^{*}(n)=\operatorname{Graph}_{\operatorname{depth},\dim}(n)$ hold?

In [6, Example 2.2], it is mentioned that
$\operatorname{Graph}_{\operatorname{depth},\dim}(n)$ is computed by using
computer. As a consequence of the computations, we can verify that
$C^{*}(n)=\operatorname{Graph}_{\operatorname{depth},\dim}(n)$ holds when
$n\leq 9$.

In this paper, we develop more theoretical considerations. The first main
result of this paper is the following:

######  Theorem 1.3 (See Remark 2.3, Propositions 3.2 and 3.3).

Let $(a,b)\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$. If $b\geq
n-3$, then $(a,b)\in C^{*}(n)$.

This theorem enables us to show the equality
$C^{*}(n)=\operatorname{Graph}_{\operatorname{depth},\dim}(n)$ when $n$ is
relatively small as follows:

######  Corollary 1.4.

The equality $C^{*}(n)=\operatorname{Graph}_{\operatorname{depth},\dim}(n)$
holds if $n\leq 12$.

For a graph $G$, we say that $G$ is chordal if any cycle in $G$ of length at
least $4$ has a chord. Let

$\displaystyle\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)=\left\\{(\operatorname{depth}S/I(G),\dim
S/I(G)):G\in\operatorname{Graph}(n),\;\text{$G$ is chordal}\right\\}.$

Namely, we restrict $\operatorname{Graph}_{\operatorname{depth},\dim}(n)$ into
chordal graphs.

The second main result of this paper is the following:

######  Theorem 1.5.

Let $n\geq 2$. Then we have
$C^{*}(n)=\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)$.

Note that in [6, Section 3], the possible pairs of dimensions and depths of
the edge ideals of Cameron–Walker graphs are determined. See [6, Theorem
3.15]. We can observe that such pairs for Cameron–Walker graphs are quite
small compared with $C^{*}(n)$.

A brief structure of this paper is as follows. In Section 2, we recall some
fundamental materials used for the proofs of Theorems 1.3 and 1.5. In Section
3, we prove Theorem 1.3 and Corollary 1.4. In Section 4, we prove Theorem 1.5.

### Acknowledgements

The first named author is partially supported by JSPS Grant-in-Aid for
Scientists Research (C) JP20K03513. The second named author is partially
supported by Grant-in-Aid for JSPS Fellows JP21J21603.

##  2\. Preliminaries

In this section, we prepare some terminologies, notions and statements for the
proofs of our main theorems.

###  2.1. Terminologies on graphs

Let $G$ be a graph on the vertex set $V(G)$ with the edge set $E(G)$.

  * •

We call a subset $W$ of $V(G)$ an independent set (resp. clique) if
$\\{v,w\\}\not\in E(G)$ (resp. $\\{v,w\\}\in E(G)$) for any $v,w\in W$. Let

$\displaystyle\alpha(G)=\max\\{|W|:W\text{ is an independent set of }G\\},$

called the independence number of $G$. It is well known that

$\dim S/I(G)=\alpha(G).$

  * •

Let $M\subset E(G)$ be a subset of the edge set. We say that $M$ is an induced
matching of $G$ if $M$ satisfies that

    * –

$e\cap e^{\prime}=\emptyset$ for any $e,e^{\prime}\in M$ with $e\neq
e^{\prime}$; and

    * –

there is no $e^{\prime\prime}\in E(G)$ such that $e\cap
e^{\prime\prime}\neq\emptyset$ and $e^{\prime}\cap
e^{\prime\prime}\neq\emptyset$.

  * •

For a vertex $v$ of $G$, let $N(v)=\\{w\in V:\\{v,w\\}\in E(G)\\}$, let
$N[v]=N(v)\cup\\{v\\}$ and let $\operatorname{deg}v=|N(v)|$.

  * •

For a vertex $v$ of $G$, let $G\setminus v$ denote the induced subgraph of $G$
by $V(G)\setminus\\{v\\}$. Similarly, for $U\subset V(G)$, let $G\setminus U$
denote the induced subgraph of $G$ by $V(G)\setminus U$.

  * •

We say that $C\subset V(G)$ is a vertex cover if $C\cap e\neq\emptyset$ for
any $e\in E(G)$. We call a vertex cover $C$ minimal if $C\setminus\\{v\\}$ is
not a vertex cover of $G$ for any $v\in C$.

  * •

A perfect elimination ordering of $G$ is an ordering $v_{n},\ldots,v_{1}$ of
the vertices of $G$ such that $\\{v_{1},\ldots,v_{k-1}\\}\cap N(v_{k})$ forms
a clique for each $k=2,\ldots,n$. It is known that $G$ is chordal if and only
if $G$ has a perfect elimination ordering ([2]).

In this paper, given a graph $G$, we use the notation $\dim G$,
$\operatorname{depth}G$ and $\operatorname{pdim}G$ instead of $\dim S/I(G)$,
$\operatorname{depth}S/I(G)$ and $\operatorname{pdim}S/I(G)$ for short,
respectively, where $\operatorname{pdim}$ denotes the projective dimension.

###  2.2. Bouquets and projective dimensions of $S/I(G)$

We recall the terminology and the notation used in [9]. The graph $B$ on the
vertex set $V(B)=\\{w,z_{1},\ldots,z_{d}\\}$ with the edge set
$\\{\\{w,z_{i}\\}:i=1,\ldots,d\\}$ is called a bouquet. Namely, a bouquet is
nothing but a star graph. A vertex $w$ is called a root, the vertices $z_{i}$
are called flowers and the edges $\\{w,z_{i}\\}$ are called stems of $B$.
These words were originally used in [10, Definition 1.7].

