# Electric Hall Effect and Quantum Electric Hall Effect

Chaoxi Cui  Centre for Quantum Physics, Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics,
Beijing Institute of Technology, Beijing 100081, China  Beijing Key Lab of
Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing
Institute of Technology, Beijing 100081, China  Run-Wu Zhang
<EMAIL_ADDRESS>Centre for Quantum Physics, Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics,
Beijing Institute of Technology, Beijing 100081, China  Beijing Key Lab of
Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing
Institute of Technology, Beijing 100081, China  Yilin Han  Centre for Quantum
Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and
Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing
100081, China  Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic
Systems, School of Physics, Beijing Institute of Technology, Beijing 100081,
China  Zhi-Ming Yu<EMAIL_ADDRESS>Centre for Quantum Physics, Key
Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement
(MOE), School of Physics, Beijing Institute of Technology, Beijing 100081,
China  Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems,
School of Physics, Beijing Institute of Technology, Beijing 100081, China
Yugui Yao<EMAIL_ADDRESS>Centre for Quantum Physics, Key Laboratory of
Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of
Physics, Beijing Institute of Technology, Beijing 100081, China  Beijing Key
Lab of Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics,
Beijing Institute of Technology, Beijing 100081, China

###### Abstract

Exploring new Hall effect is always a fascinating research topic. The ordinary
Hall effect and the quantum Hall effect, initially discovered in two-
dimensional (2D) non-magnetic systems, are the phenomena that a transverse
current is generated when a system carrying an electron current is placed in a
magnetic field perpendicular to the currents. In this work, we propose the
electric counterparts of these two Hall effects, termed as electric Hall
effect (EHE) and quantum electric Hall effect (QEHE). The EHE and QEHE emerge
in 2D magnetic systems, where the transverse current is generated by applying
an electric gate-field instead of a magnetic field. We present a symmetry
requirement for intrinsic EHE and QEHE. With a weak gate-field, we establish
an analytical expression of the intrinsic EHE coefficient. We show that it is
determined by intrinsic band geometric quantities: Berry curvature and its
polarizability which consists of both intraband and interband layer
polarization. Via first-principles calculations, we investigate the EHE in the
monolayer Ca(FeN)2, where significant EHE coefficient is observed around band
crossings. Furthermore, we demonstrate that the QEHE can appear in the
semiconductor monolayer $\rm BaMn_{2}S_{3}$, of which the Hall conductivity
exhibits steps that take on the quantized values $0$ and $\pm 1$ in the unit
of $e^{2}/h$ by varying the gate-field within the experimentally achievable
range. Due to the great tunability of the electric gate-field, the EHE and
QEHE proposed here can be easily controlled and should have more potential
applications.

As a fundamental transport phenomenon, the Hall effect, discovered over a
century ago, remains an actively expanding area of condensed matter physics
[1, 2, 3, 4, 5, 6, 7, 8, 9]. Various types of Hall effects, including ordinary
and planar (in-plane), intrinsic and extrinsic, linear and nonlinear Hall
effects, continue to be proposed and investigated [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26]. The ordinary Hall effect occurs
in systems with crossed electric and magnetic fields, where the transverse
current is perpendicular to both fields. Since the transverse current is
generated by the magnetic field ($\mathcal{B}$-field), the Hall conductivity
vanishes when the strength of the $\mathcal{B}$-field approaches zero.
Moreover, when the $\mathcal{B}$-field is large enough, the Hall conductivity
of a metal can be quantized, leading to the quantum Hall effect [4, 27, 28,
29].

Table 1: Symmetry requirements of the intrinsic EHE. “✓” and “✗” denote that
the element is symmetry allowed and forbidden, respectively. The magnetic
layer-point-group symmetry operators are divided into four parts: ${\cal{S}}$,
$\mathcal{T}{\cal{S}}$, $M_{z}{\cal{S}}$ and $M_{z}\mathcal{T}{\cal{S}}$.
Notice that $S_{2,z}={\cal{P}}$.

.  ${\cal{S}}$ $\mathcal{T}{\cal{S}}$ $M_{z}{\cal{S}}$
$M_{z}\mathcal{T}{\cal{S}}$ Elements
$\left\\{E,M_{\|}\mathcal{T},C_{n,z}\right\\}$
$\left\\{\mathcal{T},M_{\|},C_{n,z}\mathcal{T}\right\\}$
$\left\\{M_{z},C_{2,\|}\mathcal{T},S_{n,z}\right\\}$
$\left\\{M_{z}\mathcal{T},C_{2,\|},S_{n,z}\mathcal{T}\right\\}$ $\sigma_{ab}$
✓ ✗ ✓ ✗ $\chi_{ab}$ ✓ ✗ ✗ ✓ Requirement optional forbidden forbidden requisite

In addition to the $\mathcal{B}$-field, the electric gate-field
($\mathcal{E}$-field) is another common external field used to manipulate the
physical properties of systems [30, 31, 32, 33, 34, 35, 36, 37]. Particularly
in 2D materials, the $\mathcal{E}$-field is directly coupled to the layer
polarization of electron bands, providing an efficient and predictable method
to control the band structure and many physical observations of the systems
[38, 39, 40, 41]. Furthermore, electric control of spin and valley
polarization has been extensively investigated in both magnetic and non-
magnetic systems [42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. However, the use of
a static $\mathcal{E}$-field instead of a $\mathcal{B}$-field to generate a
net Hall effect and the quantum Hall effect has not been well investigated.

Figure 1: Schematics for (a) ordinary and (c) quantum Hall effects. (b) and
(d) illustrate EHE and QEHE, which can be considered as the electric
counterparts of ordinary and quantum Hall effects.

To address this important task, we in this work explore the electric
counterparts of the ordinary and quantum Hall effects in 2D magnetic systems,
where the Hall current arises from a perpendicular static $\mathcal{E}$-field
instead of $\mathcal{B}$-field, as illustrated in Fig. 1. Through symmetry
analysis, we find that the intrinsic EHE can exist in a wide range of 2D
magnetic systems, including ferromagnetic, antiferromagnetic, and
altermagnetic materials. We derive the analytical expression of the intrinsic
EHE, and show that it is closely related to Berry curvature and its
polarizability consisting of both intraband and interband layer polarization.
This means that the EHE will be pronounced around band degeneracies and the
band edges of systems with small band gaps. We validate the existence of EHE
by first-principle calculations on monolayer Ca(FeN)2, where significant EHE
is observed when the Fermi energy is around the Weyl points. Furthermore, we
predict that an experimentally accessible $\mathcal{E}$-field can induce QEHE
in monolayer $\rm BaMn_{2}S_{3}$. Remarkably, the quantized Hall conductivity
can be switched by inverting the $\mathcal{E}$-field. Therefore, our work not
only proposes a new mechanism for the Hall effect but also offers an efficient
and convenient method for generating and controlling it.

_Intrinsic EHE and general analysis.–_ Consider a 2D system within the $x$-$y$
plane. The setup for the EHE proposed here is similar to that of the ordinary
Hall effect, as illustrated in Fig. 1(b). For EHE (ordinary Hall effect), the
applied electric gate-field ${\mathcal{E}}_{z}$ (magnetic field
${\mathcal{B}}_{z}$) is perpendicular to the sample, and the transverse
response current ${\bm{j}}$ is normal to the driving ${\bm{E}}$ field. The
intrinsic EHE coefficient $\chi_{ab}$ (irrelevant to relaxation time $\tau$)
to the leading order can be expressed as

$j_{a}=\chi_{ab}{\mathcal{E}}_{z}E_{b}=\sigma_{ab}E_{b},$ (1)

where $\sigma_{ab}=\chi_{ab}{\mathcal{E}}_{z}$ denotes the electric Hall
conductivity, and the indices $\\{a,b\\}\in\\{x,y\\}$ with $a\neq b$. Notice
that ${\mathcal{E}}_{z}$ and ${\bm{E}}$ are two _independent_ electric fields.
Similar to ordinary Hall effect, the Hall current in EHE vanishes when
${\mathcal{E}}_{z}=0$. However, the symmetry conditions for the existence of
EHE and the ordinary Hall are completely different, due to the distinct
symmetry constraints on ${\mathcal{E}}_{z}$ and ${\mathcal{B}}_{z}$.

