Three-state Potts nematic order in stacked frustrated spin models with SO3 symmetry Ana-Marija Nedi c1 2Victor L. Quito1 2Yuriy Sizyuk1 2and Peter P. Orth1 2

2025-05-06 0 0 4.51MB 17 页 10玖币

侵权投诉

Three-state Potts nematic order in stacked frustrated spin models with SO(3)

symmetry

Ana-Marija Nedi´c,1, 2 Victor L. Quito,1, 2 Yuriy Sizyuk,1, 2 and Peter P. Orth1, 2

1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA

2Ames National Laboratory, Ames, Iowa 50011, USA

(Dated: May 5, 2023)

We propose stacked two-dimensional lattice designs of frustrated and SO(3) symmetric spin mod-

els consisting of antiferromagnetic (AFM) triangular and ferromagnetic (FM) sixfold symmetric

sublattices that realize emergent Z3Potts nematic order. Considering bilinear-biquadratic spin

interactions, our models describe an SO(3)-symmetric triangular lattice AFM subject to a ﬂuctu-

ating magnetization arising from the FM coupled sublattice. We focus on the classical AFM-FM

windmill model and map out the zero- and ﬁnite-temperature phase diagram using Monte Carlo

simulations and analytical calculations. We discover a state with composite Potts nematic order

above the ferrimagnetic three-sublattice up-up-down ground state and relate it to Potts phases in

SO(3)-broken Heisenberg and Ising AFMs in external magnetic ﬁelds. Finally, we show that the

biquadratic exchange in our model is automatically induced by thermal and quantum ﬂuctuations

in the purely bilinear Heisenberg model, easing the requirements for realizing these lattice designs

experimentally.

I. INTRODUCTION

Frustrated triangular-lattice Heisenberg and Ising anti-

ferromagnets that are subject to external magnetic ﬁelds

exhibit rich phase diagrams, including ﬁnite-temperature

Z3Potts phase transitions into magnetically ordered

states [1–6]. The classical triangular Heisenberg anti-

ferromagnet in a magnetic ﬁeld, which explicitly breaks

SO(3) symmetry, exhibits an extensive and continuous

ground-state degeneracy that is lifted by thermal [7] and

quantum ﬂuctuations [8] via an order-by-disorder mech-

anism [9–12]. In an intermediate ﬁeld range, a collinear

three-sublattice up-up-down (UUD) state is selected that

spontaneously breaks Z3symmetry (corresponding to the

position of the minority down spin). This results in the

stabilization of a one-third magnetization plateau both at

zero and ﬁnite temperatures T. The magnetic transition

at ﬁnite Tlies in the Z3Potts universality class [3]. Sim-

ilarly, the triangular Ising AFM in a magnetic ﬁeld de-

velops long-range UUD magnetic order at ﬁnite tempera-

ture via a Potts phase transition [5,13,14]. At T= 0 the

transition requires a nonzero critical ﬁeld value and lies in

the Kosterlitz-Thouless (KT) universality class [14,15].

Two-dimensional (2D) spin models with continuous

SO(3) symmetry, on the other hand, cannot develop long-

range magnetic order at T > 0 due to the Hohenberg-

Mermin-Wagner theorem [16,17]. However, since dis-

crete lattice symmetries can be broken at ﬁnite tem-

peratures, Potts universality can still occur via order-

ing of a composite magnetic order parameter that pre-

serves SO(3) but breaks a lattice symmetry. This leads

to the intriguing situation that the appearance of long-

range discrete order is driven by ﬂuctuations of an un-

derlying continuous degree of freedom. Phase transi-

tions involving vestigial order parameters that survive

partial melting of a primary order have been widely stud-

ied in magnetic, charge density wave and superconduct-

ing systems [18–22], oﬀering a natural explanation for

the complexity of phase diagrams based on symmetry

alone. Emergent Z3Potts order was previously found in

fully antiferromagnetic SO(3)-invariant spin models on

the honeycomb, kagome and triangular lattice [23–26],

and it was also reported in coupled clock models [27],

cold atomic gases [28], twisted bilayer graphene [29,30]

and other unconventional superconductors [31–34]. Here,

we show that it also emerges in SO(3) symmetric mag-

nets with mixed ferro- and antiferromagnetic interactions

deﬁned on various stacked lattice designs. Our work thus

largely extends the material space for the experimental

realizations of this phenomenon.