Given a graph $G$, we call $B$ a bouquet of $G$ if $B$ is a (not necessarily
induced) subgraph of $G$. Let $\mathcal{B}=\\{B_{1},\ldots,B_{p}\\}$ be a set
of bouquets of $G$. We define the following:

$\displaystyle F(\mathcal{B})$ $\displaystyle:=\bigcup_{i=1}^{p}\\{z\in
V(B_{i}):z\text{ is a flower of }B_{i}\\};$ $\displaystyle R(\mathcal{B})$
$\displaystyle:=\bigcup_{i=1}^{p}\\{z\in V(B_{i}):z\text{ is a root of
}B_{i}\\};$ $\displaystyle S(\mathcal{B})$
$\displaystyle:=\bigcup_{i=1}^{p}\\{s\in E(B_{i}):s\text{ is a stem of
}B_{i}\\}.$

Let $G$ be a graph and let $\mathcal{B}=\\{B_{1},\ldots,B_{p}\\}$ be a set of
bouquets of $G$. Assume that $V(B_{i})\cap V(B_{j})=\emptyset$ holds for any
$1\leq i<j\leq p$.

  * •

We say that $\mathcal{B}$ is strongly disjoint if there are stems $s_{i}\in
E(B_{i})$ for each $i$ such that $\\{s_{1},\ldots,s_{p}\\}$ forms an induced
matching of $G$.

  * •

We say that $\mathcal{B}$ is semi-strongly disjoint if $R(\mathcal{B})$ forms
an independent set.

Note that if $\mathcal{B}$ is strongly disjoint, since any stem in $B_{i}$ is
incident to the root of $B_{i}$, we see that any two roots should not be
adjacent. Namely, $\mathcal{B}$ becomes semi-strongly disjoint.

######  Theorem 2.1 ([9, Theorem 3.1]).

Given any graph $G$, we have

$\operatorname{pdim}G\geq\max\\{|F(\mathcal{B})|:\mathcal{B}\text{ is a
strongly disjoint set of bouquets of }G\\}.$

By Theorem 2.1, we see that given a graph $G$, we have

$\displaystyle\operatorname{deg}v\leq\operatorname{pdim}G\text{ for any }u\in
V(G)$ (2.1)

since $N(u)\cup\\{u\\}$ forms a (set of a single) bouquet with a root $u$.

######  Theorem 2.2 ([9, Theorem 5.3, Corollary 5.6]).

Let $G$ be chordal. Then we have

$\displaystyle\operatorname{pdim}G$
$\displaystyle=\max\\{|F(\mathcal{B})|:\mathcal{B}\text{ is a semi-strongly
disjoint set of bouquets of }G\\}$ $\displaystyle=\max\\{|C|:C\text{ is a
minimal vertex cover of }G\\}.$

###  2.3. Auslander–Buchsbaum formula

Let $G$ be a graph on $[n]$. Then $S/I(G)$ is an $S$-module. Since $S$ is a
polynomial ring, $\operatorname{pdim}G<\infty$ holds. Hence, by the
Auslander–Buchsbaum formula (see, e.g., [1, Theorem 1.3.3]), we have

$\operatorname{pdim}G+\operatorname{depth}G=n.$

We will often use this in the sequel.

###  2.4. On the sets $C^{*}(n)$ and $C^{-}(n)$

Let us recall the set $C^{-}(n)$ defined in [6, Section 2]. For $n\geq 2$, let

$C^{-}(n):=\\{(1,n-1)\\}\cup\left\\{(a,b)\in\mathbb{Z}^{2}:1\leq a\leq
b,\;a\leq\left\lfloor\frac{n}{2}\right\rfloor,\;b\leq n-2\right\\}.$

Then, by the proof of [6, Theorem 2.9], we see that $C^{-}(n)\subset
C^{*}(n)$. The difference of $C^{-}(n)$ and $C^{*}(n)$ will be mentioned
below. See Example 3.5.

######  Remark 2.3 (cf. [6, Proposition 1.3]).

Let $G\in\operatorname{Graph}(n)$ and assume that $\alpha(G)=n-1$. Then $G$
itself must be a bouquet. In particular, $G$ is chordal. Thus, by Theorem 2.2,
we have $\operatorname{pdim}G=n-1$. Hence, $\operatorname{depth}G=1$. These
imply that if $(a,n-1)\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$,
then $a=1$. In particular, $(1,n-1)\in C^{-}(n)\subset C^{*}(n)$.

######  Remark 2.4.

Given integers $a,b$ with $1\leq a\leq b\leq n-1$, assume that $a+b\leq n$.
Then $(a,b)\in C^{-}$ automatically holds. In fact, if $b=n-1$, then $a=1$; if
$b\leq n-2$, since $2a\leq a+b\leq n$ holds, we have $a\leq\lfloor
n/2\rfloor$.

##  3\. In the case of graphs with large independence number

In this section, we consider the inclusion
$\operatorname{Graph}_{\operatorname{depth},\dim}(n)\subset C^{*}(n)$ for some
special cases. In particular, we discuss when
$(a,b)\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$ belongs to
$C^{*}(n)$.

First, we discuss the case where $a=b$, i.e., the corresponding graph is
Cohen–Macaulay.

######  Proposition 3.1 (cf. [3, Theorem 3.3]).

Assume that $(b,b)\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$.
Then $(b,b)\in C^{*}(n)$.