From Eq. (1), one knows that for a system exhibiting EHE, it should not have a
symmetry operator that leads to $\chi_{ab}=0$, but must exhibit at least one
operator that enforces $\sigma_{ab}=0$ in the absence of ${\mathcal{E}}_{z}$.
Since the difference between $\chi_{ab}$ and $\sigma_{ab}$ is
${\mathcal{E}}_{z}$, and the mirror symmetry $M_{z}$ can reverse
${\mathcal{E}}_{z}$ while keeps the driving ${\bm{E}}$ field and $\bm{j}$
unchanged, we can decompose the magnetic layer-point-group symmetry operators
into two parts: ${\cal{O}}$ and $M_{z}{\cal{O}}$, where
${{\cal{O}}=\\{E,C_{n,z},M_{\|}\\}\otimes\\{E,\mathcal{T}\\}}$ (with
$n=2,3,4,6$) preserves ${\mathcal{E}}_{z}$ and $M_{z}{\cal{O}}$ reverse
${\mathcal{E}}_{z}$. Here, $E$ represents identity element and $M_{\|}$
denotes any mirror operator perpendicular to the plane. Under this
decomposition, it is obvious that the operators in ${\cal{O}}$ allows or
annihilate both $\sigma_{ab}$ and $\chi_{ab}$. In contrast, an operator in
$M_{z}{\cal{O}}$ that allows (annihilates) $\sigma_{ab}$ will annihilate
(allow) $\chi_{ab}$. As $\mathcal{T}$ can reverse $\bm{j}$ while keeps
$\bm{E}$ and ${\mathcal{E}}_{z}$ unchanged, ${\cal{O}}$ can be further
subdivided into ${\cal{O}}={\cal{S}}+{\cal{TS}}$, where
${\cal{S}}=\\{E,M_{\|}\mathcal{T},C_{n,z}\\}$ allows $\sigma_{ab}$ and
$\chi_{ab}$, while ${\cal{TS}}=\\{\mathcal{T},M_{\|},C_{n,z}\mathcal{T}\\}$
annihilates them. Accordingly, $M_{z}{\cal{O}}$ is separated into
$M_{z}{\cal{O}}=M_{z}{\cal{S}}+M_{z}{\cal{TS}}$, where
$M_{z}{\cal{S}}=\\{M_{z},C_{2,\|}\mathcal{T},S_{n,z}\\}$ and
$M_{z}\mathcal{T}{\cal{S}}=\\{M_{z}\mathcal{T},C_{2,\|},S_{n,z}\mathcal{T}\\}$
with $C_{2,\|}$ denoting any in-plane two-fold rotation. As earlier discussed,
since ${\cal{S}}$ permits both $\sigma_{ab}$ and $\chi_{ab}$, $M_{z}{\cal{S}}$
will also allows $\sigma_{ab}$ but annihilates $\chi_{ab}$. Similarly,
$M_{z}\mathcal{T}{\cal{S}}$ annihilates $\sigma_{ab}$ but allows $\chi_{ab}$.
Consequently, for a system hosting finite EHE, it should not have
$\mathcal{T}{\cal{S}}$ and $M_{z}{\cal{S}}$ as they can annihilates
$\chi_{ab}$, while it must have at least one symmetry in
$M_{z}\mathcal{T}{\cal{S}}$ to annihilates $\sigma_{ab}$. The symmetries in
${\cal{S}}$ are optional, as they have no influence on the existence of
$\sigma_{ab}$ and $\chi_{ab}$.

The results of the symmetry analysis are summarized in Table 1, showing that
the EHE can manifest in a wide range of magnetic materials. For example, the
${\cal{PT}}(=S_{2,z}{\cal{T}})$ symmetry in $M_{z}\mathcal{T}{\cal{S}}$
indicates that the EHE can appear in antiferromagnetic materials. Besides,
$C_{2,\|}$ can be the symmetry operator of a ferromagnetic material with in-
plane spin polarization, and that of an altermagnetic material with out-of-
plane Néel vector.

Furthermore, the magnetic materials with negligible spin-orbit coupling (SOC)
respect the spin group symmetry [52, 53, 54, 55, 56, 57], and exhibit some
emergent symmetries that are somehow counterintuitive. The most important one
is the effective $\cal{T}$ symmetry, which surprisingly exists in the
collinear or coplanar magnetic systems materials without SOC [58]. This
effective $\cal{T}$ symmetry will eliminate both anomalous Hall effect and
EHE. This means that the EHE may be sensitive to the SOC strength of the
systems, as it can be realized in the collinear or coplanar magnetic materials
with strong SOC but disappears when SOC is absent.

_Expression of intrinsic $\chi_{ab}$.–_ The expression of the intrinsic EHE
coefficient $\chi_{ab}$ can be established by perturbation theory. Without
${\mathcal{E}}_{z}$ field, the intrinsic Hall conductivity of the system is
zero. When ${\mathcal{E}}_{z}$ is finite but weak, the Hamiltonian of the
perturbed system reads

$\hat{H}=\hat{H}_{0}+g_{E}\mathcal{E}_{z}\hat{z},$ (2)

with $\hat{H}_{0}$ the unperturbed Hamiltonian, $\hat{z}$ the position
operator, and $g_{E}$ the $g$-factor-like coefficient, which is affected by
screening effect and can be determined by DFT calculation. The intrinsic Hall
conductivity of the perturbed system is obtained from the integral of Berry
curvature of all occupied bands [8, 59, 22]

$\sigma_{xy}=\frac{e^{2}}{\hbar}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}f(\tilde{\varepsilon}_{n,k})\tilde{\Omega}_{n}(\bm{k}),$
(3)

where $f$ is the Fermi-Dirac distribution, $\tilde{\varepsilon}_{n}$ and
$\tilde{\Omega}_{n}$ are the perturbed band dispersion and Berry curvature,
respectively. To the first order of ${\mathcal{E}}_{z}$, we have

$\displaystyle\tilde{\varepsilon}_{n}(\bm{k})$ $\displaystyle=$
$\displaystyle\varepsilon_{n}(\bm{k})+P_{nn}(\bm{k})\mathcal{E}_{z},$ (4)
$\displaystyle\tilde{\Omega}_{n}(\bm{k})$ $\displaystyle=$
$\displaystyle\Omega_{n}(\bm{k})+\Lambda_{n}(\bm{k})\mathcal{E}_{z},$ (5)

where $\varepsilon_{n}$ and $\Omega_{n}$ respectively are the dispersion and
Berry curvature of the original state $|u_{n}(\bm{k})\rangle$,
$P_{nm}=g_{E}\langle u_{n}(\bm{k})|\hat{z}|u_{m}(\bm{k})\rangle$ denotes the
effective layer polarization matrix element, and
$\Lambda_{n}(\bm{k})=\partial_{\mathcal{E}_{z}}\Omega_{n}(\bm{k})|_{\mathcal{E}_{z}=0}$
is the Berry curvature polarizability, expressed as

