Stacking-induced magnetic frustration and spiral spin liquid Jianqiao Liu1Xu-Ping Yao2and Gang Chen2 3 1State Key Laboratory of Surface Physics and Department of Physics Fudan University Shanghai 200433 China

2025-04-26 0 0 3.98MB 22 页 10玖币

侵权投诉

Stacking-induced magnetic frustration and spiral spin liquid

Jianqiao Liu,

1, ∗

Xu-Ping Yao,

2, ∗

and Gang Chen

2, 3, †

State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China

Department of Physics and HKU-UCAS Joint Institute for Theoretical and

Computational Physics at Hong Kong, The University of Hong Kong, Hong Kong, China

The University of Hong Kong Shenzhen Institute of Research and Innovation, Shenzhen 518057, China

(Dated: January 3, 2023)

Like the twisting control in magic-angle twisted bilayer graphene, the stacking control is another mechanical

approach to manipulate the fundamental properties of solids, especially the van der Waals materials. We

explore the stacking-induced magnetic frustration and the spiral spin liquid on a multilayer triangular lattice

antiferromagnet where the system is built from ABC stacking with competing intralayer and interlayers couplings.

By combining the nematic bond theory and the self-consistent Gaussian approximation, we establish the phase

diagram for this ABC-stacked multilayer magnet. It is shown that, the system supports a wide regime of spiral spin

liquid with multiple degenerate spiral lines in the reciprocal space, separating the low-temperature spiral order

and the high-temperature featureless paramagnet. The transition to the spiral order from the spiral spin liquid

regime is ﬁrst order. We further show that the spiral-spin-liquid behavior persists even with small perturbations

such as further neighbor intralayer exchanges. The connection to the ABC-stacked magnets, the eﬀects of Ising

or planar spin anisotropy, and the outlook on the stacking-engineered quantum magnets are discussed.

Since the discovery of superconductivity [

], quantum

anomalous Hall eﬀect [

] and other phenomena [

–

] in

twisted bilayer graphene, twistronics has emerged as an im-

portant and popular ﬁeld in the study of two-dimensional (2D)

materials. The crystal twisting provides an important control

knob to manipulate the electronic properties of quantum mate-

rials and also to induce exotic quantum phases of matter in the

underlying electronic systems. Like the more popular twist-

ing scheme, the stacking control is another useful structural

manipulation of the stacking orders of 2D materials through

rotation and translation between the layers. The stacking pro-

cedure has been successfully used to manipulate the electronic

and optical properties of layered van der Waals (vdW) ma-

terials [

–

], and the application to the 2D magnetism has

recently been explored [

–

]. Modern fabrication techniques

such as mechanical exfoliation [

–

] and molecular beam

epitaxy [

] make such a stacking control of magnetism

feasible. It was shown that, the interlayer coupling depends

strongly on the stacking, allowing the manipulation of the

magnetic properties of the stacked magnets [

–

]. While

existing works focus on the diﬀerent magnetic orders resulting

from the stacking, in this Letter we explore the possibility of

stacking-induced magnetic frustration as well as liquid-like

ﬂuctuating regimes from frustration.

