# Dynamic generation of spin spirals of moiré trapped carriers via exciton
mediated spin interactions

Chengxin Xiao  Department of Physics, The University of Hong Kong, Hong Kong,
China  HKU-UCAS Joint Institute of Theoretical and Computational Physics at
Hong Kong, China  Yong Wang  School of Physics, Nankai University, Tianjin,
China  Department of Physics, The University of Hong Kong, Hong Kong, China
Wang Yao<EMAIL_ADDRESS>Department of Physics, The University of Hong Kong,
Hong Kong, China  HKU-UCAS Joint Institute of Theoretical and Computational
Physics at Hong Kong, China

###### Abstract

Stacking transition metal dichalcogenides (TMDs) to form moiré superlattices
has provided exciting opportunities to explore many-body correlation phenomena
of the moiré trapped carriers. TMDs bilayers, on the other hand, host long-
lived interlayer exciton (IX), an elementary excitation of long spin-valley
lifetime that can be optically or electrically injected. Here we find that,
through the Coulomb exchange between mobile IXs and carriers, the IX bath can
mediate both Heisenberg and Dzyaloshinskii-Moriya type spin interactions
between moiré trapped carriers, controllable by exciton density and exciton
spin current respectively. We show the strong Heisenberg interaction, and the
extraordinarily long-ranged Dzyaloshinskii-Moriya interaction here can jointly
establish robust spin spiral magnetic orders in Mott-Wigner crystal states at
various filling factors, with spiral direction controlled by exciton current.

Introduction – Stacking transition metal dichalcogenides (TMDs) monolayers to
form moiré superlattice has provided an arena to explore many-body correlation
phenomena of the spin-valley locked massive Dirac fermions Regan  _et al._
(2020); Xu  _et al._ (2020); Tang  _et al._ (2020); Huang  _et al._ (2021);
Miao  _et al._ (2021); Liu  _et al._ (2021); Shimazaki  _et al._ (2020); Zhou
_et al._ (2021); Ghiotto  _et al._ (2021); Li  _et al._ (2021). Experiments
have discovered correlated insulating states at various integer and fractional
fillings Regan  _et al._ (2020); Xu  _et al._ (2020); Tang  _et al._ (2020);
Huang  _et al._ (2021); Miao  _et al._ (2021); Liu  _et al._ (2021); Shimazaki
_et al._ (2020); Zhou  _et al._ (2021), attributed to charge ordering by
strong Coulomb interaction in the moiré energy landscape, namely Mott and
generalized Wigner crystal states. Their magnetic properties have also been
explored using magneto-optical measurements in WSe2/WS2 heterobilayers Tang
_et al._ (2020); Wang  _et al._ (2022). Super-paramagnetic response is
observed at integer hole filling Tang  _et al._ (2020); Wang  _et al._ (2022),
attributed to antiferromagnetic coupling between nearest-neighbour moiré sites
Tang  _et al._ (2020). Remarkably, reflective magnetic circular dichroism
(RMCD) measurement shows evidences of spontaneous ferromagnetic order at
fractional hole filling when and only when the optical excitation power
exceeds a modest threshold Wang  _et al._ (2022), implying spin interaction
enabled by the optical excitation Piermarocchi  _et al._ (2002); Fernández-
Rossier  _et al._ (2004).

Another topic of intensive interest in TMDs heterostructures is the long-
living interlayer excitons (IXs) with electron and hole constituents separated
to adjacent layers Rivera  _et al._ (2015, 2016); Unuchek  _et al._ (2018);
Seyler  _et al._ (2019); Tran  _et al._ (2019); Jauregui  _et al._ (2019);
Karni  _et al._ (2019); Brotons-Gisbert  _et al._ (2020); Li  _et al._ (2020);
Bai  _et al._ (2020); Tan  _et al._ (2021). In the type-II band alignment,
optical excitation can establish large population and spin-valley polarization
of these low energy excitons, utilizing valley selection rule in individual
layer Xiao  _et al._ (2012), followed by ultrafast interlayer charge transfer
Ceballos  _et al._ (2014); Lee  _et al._ (2014). And ultralong lifetime and
spin-valley lifetime of IXs are observed Rivera  _et al._ (2016); Tan  _et
al._ (2021). Like the charge carriers, IXs can also experience a periodic
potential from the moiré landscape. Experiments in MoSe2/WSe2 moiré show that
at very low excitation power and low temperature, IX can get trapped by the
moiré potential, exhibiting quantum dot like narrow resonances in
photoluminescence Seyler  _et al._ (2019); Brotons-Gisbert  _et al._ (2020);
Li  _et al._ (2020). At modest excitation or higher temperature (30K), the
narrow-line emission is quenched and dominated by broad peaks, implying that
traps are rather shallow and IXs become mobile under these conditions Seyler
_et al._ (2019); Bai  _et al._ (2020). The ability to drive IX flow by
electrical control has been demonstrated Unuchek  _et al._ (2018); Liu  _et
al._ (2020). Moreover, condensation of electrically injected IXs Wang  _et
al._ (2019), and equilibrium IXs in excitonic insulating states Chen  _et al._
(2022); Zhang  _et al._ (2022) have also been reported in TMDs
heterostructures.