$\displaystyle\Lambda_{n}(\bm{k})$ $\displaystyle=$
$\displaystyle-2\textrm{Im}\sum_{m\neq
n}\left[-2\frac{P_{mm}-P_{nn}}{\left(\varepsilon_{m}-\varepsilon_{n}\right)^{3}}v_{x}^{nm}v_{y}^{mn}\right.$
(6) $\displaystyle+\sum_{l\neq
m}\frac{P_{lm}^{*}v_{y}^{ln}v_{x}^{nm}+P_{lm}v_{x}^{nl}v_{y}^{mn}}{\left(\varepsilon_{n}-\varepsilon_{m}\right)^{2}\left(\varepsilon_{m}-\varepsilon_{l}\right)}$
$\displaystyle\left.+\sum_{l\neq
n}\frac{P_{ln}v_{y}^{ml}v_{x}^{nm}+P_{ln}^{*}v_{x}^{lm}v_{y}^{mn}}{\left(\varepsilon_{n}-\varepsilon_{m}\right)^{2}\left(\varepsilon_{n}-\varepsilon_{l}\right)}\right],$

with $v^{nm}_{x(y)}=\langle u_{n}(\bm{k})|\hat{v}_{x(y)}|u_{m}(\bm{k})\rangle$
the velocity matrix element. Similarly, one has
$f(\tilde{\varepsilon}_{n})=f(\varepsilon_{n})+\frac{\partial
f}{\partial\varepsilon_{n}}P_{nn}(\bm{k})\mathcal{E}_{z}$. Substituting
$f(\tilde{\varepsilon}_{n})$ and $\tilde{\Omega}_{n}(\bm{k})$ into Eq. (3),
the intrinsic EHE coefficient $\chi_{xy}$ (up to the leading order) is
obtained as [58]

$\displaystyle\chi_{xy}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\hbar}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}$ (7)
$\displaystyle\times\left[\frac{\partial
f(\varepsilon_{n})}{\partial\varepsilon_{n}}P_{nn}(\bm{k})\Omega_{n}(\bm{k})+f(\varepsilon_{n})\Lambda_{n}(\bm{k})\right],$

indicating that the intrinsic EHE results from the change of the band
structure [Eq. (4)] and the Berry curvature [Eq. (5)] of the systems induced
by $\mathcal{E}_{z}$. Besides, we identify the following important features of
$\chi_{ab}$.

First, the first term in Eq. (7) is a Fermi surface property, but the second
term is calculated by the integral of all the occupied bands. Both terms are
solely determined by the intrinsic band quantities of systems, and can thus be
easily implemented in the first-principles calculations. Moreover, due to the
presence of Berry curvature and its polarizability in Eq. (7), the EHE would
be pronounced around band crossing points and the band edges of the systems
with narrow band gap.

Second, while the first term of $\chi_{xy}$ is in direct proportion to the
intraband layer polarization $P_{nn}$, the second term has contribution from
interband layer polarization $P_{nm}$ ($n\neq m$), which can be finite even if
$P_{nn}=P_{mm}=0$. This means that the appearance of EHE does not require the
bands around Fermi surface to be (intraband) layer polarized.

Third, the fully occupied bands do not make any contribution to $\chi_{xy}$,
because the intrinsic Hall conductivity of fully occupied bands in a 2D system
is a topological quantity, characterized by Chern number, which is robust
against small perturbations [22]. Thus, the $\chi_{xy}$ discussed here should
be zero for semiconductors and insulators. However, when ${\mathcal{E}}_{z}$
is strong enough to cause band inversion, a QEHE may appear as discussed
below.

At last, it would be instructive to discuss the relationship between EHE and
the recently proposed layer Hall effect [60, 61, 62, 63, 64]. Again, taking
the ordinary Hall effect as a comparison, one can find that this relationship
is completely similar to that between ordinary Hall and spin Hall effects, as
${\cal{E}}$-field and ${\cal{B}}$-field are respectively coupled to layer and
spin degrees of freedom. Thus, it is apparent that the systems having layered
Hall effect would have EHE, but not vice versa, which also can be inferred
from the second feature of $\chi_{xy}$. Besides, the EHE may appear in 2D
buckled systems, as their band and Berry curvature can be affected by
${\cal{E}}$-field. However, for such systems it is not meaningful to introduce
the layer degree of freedom, and then the layer Hall effect.

Figure 2: (a) Structure and (b) BZ of monolayer $\rm Ca(FeN)_{2}$. (c)
Orbital-projected band structure of $\rm Ca(FeN)_{2}$ with SOC. (d)
$\chi_{xy}$ as a function of $E_{F}$. The inset shows a small peak resulting
from the LWP at M point. Here $g_{E}=0.23$ is used, which is obtained from DFT
calculations.

_EHE and material candidate.–_ Guided by symmetry analysis and the expression
of EHE coefficient, we propose monolayer $\rm Ca(FeN)_{2}$ as a material
candidate hosting EHE. Monolayer $\rm Ca(FeN)_{2}$ has a square lattice
structure with optimized lattice constant $a=b=3.65$ Å [58], as shown in Fig.
2(a). Its ground state exhibits a collinear altermagnetic configuration with
Néel vector pointing out of the plane, and the magnetic moments are mainly on
the two inequivalent Fe sites with a magnitude about $\pm 3.54\mu_{B}$.
Importantly, monolayer $\rm Ca(FeN)_{2}$ belongs to magnetic layer group (MLG)
No. 414 (59.5.414), which has $S_{4z}{\cal{T}}$ symmetry satisfying the
symmetry requirement listed in Table 1. Moreover, the SOC effect in $\rm
Ca(FeN)_{2}$ is not negligible.

The electronic band structure of the monolayer $\rm Ca(FeN)_{2}$ with SOC is
plotted in Fig. 2(c), where the orbital component of the electronic states is
also presented. A key observation is that there exist two band crossings at
$\Gamma$ and $M$ points, respectively. Since both $\Gamma$ and $M$ points of
MLG No. 414 have only one 2D irreducible band representation, it is easy to
know that the two band crossing are linear Weyl points (LWPs) [65]. One can
expect that the intrinsic EHE coefficient $\chi_{xy}$ would be sizable around
these two LWPs.

Based on the first-principle calculations and Eq. (7), we calculate
$\chi_{xy}$ of the monolayer $\rm Ca(FeN)_{2}$ as a function of Fermi energy
$E_{F}$. The result is shown in Fig. 2(d), in which two peaks can be found
around $0$ and $-0.38$ eV, which are exactly the energy position of the two
LWPs at $\Gamma$ and $M$ points. As discussed above, for the collinear
magnetic systems, the EHE has a strong dependence on the SOC strength, as it
will vanish when SOC is absent. Here, the LWP at $\Gamma$ ($M$) point is
mainly composed of the $\rm Fe$ ($\rm N$) atoms [see Fig. 2(c)]. The SOC
strength of Fe atom is much stronger than that of N atoms. Thus, the peak from
the LWP at $\Gamma$ point is much more pronounced than that of $M$ point, as
shown in Fig. 2(d). Besides, due to the large local band gap at $X$ and $Y$
points, $\chi_{xy}$ is rather small at the valence band edge.

From the calculations of the intrinsic EHE coefficient, one knows that under a
${\cal{E}}_{z}$ field of $0.01$ eV/Å, the intrinsic electric Hall conductivity
can reach $\sim 0.3~{}e^{2}/h$ for lightly electron doping. This value is
comparable to the Hall conductivity in many 2D materials, and is detectable in
experiment [66, 67, 60, 68].

_QEHE and material candidate.–_ Similar to the ordinary Hall effect, the EHE
can also have its quantized version. The quantum Hall effect appears in 2D
systems that are subjected to strong ${\cal{B}}$ field [5, 6]. However, with
suitable 2D materials, the QEHE can occur with moderate ${\cal{E}}_{z}$ field.