###### Proof.

We see that $(b,b)\in C^{*}(n)$ holds if and only if $\lceil b/(n-b)\rceil\leq
1$ holds. Here, it follows from [3, Theorem 3.3] that if $G$ is
Cohen–Macaulay, then $b\leq n/2$. This implies that when $G$ is Cohen–Macaulay
with $b=\dim G$, we have $b\leq n/2$, i.e., $(b,b)\in C^{*}(n)$. ∎

Next, we discuss the case where $b$ is relatively large. Note that we always
have $1\leq b\leq n-1$ and the case $b=n-1$ has been already discussed in
Remark 2.3.

######  Proposition 3.2.

Let $n\geq 4$. If $(a,n-2)\not\in C^{*}(n)$, then
$(a,n-2)\not\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$.

###### Proof.

Let

$f(a,b):=b+1-\left\lceil\frac{b}{n-b}\right\rceil-a.$

Note that for integers $a,b$ with $1\leq a\leq b\leq n-1$, $f(a,b)\geq 0$
holds if and only if $(a,b)\in C^{*}(n)$.

In what follows, we lead a contradiction if there exists a graph
$G\in\operatorname{Graph}(n)$ with $(\operatorname{depth}G,\dim G)=(a,n-2)$
such that $f(a,n-2)<0$.

Since $f(a,n-2)=\lfloor n/2\rfloor-a$, we have $\lfloor
n/2\rfloor<\operatorname{depth}G=n-\operatorname{pdim}G$, i.e.,
$\operatorname{pdim}G<\lceil n/2\rceil$. On the other hand, since $\dim
G=n-2$, there is an independent set $W$ of $G$ with $|W|=n-2$. Let
$v_{1},v_{2}$ be the remaining vertices, i.e.,
$\\{v_{1},v_{2}\\}=V(G)\setminus W$. Since $G$ is connected and $W$ is an
independent set, we see that

$\displaystyle n-2$ $\displaystyle\leq\sum_{w\in
W}\operatorname{deg}w=|N(v_{1})\cap W|+|N(v_{2})\cap W|$
$\displaystyle=\begin{cases}\operatorname{deg}v_{1}+\operatorname{deg}v_{2}-2,&\text{
if }\\{v_{1},v_{2}\\}\in E(G),\\\
\operatorname{deg}v_{1}+\operatorname{deg}v_{2},&\text{ if
}\\{v_{1},v_{2}\\}\not\in E(G).\end{cases}$

On the other hand, since $\operatorname{pdim}G<\lceil n/2\rceil$, we see that
$\operatorname{deg}v_{i}<\lceil n/2\rceil$ for $i=1,2$ by (2.1). If
$\\{v_{1},v_{2}\\}\in E(G)$, then we see that

$n-2\leq\operatorname{deg}v_{1}+\operatorname{deg}v_{2}-2\leq\frac{n-1}{2}+\frac{n-1}{2}-2=n-3,$

a contradiction. Hence, $\\{v_{1},v_{2}\\}\not\in E(G)$.

Suppose that $n$ is even. Then we see that

$n-2\leq\operatorname{deg}v_{1}+\operatorname{deg}v_{2}\leq\frac{n-2}{2}+\frac{n-2}{2}=n-2,$

i.e., we have $\operatorname{deg}w=1$ for each $w\in W$. Since
$\\{v_{1},v_{2}\\}\not\in E(G)$, we see that there is no path in $G$
connecting $v_{1}$ and $v_{2}$, a contradiction to the connectedness of $G$.
Hence, $n$ should be odd. Moreover, there should be at least one vertex
$w_{0}$ in $W$ with $\operatorname{deg}w_{0}\geq 2$. Since $\sum_{w\in
W}\operatorname{deg}w\geq n-1$ and $\operatorname{deg}v_{i}\leq(n-1)/2$ hold,
we conclude that $G$ should look like as follows:

$v_{1}$$v_{2}$$w_{0}$

Note that $\operatorname{deg}v_{1}=\operatorname{deg}v_{2}=(n-1)/2$. In
particular, $G$ is chordal. By taking a set of bouquets consisting of two
bouquets whose roots are $v_{1}$ and $v_{2}$, we see from Theorem 2.2 that

$\left\lceil\frac{n}{2}\right\rceil>\operatorname{pdim}G\geq n-2,$

a contradiction to $n\geq 4$. ∎

######  Proposition 3.3.

Let $n\geq 8$. If $(a,n-3)\not\in C^{*}(n)$, then
$(a,n-3)\not\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$.

###### Proof.

Let $f(a,b)$ be the same thing as above. Then $f(a,n-3)=\lfloor
2n/3\rfloor-a-1$.

Suppose that there exists a graph $G\in\operatorname{Graph}(n)$ with
$(\operatorname{depth}G,\dim G)=(a,n-3)$ such that $f(a,n-3)<0$. Since $\dim
G=n-3$, there is an independent set $W$ with $|W|=n-3$. Let
$v_{1},v_{2},v_{3}$ be the remaining vertices. By (2.1), we have
$\operatorname{deg}v_{i}\leq n-a<n-\lfloor 2n/3\rfloor+1=\lceil n/3\rceil+1$,
i.e.,

$\displaystyle\operatorname{deg}v_{i}\leq\left\lceil\frac{n}{3}\right\rceil\text{
for }i=1,2,3.$ (3.1)

Similarly, for any strongly disjoint set $\mathcal{B}$ of bouquets of $G$, we
have

$\displaystyle|F(\mathcal{B})|\leq\left\lceil\frac{n}{3}\right\rceil.$ (3.2)

Here, we have the following four possibilities on the adjacencies of
$v_{1},v_{2},v_{3}$:

(i)$v_{1}$$v_{2}$$v_{3}$$v_{1}$(ii)$v_{2}$$v_{3}$(iii)$v_{1}$$v_{2}$$v_{3}$(iv)$v_{1}$$v_{2}$$v_{3}$

We divide our discussions into these four cases.