Here we discover a novel form of spin-spin interaction between moiré trapped
carriers through the mediation by a bath of mobile IXs in bilayer TMDs. The
exchange between trapped electron (hole) and the electron (hole) constituent
of IX forms the basis of a Ruderman-Kittel-Kasuya-Yosida (RKKY) type
interaction between trapped carriers through the mediation by mobile IXs. With
an IX bath in Bose distribution, the mediated interaction between trapped
carriers has a Heisenberg form with interaction range and strength controlled
by density and temperature of IXs, capable of establishing ferromagnetic order
at carrier-carrier distance up to $10-20$ nm as observed in Ref. Wang  _et
al._ (2022). Non-equilibrium part of IX distribution that corresponds to an
excitonic spin current can further introduce Dzyaloshinskii-Moriya interaction
(DMI) of extraordinarily long range, practically limited by exciton’s
coherence length only. By this long-range and anisotropic DMI, albeit weak,
the ferromagnetic order gets pinned in-plane and spirals along the excitonic
current direction with sub-micron wavelength, where the critical temperature
of forming the magnetic spiral is determined by the strong Heisenberg part.
Our finding points to an exciting possibility to dynamically establish and
manipulate magnetization textures in Mott-Wigner crystal states in moiré.

Figure 1: (a) Through the mediation by mobile interlayer excitons (IX), a
Heisenberg interaction $\mathcal{J}\bm{S}_{i}\cdot\bm{S}_{j}$ is realized for
moiré trapped holes (red spheres on triangular lattice sites). Wavy lines
denote Coulomb exchange between trapped holes and the hole constituent of IX.
(b) In presence of an IX spin current, a long-range Dzyaloshinskii-Moriya
interaction $\bm{\mathcal{D}}\cdot(\bm{S}_{i}\times\bm{S}_{j})$ is also
introduced between the trapped holes, with an out-of-plane $\bm{\mathcal{D}}$.
The schematic illustrates optical injection of spin polarized IX whose lateral
motion is driven by electrical current.

We consider a TMD heterobilayer doped with holes, which are trapped in the
triangular moiré superlattice potential formed in the top layer due to the
twisting and/or lattice mismatch (Fig. 1). A bath of spin polarized mobile IXs
coexist in the system either by optical or electrical injection. The trapped
hole can exchange with the hole constituent of the IX residing in their common
layer, leading to a hole-exciton interaction dependent on their spin
configurations. The Hamiltonian for the trapped holes and mobile IXs can be
generally written as,

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k}\sigma_{e}\sigma}\epsilon_{\bm{k}\sigma_{e}\sigma}a_{\bm{k}\sigma_{e}\sigma}^{\dagger}a_{\bm{k}\sigma_{e}\sigma}+\sum_{j\sigma}\epsilon_{d}d_{j\sigma}^{\dagger}d_{j\sigma},$
$\displaystyle H_{v}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}j\bm{k}\bm{k}^{\prime}\sigma\sigma^{\prime}\sigma_{e}\end{subarray}}Ie^{-i(\bm{k}-\bm{k}^{\prime})\cdot\bm{R}_{j}}d^{\dagger}_{j\sigma^{\prime}}a_{\bm{k}\sigma_{e}\sigma}^{\dagger}a_{\bm{k}^{\prime}\sigma_{e}\sigma^{\prime}}d_{j\sigma}.$

$a^{\dagger}_{\bm{k}\sigma_{e}\sigma}$ creates an IX with center-of-mass
momentum $\bm{k}$ and spin indices $\sigma_{e}$ and $\sigma$ for its electron
and hole constituents respectively, and $d^{\dagger}_{j\sigma}$ creates a
trapped hole of spin index $\sigma$ at lattice site $j$. $H_{v}$ describes the
Coulomb exchange between the IXs and trapped holes. The magnitude and sign of
$I$ are determined by the competition of three scattering channels Combescot
_et al._ (2008); Shiau  _et al._ (2012), depending on the wavefunction overlap
of IX and trapped hole (details in Supplementary supplementary ).