To directly demonstrate it, we search for the materials having narrow direct
band gap and at least one symmetry operator in $M_{z}{\cal{TS}}$ (see Table
1), as the operator in $M_{z}{\cal{TS}}$ guarantees $\sigma_{ab}=0$ but is
broken by ${\cal{E}}_{z}$ field. We find that the monolayer $\rm
BaMn_{2}S_{3}$ [58] belongs to MLG No. 514 (78.5.514) and then falls into this
requirement. Monolayer $\rm BaMn_{2}S_{3}$ is structured in a triangular
lattice with lattice constant $a=b=6.74$ Å [58]. The magnetic configuration of
monolayer $\rm BaMn_{2}S_{3}$ in ground state are depicted in Fig. 3 (a) with
Néel vector oriented perpendicular to the plane. The magnetic moments
primarily located on two Mn atoms with the magnitude about $\pm 4.33\mu_{B}$.
The monolayer has $M_{z}\cal{T}$ symmetry, which forbids the intrinsic Hall
conductivity but can be broken by ${\cal{E}}_{z}$. Besides, it has $C_{3z}$
rotation symmetry, which still preserves when ${\cal{E}}_{z}\neq 0$. The band
structure of monolayer $\rm BaMn_{2}S_{3}$ with SOC is plotted in the middle
figure of Fig. 3(c), showing it is a semiconductor with a direct gap of $54.5$
meV at $\Gamma$ point.

Figure 3:  (a) Structure and (b) BZ of monolayer $\rm BaMn_{2}S_{3}$. (c) Band
structure of $\rm BaMn_{2}S_{3}$ with SOC for ${\cal{E}}_{z}=0,\pm 0.2$ eV/Å.
$C_{3,z}$ eigenvalue of states at $\Gamma$ point is highlighted in red with
$W=e^{-i\pi/3}$. (d) Intrinsic Hall conductivity $\sigma_{ab}$ (left axis) of
$\rm BaMn_{2}S_{3}$ as a function of ${\cal{E}}_{z}$ (down axis). The (10)
edge states for ${\cal{E}}_{z}=0,\pm 0.2$ eV/Å (up and right axis) is plotted
as background.

By applying ${\cal{E}}_{z}$ field and increasing its strength, the band gap of
system becomes small and closes at a moderate value
${\cal{E}}_{z}=\pm{\cal{E}}_{c}$ with ${\cal{E}}_{c}=0.135$ eV/Å, which is
experimentally accessible [69]. Further increasing $|{\cal{E}}_{z}|$, the
system becomes an insulator again [see Fig. 3(c)]. When ${\cal{E}}_{z}=0$, the
$C_{3z}$ eigenvalues of the four low-energy bands are $W=e^{-i\pi/3}$,
$W^{*}$, $-1$ and $-1$, respectively [see Fig. 3(c)]. However, when
${\cal{E}}_{z}>{\cal{E}}_{c}$ (${\cal{E}}_{z}<-{\cal{E}}_{c}$), a band
inversion between the conduction band with $C_{3z}=W$ ($C_{3z}=W^{*}$) and the
valence band with $C_{3z}=-1$ occurs. According to the theory of symmetry
index [70], this kind of band inversion indicates that the monolayer $\rm
BaMn_{2}S_{3}$ has changes from a trivial insulator to a Chern insulator with
Chern number ${\cal{C}}=\pm 1$ when ${\cal{E}}_{z}>{\cal{E}}_{c}$
(${\cal{E}}_{z}<-{\cal{E}}_{c}$). Consequently, the Hall conductivity of
system changes from $0$ to $\pm e^{2}/h$, depending on the direction of
${\cal{E}}_{z}$, and can form three Hall conductivity plateaus by varying
${\cal{E}}_{z}$, as shown in Fig. 3(d). This electric generation and control
of Hall conductivity may serve as a quantized version of EHE and the electric
counterpart of quantum Hall effect.

Similar to quantum Hall effect, the quantized Hall conductivity in QEHE
results from finite Chern number of the system and is manifested in the
appearance of the chiral edge state [29]. The spectrum of the (10) edge state
with ${\cal{E}}_{z}=0,\pm 0.2$ eV/Å is plotted as background of Fig. 3(d),
where the number and chirality of the chiral edge state are consistent with
above analysis.

_Conclusions.–_ In this work, we propose a new type of Hall effect: the EHE in
a two-dimensional magnetic system. By symmetry analysis, we show that this
previously overlooked effect can exist in a variety of magnetic materials,
including ferromagnetic, antiferromagnetic and altermagnetic materials. We
also establish an analytical expression for the intrinsic EHE coefficient, and
predict specific material candidate for EHE. In addition, based on monolayer
$\rm BaMn_{2}S_{3}$, we propose a quantized version of EHE, namely QEHE. Our
work not only broadens the concept of Hall effect, but also provides a new
mechanism for generating and manipulating Hall current by electric methods.

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# Fuzzification of Fractal Calculus

Alireza Khalili Golmankhaneh 111Corresponding author
Department of Physics, Urmia Branch, Islamic Azad University, Urmia 63896,
Iran
Kerri Welch
Faculty at California Institute of Integral Studies, San Francisco, CA, U.S.A.
Cristina Serpa
Instituto Superior de Engenharia de Lisboa (ISEL),
Instituto Politécnico de Lisboa, Lisbon, Portugal
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional
(CMAFcIO),
Faculdade de Ciências, Universidade de Lisboa, Lisbon, Portugal
Palle E. T. Jørgensen
Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419,
U.S.A

###### Abstract

In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus
in terms of fractal limit, continuity, its derivative, and integral are
formulated. The fractal fuzzy calculus is a new framework that includes
fractal fuzzy derivatives and fractal fuzzy integral. In this framework, fuzzy
number-valued functions with fractal support are the solutions of fractal
fuzzy differential equations. Different kinds of fractal fuzzy differential
equations are given and solved.

Keywords: Fractal fuzzy differential equations, fuzzy number-valued functions,
fractal fuzzy derivatives, fractal fuzzy integral
2010 Mathematics Subject Classification: 26E50, 34A07, 28A80

##  1 Introduction

Fractal geometry mathematically describes complex shapes that are not
described by Euclidean geometry [1]. These shapes are found in nature such as
clouds, mountains, lightning and etc. which are called fractals [2]. The most
important properties of fractals are self-similarity, and have non-integer
dimensions. Fractals are non-differentiable in the sense of ordinary calculus
since they have a rough structure rather smooth. Their fractal dimensions
exceed their topological dimensions and appear similar at various scales [3,
4]. Fractals have different measures like Hausdorff’s measure. In this
context, ordinary calculus which is based on length, area, and volume fails to
define derivatives and integrals on them [5].
Many researchers have tried to formulate analysis on fractals in order to
explain their physical properties [6], i.e. harmonic analysis [7, 8, 9],
measure theory [10], fractional Brownian motion and probability-theoretical
approaches [11, 12], fractional space [13], fractional calculus [14, 15].
In seminal papers, ordinary calculus was adopted to define their derivatives
and integrals of functions with fractal support, like Cantor sets and Koch
curves [16, 17, 18]. This new framework which is a generalization of ordinary
calculus is called fractal calculus or $F^{\alpha}$-calculus. Fractal calculus
is simple, constructive, and algorithmic and applied in physics [19]. Fractal
calculus was developed in different branches such as stability of solutions of
fractal differential equations, nonlocal reverse Minkowski’s fractal integral
inequalities, and properties of staircase function [20, 21, 22, 23, 24, 25].
Fractal calculus was generalized on fractal cubes and tartan Cantor spaces and
Laplace equations on fractal cubes were solved [26, 27]. Fractal derivatives
and integrals were worked out fractal interpolation functions and Weierstrass
functions [28]. Random variables, stochastic process and stable distributions
on fractal were defined, and corresponding stochastic differential equations
were solved [29, 30]. Fractal Laplace, Fourier and Sumudu transforms were
defined in order to solve fractal differential equation and applied in
electrical circuits, economy and in dynamics [31, 32, 33, 34, 35, 36, 37].

Fractal anomalous diffusion has been formulated as a diffusion process in
fractal media and it has a power law relationship between the mean squared
displacement and time [38, 39, 40].
Fuzzy sets, numbers, fuzzy-valued functions, fuzzy derivatives, and integrals
were introduced and applied to model the processes with uncertainty in
science, physical science, engineering, and social science [41, 42, 43, 44]. A
linear second-order differential equation with constant coefficients with
boundary values expressed by fuzzy numbers have been solved [45]. The fuzzy
optimal control problem has been considered to optimize the expected values of
the appropriate objective fuzzy functions [46]. The differentiability of fuzzy
number-valued functions based on the Hausdorff distance between fuzzy numbers
has been suggested [47]. First order linear fuzzy differential equations under
differential inclusions and strongly generalized differentiability approaches
have been studied [48]. Linear fuzzy differential equations appying the
concept of generalized differentiability and conditions for the existence of
solutions have been investigated [49].