(i) In this case, we see from (3.1) that

$n-3\leq\sum_{w\in
W}\operatorname{deg}w=\sum_{i=1}^{3}(\operatorname{deg}v_{i}-2)\leq
3\left(\left\lceil\frac{n}{3}\right\rceil-2\right)\leq
3\left(\frac{n+2}{3}-2\right)=n-4,$

a contradiction.

(ii) By the similar computation to (i), we have that

$\displaystyle n-3\leq\sum_{w\in
W}\operatorname{deg}w=\sum_{i=1}^{3}\operatorname{deg}v_{i}-4\leq
3\left\lceil\frac{n}{3}\right\rceil-4=\begin{cases}n-4\;\;\text{ if }n\equiv
0\pmod{3},\\\ n-3\;\;\text{ if }n\equiv 2\pmod{3},\\\ n-2\;\;\text{ if
}n\equiv 1\pmod{3}.\end{cases}$

Thus, we get the following possibilities:

  * •

$n\equiv 2\pmod{3}$ and $\operatorname{deg}w=1$ for each $w\in W$;

  * •

$n\equiv 1\pmod{3}$ and $\operatorname{deg}w=1$ for each $w\in W$;

  * •

$n\equiv 1\pmod{3}$, $\operatorname{deg}w_{0}=2$ for some $w_{0}\in W$ and
$\operatorname{deg}w=1$ for each $w\in W\setminus\\{w_{0}\\}$.

In the first two cases (resp. the third case), $G$ should look like the left-
most one (resp. the central one or the right-most one) of the following
figure:

$v_{1}$$v_{2}$$v_{3}$$v_{1}$$v_{2}$$v_{3}$$w_{0}$$v_{1}$$v_{2}$$v_{3}$$w_{0}$

Note that we have $|N(v_{i})\cap W|\geq\lceil n/3\rceil-2>1$ for $i=1,3$ and
$|(N(v_{1})\cup N(v_{3}))\cap W|\geq 2\lceil n/3\rceil-3$ in any cases. Hence,
we can always take a strongly disjoint set of bouquets $\mathcal{B}$
consisting of two bouquets whose roots are $v_{1}$ and $v_{3}$ satisfying
$|F(\mathcal{B})|\geq 2\lceil n/3\rceil-3+1.$ (Note that “$+1$” comes from a
stem $\\{v_{1},v_{2}\\}$.) By (3.2), we obtain that

$\left\lceil\frac{n}{3}\right\rceil\geq
2\left\lceil\frac{n}{3}\right\rceil-2\;\Longleftrightarrow\;2\geq\left\lceil\frac{n}{3}\right\rceil\;\Longleftrightarrow\;6\geq
n,$

a contradiction.

(iii) Take bouquets $B_{1},B_{2},B_{3}$ of $G$ whose roots are
$v_{1},v_{2},v_{3}$, respectively, such that $V(B_{i})\cap V(B_{j})=\emptyset$
($i\neq j$) and $F(\\{B_{1},B_{2},B_{3}\\})=W$ hold.

  * •

Assume that $N(v_{3})\setminus N(v_{2})\neq\emptyset$. Let $w\in
N(v_{3})\setminus N(v_{2})$. Then, by taking a strongly disjoint set
$\mathcal{B}^{\prime}$ of bouquets consisting of $B_{2}^{\prime}$ and a
bouquet $\\{v_{3},w\\}$, where $B_{2}^{\prime}$ is constructed by attaching a
new flower $v_{1}$ to $B_{2}$, we see from (3.2) that

$\displaystyle\left\lceil\frac{n}{3}\right\rceil\geq|F(\mathcal{B}^{\prime})|=|F(B_{2})|+2=n-3-|F(\\{B_{1},B_{3}\\})|+2\geq
n-1-\left\lceil\frac{n}{3}\right\rceil.$

Hence, we obtain the inequality

$\left\lceil\frac{n}{3}\right\rceil\geq
n-1-\left\lceil\frac{n}{3}\right\rceil,$

i.e., $n\leq 7$, a contradiction.

  * •

Assume that $N(v_{3})\setminus N(v_{2})=\emptyset$, i.e., $N(v_{3})\subset
N(v_{2})$. Take $w\in N(v_{3})$ and let $B^{\prime}$ be a new bouquet with its
root $w$ and flowers $v_{2},v_{3}$. Then $\\{B_{1},B^{\prime}\\}$ becomes
strongly disjoint. Thus, by (3.2), we have

$\left\lceil\frac{n}{3}\right\rceil\geq|F(\\{B_{1},B^{\prime}\\})|=|F(B_{1})|+2\geq
n-3-|F(\\{B_{2}\\})|+2\geq n-1-\left\lceil\frac{n}{3}\right\rceil,$

a contradiction.