We design SO(3)-symmetric spin models on stacked

2D lattices that exhibit emergent Z3Potts phase transi-

tions at ﬁnite temperatures. The models include nearest-

neighbor bilinear and biquadratic exchange interactions,

where the bilinear interactions take opposite signs on dif-

ferent sublattices. As shown in Fig. 1, the models con-

FIG. 1. (a) Stacked 2D lattice designs of SO(3)-invariant spin

models with nearest-neighbor couplings [see Eq. (1)]. They

contain an AFM triangular layer (black, Jtt >0) coupled to

a sixfold symmetric FM layer (red, Jhh <0). Diﬀerent lay-

ers are coupled via bilinear Jth and biquadratic Kth interac-

tions (green). Lattice designs from left to right are triangular-

honeycomb, triangular-kagome and ABC stacked triangular.

(b) Unit cell of the triangular-honeycomb (windmill) lattice

with couplings indicated.

arXiv:2210.04900v2 [cond-mat.str-el] 3 May 2023

tain a frustrated AFM triangular sublattice coupled to

a sixfold symmetric FM layer. As a function of the bi-

quadratic coupling, our models eﬀectively interpolate be-

tween AFM Heisenberg and Ising models on a triangular

lattice in a magnetic ﬁeld, yet in a fully SO(3) invariant

setup.

Materials that realize 2D Z3Potts order in elemen-

tary discrete degrees of freedom have been the subject of

intense studies over the years. Examples are inert gases

that are adsorbed on graphite substrates [35–41] and real-

izations in liquid crystals [42–47]. More recent directions

include synthetic matter platforms such as Rydberg atom

lattices [48] and cold atoms [28]. Another possibility to

experimentally realize three-state Potts models are sys-

tems that exhibit emergent Potts-nematic order, where

the order parameter is a composite object [25,29,32,49–

51]. Here, we identify a new family of continuous SO(3)-

invariant Heisenberg spin models on stacked lattice de-

signs that host three-state Potts-nematic order.

Recently, a number of stacked magnetic materi-

als realizing the required lattice geometry and con-

taining frustrated triangular layers have been found,

including the triangular-honeycomb lattice material

K2Mn3(VO4)2CO3[52], the triangular-kagome lattice

ﬁlm MgCr2O4[53], and the stacked triangular lattice ma-

terials CaMn2P2[51] and Fe1/3NbS2[25,26] with tun-

able magnetic phases under intercalation [54]. While the

stacked lattices in these materials agree with the ones we

propose in Fig. 1, the exchange interactions in these ma-

terials are diﬀerent, for example, the honeycomb layer in

K2Mn3(VO4)2CO3exhibits AFM rather than FM inter-

actions. We are not aware of a material candidate that

exactly realizes the spin model we introduce and study

below. Since materials with mixed ferro- and antiferro-

magnetic interactions are rather common, we hope, how-

ever, that our work will stimulate eﬀorts to ﬁnd material

candidates with stacked AFM triangular and FM sixfold

symmetric layers. In addition, stacking two-dimensional

van der Waals magnets [55–57] poses an alternative route

to the realization of the proposed spin models.

This paper is organized as follows. In Section II, we in-

troduce the model we consider, a rotationally-symmetric

bilayer spin Hamiltonian with bilinear and biquadratic

exchange interactions, which we investigate for the case

of classical spins here. The zero-temperature phase dia-

gram is mapped out in Section III, showing a large re-

gion in which the UUD state is the ground state. Our

key results are presented in Section IV, which includes

the ﬁnite-temperature phase diagram that hosts a region

with Potts-nematic order, as well as a scaling analysis to

establish the Potts criticality of the phase transition. It

also includes a discussion of domain walls, that are re-

sponsible for disordering the Z3Potts UUD phase. In

Section V, we discuss the role of quantum and thermal

ﬂuctuations in promoting biquadratic exchange via an

order-by-disorder mechanism in a model with purely bi-

linear Heisenberg interactions. Finally, we present con-

clusions in Section VI and the details of a few calculations

in the Appendices.

II. BILAYER SPIN MODEL

We consider SO(3)-symmetric bilayer spin Hamiltoni-

ans of the form

H=X

hi,jiαβ

JαβSαi ·Sβj +X

hi,jith

Kth (Sti ·Shj )2.(1)

Here, α, β ∈ {t, h}labels the diﬀerent layers with tre-

ferring to the AFM triangular layer (Jtt >0) and hto

the FM coupled sixfold symmetric layer (Jhh <0). We

consider both signs of the sublattice couplings Jth and

Kth. The summation runs over nearest-neighbor pairs

of spins on sublattices α, β. In the following, we focus

on classical spin models for which Sαi are unit-length

vectors. While we include a biquadratic interlayer inter-

action Kth in the model, we show later in Sec. Vthat the

eﬀects of such a coupling emerge naturally in a purely bi-

linear model from quantum and thermal ﬂuctuations via

an order-by-disorder mechanism [11,12,58,59]. This re-

duces the requirements for the experimental realization

of the model.