A First-order fuzzy differential equation (FDE) with fuzzy initial value was
solved [50].
In this paper, we introduce a new framework which is a generalization of
fractal calculus to include fuzzy-valued functions.
The plan of the paper is as follows:
In Section 2, we summarize the fractal calculus and fuzzy calculus. Fractal
fuzzy calculus is formulated and defined in Section 3. In Section 4,
$\alpha$-order fractal fuzzy differential equations are suggested and solved.
Section 5 is devoted to conclusion.

##  2 Preliminaries

In this section we summarize the fractal calculus on fractal curves [16, 17,
18, 19].

###  2.1 Fractal calculus on fractal curves

######  Definition 1.

For a fractal curve $F$ and a subdivision
$P_{[a,b]},[a,b]\in[a_{0},b_{0}]\in\mathbb{R}$, the mass function is defined
by

$\gamma^{\alpha}(F,a,b)=\lim_{\delta\rightarrow
0}\inf_{|P|\leq\delta}\sum_{i=0}^{n-1}\frac{|\mathbf{w}(t_{i+1})-\mathbf{w}(t_{i})|^{\alpha}}{\Gamma(\alpha+1)},$
(1)

where $|.|$ denotes the Euclidean norm on $\mathbb{R}^{n}$, $1\leq\alpha\leq
n$, $P_{[a,b]}=\\{a=t_{0},...,t_{n}=b\\}$, and $|P|=\max_{0\leq i\leq
n-1}(t_{i+1}-t_{i})$ for a subdivision $P$.

######  Definition 2.

The $\gamma$-dimension of $F$ is defined by

$\displaystyle\dim_{\gamma}(F)$
$\displaystyle=\inf\\{\alpha:\gamma^{\alpha}(F,a,b)=0\\}$
$\displaystyle=\sup\\{\alpha:\gamma^{\alpha}(F,a,b)=\infty\\}$ (2)

######  Definition 3.

The rise function of a fractal curve $F$ is defined by

$S_{F}^{\alpha}(u)=\left\\{\begin{array}[]{ll}\gamma^{\alpha}(F,p_{0},u),&u\geq
p_{0};\\\ -\gamma^{\alpha}(F,u,p_{0}),&u<p_{0}.\end{array}\right.$ (3)

where $u\in[a_{0},b_{0}]$, and $S_{F}^{\alpha}(u)$ gives the mass of the
fractal curve $F$ upto point $u$.

######  Definition 4.

Let be a function $f:F\rightarrow\mathbb{R}$. Then $F$-limit of $f$ as
$\theta^{\prime}\rightarrow\theta$ through points of $F$ is $l$, if for given
$\epsilon$ there exists $\delta>$ such that

$\theta^{\prime}\in F\leavevmode\nobreak\ \leavevmode\nobreak\
and\leavevmode\nobreak\ \leavevmode\nobreak\
|\theta^{\prime}-\theta|<\delta\Rightarrow|f(\theta^{\prime})-l|<\epsilon$ (4)

or

$\underset{\theta^{\prime}\rightarrow\theta}{F_{-}lim}f(\theta^{\prime})=l.$
(5)

######  Definition 5.

A function $f:F\rightarrow\mathbb{R}$ is said to be $F$-continuous at $\theta$
if

$\underset{\theta^{\prime}\rightarrow\theta}{F_{-}lim}f(\theta^{\prime})=f(\theta).$
(6)

######  Definition 6.

The fractal derivative $F^{\alpha}$-derivative is defined by

$D_{F}^{\alpha}f(\theta)=\underset{\theta^{\prime}\rightarrow\theta}{F_{-}lim}\leavevmode\nobreak\
\frac{f(\theta^{\prime})-f(\theta)}{J(\theta^{\prime})-J(\theta)},$ (7)

where $F_{-}lim$ indicates the fractal limit (see in [18]),
$\mathbf{w}(u)=\theta$ and $S_{F}^{\alpha}(u)=J(\theta)$.

######  Remark 1.

We note that the Euclidean distance from origin upto a point
$\theta=\mathbf{w}(u)$ is given by
$L(\theta)=L(\mathbf{w}(u))=|\mathbf{w}(u)|.$

######  Definition 7.

The fractal integral or $F^{\alpha}$-integral is defined by

$\displaystyle\int_{C(a,b)}f(\theta)d_{F}^{\alpha}\theta$
$\displaystyle=\sup_{P[a,b]}\sum_{i=0}^{n-1}\inf_{\theta\in
C(t_{i},t_{i+1})}f(\theta)(J(\theta_{i+1})-J(\theta_{i}))$
$\displaystyle=\inf_{P[a,b]}\sum_{i=0}^{n-1}\sup_{\theta\in
C(t_{i},t_{i+1})}f(\theta)(J(\theta_{i+1})-J(\theta_{i})),$ (8)

where $t_{i}=\mathbf{w}^{-1}(\theta_{i})$, and $C(a,b)$ is the section of the
curve lying between points $\mathbf{w}(a)$ and $\mathbf{w}(b)$ on the fractal
curve $F$ [18].

###  2.2 Fuzzy calculus on real-line

In this section, we review fuzzy calculus which will be used to fuzzification
of the fractal calculus [41, 42, 43, 44].
A generalized Hukuhara difference for fuzzy sets and a new generalized
differentiability concepts for fuzzy valued functions were given in [51, 52].

######  Definition 8.

Let $X\neq\emptyset$. Then, a set $A\subset X$ is characterized by its
membership function $u_{A}(x):X\rightarrow[0,1]$. Thus $u_{A}(x)$ is the
degree of membership of element $x$ in the fuzzy set $A$ for each $x\in X$.

######  Definition 9.

Let $A$ be a fuzzy subset of a real number
$u_{A}(x):\mathbb{R}\rightarrow[0,1]$. Then $A$ is called fuzzy number if it
satisfies the following axioms

  1. 1.

$A$ is normal. It means that there exists $x_{0}$ in $\mathbb{R}$, such that
$u_{A}(x_{0})=1$.

  2. 2.

$A$ is convex, namely,

$u_{A}(tx+(1-t)y)\geq\min\\{u_{A}(x),u_{A}(y)\\},\leavevmode\nobreak\ \forall
t\in[0,1],\leavevmode\nobreak\ x,\leavevmode\nobreak\ y\in\mathbb{R}.$ (9)

  3. 3.

$u_{A}(x)$ is upper semi continuous on $\mathbb{R}$, viz, for given
$\epsilon>0$, there exists $\delta>0$ such that

$|x-x_{0}|<\delta\Rightarrow u_{A}(x)-u_{A}(x_{0})<\epsilon.$ (10)

  4. 4.

The support of $u_{A}(x)$ is compact. e.g.

$supp(u_{A}(x))=cl_{\mathbb{R}}\\{x\in\mathbb{R};u_{A}(x)>0\\}$ (11)

is compact.

######  Definition 10.

A fuzzy number $A$ is determined by a pair of functions
$A=(A^{-}(r),A^{+}(r))$, with $A^{-}(r),A^{+}(r):[0,1]\rightarrow\mathbb{R}$
that satisfies the following condition:

  1. 1.

$A^{-}(r)=A_{r}^{-}\in\mathbb{R}$ is a bounded, monotonic, increasing, left
continuous function in $(0,1]$ and it is right-continuous at $0$.

  2. 2.

$A^{+}(r)=A_{r}^{+}\in\mathbb{R}$ is a bounded, monotonic, decreasing, left
continuous function in $(0,1]$ and it is right-continuous at $0$.

  3. 3.