(iv) Take bouquets $B_{1},B_{2},B_{3}$ of $G$ whose roots are
$v_{1},v_{2},v_{3}$, respectively, such that $V(B_{i})\cap V(B_{j})=\emptyset$
($i\neq j$) and $F(\\{B_{1},B_{2},B_{3}\\})=W$ hold. If the situation (iv)
happens, then there are $j,k$ such that $\\{B_{j},B_{k}\\}$ is strongly
disjoint. In fact, if not, then either $F(B_{j})\subset F(B_{k})$ or
$F(B_{k})\subset F(B_{j})$ holds for each $1\leq j<k\leq 3$. This implies that
we have $F(B_{1})\subset F(B_{2})\subset F(B_{3})$ after properly changing the
indices of $B_{1},B_{2},B_{3}$. In particular, $F(B_{3})=W$. Thus,
$n-3=\operatorname{deg}v_{3}\leq\lceil n/3\rceil\leq(n+2)/3$, i.e., $n\leq 5$,
a contradiction.

Let, say, $\mathcal{B}=\\{B_{1},B_{2}\\}$ be a strongly disjoint set of
bouquets of $G$.

  * •

Suppose that $|F(\mathcal{B})|\geq\lfloor 2n/3\rfloor-1$. Then we see that

$\left\lceil\frac{n}{3}\right\rceil\geq|F(\mathcal{B})|\geq\left\lfloor\frac{2n}{3}\right\rfloor-1\;\Longrightarrow\;\frac{n+2}{3}\geq\frac{2n-2}{3}-1\;\Longleftrightarrow\;7\geq
n,$

a contradiction.

  * •

Suppose that $|F(\mathcal{B})|\leq\lfloor 2n/3\rfloor-2$. Since
$|W|=|F(\mathcal{B})|+|F(B_{3})|=n-3$ and
$|F(B_{3})|\leq\operatorname{deg}v_{3}-1$ by the connectedness of $G$, we see
that

$\left\lceil\frac{n}{3}\right\rceil-1\geq\operatorname{deg}v_{3}-1\geq|F(B_{3})|\geq
n-3-\left\lfloor\frac{2n}{3}\right\rfloor+2=\left\lceil\frac{n}{3}\right\rceil-1.$

Hence, all the equalities of these inequalities are satisfied. In particular,
we have $|F(B_{3})|=\operatorname{deg}v_{3}-1=\lceil n/3\rceil-1$. Since
$n-3>\lceil n/3\rceil$ and $G$ is connected, there is $w\in W\setminus
N(v_{3})$ adjacent to $v_{1}$ or $v_{2}$, say, $v_{1}$. Then we can construct
a new strongly disjoint set of bouquets $\mathcal{B}^{\prime}$ consisting of
bouquets $N(v_{3})\cup\\{v_{3}\\}$ and $\\{v_{1},w\\}$ such that
$|F(\mathcal{B}^{\prime})|=\lceil n/3\rceil+1$, a contradiction to (3.2).

∎

By using Propositions 3.1, 3.2 and 3.3, we can prove Corollary 1.4.

###### Proof of Corolalry 1.4.

It is enough to show that
$\operatorname{Graph}_{\operatorname{depth},\dim}(n)\subset C^{*}(n)$ for
$n\leq 12$.

Take $(a,b)\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$
arbitrarily. Then the inequalities $1\leq a\leq b\leq n-1$ automatically hold.

  * •

If $a=b$, then we see that $(a,b)\in C^{*}(n)$ by Proposition 3.1.

  * •

The case of $b=n-1$ has been already discussed in Remark 2.3.

Hence, in what follows, we assume that $1\leq a<b\leq n-2$. Here, we consider
the set

$C^{\prime}(n):=\\{(a,b)\in\mathbb{Z}^{2}:1\leq a<b\leq n-2\\}\setminus
C^{*}(n).$

Then the direct computations show the following:

$\displaystyle C^{\prime}(n)=\emptyset\text{ for }2\leq n\leq
6,\;\;C^{\prime}(7)=\\{(4,5)\\},\;\;C^{\prime}(8)=\\{(5,6)\\},\;\;C^{\prime}(9)=\\{(5,7),(6,7)\\},$
$\displaystyle
C^{\prime}(10)=\\{(6,7),(6,8),(7,8)\\},\;\;C^{\prime}(11)=\\{(6,9),(7,8),(7,9),(8,9)\\},$
$\displaystyle C^{\prime}(12)=\\{(7,10),(8,9),(8,10),(9,10)\\}.$

Let $n\leq 12$. Suppose that $(a,b)\not\in C^{*}(n)$. Then $(a,b)\in
C^{\prime}(n)$. By the above computations, we see that all elements of
$\bigcup_{n=2}^{12}C^{\prime}(n)$ satisfy $b=n-2$ or $b=n-3$. Hence, it
follows from Propositions 3.2 and 3.3 that
$(a,b)\not\in\operatorname{Graph}_{\operatorname{depth},\dim}(n)$, a
contradiction.

Therefore, $C^{*}(n)=\operatorname{Graph}_{\operatorname{depth},\dim}(n)$
holds if $n\leq 12$. ∎

######  Remark 3.4.

We see that

$C^{\prime}(13)=\\{(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)\\}.$

Since we cannot claim that
$(8,9)\not\in\operatorname{Graph}_{\operatorname{depth},\dim}(13)$ only by
Propositions 3.2 and 3.3, the above proof of Corollary 1.4 is not available if
$n=13$. Instead, we can claim that
$\operatorname{Graph}_{\operatorname{depth},\dim}(13)=C^{*}(13)$ or
$C^{*}(13)\cup\\{(8,9)\\}$.