Figure 2: Spin interactions controlled by density ($n$) and spin current of
the exciton bath. (a) The isotropic Heisenberg interaction strength
$\mathcal{J}(\bm{r})$, with the exciton bath in Bose distribution
$f_{0}^{n,T}$ of various density $n$ at $T=30$K. (b, c) The Dzyaloshinskii-
Moriya interaction set on by an exciton spin current, the latter described by
a shifted distribution $f({\bm{k}})=f_{0}^{n,T}({\bm{k}}+\delta\bm{k})$, with
$n=10^{11}$cm-2 and $T=30$K. In (b), the anisotropic $\mathcal{D}(\bm{r})$ is
shown for a shift in the $x$ direction by $\delta k=2\mu$m-1 and $\delta
k=10\mu$m-1 respectively. In (c), $\mathcal{D}(x,y=0)$, the interaction along
the current direction, is plotted as a function of $\delta k$.

Exciton mediated spin-spin Interaction – We follow the standard approach to
derive the indirect RKKY type interaction between the trapped holes mediated
by the mobile IXs. With the interaction $H_{v}$ treated as perturbation, the
effective spin-spin interaction between the trapped holes can be obtained by
performing the Schrieffer-Wolff transformation
$e^{\mathcal{S}}He^{-\mathcal{S}}$, where $\mathcal{S}$ satisfies
$[H_{0},\mathcal{S}]=H_{v}$. Tracing out the IX degrees of freedom and keeping
to the second order of $H_{v}$, we obtain the effective Hamiltonian for
trapped holes,

$H_{h}=\sum_{i,j}\mathcal{J}(\bm{r}_{ij})\bm{S}_{i}\cdot\bm{S}_{j}+\eta\bm{\mathcal{D}}(\bm{r}_{ij})\cdot(\bm{S}_{i}\times\bm{S}_{j}).$
(1)

The two terms are of Heisenberg and DMI form respectively Yu  _et al._ (2021),
with coupling coefficients,

$\displaystyle\mathcal{J}(\bm{r}_{ij})$ $\displaystyle=$
$\displaystyle\sum_{\bm{k}\bm{k}^{\prime}}\dfrac{I^{2}}{2}\cos\left[(\bm{k}^{\prime}-\bm{k})\cdot\bm{r}_{ij}\right]\dfrac{f({\bm{k}})-f({\bm{k}^{\prime}})}{\epsilon({\bm{k}})-\epsilon({\bm{k}^{\prime}})},$
$\displaystyle\bm{\mathcal{D}}(\bm{r}_{ij})$ $\displaystyle=$
$\displaystyle\hat{\bm{z}}\sum_{\bm{k}\bm{k}^{\prime}}\dfrac{I^{2}}{2}\sin\left[(\bm{k}^{\prime}-\bm{k})\cdot\bm{r}_{ij}\right]\dfrac{f({\bm{k}})+f({\bm{k}^{\prime}})}{\epsilon({\bm{k}})-\epsilon({\bm{k}^{\prime}})}.$

Here $S_{j}^{+}=d^{\dagger}_{j\uparrow}d_{j\downarrow}$,
$S_{j}^{-}=d^{\dagger}_{j\downarrow}d_{j\uparrow}$,
$S_{j}^{z}=\frac{1}{2}(d^{\dagger}_{j\uparrow}d_{j\uparrow}-d^{\dagger}_{j\downarrow}d_{j\downarrow})$
are the spin operators of the trapped holes.
$\bm{r}_{ij}\equiv\bm{R}_{j}-\bm{R}_{i}$ is the displacement vector from site
$i$ to $j$. Without losing generality, we consider a bath of spin singlet IX
in distribution $f_{\uparrow}=\frac{1+\eta}{2}f({\bm{k}})$,
$f_{\downarrow}=\frac{1-\eta}{2}f({\bm{k}})$, $\eta$ denoting the exciton spin
polarization. The exchange with a trapped hole of a different spin-valley
index can turn the singlet to the intervalley spin triplet. With electron-hole
exchange quenched by their layer separation, the energy splitting between
these two species is negligible. The spin index can therefore be dropped in
the exciton dispersion
$\epsilon_{\bm{k}\sigma_{e}\sigma}=\epsilon({\bm{k}})=\frac{\hbar^{2}k^{2}}{2m_{X}}$.