For concreteness, we study the triangular-honeycomb

lattice design (also known as the windmill lattice [60,61])

in the following, which is shown on the left in Fig. 1(a).

We note that the fully AFM version of the windmill

model has previously been studied and found to host

an emergent Z6order parameter [60–62]. Our gen-

eral results also apply to the other lattice geometries,

triangular-kagome and ABC-stacked triangular lattices,

if one applies a simple rescaling of the FM Jhh cou-

pling. The triangular-kagome lattice model [middle panel

in Fig. 1(a)] is described by Eq. (1) with rescaled cou-

pling J0

hh =3

4Jhh. Here, J0

hh denotes the FM coupling

in the kagome layer). The ABC stacked triangular lat-

tice [right panel in Fig. 1(a)] is described by Eq. (1) with

hh =1

2Jhh with J0

hh being the coupling in each FM

triangular layer. The other couplings in the triangular-

kagome and ABC-triangular models are the same as in

the windmill lattice model.

We note that here we focus on the minimal classical mi-

croscopic model that leads to the emergent Potts physics

in the lattices we are studying. We, therefore, do not

include further-range spin-spin interactions or additional

intralayer nearest-neighbor biquadratic exchange interac-

tions. A biquadratic coupling on the honeycomb layer,

Khh <0, would not add anything new to the model as a

collinear arrangement of the spins is already preferred by

the FM interaction Jhh <0. In contrast, an easy-plane

biquadratic exchange Khh >0 would compete with Jhh

and lead to a more complex phase diagram if it is suf-

ﬁciently strong. A biquadratic term on the triangular

lattice, Ktt, would either favor the collinear UUD state

for Ktt <0 or a non-coplanar umbrella state for Ktt >0.

Since the parameter Kth plays a similar role as we show

below, we do not include Ktt in our minimal model.

III. ZERO-TEMPERATURE CLASSICAL

PHASE DIAGRAM

In this Section, we establish the zero-temperature

phase diagram of the model in Eq. (1) as a starting point

for the ﬁnite temperature study. We focus on the regime

of FM Jhh <0 and AFM Jtt >0 and determine the

ground state (GS) phase diagram using two complemen-

tary techniques: we ﬁrst employ a variational ansatz,

which assumes a three-sublattice periodicity on the tri-

angular lattice and a uniform state on the honeycomb

lattice. In addition to the ground state, we obtain ana-

lytical expressions for their energies and the phase bound-

aries. We then conﬁrm the variational results using low-

temperature classical Monte Carlo (MC) simulations.

A. Variational phase diagram

We determine the classical ground state phase dia-

gram for FM Jhh <0 and AFM Jtt >0 assuming

a variational ansatz that considers one honeycomb (h)

and three triangular (A,B,C) sublattices [see inset in

Fig. 2(a)]. This ansatz is well justiﬁed in the regime

|Jhh|  Jtt >|Jth|,|Kth|. The analytical form of the

ansatz is given in Appendix A. To obtain the GS phase di-

agram, we ﬁrst numerically minimize the variational en-

ergy for ﬁxed interaction parameters Jth/Jtt and Kth/Jtt.

The contribution from parameter Jhh is constant in the

variational manifold. We start the minimization from

20 diﬀerent random initial sets of variational angles de-

scribing the spin directions and keeping the lowest energy

solution. We identify ﬁve classes of ground states that ap-

pear in the phase diagram: FM, UUD, umbrella, Y, and

V state. Considering the symmetries of these phases, we

ﬁnd simpler expressions for the variational energies using

fewer angles (see Appendix A). By comparing diﬀerent

energies, we also obtain an analytical expression of the

phase boundaries. Details of this calculation and the ex-

plicit expressions are provided in Appendix A.

The resulting ground state phase diagram as a func-

tion of nonzero Jth/Jtt and Kth/Jtt is shown in Fig. 2(a).

The form of the ansatz assumes that |Jhh|&Jtt and we

conﬁrm below using MC simulations that it holds also

for |Jhh|=Jtt. The phase diagram contains ﬁve diﬀer-

ent phases with spin arrangements sketched in Fig. 2(a),

where the red spin refers to the direction of the uniform

honeycomb layer and the three black spins denote the

directions of the triangular spins on the three sublattices

A, B, C. The symmetries that are broken in the ordered

states are described by the order parameter manifolds of

degenerate ground states, which are given in Table I. To

discuss the phase diagram, we ﬁrst notice that Kth <0

favors coplanar and collinear phases (UUD, V, Y, FM),

while Kth >0 prefers non-coplanar phases (umbrella).