######  Example 3.5.

For small $n$’s, the direct computations show the following:

$\displaystyle C^{*}(n)$ $\displaystyle=C^{-}(n)\text{ for
}n=2,3,\ldots,8,10,$ $\displaystyle C^{*}(9)$
$\displaystyle=C^{-}(9)\cup\\{(5,6)\\},$ $\displaystyle C^{*}(11)$
$\displaystyle=C^{-}(11)\cup\\{(6,6),(6,7),(6,8)\\},$ $\displaystyle
C^{*}(12)$ $\displaystyle=C^{-}(12)\cup\\{(7,8),(7,9),(8,9)\\}.$

Note that [6, Example 2.11] also mentions the similar equalities in the case
$3\leq n\leq 9$.

##  4\. In the case of chordal graphs

In this section, we discuss the behavior of pairs of dimensions and depths of
edge ideals of chordal graphs.

###  4.1. The inclusion
$C^{*}(n)\subset\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)$

First, we discuss the inclusion
$C^{*}(n)\subset\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)$.
Here, we remark that since we only know that
$\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)\subset\operatorname{Graph}_{\operatorname{depth},\dim}(n)$
by definition, we cannot immediately claim
$C^{*}(n)\subset\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)$
from Theorem 1.1.

Let $(a,b)\in C^{*}(n)$. We divide the discussions into two cases:

  * (i)

$a+b>n$; or

  * (ii)

$a+b\leq n$.

(i) In this case, we can apply the same proof as that of [6, Theorem 2.8]. In
that proof, the graphs $G(m;s_{1},\ldots,s_{m})$ given in [6, Construction
2.5] play the essential role and we see that $G(m;s_{1},\ldots,s_{m})$ is
chordal. Hence, we obtain the inclusion by the same discussion.

(ii) In this case, we cannot apply the same proof. In fact, Lemmas 2.3 and 2.4
are used for the proof in the case $a+b\leq n$. Especially, Lemma 2.3 uses the
technique of $S$-cone (also known as $S$-suspension). In general, the
chordality is not preserved by taking $S$-cone. Hence, Lemma 2.3 is not
immediately available for chordal graphs.

Instead, we directly prove the following:

######  Proposition 4.1.

Let $a,b,n$ be integers satisfying $1\leq a\leq b\leq n-1$ and $a+b\leq n$.
Then there exists a chordal graph $G\in\operatorname{Graph}(n)$ such that
$(\operatorname{depth}G,\dim G)=(a,b)$. Namely, we have
$(a,b)\in\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)$.

Note that the assumption $a+b\leq n$ immediately implies $(a,b)\in C^{*}(n)$
(see Remark 2.4).

###### Proof.

Let $G$ be the graph defined as follows:

$\displaystyle V(G)$ $\displaystyle=U\sqcup W,\;\text{ where
}\;U=\\{u_{1},\ldots,u_{n-b}\\}\text{ and }W=\\{w_{1},\ldots,w_{b}\\};$
$\displaystyle E(G)$ $\displaystyle=\\{\\{u,u^{\prime}\\}:u,u^{\prime}\in
U\\}\cup\\{\\{u_{i},w_{i}\\}:1\leq i\leq a-1\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\cup\\{\\{u_{j},w_{k}\\}:a\leq
j\leq n-b,\;a\leq k\leq b\\}.$

It is straightforward to check that the ordering
$w_{1},\ldots,w_{b},u_{1},\ldots,u_{n-b}$ becomes a perfect elimination
ordering of $G$. Hence, $G$ is chordal. Moreover, we can easily see that $W$
is an independent set with $|W|=\alpha(G)$. Hence, $\dim G=b$.

In what follows, we prove that $\operatorname{depth}G=a$, i.e.,
$\operatorname{pdim}G=n-a$.

Regarding the minimal vertex covers of $G$, we observe that any minimal vertex
cover contains at least $(n-b-1)$ vertices of $U$ since $U$ forms a clique. In
particular, we see that $U$ itself becomes a minimal vertex cover. Fix
$u_{j}\in U$ and let $C$ be a minimal vertex cover not containing $u_{j}$.
Then $U\setminus\\{u_{j}\\}\subset C$.

  * •

If $1\leq j\leq a-1$, then $w_{j}\in C$, so
$C=(U\setminus\\{u_{j}\\})\cup\\{w_{j}\\}$.

  * •

If $a\leq j\leq n-b$, then $w_{a},\ldots,w_{b}\in C$, so
$C=(U\setminus\\{u_{j}\\})\cup\\{w_{a},\ldots,w_{b}\\}$.

Therefore, by Theorem 2.2, we conclude that

$\operatorname{pdim}G=\max\\{|C|:C\text{ is a minimal vertex cover of
}G\\}=\max\\{|U|,|U|+b-a\\}=n-a.$

∎

###  4.2. The inclusion
$\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)\subset
C^{*}(n)$

For the proof of this inclusion, we prepare the following lemma.

######  Lemma 4.2.

(1) For any vertex $v$ of a graph $G$, we have

$\dim G=\max\\{\dim(G\setminus N[v])+1,\dim(G\setminus v)\\}.$

(2) Let $G$ be a chordal graph and let $v_{n},\ldots,v_{1}$ be a perfect
elimination ordering of $G$. Then we have

$\operatorname{depth}G=\min\\{\operatorname{depth}(G\setminus
N[v_{n}])+1,\operatorname{depth}(G\setminus v_{n})\\}.$

###### Proof.

(1) Let $W$ be an independent set of $G$ with $|W|=\alpha(G)$. Then the
required equality directly follows from the following observations:

  * •

If $v\in W$, then $W\setminus\\{v\\}\subset V(G\setminus N[v])$ becomes an
independent set of $G\setminus N[v]$ with
$|W\setminus\\{v\\}|=\alpha(G\setminus N[v])$.