We start from the 2D magnet with the simplest frustrated

structure, i.e., the triangular lattice, and stack the triangular lay-

ers along the

direction to form a multilayer three-dimensional

(3D) system. The stacking order was known to be crucial

in determining the electronic states [

–

]. For multilayer

graphene, it was shown that diﬀerent (chiral) stacking cre-

ates rather distinct low-energy descriptions for the electron

bands [

–

], and thus leads to distinct and interesting elec-

tronic properties [

–

]. In the electronic systems, the

stacking order changes the electronic properties by modifying

the electron tunneling channels and the electron interactions. In

magnets, the stacking order of the magnetic layers inﬂuences

the lattice structure and then the magnetic interaction. Among

many diﬀerent possible stacking orders, we here choose an

ABC stacking of the triangular layers. This choice turns out to

be one of the simplest stackings that could generate magnetic

frustration and non-trivial magnetic physics. Clearly, the AA

stacking is a simple uniform stacking along the

direction and

does not really lead to anything interesting if only the nearest-

neighbor (NN) interaction is considered. The AB stacking,

where the reference site of the B layer is projected to the center

of the triangular plaquette on the A layer, generates interesting

magnetic correlations and belongs to the extensively studied bi-

partite lattices. The ABC stacking in Fig. 1(a), that seemingly

triples the crystal unit cell, is in fact a 3D Bravais lattice. By

creating a corner-shared tetrahedral structure along the

axis,

the ABC stacking drastically enhances the magnetic frustration

and can induce a classical spin-liquid regime at low temper-

atures even for Ising spins [

]. Together with the intralayer

interaction from the ABC-stacked structure, the interlayer in-

teractions generate rich and interesting magnetic behaviors

including the subextensive ground-state degeneracy, thermal

order-by-disorder, magnetic transition to spiral orders, thermal

crossover and spiral spin liquid (SSL) regimes. We reveal

these behaviors with the intralayer and interlayer Heisenberg

interactions using a set of analytical techniques.

For each site of the ABC-stacking triangular multilayers,

there exist six NN sites within the same layer and three in each

of the two adjacent layers. Distinct from the AB-stacking case,

the triangular layer is no longer a mirror plane in the ABC-

stacked case. Instead, the lattice site becomes an inversion cen-

ter. The primitive lattice vectors are chosen as

a1=(1,0,0)

a2=(−1/2,√3/2,0)

a3=(1/2,√3/6,h)

, where the inter-

layer separation

varies for diﬀerent materials. In this Letter,

we take a unit layer distance

h=1

for convenience. Starting

from the NN antiferromagnetic Heisenberg model on the trian-

gular lattice, we incorporate the NN interlayer spin interactions

arXiv:2210.06372v3 [cond-mat.str-el] 1 Jan 2023

(a) (b) (c) (d) (e)

J⊥

FIG. 1. (a) The multilayer triangular lattice with the ABC stacking. The dashed line along the

direction indicates the projection of a site from

the top layer to the centers of unequivalent triangles within the lower two layers. The intralayer and interlayer interactions are denoted by

and

J⊥

, respectively. The spiral manifolds (blue) and their projections (red) on the

plane are presented for (b)

J⊥/J1=0.3

, (c)

1.0

, (d)

1.5

, and

(e) 3.0. The BZ boundaries for a monolayer triangular lattice are plotted in gray.

with the Hamiltonian

H=J1X

hi jik

Si·Sj+J⊥X

hi ji⊥

Si·Sj.(1)

Here

hi jik

and

hi ji⊥

refer to intra- and interlayer NN pairs,

respectively. The antiferromagnetic interactions are denoted

and

J⊥

[see Fig. 1(a)]. In the decoupling limit where

J⊥/J1=0

, the ground state on the monolayer triangular lattice

is the well-known 120

◦

state. As we demonstrate below, the

ABC stacking drastically enhances the magnetic frustration and

suppresses the magnetic ordering once the interlayer coupling

is considered.

Zero-temperature classical ground states.—By perform-

ing the Fourier transformation on the spin operator

Si=1

√NsPkSkeık·ri

, the spin Hamiltonian can be recast in

the reciprocal space as

H=PkS−kJ(k)Sk

, where

the total number of spins,

J(k)=Pdi j Ji jeık·di j

is the ex-

change interaction, and

di j ≡ri−rj

denotes the NN vectors

for both intra and interlayer bonds. Following the recipe of

the Luttinger-Tisza method, this local unit-length constraint

|Si|=1

for each spin is softened and replaced by a global one

Pi|Si|=Ns

. The classical ground state of the spin Hamil-

tonian can be obtained by searching the minimum eigenval-

ues of

(

) and verifying the satisfaction of the local con-

straints. It is convenient to introduce a complex parameter

ξ(k)≡Λ(k)eıθ(k)=1+eık·a1+eık·(a1+a2)

, where its modulus

and argument have been assigned to be Λ(

) and

(

), respec-

tively. The exchange interaction is further rewritten as

J(k)=1

2J1[Λ(k)2−3] +J⊥Λ(k) cos[k·a3−θ(k)].(2)

At this stage, the minima of

(

) are simply characterized

ξ(k)=−eık·a3J⊥/J1

. By solving the equation about

(

the propagation vectors of the eigenvalue minima form several

1D manifolds in the reciprocal space for

0<J⊥/J1<3

shown in Figs. 1(b-d). In particular, a spin-spiral state can be

constructed through these propagation vectors and satisﬁes the

local constraints strictly. Therefore, the spiral manifolds with

a subextensive degeneracy from the Luttinger-Tisza method

are the physical ground states. They are responsible for the

formation of the SSL of the (

ds,dc

)=(1

2) type [

] at

ﬁnite temperatures when thermal ﬂuctuations are introduced.

Here

and

refer to the dimension and codimension of spiral

manifolds, respectively.

The degenerate spiral manifold evolves with

J⊥/J1

. In the

weak interlayer coupling regime where

J⊥/J1<1

, the spiral

manifolds manifest as six helices in Fig. 1(b). Their projections

onto the

plane are comprised of six disconnected contours

around the

points in the Brillouin zone (BZ) for the mono-

layer triangular system. As

J⊥/J1

increases from 0 to 1, the

helices and their projected contours expands concurrently. For

J⊥/J1=1

, the spiral manifolds cross each other and become

intersected lines in Fig. 1(c). The degeneracy of the ground

states reaches its maximum as well and indicates the strongest

magnetic frustration. In the strong interlayer coupling regime

with

1<J⊥/J1<3

, the degenerate spiral manifold is further

reduced into discrete and distort contours as shown in Fig. 1(d).

Their contours decrease with increasing

J⊥/J1

. Finally, they

shrink into the points at (0

,±π

) when

J⊥/J1≥3

. The ground

state turns out to be the antiferromagnetic (ferromagnetic) or-

der between (within) the triangular layers.

Thermal order-by-disorder.—As the temperature increases

from absolute zero, the thermal ﬂuctuations enter into the sys-

tem and could lift the subextensive ground-state degeneracy.

For weak thermal ﬂuctuations at low temperatures, this induces

a discrepancy in the entropy for the spin-spiral wavevector on

the the spiral manifold, despite the fact that diﬀerent spin spiral

conﬁgurations share the same energy. The one that possesses

the highest entropy would be stabilized. This mechanism for

the establishment of the long-range orders is known as the

thermal order-by-disorder [

–

]. To formulate this eﬀect

for our case, we perform the low-temperature free energy and

entropy calculation, and the details can be found in the Sup-

plemental Material (SM) [

]. In Fig. 2(a), we further depict

the phase diagram and mark the regimes of thermal order by

disorder. The ﬁnite temperature SSL regime is discussed in the

later part of the Letter.

We sketch the thermal order-by-disorder eﬀect here. At

low temperatures, the thermal ﬂuctuations of the spins are

around the ground-state manifold. To characterize the ther-

mal ﬂuctuation of the spins, it is more convenient to pa-

rameterize the ﬂuctuating spins based on the spin conﬁg-

urations from the ground-state manifold. For an arbitrary

spin-spiral order with wavevector

, the spins would deviate

from their ordered orientations

Si=[cos(Q·ri),sin(Q·ri),0]

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

J⊥/J1

Spiral order

(from thermal disorder)

Spiral spin liquid regime

Featureless paramagnet

Low

High

0.2

0.4

0.6

0.8

1.0

Low

High

⟨S−k·Sk⟩

(a)

(b) (c)

(d) (e)

FIG. 2. (a) The classical phase diagram for the

J⊥

Heisenberg model on an ABC-stacked triangular lattice. The crossover (ﬁrst-order phase

transition) is outlined by the dashed (solid) line. (b) The distribution of free energy

on the spiral manifolds for

J⊥/J1=0.5

. The NBT results

hS−k·Ski

in (c) the spiral ordered phase with

T=0.229

, (d) the SSL regime with

T=0.429

, and (e) the high-temperature paramagnet with

T=1.589

. The regions with the lower density are set to be more transparent. The arrows in (b) indicates the positions where

hS−k·Ski

are

highly concentrated on the spiral manifolds (blue). The system size is 50 ×50 ×50.

due to the thermal ﬂuctuations. This deviation can be de-

scribed by a perpendicular vector

φi

Si=φi+¯

Si(1 −φ2

i)1/2

and

|φi|  1

at very low temperatures. To capture the low-

temperature properties, it is suﬃcient to expand the Hamilto-

nian up to the quadratic order of the in-plane and out-of-plane

components

φi

and

φo

with

Hφ=Pi j ˜

Ji jφo

iφo

j+˜

Ji j(¯

Si·¯

Sj)φi

iφi

and

Ji j =Ji j −δi jJ(Q)

. Under this approximation, the low-

temperature free energy is given by

FQ∼TZk

ln WQ(k)+C,(3)

where

(

−J

(

Pdi j Ji j

ık·di j cos

(

Q·di j

) and

is a

constant. In Fig. 2(b), we plot the distribution of

-dependent

free energy

on the spiral manifolds for

J⊥/J1=0.5

. The

relative strength of

is encoded into the color gradient, and

the darkest points represent the selected wave vectors whose

exact coordinates have been listed in the SM [41].

The ﬁnite-temperature behaviors.—Upon further increasing

the temperatures, the selected spin spiral orders via the thermal

order-by-disorder would melt under the strong thermal ﬂuc-

tuations. Before entering into a featureless paramagnet, the

SSL could be revived at intermediate temperatures. To fully

reveal the ﬁnite-temperature behaviors, we here implement

a nematic bond theory (NBT) [

] and the conventional self-

consistent Gaussian approximation (SCGA) to construct the

classical phase diagram, which has been shown in Fig. 2(a).

Both methods start from the partition function in the form of

an imaginary-time functional integral

Z=ZD[S]D[χ] e−βHe−ıβ Piχi(|Si|2−1),(4)

where the Lagrange multiplier

χi

serves as an auxiliary ﬁeld to

impose the local constraint and

is the inverse of temperature.

In the NBT framework, the auxiliary constraint ﬁeld

χk−k0

divided into the static sector

∆(T)=ıχk=0

and the ﬂuctuating

sector

Xk,k0=−ıχk−k0(1 −δk,k0)

after the Fourier transforma-

tion. The separation of variables yields the action

S=βX

k,k0

S−k(Kk,k0−Xk,k0)·Sk0−βV∆(T),(5)

where

Kk,k0≡K0,kδk,k0=[J(k)+ ∆(T)]δk,k0

. An eﬀective

partition function

Z=Rd∆eβV∆(T)Z[∆]

can be obtained af-

ter the integration over the spin components in the large-

limit [

]. The eﬀective action in

[∆] is in the power

of the ﬁeld X. To integrate the ﬂuctuating sector X out,

the self-consistent equations should be established for the

bare spin propagators

hS−k·Ski=(2β)−1NK−1

0,k

and the in-

verse constraint ﬁeld propagators

hχ−kχki−1

D−1

0,k

Pk0K−1

0,k+k0K−1

0,k0

. They are renormalized perturbatively

by the higher order X terms in

[∆] and thus dressed by the

a proper self-energy Σand polarization Π, respectively. The

resulting Dyson equations are

Keﬀ,k=K0,k−Σk,(6)

D−1

eﬀ,k=D−1

0,k−Πk.(7)

As suggested in Ref. [

], at the cost of omitting all vertex cor-

rections, the Dyson equations can be solved self-consistently

with

Σk=−X

k0,0

K−1

eﬀ,k−k0Deﬀ,k0,(8)

Πk=D−1

0,k−N

K−1

eﬀ,k+k0K−1

eﬀ,k0,(9)

and are depicted as the diagrams in Fig. 3(a). With these ap-

proximations, the ﬁnal integral in

over the static sector ∆can

(a) (b)

=−

FIG. 3. (a) Self-consistent equations for the self-energy Σand po-

larization Π. (b) The derivative loop diagrams in the free energy

density.

be evaluated at the saddle point where

NT/(2V)PkK−1

eﬀ,k=1

The free-energy density, that includes the loop diagrams in

Fig. 3(b), are derived explicitly [41].

For the concerned parameter regime

0<J⊥/J1<3

where

there exists a subextensive degeneracy, the concentrated

weights of spin structure factors

hS−k·Ski

, that are calcu-

lated with the NBT, are found in the low-temperature regime

at the discrete momentum points, indicating the spin-spiral

orders. As shown in Fig. 2(c), the positions of these high

weights are identical to the results based on the entropy and the

thermal order-by-disorder calculations. Moreover, the free en-

ergy density manifests a ﬁrst-order phase transition above the

ordered states at the temperatures shown in Fig. 2(a). The dis-

tribution of

hS−k·Ski

also changes drastically. Right above

the transition temperature

, the point-like concentrations

hS−k·Ski

disappear immediately. Instead, there are clear

spectral weight enhancements around the spiral manifold, and

they decay rapidly away from it as shown in Fig. 2(d). These

features are characteristic to the SSL [

] and persist within

a broad temperature window [see Fig. 2(a)].

The SSL behaviors are gradually overwhelmed with the

prevailing thermal ﬂuctuations. At higher temperatures, the

spectral weights of

hS−k·Ski

tend to spread throughout the

whole BZ as shown in Fig. 2(e). Eventually, the spectral peaks

around the spiral manifolds would become indiscernible when

the system is deeply in the featureless paramagnet. The sys-

tem experiences a crossover from the SSL to the featureless

paramagnet. In the description of the NBT, the ﬂuctuating

sector X

k,k0

of the constraint ﬁeld becomes insigniﬁcant and

can be neglected in Eq.

(5)

. This simpliﬁcation in the NBT

leads to the well-known SCGA, which can qualitatively de-

scribe this thermal crossover [

]. In the phase diagram of

Fig. 2(a), the crossover temperatures are outlined based on

the “smoothening” of the spectral peaks [

]. Physically, this

thermal crossover from higher temperatures to lower tempera-

tures corresponds to the growth of the spin correlation. At a

temperature much above Curie temperature, all the spins are

ﬂuctuating thermally and there is not much correlation between

the spins. At the order of the Curie temperature, the spins be-

come gradually correlated. At even lower temperatures in the

SSL regime, the spin correlation in the the momentum space

reveals the structures of the degenerate spiral manifold. In the

SSL regime, the thermal ﬂuctuations are mainly around the

spiral manifold, which may resemble the thermal ﬂuctuation

near a critical point to some extent, and a semi-universal ther-

modynamic property is expected. It is found that, the speciﬁc

heat behaves like

CV=c1+c2T

in the SSL regime, where

c1,2

are constants [41].

Subleading spin interactions.—While the thermal order-by-

disorder and the entropy eﬀect could lift the degeneracy of

the spiral manifold at low temperatures, it is well-known that,

other subleading spin interactions could enter and break the de-

generacy. For instance, in the presence of the second- and third-

nearest spin interactions (denoted as

and

, respectively),

the spiral manifolds only exist at a special point

J2/J1=2J3/J1

and

0<J⊥/J1≤3+30J3/J1

[

]. While this eﬀect is clearly

important at low temperatures, especially in the relevant

ACrO2

antiferromagnets [

], the more tempting question is about the

stability of the SSL regime that is connected to the degenerate

spiral manifold. Or, more experimentally, can the degenerate

spiral manifold still manifest itself in the ﬁnite-temperature

spin correlation? Certainly, when the subleading interaction

is rather weak, this is expected. To what extent the spin cor-

relation is modiﬁed by the subleading interaction, however,

depends on the several competing energy scales and could vary

from material to material. It is, therefore, more appropriate to

simply demonstrate this for the speciﬁc interactions that are rel-

evant to certain materials. We have performed the NBT calcu-

lations for

(J1,J2,J3,J⊥)=(1.0,0.0,0.13,0.1)

that are closely

relevant to the ﬁrst-principles results for

HCrO2

[

]. The

spin-spiral orders at low-temperatures are conﬁrmed through

the magnetic Bragg peak of

hS−k·Ski

[see Fig. 4(a)]. A ﬁrst-

order transition is evidenced at TC≈0.470 [41].

The spectral weights of

hS−k·Ski

become pronounced

along the degenerate spiral manifolds once the temperature ex-

ceeds

. Its speciﬁc thermal evolution, however, carries a bit

more structure. Within a narrow window

TC<T.0.573

, the

most prominent weights appear near the ordered wave vectors

[indicated by arrows in Fig. 4(b)]. With increasing temperature,

two consecutive crossovers can be identiﬁed. First, the inho-

mogeneity of

hS−k·Ski

along the degenerate spiral manifold

is quickly ﬂattened with the growing thermal ﬂuctuations. A

more homogeneous distribution is recovered when

T&0.573

as shown in Fig. 4(c). Finally, the system undergoes another

crossover into the featureless paramagnet, as indicated by the

spreading of hS−k·Skiin Fig. 4(d).

Discussion.—The

J⊥

Heisenberg model for the SSL

physics is quite distinct from previous studies based on bi-

partite lattices [

]. Due to the geometric frustrations that

are naturally induced by the ABC stacking, an inﬁnitesimal

interlayer coupling is suﬃcient to spawn the SSL. For bipartite

lattice models, a ﬁnite interaction threshold is required for the

SSL. For example, the criteria are

J2/J1>1/6

for the honey-

comb lattice [

J2/J1>1/8

for the diamond lattice [

], and

more strictly J2/J1=2J3/J1>1/4 for the square lattice [51].

This restriction may challenge the realization of SSLs because

further exchange interactions can be relatively weak in real

materials. The SSL condition

0<J⊥/J1<3

for our model is

immediately realized once the stacking structure is fabricated

to a suﬃcient number of layers.

Even for few layers, the SSL physics is still expected. When

descending to a bilayer, our model is equivalent to a

0.2

0.4

0.6

0.8

1.0

(a) (b) (c) (d)

T= 0.469 T= 0.474 T= 0.580 T= 0.890

⟨S−kSk⟩

Low

High

FIG. 4. The NBT results of

hS−k·Ski

(J1,J⊥,J2,J3)=(1.0,0.1,0.0,0.13)

for the system size

50 ×50 ×50

. The ﬁrst two temperatures

are very close to the ﬁrst-order transition temperature

TC≈0.470

. The arrows indicate the (a) point-like and (b) arc-shape concentrations of

hS−k·Ski, respectively.

Heisenberg model on a honeycomb lattice. Furthermore, for

even numbers of layers, the ABC-stacked triangular lattice

can be viewed as a multilayer honeycomb lattice still with the

ABC stacking despite a displacement of two sublattices along

the

direction. Very recently, a 2D SSL has been advocated

by neutron scattering measurements in a vdW honeycomb

magnet

FeCl3

with the same stacking [

]. It is also immune

to intricate interlayer couplings. Although the interlayer spin

exchanges are diﬀerent here, a similar SSL is promising, e.g.,

through appropriate stacking controls. The nature of a few-

layer version of our model is worthy of further study.

Besides the stacking fabrication of vdW materials, ABC-

stacked triangular multilayer magnets actually exist in nature.

There are a family of magnets with the formula

AMX2

where

is a monovalent metal,

is a trivalent metal such as the

transition metal ion Cr [

–

] or the rare-earth ion [

–

], and

is a chalcogen, and the rhombohedral vdW com-

pounds

MX2

such as

NiBr2

and

NiI2

[

–

]. Both families

of magnets could experience extra magnetic anisotropies be-

yond the simple Heisenberg model. The simplest and common

anisotropy for the transition metal ions such as

Cr3+

and

Ni2+

ions is the single-ion spin anisotropy. In the presence of the

easy-plane anisotropy, it is still possible to construct the spi-

ral orders within the XY plane, and the SSL physics is still

expected. With the easy-axis spin anisotropy, one cannot con-

struct spiral orders with Ising spins and thus the ground-state

conﬁgurations are completely diﬀerent. The thermal ﬂuctua-

tions, however, could violate the Ising constraint and induce

the SSL regime [

]. Besides the characteristics as shown

in Figs. 2(c-e), the spin structure factors could possess a re-

ciprocal kagom

e-like structure from the competition between

frustration and spin stiﬀness [

]. The magnetic anisotropy

for the rare-earth chalcogenides

AMX2

is mainly the exchange

anisotropy from the strong spin-orbit coupling. Because of

the short-range orbitals of the 4

electrons, the spin exchange

is most likely to be dominated by the intralayer interactions,

and the SSL physics due to the interlayer coupling is probably

less relevant over there. The mechanical control such as twist-

ing, bending, and stacking is an uprising control knob of the

physical properties of quantum materials. We hope our work

to stimulate some interest in the stacking control of quantum

magnets and materials.

We thank Chun-Jiong Huang for useful discussions. This

work is supported by the National Science Foundation of China

with Grant No. 92065203, the Ministry of Science and Tech-

nology of China with Grants No. 2021YFA1400300, by the

Shanghai Municipal Science and Technology Major Project

with Grant No. 2019SHZDZX01, by NNSF of China with

No. 12174067, and by the Research Grants Council of Hong

Kong with General Research Fund Grant No. 17306520.

∗These authors contributed equally.

†gangchen@hku.hk

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Stacking-inducedmagneticfrustrationandspiralspinliquidJianqiaoLiu,1,Xu-PingYao,2,andGangChen2,3,y1StateKeyLaboratoryofSurfacePhysicsandDepartmentofPhysics,FudanUniversity,Shanghai200433,China2DepartmentofPhysicsandHKU-UCASJointInstituteforTheoreticalandComputationalPhysicsatHongKong,TheUniversityo...

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Stacking-induced magnetic frustration and spiral spin liquid Jianqiao Liu1Xu-Ping Yao2and Gang Chen2 3 1State Key Laboratory of Surface Physics and Department of Physics Fudan University Shanghai 200433 China

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