For $r\in(0,1]$ we have $A^{-}(r)\leq A^{+}(r)$.

The Definition 10 is called parametric form of fuzzy numbers.

######  Definition 11.

The $r$-cut of fuzzy number $A$ is defined and called level wise form by

$[A]_{r}=A_{r}=\\{x\in\mathbb{R};u_{A}(x)\geq r\\}$ (12)

where $A_{r}$ is closed interval $A_{r}=[A_{r}^{-},A_{r}^{+}]$ for any
$r\in[0,1]$. We note that $[A]_{0}=supp(u_{A}(x))$ and $F_{\mathbb{R}}$ denote
space of fuzzy number.

######  Definition 12.

For every $A,B\in F_{\mathbb{R}}$ and
$\lambda\in\mathbb{R},\leavevmode\nobreak\ r\in[0,1]$ the addition and scalar
multiplication is defined by

$(A\oplus B)_{r}=A_{r}+B_{r},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\lambda\odot A)_{r}=\lambda A_{r}.$ (13)

######  Definition 13.

The Hausdorff distance between two fuzzy numbers $A,\leavevmode\nobreak\ B$
using their $r$-cuts is defined by

$d_{H}(A,B)=\sup_{0\leq r\leq
1}\max\\{|A_{r}^{-}-B_{r}^{-}|,|A_{r}^{+}-B_{r}^{+}|\\}$ (14)

######  Remark 2.

The set of fuzzy numbers $(F_{\mathbb{R}},d)$ with addition and scalar
multiplication given in Definition 12, is a complete metric space.

######  Definition 14.

Consider $A,\leavevmode\nobreak\ B\in F_{\mathbb{R}}$. The Hukuhara difference
of $A,B$ is defined by

$C=A\ominus B,$ (15)

if $A=B\oplus C$.

######  Definition 15.

Consider a fuzzy number valued function $f:\mathbb{R}\rightarrow
F_{\mathbb{R}}$ and $x_{0}\in\mathbb{R}$, then $l$ is called limit of $f$ at
point $x_{0}$ if for every given $\epsilon>0$, there exist $\delta>0$ such
that [43, 53, 42]

$0<|x-x_{0}|<\delta\Rightarrow d_{H}(f(x),l)<\epsilon,$ (16)

or,

$\lim_{x\rightarrow x_{0}}f(x)=l,$ (17)

if it exists, where $d_{H}$ is the Hausdorff distance.

######  Definition 16.

The fuzzy function $f$ is called fuzzy continuous if [43, 53]

$\lim_{x\rightarrow x_{0}}f(x)=f(x_{0}).$ (18)

######  Definition 17.

A fuzzy number valued function $f:\mathbb{R}\rightarrow F_{\mathbb{R}}$ is
called Hukuhara differentiable if there exist $f^{\prime}(x)\in
F_{\mathbb{R}}$ such that

  * •

Case 1.($I$-differentiable)

$f^{\prime}(x)=\lim_{y\rightarrow x}\frac{f(y)\ominus
f(x)}{y-x},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ y>x$
(19)

  * •

Case 2.($II$-differentiable)

$f^{\prime}(x)=\lim_{y\rightarrow x}\frac{f(x)\ominus
f(y)}{y-x},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ y>x$
(20)

where $f^{\prime}(x)$ is called the fuzzy derivative of $f$ at $x$.

######  Theorem 1.

Let a fuzzy number valued function $f:\mathbb{R}\rightarrow F_{\mathbb{R}}$ be
denoted by $f(x)=(\underline{f}(x,r),\overline{f}(x,r))$ for each $r\in[0,1]$
[54]. Then [55, 49]

  1. 1.

If $f$ is $I$-differentiable, then we have

$f^{\prime}(x)=(\underline{f}^{\prime}(x,r),\overline{f}^{\prime}(x,r)).$ (21)

  2. 2.

If $f$ is $II$-differentiable, then we have

$f^{\prime}(x)=(\overline{f}^{\prime}(x,r),\underline{f}^{\prime}(x,r)).$ (22)

######  Definition 18.

Let $f(x)$ be a fuzzy number-valued function. Then the fuzzy Riemann integral
is defined as [42]

$J=FR\int_{a}^{b}f(x)dx=\oplus\sum_{i=0}^{n}\Delta x_{i}\odot f(x_{i}),$ (23)

where $\Delta x_{i}=x_{i+1}-x_{i}$ and $\\{a=x_{0}<x_{1}<...<x_{n}=b\\}$ is a
partition of $I=[a,b]$.
The fuzzy Riemann integral of $f(x)$ is $J$ if for every given $\epsilon>0$,
there exist $\delta>0$ such as

$d_{H}\bigg{(}\oplus\sum_{i=0}^{n}\Delta x_{i}\odot
f(x_{i}),J\bigg{)}<\epsilon.$ (24)

where $J$ is a fuzzy number.

######  Definition 19.

Let $f:I\rightarrow F_{\mathbb{R}}$ be a triangular number-valued function and
$f(x)=(f_{1}(x),f_{2}(x),f_{3}(x))$ and $x_{0}\in I$. Then the fuzzy integral
is defined by [53]

$\int_{a}^{b}f(x)dx=\bigg{(}\int_{a}^{b}f_{1}(x)dx,\int_{a}^{b}f_{2}(x)dx,\int_{a}^{b}f_{3}(x)dx\bigg{)}$
(25)

##  3 Fuzzy fractal calculus on fractal curves

In this section, we introduce fractal fuzzy calculus.

######  Definition 20.

Let $f(\theta):F\rightarrow F_{\mathbb{R}}$ be a number-valued function on a
fractal curve $F$. Then the fuzzy $F$-limit of $f$ at $\theta_{0}$ through $F$
is $l$, if for a given $\epsilon>0$, there exist $\delta>0$, such that

$\theta\in F,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
|\theta-\theta_{0}|<\delta\Rightarrow d_{H}(f(\theta),l)<\epsilon,$ (26)

or

$\underset{\theta\rightarrow\theta_{0}}{FF_{-}lim}f(\theta^{\prime})=l$ (27)

where $d_{H}$ is the Hausdorff distance.

######  Definition 21.

Let $f(\theta):F\rightarrow F_{\mathbb{R}}$ be a number-valued function on
$F$. Then, $f$ is called fuzzy $F$-continuous if

$\underset{\theta\rightarrow\theta_{0}}{FF_{-}lim}f(\theta)=f(\theta_{0}).$
(28)

######  Definition 22.

Let $f(\theta):F\rightarrow F_{\mathbb{R}}$ be a number-valued function. Then
fractal Hukuhara differentive at $\theta_{0}\in F$ is defined by

  * •

Case 1. ($I$-$F^{\alpha}$-differentiable)

$D_{F,H}^{\alpha}f(\theta_{0})=\underset{\theta\rightarrow\theta_{0}}{FF_{-}lim}\leavevmode\nobreak\
\frac{f(\theta)\ominus
f(\theta_{0})}{J(\theta)-J(\theta_{0})},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \theta>\theta_{0}.$ (29)

  * •

Case 2. ($II$-$F^{\alpha}$-differentiable)

$D_{F,H}^{\alpha}f(\theta_{0})=\underset{\theta\rightarrow\theta_{0}}{FF_{-}lim}\leavevmode\nobreak\
\frac{f(\theta_{0})\ominus
f(\theta)}{J(\theta)-J(\theta_{0})},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \theta>\theta_{0}.$ (30)

where $D_{F,H}^{\alpha}f(\theta_{0})$ is a fuzzy number.