  * •

If $v\not\in W$, then $W\subset V(G\setminus v)$ becomes an independent set of
$G\setminus v$ with $|W|=\alpha(G\setminus v)$.

(2) We prove the equality by induction on $|V(G)|$.

Let $v=v_{n}$ and $\operatorname{deg}v=m$. Let $C$ be a minimal vertex cover
of $G$ with $|C|=\operatorname{pdim}G$ (cf. Theorem 2.2).

  * •

If $v\in C$, then $C^{\prime}=C\setminus\\{v\\}$ becomes a minimal vertex
cover of $G\setminus v$ with $\operatorname{pdim}(G\setminus v)=|C^{\prime}|$.
Hence, by the hypothesis of induction, we see that

$\displaystyle\operatorname{depth}G$
$\displaystyle=|V(G)|-\operatorname{pdim}G=|V(G)|-(|C^{\prime}|+1)=|V(G\setminus
v)|-|C^{\prime}|$ $\displaystyle=|V(G\setminus
v)|-\operatorname{pdim}(G\setminus v)=\operatorname{depth}(G\setminus v).$

  * •

If $v\not\in C$, since $N[v]$ forms a clique by definition of perfect
elimination ordering, we see that $N(v)\subset C$. Thus,
$C^{\prime\prime}=C\setminus N[v]$ becomes a minimal vertex cover of
$G\setminus N[v]$ with $\operatorname{pdim}(G\setminus
N[v])=|C^{\prime\prime}|$. Hence, we see that

$\displaystyle\operatorname{depth}G$
$\displaystyle=|V(G)|-\operatorname{pdim}G=|V(G)|-(|C^{\prime\prime}|+m)=(|V(G\setminus
N[v])|+1)-|C^{\prime\prime}|$ $\displaystyle=|V(G\setminus
N[v])|-\operatorname{pdim}(G\setminus N[v])+1=\operatorname{depth}(G\setminus
N[v])+1.$

Here, Theorem 2.2 says that if $C_{0}$ is a minimal vertex cover with
$|C_{0}|\neq\operatorname{pdim}G$, then $|C_{0}|<\operatorname{pdim}G$. Thus,
the required equality follows. ∎

Now, we prove the inclusion
$\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)\subset
C^{*}(n)$. Take
$(a,b)\in\operatorname{Graph}_{\operatorname{depth},\dim}^{\mathrm{chordal}}(n)$
arbitrarily. Then there is a chordal graph $G$ such that
$(\operatorname{depth}G,\dim G)=(a,b)$. To prove $(a,b)\in C^{*}(n)$, it is
enough to show that

$\displaystyle b\leq(n-b)(b-a+1).$ (4.1)

In fact, (4.1) is equivalent to the following:

$\displaystyle\frac{b}{n-b}\leq
b-a+1\;\Longleftrightarrow\;\left\lceil\frac{b}{n-b}\right\rceil\leq
b-a+1\;\Longleftrightarrow\;a\leq b+1-\left\lceil\frac{b}{n-b}\right\rceil.$

This means that $(a,b)\in C^{*}(n)$.

Let $v_{n},\ldots,v_{1}$ be a perfect elimination ordering of $G$. We prove
the inequality (4.1) by induction on $n$.

Let $v=v_{n}$ and let

$a^{\prime}=\operatorname{depth}(G\setminus v),\;\;b^{\prime}=\dim(G\setminus
v),\;\;a^{\prime\prime}=\operatorname{depth}(G\setminus N[v])\text{ and
}b^{\prime\prime}=\dim(G\setminus N[v]).$

By Lemma 4.2 (1), either $b=b^{\prime}$ or $b=b^{\prime\prime}+1$ holds.

  * •

Assume that $b=b^{\prime}$. Since $a\leq a^{\prime}$ holds by Lemma 4.2 (2),
we see from the hypothesis of induction that

$\displaystyle
b=b^{\prime}\leq((n-1)-b^{\prime})(b^{\prime}-a^{\prime}+1)\leq(n-b)(b-a+1).$

  * •

Assume that $b=b^{\prime\prime}+1$. Since $a\leq a^{\prime\prime}+1$, we see
that

$\displaystyle b$
$\displaystyle=b^{\prime\prime}+1\leq((n-\operatorname{deg}v-1)-b^{\prime\prime})(b^{\prime\prime}-a^{\prime\prime}+1)+1$
$\displaystyle\leq(n-\operatorname{deg}v-b)(b-a+1)+1=(n-b)(b-a+1)-(\operatorname{deg}v(b-a+1)-1)$
$\displaystyle\leq(n-b)(b-a+1).$

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the thermodynamics of the Kerr black hole (focusing on $D=4$). This can be
dervied by simply evaluating the leading $\alpha^{\prime}$ correction to the
action on the uncorrected black hole background. The end results, in the
bosonic/heterotic cases, are

$\tilde{\beta}=\frac{4\pi
r_{+}(a^{2}+r_{+}^{2})}{r_{+}^{2}-a^{2}},\quad\tilde{\Omega}=\frac{a}{r_{+}^{2}+a^{2}},$
(148)
$\tilde{M}^{\prime}=\frac{r_{+}^{2}+a^{2}}{2r_{+}},\quad\tilde{S}^{\prime}=\pi(r_{+}^{2}+a^{2})+2\pi\lambda,\quad\tilde{J}^{\prime}=a\tilde{M}^{\prime}.$
(149)

Here $\lambda=\alpha^{\prime}/2,\alpha^{\prime}/4$ for bosonic and heterotic
string theory, respectively. $\lambda$ vanishes for Type II theories and the
leading order correction happens at $\alpha^{\prime 3}$, therefore we won’t
discuss it here. We emphasize that $r_{+},a$ in (148) and (149) are only
parameters to parametrize $\tilde{\beta}$ and $\tilde{\Omega}$ and they differ
from the corresponding geometric quantities by $\alpha^{\prime}$ corrections.

For bosonic string theory, we can then plug (148), (149) into (143) - (147)
can get the leading $\alpha^{\prime}$ corrections to the rotating two charge
system:

$\frac{1}{T}=\frac{4\pi
r_{+}(r_{+}^{2}+a^{2})}{r_{+}^{2}-a^{2}}\frac{\cosh\upalpha+\cosh\upbeta}{2},\quad\Omega=\frac{a}{r_{+}^{2}+a^{2}}\frac{2}{\cosh\upalpha+\cosh\upbeta},$
(150)
$Q_{L}=\sinh\upalpha\cosh\upbeta\left[\frac{\pi(r_{+}^{2}+a^{2})}{2r_{+}}+\lambda\frac{\pi(r_{+}^{2}-a^{2})}{r_{+}(r_{+}^{2}+a^{2})}\right],$
(151)
$Q_{R}=\cosh\upalpha\sinh\upbeta\left[\frac{\pi(r_{+}^{2}+a^{2})}{2r_{+}}+\lambda\frac{\pi(r_{+}^{2}-a^{2})}{r_{+}(r_{+}^{2}+a^{2})}\right],$
(152) $J=\frac{\cosh\upalpha+\cosh\upbeta}{2}a\frac{r_{+}^{2}+a^{2}}{2r_{+}},$
(153) $2\pi
M=\frac{\pi(r_{+}^{2}+a^{2})}{2r_{+}}(1+\cosh\upalpha\cosh\upbeta)+\lambda\frac{\pi(r_{+}^{2}-a^{2})}{r_{+}(r_{+}^{2}+a^{2})}(\cosh\upalpha\cosh\upbeta-1),$
(154)
$S=\frac{\cosh\upalpha+\cosh\upbeta}{2}(\pi(r_{+}^{2}+a^{2})+2\pi\lambda),$
(155)

where we’ve set $G_{N}=1$ in the above expressions.

For the case of the heterotic string theory, the correct solution generating
transformation in fact differs from the bosonic and type II theory by terms
proportional to $\alpha^{\prime}$. The correct transformation was motivated by
studying the Horowitz-Polchinski solution in Chen:2021dsw , but was later
independently verified by studying the $\alpha^{\prime}$ corrections to the
non-rotating two-charge system Massai:2023cis . The derivation here follows
that of Chen:2021dsw , with the main new ingredient being that the angular
potential $\Omega$ is related to $\tilde{\Omega}$ of the seed by
$R\Omega=\tilde{R}\tilde{\Omega}$. We omit the intermediate steps and present
the results analogous to (143) to (147).

$R=\frac{\tilde{R}}{2}\left[\left(1+\frac{\alpha^{\prime}}{2\tilde{R}^{2}}\right)\cosh\upalpha+\left(1-\frac{\alpha^{\prime}}{2\tilde{R}^{2}}\right)\cosh\upbeta\right],\quad\Omega=\frac{\tilde{R}}{R}\tilde{\Omega},$
(156) $2\pi
Q_{L}=e^{-\Phi_{D}}\cosh\upbeta\sinh\upalpha\left[\frac{\tilde{S}^{\prime}}{\tilde{R}}+2\pi\tilde{\Omega}\tilde{J}^{\prime}\right],\quad
2\pi
Q_{R}=e^{-\Phi_{D}}\cosh\upalpha\sinh\upbeta\left[\frac{\tilde{S}^{\prime}}{\tilde{R}}+2\pi\tilde{\Omega}\tilde{J}^{\prime}\right],$
(157)
$J=e^{-\Phi_{D}}\frac{\tilde{J}^{\prime}}{2}\left[\left(1+\frac{\alpha^{\prime}}{2\tilde{R}^{2}}\right)\cosh\upalpha+\left(1-\frac{\alpha^{\prime}}{2\tilde{R}^{2}}\right)\cosh\upbeta\right],$
(158)
$S=e^{-\Phi_{D}}\left[\frac{\cosh\upalpha+\cosh\upbeta}{2}\tilde{S}^{\prime}-\frac{\alpha^{\prime}(\cosh\upalpha-\cosh\upbeta)}{4\tilde{R}^{2}}(\tilde{S}^{\prime}+2\pi\tilde{R}\tilde{\Omega}\tilde{J}^{\prime})\right],$
(159) $2\pi M=2\pi
e^{-\Phi_{D}}\tilde{M}^{\prime}+e^{-\Phi_{D}}(\cosh\upalpha\cosh\upbeta-1)\left(\frac{\tilde{S}^{\prime}}{\tilde{R}}+2\pi\tilde{\Omega}\tilde{J}^{\prime}\right).$
(160)

We can further plug (148) and (149) and get the explicit formulas for the
$\alpha^{\prime}$ corrected thermodynamic quantities for rotating two-charge
solutions in heterotic string theory.

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