With the isotropic IX dispersion, it is obvious that the DMI coefficient
$\mathcal{D}$ will vanish if the distribution is an even function
$f({\bm{k}})=f({-\bm{k}})$, as in a thermal distribution. In this case, there
only exists a Heisenberg-type spin-spin interaction in Eq. (1), as
schematically shown Fig. 1(a), which favors ferromagnetic order of the trapped
holes. The DMI-type term in (1) can emerge for a non-equilibrium IX bath with
$\eta\neq 0$ and $f({\bm{k}})\neq f(-{\bm{k}})$, which corresponds to a spin-
polarized exciton current. Flipping the spin polarization and the current
direction simultaneously will not change $\mathcal{D}$, meaning that it can
also be induced by a pure spin current of exciton without net spin
polarization. In contrast to the DMI interaction of magnetic impurities
mediated by equilibrium electrons in spin-momentum locked bands Zhu  _et al._
(2011); Hosseini and Askari (2015), we note the DMI vector here is out-of-
plane $\bm{\mathcal{D}}=\hat{\bm{z}}\mathcal{D}(\bm{r}_{ij})$, so this DMI
tends to align spins in-plane in orthogonal directions.

Fig. 2 presents examples of the numerically calculated Heisenberg and DMI
coefficients $\mathcal{J}(\bm{r})$ and $\mathcal{D}(\bm{r})$. We have set the
distribution function as $f({\bm{k}})=f_{0}^{n,T}({\bm{k}}+\delta\bm{k})$,
where $f_{0}^{n,T}$ denotes a Bose distribution at temperature $T$ and density
$n$, and the momentum space shift $\delta\bm{k}$ is linearly proportional to
the IX current or drift velocity in the range concerned. The isotropic
Heisenberg coefficient $\mathcal{J}$ as a function of IX density $n$ and hole-
hole separation is shown in Fig. 2(a), at $T=30$K. In contrast to the RKKY
mediated by fermions, the exciton mediated $\mathcal{J}$ is always negative,
i.e. ferromagnetic coupling, in the concerned parameter range, due to the
nature of the Bose distribution. A ferromagnetic order is thus favored for the
trapped holes. Different from the oscillating feature in fermion-mediated RKKY
interaction, the magnitude of $\mathcal{J}$ decreases monotonically with
increased distance. The exchange strength can reach $\sim 0.1$meV at a hole-
hole separation of 5 nm, under IX density of $10^{11}$cm-2, consistent with
the observed $T_{c}$ of the ferromagnetic order and the estimated IX density
as reported in Ref. Wang  _et al._ (2022). We note that $\mathcal{J}$ has no
significant change for the range of $\delta k$ considered.

Fig. 2(b) plots $\mathcal{D}(\bm{r})$ at $\delta k=2\mu$m-1 and $10\mu$m-1
respectively, the former corresponding to a drift velocity of 600 m/s
(observed for monolayer exciton driven by dynamical strain Datta  _et al._
(2022)). The DMI coefficient is anisotropic, which oscillates along the
current direction ($x$) and decay in $y$ direction. Remarkably, the DMI is
extraordinarily long range in the current direction, without noticeable decay
in the range shown. This is a characteristic from the Bose distribution in the
mediation channel, in sharp contrast to the electron mediated RKKY Zhu  _et
al._ (2011); Hosseini and Askari (2015). Fig. 2(c) shows $\mathcal{D}$ along
the current direction, at various $\delta k$.

$\mathcal{D}(\bm{r})$ calculated using Eq. (LABEL:DMI) has assumed that IX
retains its phase coherence between successive scatterings with trapped holes.
With a finite phase coherence length $l_{\phi}$ in reality,
$\mathcal{D}(\bm{r})$ shall be multiplied with a factor $e^{-r/l_{\phi}}$,
which determines its practical interaction range along the current direction.
$l_{\phi}$ of hundreds of nm has been measured for quantum well excitons Zhao
_et al._ (2002). This suggests that even with significant decoherence, the DMI
interaction range is still much larger than the Heisenberg. While the peak
value of $\mathcal{D}$ is nearly two orders of magnitude smaller than that of
$\mathcal{J}$, the anisotropic DMI here can nevertheless have considerable
contribution to the total energy in determining the magnetic order.

Figure 3: (a-g) Spin textures of an ordered $\nu=1/3$ hole filling, from the
DMI mediated by an exciton bath in distribution
$f({\bm{k}})=f_{0}({\bm{k}}+\delta\bm{k})$, with the shift $\delta
k=10\mu$m-1. $f_{0}$ is a Bose distribution of $n=10^{11}$cm-2 and $T=30$K.
$x$-component of the spins are color coded. The Heisenberg interaction is
gradually switched on from zero in (a) to full strength in (c) and (d), as
denoted by parameter $\alpha$ (c.f. Eq. (3)). (d) has a finite magnetic field
$H=0.135$T applied out-of-plane. (e) shows the out-of-plane magnetization
curve, and $H_{c}$ denotes the critical field of its saturation. (f) and (g)
show zoom-in plots of the three-dimensional spin configurations within dashed
box in (c) and (d) respectively. (h) The spiral wavevector $\lambda^{-1}$ in
units of $\mu m^{-1}$, and (i) the critical field $H_{c}$ in units of Tesla,
as functions of $\delta k$ and filling factor $\nu$ of the charge order. In
(h), red and blue bars are for exciton current in the $x$ and $y$ directions
respectively.

Current controlled magnetic spirals – Conventional DMI tends to induce non-
collinear magnetization textures such as spin spirals or skyrmions in chiral
magnetic materials Yu  _et al._ (2021). In the Mott-Wigner crystal states of
the trapped holes, we show that the two distinct parts of IX mediated
interaction together establishes robust long wavelength magnetic spirals along
the current direction.

In search of the ground state spin configuration, we exploited the classical
Monte Carlo simulation, performed on a $500\times 500$ nm triangular lattice
with open boundary condition, where the moiré superlattice constant is $5$ nm
(see Supplementary supplementary ). The decoherence of IX is taken into
account by multiplying $\mathcal{J}$ and $\mathcal{D}$ by the factor
$e^{-r/l_{\phi}}$ with coherence length $l_{\phi}=30$ nm. A perpendicular
magnetic anisotropy of $3\mu$eV is extracted from the coercive field measured
in Ref. Wang  _et al._ (2022). In examining the magnetic field effect, an out-
of-plane g-factor of 10 is taken for the holes Aivazian  _et al._ (2015). The
numerical simulations are performed with the following Hamiltonian,

$H_{h}^{\prime}=\sum_{i,j}\alpha\mathcal{J}(\bm{r}_{ij})\bm{S}_{i}\cdot\bm{S}_{j}+\beta\eta{\mathcal{D}}(\bm{r}_{ij})(\bm{S}_{i}\times\bm{S}_{j})_{z}.$
(3)

where the parameters $\alpha$ and $\beta$ are introduced to artificially
switch on/off the two terms for examining their individual roles at given IX
bath parameters.

Fig. 3(a-c) shows the ground state magnetization texture at hole filling
$\nu=1/3$, where the IX bath has a density $n=10^{11}$ cm-2 and $\delta
k=10\mu$m-1. The Heisenberg interaction is progressively switched on, from
zero in Fig. 3(a), to full strength in 3(c) ($\alpha=1$), with full strength
of DMI in all three plots ($\beta=1$). One can clearly see that by the DMI the
spins get fully pinned in-plane (see also Fig. 3(f)), and spiral along the
current direction, while the strong Heisenberg term renders the spiral
wavelength longer. $S_{x}$ can be fitted perfectly by a single-period
trigonometric function. The net magnetization averages out in all directions
when the system size is large enough compared to the spiral wavelength. Fig.
3(d) shows the spin configurations under a modest magnetic field $H=0.135$T,
which is sufficient to tilt the spin out-of-plane, leading to a net
magnetization $\overline{S_{z}}$ out-of-plane, while the in-plane spiral
texture is unchanged (c.f. Fig. 3(g)). Fig. 3(e) plots $\overline{S_{z}}$ as a
function of out-of-plane magnetic field. The modest critical field $H_{c}$ for
the saturation of $\overline{S_{z}}$ reflects the small magnitude of DMI.