Our focus in the following is on the Kth <0 regime, in

particular the collinear UUD region, which is the ground

state for 0 > Jth/Jtt >−1 and suﬃciently negative Kth.

For small |Kth|/Jtt, the system undergoes a sequence of

transitions as |Jth|/Jtt increases from Y to UUD to V and

ﬁnally to the FM phase. While we focus on FM Jth <0

here, we note that the phase diagram for AFM Jth is

easily obtained by inverting the spins on the honeycomb

lattice Sh→ −Sh.

B. MC low-temperature phase diagram

We conﬁrm the variational phase diagram in Fig. 2(a)

using classical Monte Carlo (MC) simulations at Jhh =

−Jtt. We determine the phase boundaries between the Y,

V and UUD phases as a function of Jth and Kth <0 and

ﬁnd that the MC phase boundaries precisely agree with

the variational ones (see the MC boundaries as white

dashed lines in the variational phase diagram Fig. 2(a)).

We do not ﬁnd evidence for new phases with a larger

periodicity.

The classical Monte Carlo calculations are performed

for a series 400 temperatures, grouped into two logarith-

mically spaced sets between 0.05 ≤T/Jtt ≤5.0 and be-

tween 0.3≤T/Jtt ≤5.0. Each MC step includes a local

Metropolis MC update and a parallel-tempering move

between the ensembles at diﬀerent temperatures, using

5×105MC steps in total. The ﬁrst half is discarded

for thermalization and we perform measurements only

in the second half. To determine the low-temperature

phase diagram in the range −0.5≤Kth/Jtt ≤0 and

−1.0≤Jth/Jtt ≤0, we run MC simulations on a detailed

coupling parameter grid with spacings ∆Kth/Jtt = 0.01

and ∆Jth/Jtt = 0.05, and use ﬁxed Jhh =−1, Jtt = 1.

The calculations are obtained for linear system sizes of

L= 24 unit cells, corresponding to a total number of

spins N= 3L2= 1728.

To distinguish the diﬀerent phases, we analyze in Fig. 3

the absolute value of the magnetization on the triangular

sublattice

h|mt|i =*1

NtX

Sti+.(2)

Here, iruns over the Nt=L2spins on the triangular

sublattice, and h·i denotes both the MC average and av-

Classical ground state Order parameter manifold

FM SO(3)

UUD SO(3) ×Z3

Y SO(3) ×U(1) ×Z3

V SO(3) ×U(1) ×Z3

Umbrella SO(3) ×U(1) ×Z2

TABLE I. Order parameter manifolds describing the symme-

tries that are broken in the diﬀerent classical ground states.

It also corresponds to the ground state degeneracy. Apart

from the additional SO(3) symmetry, these are identical to

ones found for the triangular Heisenberg AFM in a magnetic

ﬁeld [1].

FIG. 2. (a) Zero-temperature variational phase diagram for FM Jhh and AFM Jtt in the regime |Jhh|&Jtt as a function of

Kth and Jth. The phase diagram is obtained using a variational ansatz with three types of triangular spins (A,B,C) (black)

and one type of honeycomb spins (h) (red) [see inset]. Arrows illustrate classical ground states. The dashed white line is

obtained from classical MC calculations for Jhh =−Jtt =−1. Colored lines correspond to one-dimensional cuts shown at ﬁnite

temperatures in panels (b-c). (b) Z3Potts-nematic transition temperature Tcas a function of Jth and Tabove UUD phase at

ﬁxed Kth/Jtt =−0.3 (dark blue) and Kth /Jtt =−0.5 (light blue). Other parameters are Jtt = 1, Jhh =−1. (c) Potts Tcas

a function of Kth for ﬁxed Jth/Jtt =−0.3 (purple) and Jth /Jtt =−0.5 (red). Other parameters are Jtt = 1, Jhh =−1. The

star shows Potts Tcin the Heisenberg triangular AFM in a magnetic ﬁeld at the UUD point [3], and empty circles are Potts Tc

for the Ising triangular AFM in corresponding magnetic ﬁelds [13,14]. (d) Potts Tcas a function of Jhh for ﬁxed Jth =−0.5,

Kth =−1 and Jtt = 1. The values for Potts Tcare obtained using MC simulations from the crossing of the Binder cumulant