######  Definition 23.

Let $f(x)$ be a fractal fuzzy number valued function. Then the fractal fuzzy
Riemann integral is defined as [42]

$J=FFR\int_{C(a,b)}f(\theta)d_{F}^{\alpha}\theta=\oplus\sum_{i=0}^{n}\Delta
J_{i}\odot f(\theta_{i}),$ (31)

The fractal fuzzy Riemann integral of $f(\theta)$ is $J$ if for a given
$\epsilon>0$, there exist $\delta>0$ such as

$d_{H}\bigg{(}\oplus\sum_{i=0}^{n}\Delta J_{i}\odot
f(\theta_{i}),J\bigg{)}<\epsilon.$ (32)

######  Definition 24.

Let $f:F\rightarrow F_{\mathbb{R}}$ be a fractal triangular number-valued
function, $f(\theta)=(f_{1}(\theta),f_{2}(\theta),f_{3}(\theta))$, and
$\theta_{0}\in F$, then

$\int_{C(a,b)}f(\theta)d_{F}^{\alpha}\theta=\bigg{(}\int_{C(a,b)}f_{1}(\theta)d_{F}^{\alpha}\theta,\int_{C(a,b)}f_{2}(\theta)d_{F}^{\alpha}\theta,\int_{C(a,b)}f_{3}(\theta)d_{F}^{\alpha}\theta\bigg{)}.$
(33)

##  4 Fractal fuzzy differential equations

First order linear fuzzy differential equations by using the generalized
differentiability concept were solved [55, 49]. In this section, a
$\alpha$-order fuzzy differential equation (F.D.E) is given. Then it is
changed by its equivalent parametric form, and a new system, which contains
two fractal differential equations, is solved.
Consider the following fractal fuzzy differential equation with initial
condition:

$D_{F,H}^{\alpha}x(\theta)=f(J(\theta),x(\theta)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{x}(\theta_{0})=\tilde{x}_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \theta\in F,$ (34)

where $f:F\times F_{\mathbb{R}}\rightarrow F_{\mathbb{R}}$ is a fuzzy-valued
function and $\tilde{x}_{0}\in F_{\mathbb{R}}$. To solve Eq.(34), first we
solve $1$-cut and $0$-cut of Eq.(34) as the following form

$\left\\{\begin{array}[]{ll}(D_{F,H}^{\alpha}x)^{[1]}(\theta)=f^{[1]}(J(\theta),x(\theta)),&\\\
&\\\ x^{[1]}(\theta_{0})=\tilde{x}_{0}^{[1]}\leavevmode\nobreak\
\leavevmode\nobreak\ \theta_{0}\in[0,\Theta].&\end{array}\right.$ (35)

and

$\left\\{\begin{array}[]{ll}(D_{F,H}^{\alpha}x)^{[0]}(\theta)=f^{[0]}(J(\theta),x(\theta)),&\\\
&\\\ x^{[0]}(\theta_{0})=\tilde{x}_{0}^{[0]}\leavevmode\nobreak\
\leavevmode\nobreak\ \theta_{0}\in[0,\Theta].&\end{array}\right.$ (36)

Then by solving Eqs.(35) and (36), we can find $\tilde{x}(\theta)$ which is
the solution of the fractal fuzzy differential equation Eq.(34).
Here we consider two cases:
Case (I): Suppose that $\tilde{x}(\theta)$ is $I$-$F^{\alpha}$-differentiable.
Then, we can write

$D_{F,H}^{\alpha}x(\theta)=[D_{F,H}^{\alpha}\underline{x}(\theta,r),D_{F,H}^{\alpha}\overline{x}(\theta,r)].$
(37)

In view of Eqs.(34) and (37) for $r\in[0,1]$, we have

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x}(\theta,r)=\underline{f}(\theta,r)&\theta_{0}\leq\theta\leq\Theta\\\
&\\\
D_{F,H}^{\alpha}\overline{x}(\theta,r)=\overline{f}(\theta,r)&\theta_{0}\leq\theta\leq\Theta.\end{array}\right.$
(38)

Hence

$\left\\{\begin{array}[]{ll}[(1-r)D_{F,H}^{\alpha}\underline{x^{[0]}}(\theta)+rD_{F,H}^{\alpha}\underline{x^{[1]}}(\theta)=(1-r)\underline{f^{[0]}}(\theta)+r(\underline{f^{[1]}})(\theta),&\\\
&\\\
(1-r)D_{F,H}^{\alpha}\overline{x^{[0]}}(\theta)+rD_{F,H}^{\alpha}\overline{x^{[1]}}(\theta)=(1-r)\overline{f^{[0]}}(\theta)+r(\overline{f^{[1]}})(\theta),&\\\
&\\\
\underline{x}(\theta_{0},r)=(1-r)\underline{x^{[0]}}(\theta_{0})+r\underline{x^{[1]}}(\theta_{0}),&\\\
&\\\
\overline{x}(\theta_{0},r)=(1-r)\overline{x^{[0]}}(\theta_{0})+r\overline{x^{[1]}}(\theta_{0}).\end{array}\right.$
(39)

It follows that

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[0]}}(\theta)=\underline{f^{[0]}}(\theta),&\\\
&\\\ D_{F,H}^{\alpha}\overline{x^{[0]}}(\theta)=\overline{f^{[0]}}(\theta)&\\\
&\\\ \underline{x^{[0]}}(\theta_{0})=\underline{x_{0}}^{[0]}&\\\ &\\\
\overline{x^{[0]}}(\theta_{0})=\overline{x_{0}}^{[0]},\end{array}\right.$ (40)

and

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[1]}}(\theta)=\underline{f^{[1]}}(\theta),&\\\
&\\\ D_{F,H}^{\alpha}\overline{x^{[1]}}(\theta)=\overline{f^{[1]}}(\theta)&\\\
&\\\ \underline{x^{[1]}}(\theta_{0})=\underline{x_{0}}^{[1]}&\\\ &\\\
\overline{x^{[1]}}(\theta_{0})=\overline{x_{0}}^{[1]},\end{array}\right.$ (41)

One can find
$\underline{x^{[0]}}(\theta),\overline{x^{[0]}}(\theta),\underline{x^{[1]}}(\theta),\overline{x^{[1]}}(\theta)$
by solving Eqs.(40) and (41). Therefore we obtain the solution of Eq.(34)
using $0$-cut and $1$-cut solutions as follows:

$\tilde{x}(\theta)=[\underline{x}(\theta,r),\overline{x}(\theta,r)]=[(1-r)\underline{x^{[0]}}(\theta)+r\underline{x^{[1]}}(\theta),(1-r)\overline{x^{[0]}}(\theta)+r\overline{x^{[1]}}(\theta)].$
(42)

Case (II). Let $\tilde{x}(\theta)$ be $II$-$F^{\alpha}$-differentiable. Then,
we can write

$D_{F,H}^{\alpha}x(\theta)=[D_{F,H}^{\alpha}\overline{x}(\theta,r),D_{F,H}^{\alpha}\underline{x}(\theta,r)].$
(43)

Likewise, Case (I), we have

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[0]}}(\theta)=\overline{f^{[0]}}(\theta),&\\\
&\\\
D_{F,H}^{\alpha}\overline{x^{[0]}}(\theta)=\underline{f^{[0]}}(\theta)&\\\
&\\\ \underline{x^{[0]}}(\theta_{0})=\overline{x_{0}}^{[0]}&\\\ &\\\
\overline{x^{[0]}}(\theta_{0})=\underline{x_{0}}^{[0]},\end{array}\right.$
(44)

and

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[1]}}(\theta)=\overline{f^{[1]}}(\theta),&\\\
&\\\
D_{F,H}^{\alpha}\overline{x^{[1]}}(\theta)=\underline{f^{[1]}}(\theta)&\\\
&\\\ \underline{x^{[1]}}(\theta_{0})=\overline{x_{0}}^{[1]}&\\\ &\\\
\overline{x^{[1]}}(\theta_{0})=\underline{x_{0}}^{[1]},\end{array}\right.$
(45)

By solving the ordinary fractal differential equations (44)) and (45), one may
obtain the solution of FDE (34) which is $II$-$F^{\alpha}$-differentiable as

$\tilde{x}(\theta)=[\overline{x}(\theta,r),\underline{x}(\theta,r)].$ (46)

######  Example 1.

Consider the fractal fuzzy differential equation as

$D_{F,H}^{\alpha}x(\theta)=x(\theta)+\tilde{c},$ (47)

with the conditions

$x(0,r)=[r,2-r],\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{c}=[r-1,1-r],r\in[0,1].$ (48)

Here we consider two cases.
Case I. Let $\tilde{x}(\theta)$ be $I$-$F^{\alpha}$-differentiable. Then by
using Eqs.(40) and (41) we arrive at

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[0]}}(\theta)=\underline{x^{[0]}}(\theta)-1&\\\
&\\\
D_{F,H}^{\alpha}\overline{x^{[0]}}(\theta)=\overline{x^{[0]}}(\theta)+1&\\\
&\\\ \underline{x^{[0]}}(\theta_{0})=0&\\\ &\\\
\overline{x^{[0]}}(\theta_{0})=2\end{array}\right.$ (49)

and

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[1]}}(\theta)=\underline{x^{[1]}}(\theta)&\\\
&\\\ D_{F,H}^{\alpha}\overline{x^{[1]}}(\theta)=\overline{x^{[1]}}(\theta)&\\\
&\\\ \underline{x^{[1]}}(\theta_{0})=1&\\\ &\\\
\overline{x^{[1]}}(\theta_{0})=1\end{array}\right.$ (50)

By solving Eqs.(49) and (50), we obtain

$\displaystyle\underline{x^{[0]}}(\theta)$
$\displaystyle=-\exp(J(\theta))+1,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \overline{x^{[0]}}(\theta)=3\exp(J(\theta))-1$
$\displaystyle\underline{x^{[1]}}(\theta)$
$\displaystyle=\exp(J(\theta))+1,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \overline{x^{[1]}}(\theta)=\exp(J(\theta))$ (51)

By substituting Eq.(1) into Eq.(42), we get

$\tilde{x}(\theta)=[\exp(J(\theta))(2r-1)-r+1,r-\exp(J(\theta))(2r-3)-1].$
(52)

Case II. Let $\tilde{x}(\theta)$ be $II$-$F^{\alpha}$-differentiable. Then, by
utilizing Eqs.(44) and (45) we get

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[0]}}(\theta)=\overline{x^{[0]}}(\theta)+1,&\\\
&\\\
D_{F,H}^{\alpha}\overline{x^{[0]}}(\theta)=\underline{x^{[0]}}(\theta)-1&\\\
&\\\ \underline{x^{[0]}}(\theta_{0})=0&\\\ &\\\
\overline{x^{[0]}}(\theta_{0})=2,\end{array}\right.$ (53)

and

$\left\\{\begin{array}[]{ll}D_{F,H}^{\alpha}\underline{x^{[1]}}(\theta)=\overline{x^{[1]}}(\theta),&\\\
&\\\
D_{F,H}^{\alpha}\overline{x^{[1]}}(\theta)=\underline{x^{[1]}}(\theta)&\\\
&\\\ \underline{x^{[1]}}(\theta_{0})=1&\\\ &\\\
\overline{x^{[1]}}(\theta_{0})=1,\end{array}\right.$ (54)

Solving Eqs.(53) and (54) gives

$\displaystyle\underline{x^{[0]}}(\theta)$
$\displaystyle=\exp(J(\theta))-r+\frac{(2r-2)}{\exp(J(\theta))}+1,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{x^{[0]}}(\theta)=r+\exp(J(\theta))-\frac{(2r-2)}{\exp(J(\theta))}-1$
$\displaystyle\underline{x^{[1]}}(\theta)$
$\displaystyle=\exp(J(\theta)),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \hskip
99.58464pt\overline{x^{[1]}}(\theta)=\exp(J(\theta)).$ (55)

By substituting Eq.(1) into Eq.(46), we find the solution of Eq.(47) as
follows

$\tilde{x}(\theta)=\bigg{[}\exp(J(\theta))-r+\frac{2r-2}{\exp(J(\theta))}+1,r+\exp(J(\theta))-\frac{2r-2}{\exp(J(\theta))}-1\bigg{]}.$
(56)

In Figure 1, we have plotted Eq.(56) for the case of $r=0.3$.

Figure 1: Graph of Eq.(56) for $r=0.3$

######  Example 2.

Consider the fractal differential equation with fuzzy boundary condition:

$\left\\{\begin{array}[]{ll}(D_{F}^{\alpha})^{2}x(\theta)-4D_{F}^{\alpha}x(\theta)+4x(\theta)=1-2J^{2}(\theta),&\\\
x(0)=(2,3,4),&\\\ x(1)=(1,2,2.5),\end{array}\right.$ (57)

Let the solution be as $x(\theta)=x_{cr}(\theta)+\tilde{x}_{un}(\theta)$,
where $x_{cr}$ is the crisp part of the solution and $\tilde{x}_{un}(\theta)$
is the uncertainty part [45]. First, we solve the following equation

$\left\\{\begin{array}[]{ll}(D_{F}^{\alpha})^{2}x(\theta)-4D_{F}^{\alpha}x(\theta)+4x(\theta)=1-2J^{2}(\theta),&\\\
x(0)=3,&\\\ x(1)=2,\end{array}\right.$ (58)

The crisp solution of Eq.(58) is

$x_{cr}(\theta)=-\frac{1}{2}(J(\theta)+1)^{2}+3.5(1-J(\theta))\exp(2J(\theta))+4J(\theta)\exp(2(J(\theta)-1))$
(59)

To determine the $\tilde{x}_{un}(\theta)$, let us consider the fractal fuzzy
homogeneous equations:

$\left\\{\begin{array}[]{ll}(D_{F}^{\alpha})^{2}x(\theta)-4D_{F}^{\alpha}x(\theta)+4x(\theta)=0,&\\\
x(0)=(-1,0,1),&\\\ x(1)=(-1,0,0.5)\end{array}\right.$ (60)

The linear independent solutions of Eq.(60) are
$x_{1}(\theta)=\exp(2J(\theta))$, and
$x_{2}(\theta)=J(\theta)\exp(2J(\theta))$

Then we have

$\displaystyle\mathbf{p}=(\exp(2J(\theta)),J(\theta)\exp(2J(\theta))),\leavevmode\nobreak\
\leavevmode\nobreak\ \mathcal{M}=\left(\begin{array}[]{cc}1&0\\\
e^{2}&e^{2}\\\ \end{array}\right),$ (63) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{q}=\mathbf{p}\mathcal{M}^{-1}=((1-J(\theta))\exp(2J(\theta)),J(\theta)\exp(2(J(\theta)-1)).$
(64)

Thus we obtain

$\tilde{x}_{un}(\theta)=(1-J(\theta))\exp(2J(\theta))(-1,0,1)+J(\theta)\exp(2(J(\theta)-1))(-1,0,0.5).$
(65)

To represent $\tilde{x}_{un}(\theta)$ by $\kappa$-cuts, we can write

$\tilde{x}_{un,\kappa}(\theta)=(1-\kappa)[\underline{x_{un,0}}(\theta),\overline{x_{un,0}}(\theta)].$
(66)

The $\kappa$-cuts representation solution of Eq.(57) is

$x_{\kappa}(\theta)=x_{cr}(\theta)+(1-\kappa)[\underline{x_{un,0}}(\theta),\overline{x_{un,0}}(\theta)].$
(67)

##  5 Conclusion

In this paper, we have formulated the fractal fuzzy calculus which is a
generalization of fractal calculus on fuzzy number-valued functions. Fractal
calculus is a generalization of ordinary calculus which involves functions
with a fractal domain such as Cantor set and Koch curves. Fractal fuzzy
differential equations can be used to model uncertainty in the initial
condition or dynamic of media with a fractal structure. The research in this
direction is in progress.

Acknowledgements: Cristina Serpa acknowledges partial funding by national
funds through FCT-Foundation for Science and Technology, project reference:
UIDB/04561/2020.

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