The dependencies of the spiral wavelength $\lambda$ and critical field on the
hole filling factor $\nu$ and the IX current are examined in Fig. 3(h, i) (see
Supplementary supplementary  for the charge orders assumed at the fractional
fillings). As expected, the spiral wavelength decreases with the increase of
IX current represented by $\delta k$. The increase of $H_{c}$ with $\nu$ is a
consequence of the long range nature of the DMI. At larger $\nu$, more
neighbors contribute to the DMI effective in-plane field on each spin, which
thus requires a larger critical magnetic field to align along $z$ direction.
We note that the perpendicular magnetic anisotropy plays a similar role as the
magnetic field Zeeman term, setting the lower bound of DMI strength for the
emergence of in-plane magnetic spiral. At small $\delta k$ and $\nu$, the
vanishing $H_{c}$ and $\lambda^{-1}$ mean the ground states become out-of-
plane ferromagnetic. Increasing the exciton coherence length (hence the
effective DMI range along current direction) can also turn the magnetic order
into in-plane spiral under these conditions.

Figure 4: (a-c) Temperature dependence of the magnetic order parameter of
moiré trapped holes, under different exciton density $n$ (a), hole filling
factor $\nu$ (b), and IX current denoted by $\delta k$ (c). The common
parameters are $n=10^{11}\text{cm}^{2},\delta k=10\mu$m-1 and $\nu=1/3$,
unless specified otherwise on the figure panels. (d) Comparison of temperature
dependencies when Heisenberg or DMI interaction is switched off. Blue dot:
both interactions on ($\alpha=1,\beta=1$). Orange square: Heisenberg switched
off ($\alpha=0$). Green diamond: DMI switched off ($\beta=0$). The insets plot
spin-spin correlation, showing diminished order parameter at finite
temperature whereas the spirals persist.

Finally, we examine the temperature dependence of the magnetic spiral. The
order parameter $\mathcal{M}$ is defined as the amplitude of the spin-spin
correlation which is found in a trigonometric form
$\langle\bm{S}_{i}\cdot\bm{S}_{j}\rangle=\mathcal{M}\cos(\bm{k}\cdot\bm{r}_{ij})$,
where the bracket means thermal average. In Fig. 4(a-c), the $\mathcal{M}$-$T$
curve is shown under varies IX density, current, and hole filling, which
exhibits a critical temperature $k_{B}T_{c}$ in the range of few meV. The
increase of $T_{c}$ with IX density $n$ as shown in Fig. 4(a) is well expected
as the latter controls the strength of mediated spin-spin interaction. $T_{c}$
is also higher at larger filling factor (Fig. 4(b)), which is another
reflection of the long interaction range. Noteworthily, $T_{c}$ has no
appreciable change with $\delta k$ (Fig. 4(c)), although the latter tunes the
DMI strength dramatically and hence the spiral wavelength. The observation
implies that $T_{c}$ is dominantly determined by the Heisenberg part which is
not sensitive to $\delta k$.

This is confirmed by the simulation where we artificially switch off the
Heisenberg or DMI part respectively by setting $\alpha$ or $\beta$ to zero in
Eq. (3). As shown in Fig. 4(d), when the DMI is turned off, the
$\mathcal{M}$-$T$ curve is essentially unchanged, except that the magnetic
order changes from the in-plane spiral to the out-of-plane ferromagnetic at
$\beta=0$. In contrast, when Heisenberg is turned off, $T_{c}$ becomes small.
Moreover, upon increasing temperature close to $T_{c}$, $\mathcal{M}$
diminishes but the magnetic spiral structure persists. These comparisons
suggest that the long-wavelength magnetic spiral is a unique ferromagnetic
texture jointly formed by the strong but shorter-range Heisenberg and the
ultralong-range DMI, in a mesoscopic superlattice. As the interactions are
fully controlled by the optically and electrically injectable IX and its spin
current, the findings point to an exciting possibility to dynamically
establish and manipulate magnetization textures in Mott-Wigner crystal states
in moiré.

###### Acknowledgements.

We thank Xu Zhang, Xi Wang, Jiayi Zhu and Xiaodong Xu for stimulating and
helpful discussions. The work is support by the Research Grant Council of Hong
Kong SAR (AoE/P-701/20, HKU SRFS2122-7S05), the Croucher Foundation, and the
National Natural Science Foundation of China (No. 12074195). W.Y. acknowledges
support by Tencent Foundation. Y.W. also wishes to thank the host of Kavli
Institute for Theoretical Sciences at UCAS.

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