U2of m3.

erage over 15 temperatures, T /Jtt ∈(0.005,0.05). Note

that while h|mt|i vanishes in the thermodynamic limit

at ﬁnite temperatures, it converges to a ﬁnite value for

ﬁnite system sizes at low temperatures. We can thus use

MC simulations at ﬁnite temperature to determine the

ground state value of h|mt|i at T= 0. We also compute

the chirality of spins on the triangular sublattice

χc=*1

NpX

(SAi ×SBi)·SCi+,(3)

where Np= 2L2is the number of the triangular plaque-

ttes, and conﬁrm that it vanishes, χc= 0, for all phases

shown in Fig. 3(a). These phases at Kth <0 are thus

either coplanar or collinear. We also conﬁrm that non-

coplanar symmetric umbrella states with ﬁnite chirality

appear for Kth >0, but these are not the focus of this

work.

Fig. 3illustrates the separation of the UUD phase

for which h|mt|i = 1/3 from other phases for which

h|mt|i 6= 1/3. The white dashed lines indicate the phase

boundary obtained from cuts at ﬁxed Jth such as the

one shown in Fig. 3(b). As noted above, the MC phase

boundaries between the UUD and the V and Y states are

also included in Fig. 2(a), validating the variational ap-

proach. Note that at Kth = 0, the model can be related

to the known case of the triangular Heisenberg AFM in a

magnetic ﬁeld [1], because the ferromagnetically ordered

honeycomb spins (at T= 0) act as an external magnetic

ﬁeld on Sti. Like the Heisenberg AFM in a magnetic ﬁeld,

our model exhibits the V state for h|mt|i >1/3, the UUD

state for h|mt|i = 1/3 and the Y state for h|mt|i <1/3,

yet in a fully SO(3)-invariant model - of course, SO(3)

is broken at T= 0 by the FM order on the honeycomb

sublattice.

IV. FINITE-TEMPERATURE PHASE

DIAGRAM AND POTTS-NEMATIC

TRANSITION

In this Section, we examine the ﬁnite-temperature

phase diagram above the UUD ground state, where the

system undergoes a Z3Potts-nematic transition. We

put an emphasis on exploring diﬀerent parameter lim-

its, where we can make connections and point out diﬀer-

ences with the Ising and Heisenberg models in magnetic

ﬁelds. We also numerically establish the universality of

the phase transition. We start from the UUD phase at

T= 0 and characterize the emergent Z3Potts-nematic

order that melts at a ﬁnite transition temperature Tc.

At ﬁnite temperature, only the discrete Z3part of the

UUD order parameter can exhibit long-range order since

the continuous SO(3) degrees of freedom remain disor-

dered. The Z3breaking corresponds to a breaking of a

discrete lattice symmetry: in this case, it is translational

symmetry, leading to a tripling of the unit cell. To iden-

tify the Z3broken phase, we construct a composite Z3

Potts-nematic order parameter m3=m3eiθ as

m3eiθ =−1

2Np

i=1 SAi +SBie2πi

3+SCie−2πi

3·Shi .

(4)

The summation runs over all Np= 2L2triangular pla-

quettes consisting of three triangular and one honeycomb

spin [see inset of Fig. 2(a)]. In the T= 0 ground state

UUD phase, one ﬁnds m3= 1 and θ= 0,±2π

3corre-

sponding to the three degenerate conﬁgurations, which

are illustrated in Fig. 4(a). Unlike the corresponding

Potts order parameter of the Heisenberg AFM in an ex-

ternal ﬁeld [3], this composite Potts nematic order pa-

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Three-statePottsnematicorderinstackedfrustratedspinmodelswithSO(3)symmetryAna-MarijaNedic,1,2VictorL.Quito,1,2YuriySizyuk,1,2andPeterP.Orth1,21DepartmentofPhysicsandAstronomy,IowaStateUniversity,Ames,Iowa50011,USA2AmesNationalLaboratory,Ames,Iowa50011,USA(Dated:May5,2023)Weproposestackedtwo-dimensi...

展开>> 收起<<

Three-state Potts nematic order in stacked frustrated spin models with SO3 symmetry Ana-Marija Nedi c1 2Victor L. Quito1 2Yuriy Sizyuk1 2and Peter P. Orth1 2.pdf

共17页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Three-state Potts nematic order in stacked frustrated spin models with SO3 symmetry Ana-Marija Nedi c1 2Victor L. Quito1 2Yuriy Sizyuk1 2and Peter P. Orth1 2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: