Table Of ContentExotic Spin Phases in Two Dimensional Spin-orbit Coupled Models: Importance of
Quantum Fluctuation Effects
Chao Wang,1,2 Ming Gong,1,2,∗ Yongjian Han,1,2 Guangcan Guo,1,2 and Lixin He1,2,†
1CAS Key Laboratory of Quantum Information, Chinese Academy of Sciences,
University of Science and Technology of China, Hefei, 230026, China
2Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, 230026, China
We investigate the phase diagrams of the effectivespin models derived from Fermi-Hubbardand
7 Bose-HubbardmodelswithRashbaspin-orbitcoupling,usingstringbondstates,oneofthequantum
1 tensornetworkstatesmethods. Wefocusontheroleofquantumfluctuationeffectinstabilizingthe
0 exotic spin phases in these models. For boson systems, and when the ratio between inter-particle
2 and intra-particle interaction λ > 1, the out-of-plane ferromagnetic (FM) and antiferromagnetic
(AFM) phases obtained from quantum simulations are the same to those obtained from classic
n
model. However, the quantum order-by-disorder effect reduces the classical in-plane XY-FM and
a
J XY-vortex phases to the quantum X/Y-FM and X/Y-stripe phase when λ < 1. The spiral phase
andskyrmionphasecanberealizedinthepresenceofquantumfluctuation. FortheFermi-Hubbard
6
model, the quantum fluctuation energies are always important in the whole parameter regime. A
2
general picture to understandthe phasediagrams from symmetry point of view is also presented.
]
l PACSnumbers: 71.10.Fd,75.10.Jm,64.60.Cn,67.85.-d
e
-
r
t
s The ultracold atoms in optical lattice[1–4] provide an agrams of the effective spin models with Rashba SOC,
t. excellenttoolboxforsimulatingvariousspinmodels,such derived from Bose-Hubbard (BH) model and Fermi-
a
as Heisenberg [5] model and Kitaev model[6] etc., and Hubbard (FH) model on a 12×12 square lattice, using
m
has been one of the central concepts in quantum simu- recently developed string bond states, one of the tensor
-
lations. Along this line some primary results have been network states (TNS) methods[49–53]. The TNS meth-
d
n obtained[7–9]. The simplest ferromagnetic (FM) or an- ods provide promising tools to investigate quantum sys-
o tiferromatic (AFM) Heisenberg spin models can be ob- tems with frustrated interactions. Details of the calcu-
c tained in the deep Mott phase regime[3] when the Hub- lations are presented in Supplementary materials [54].
[
bard model possesses rotational symmetry. The recent We find whereas in some parameters regions the classic
1 interest in the searching of exotic spin structures in op- spin model can give qualitatively correct ground states,
v ticallatticeis stimulatedby the experimentalrealization in some regions, the quantum effects are crucial to get
2
of spin-orbit coupling (SOC), which can be regarded as correct ground states. In particular for the fermion sys-
2
the simplest non-Abelian gauge potential in nature[10– tems, the quantum effects are always important.
7
7 27]. Inthesecases,theeffectivespinmodelsmaybecome Effective Spin Models. For a BH model with Rashba
0 more complicated due to the appearance of some exotic SOC, the Hamiltonian can be written as H = H +
BH 0
1. terms, e.g., the Dzyaloshinskii-Moriya (DM) [28, 29] in- U2 Pi,σniσ(niσ −1)+λUPini↑ni↓,whereU andλU are
0 teractions and their deformations. on-site intra-particle and inter-particle interactions and
7 H = −t Ψ†exp[−iαe ·(~σ×e )]Ψ . Here Ψ† =
The DM interaction has already been widely inves- 0 Phi,ji i z ij j
1
: tigated in solid materials[30–39] and now it is resur- (b†i↑,b†i↓), with b†iσ being the creationoperatorwith site i
v faced in ultracold atoms due to its flexibility in exper- and spin σ =↑,↓ and eij being the unit vector from site
i
X iments, e.g., the SOC interactions can be made much i to j. In the first Mott lobe (U ≫t), each site contains
r strongerthantheircounterpartinrealmaterials. Results onlyoneparticle,the effectivespinmodelcanbe written
a based on classical simulations[40–42], Ginzburg-Landau as,
theory[43, 44], dynamical mean-field theory[45] and spin
cos(2α) 1 cos(2α)
wave expansion[46–48] have unveiled rich phase struc- H =J X[ λ SixSjx+ λSiySjy+ λ (2λ−1)SizSjz
tures including spinspirals,skyrmionsin the presenceof hi,jix
the frustrated interactions caused by the SOC: there are 1 cos(2α)
strong competition between spin-independent tunneling −sin(2α)(SixSjz −SizSjx)]+J X[λSixSjx+ λ SiySjy
and the SOC induced spin-flipping tunneling. However, hi,jiy
the role of quantum fluctuation effect to the quantum cos(2α)
+ (2λ−1)SzSz −sin(2α)(SySz −SzSy)], (1)
phasediagramsinthesemodelshavenotbeenthoroughly λ i j i j i j
investigated. Whether andhow these phasescan survive
whereJ=−4t2/U<0,andhi,ji meansthenearestneigh-
µ
in the presence of quantum fluctuation are still unclear.
bors in the µ = x,y directions. In this model α de-
In this Letter, we investigate the quantum phase di- termines the strength of SOC, and λ represents the
2
along µ-direction, Sµ(k) has peaks at k = (0,0) and
(π,π), respectively; and in the strip phase the strongest
Z-FM Spiral-1 Z-AFM peaks happen at k =(0,π) or (π,0). We also define the
a spiralandskyrmionorderparametersinrealspaceas[55],
Skyr- Sp (i,j)=16hθiθji, Sk(i,j)=64hvivji, (3)
µ µ µ s s
c Spiral-2 mion
Spiral3 b d where θxi = (Si ×Si+ex)y and θyi = (Si ×Si+ey)x are
related to the relative planer spin angles for spins at
X/Y-FM X/Y-stripe
site i and i+e . To account for the three dimensional
µ
spin alignment effect, we define the spin volume con-
structed by the spins at the three neighboring sites as
vsi = Si·(Si+ex ×Si+ey). In the co-planar spiral phase,
vi =0 exactly, but it is nonzero in the skyrmion phases.
s
(a) Q (b) Q To determine the long-rangeorderofthe system,we cal-
culate the order parameters as
1 1
Spµ =XL2|Spµ(i,i+l)|,Sk=XL2|Sk(i,i+l)|, (4)
y i i
y x x where l = (L/2,L/2) to make |i−j| as large as possi-
(c) (d)
ble and i is averaged over the whole lattice for better
numerical accuracy.
Q
1
y y (a)
1.0
x Q2 x ers
et 0.8 Z-FM Spiral-1 Z-AFM
m
FIG. 1. (Color online). Upperpanel: Phase diagram for Eq. ara 0.6 Sz(0,0)
1fe,rcroamlcualganteetdicfr(oamntiTfeNrrSommaegthnoetdi.c)Zp-FhMase(Zw-iAthFMsp)indepnoolateriztehde der P 0.4 SSzp(x, )
alongthezdirection,andX/Ymeansthatthespinsarepolar- Or 0.2 Spy
izedalongeitherxorydirection. Foracomparisonthephase =1.5
0.0
boundaries determined by classical simulations are shown in 0.0 0.1 0.2 0.3 0.4 0.5
dashedlines. Lowerpanel: Spintexturesof(a)Spiral-1phase (b)
with spins spiral along Q = ex −ey direction; (b) Spiral-2 s 1.2 X/Y-FM Spiral-2 Skyr- X/Y-stripe
pphhaassee wspiitnhsssppinirsalspailroanlgabloontghQQ1==exeydiarnecdtiQon2; =(c)eSxpdirirael-c3- meter 01..80 Spiral-3 mion Sx/y(0,0)
tions; (d) 3×3 skyrmion phase. ara 0.6 Sx(0,)/Sy(,0)
P
er 0.4 Splg
anisotropy of the exchange interactions. Similarly in Ord 0.2 SSpksm
=0.8
the FH model, the Hamiltonian reads as HFH = H0 + 0.0
Un n whereH hasthe sameformasbosonmodel 0.0 0.1 0.2 0.3 0.4 0.5
Pi i↑ i↓ 0
with Ψ† replaced by (f†,f†), where f† is the fermion
i↑ i↓ iσ
FIG. 2. (Color online). Order parameters for the effective
creation operator at site i and spin σ = ↑,↓. The cor-
spinmodelderivedfrom Bose-Hubbardmodelat(a)λ=1.5;
responding effective spin model equals to that in Eq.1
and (b)λ=0.8, where Sp [Sp ] is the larger (smaller) one
at λ = 1 except that now J = 4t2/U >0 due to Pauli lg sm
of Sp and Sp (see text for details).
exclusion principle. Hereafter we let 4t2/U = 1 for con- x y
venience.
Phase Diagram for Boson. The phase diagram is pre-
Thefollowingorderparametersareusedtodistinguish
sented in Fig. 1 andcorrespondingorder parametersare
different phases. Firstly, the static magnetic structure
given in Fig. 2 at λ=1.5 and λ=0.8 and α ∈ [0,π/2].
factor is defined as [µ=x,y,z, i=(i ,i )],
x y
The spin model in Eq. 1 possesses some unique sym-
4 metries, which is crucial to understand this phase dia-
Sµ(k)= XhSµ·Sµiei[(ix−jx)kx+(iy−jy)ky]/L. (2)
L2 i j gram. Firstly, the Hamiltonian in Eq. 1 is invariant
i,j upon operation α→π −α and Sx,y → −Sx,y, which is
i i
on a L×L square lattice. For the FM and AFM phases equivalent to the transformation U†b U = −b , where
↑ i↑ ↑ i↑
3
U↑ =exp(iπPini↑)intheoriginalBHmodel. Thissym-
metry directly leads to U↑†H(α,λ)U↑ =H(π−α,λ), i.e., -0.3 (a) =1.3 Eq -0.4 (b) =0.8 Eq
the phase diagram should be symmetric about α = π2. -0.4 Ec -0.5 Ec
Therefore we only show the result for α∈[0,π/2].
y -0.5
We first discuss the phase diagram at four corners, rg -0.6
αwh=ere0,αi.∼e.,0inorthπe/2abasenndceλo≪f SO1Co,rtλhe≫orig1i.naWl shpeinn Ene --00..76 Efluc00..0012 -0.7 E fluc00..0012
model can be reduced to an effective XXZ spin model, -0.8 -0.8
0.00
0.00
with J = J = −1/λ, and J = −(2λ−1)/λ. When
x y z -0.9 -0.9
λ>1, |Jz|>|Jx|, the ground state is a Z-FM state, i.e., 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
all spins are ferromagneticlly aligned along the z direc- -0.5
(c) =0.3 Eq -0.3 (d) fermion Eq
tion. OurTNScalculationsshowthatforsmall α.0.15,
-1.0 Ec -0.4 Ec
the ground state is still Z-FM, as determined by the or-
der parameters shown in Fig. 2a. In this region, the y -1.5 -0.5
g
qcaltuaiaosnsnitcueffomnecest,i.msuuglagteisotninsgytiehledmthineosrarmoelegorfouqunadnsttuamteflausctthue- Ener --22..50 E fluc00..23 --00..76 Efluc00..1156
-0.8
Interestingly, at α = π, the model can be mapped 0.1 0.14
2 -3.0 -0.9
to the α=0 case via a symmetry transformation, U = 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
Qie−iπ2ixσxe−iπ2iyσy, i.e., U†Hπ2U =H0. Use this trans-
formation,weimmediatelyseethatthegroundstatenear FIG. 3. (Color online). Ground state energies from the clas-
α= π2 is a Z-AFM. We therefore see that these two lim- sical (Ec) and quantum (Eq) simulations at (a) λ = 1.3, (b)
its (α=0andα=π)shouldhavethe exactsameenergies, λ=0.8and(c)λ=0.3forboson,and(d)forfermionmodels.
2
and the quantum effects are small in both phases, which Insets give the corresponding fluctuation energy δEfluc. The
vertical lines are thephase boundaries calculated by TNS.
are confirmed by the numerical results.
However, there are dramatic difference in the case of
λ <1 where the in-plane exchange energy dominates. [cos(2α)(SySy+SzSz)+SxSx]. Obviously when
The order parameters calculated by TNS at λ=0.8 are Pα=hi,jπiy, i j i j i j
4
shown in Fig. 2. In the region of 0 < α/π < 0.13, the
ground state is a FM phase, with all spins are polarized λli→m0λHλ,α=π4 =−(X SiySjy + X SixSjx) (5)
along either x or y direction, which we denote as X/Y- hi,jix hi,jiy
FM phase. Remarkably this phase is very different from
which gives a compass model due to the strong cou-
what is obtained from the classical spin model, which
pling between the spins and directions[58]. This model
gives a rotational invariant FM state [41] with all spins
can not be solved exactly; however it can be shown ex-
layinthe x-y plane(dubbed as XY-FM).To understand actly that the ground state is 2L+1-fold degenerated for
this difference,we note that the in-plane rotationalsym-
a L×L square lattice[59–61]. It therefore corresponds
metry is not inherent of the originalHamiltonian, which
to a critical boundary between the X/Y-FM and X/Y
possesses only C symmetry. The rotational invariance
4 stripe phases since any deviation from this critical point
of the ground state in the classic model is due to the
byvaryingtheparameters(λandα)canbreakthedegen-
accidental degeneracy because the ground state of clas-
eracy and open an energy gap. The classical and quan-
sic model happen to has S =0. When quantum fluctu-
z tum simulations yield the same critical point.
ation is introduced, it breaks the accidental degeneracy
We next try to understand the spiral and skyrmion
andrestoretheC symmetryoftheoriginalHamiltonian,
4 phasesinthepresenceofstrongDMinteraction. Theor-
which therefore single out a ground state with lower en-
der parameters are shown in Eq. 4 (and the correspond-
ergy than the classical solution. This is the known as
ingspintexturesareshowninthe lowerpanelofFig. 1).
order-by-disorder mechanism[56, 57]. Again, we can ap-
Thespiral-1phasehastwodegeneratestatesspiralalong
ply symmetry transformation U = Qie−iπ2ixσxe−iπ2iyσy either ex+ey or ex−ey direction. For these two cases
near α = π/2, which yields a X/Y-stripe phase (as con- the strongest peaks in the structure factor S(k) appear
firmed by numerical results) for quantum spin model, in atk=±(k ,k )andk=±(k ,−k ),respectively,where
0 0 0 0
contrast to the 2×2 vortex state obtained from classical
k can be smoothly tuned by α and λ. However, due to
0
simulations. the finite size used in the simulation, only k =2π, π, π
0 3 2 3
The line λ = 0 in principle can not he achieved and π are observed, which are commensurate with the
6
due to the energy-costless double occupation. How- system size. In this phase, the skyrmion order Sk ∼ 0,
ever this limit can still be defined in the sense of whereas Sp = Sp 6= 0 are strongest among all the or-
x y
limλ→0λHλ =−Phi,jix[cos(2α)(SixSjx+SizSjz)+SiySjy]− der parameters. The spiral-2 phase has two degenerate
4
states, one is a spin spiral along x direction, and other
oneis alongy direction. Therefore,onlyone ofthe order
1.2
parameters,eitherSp orSp (seeFig. 2b)isnonzero. In Skyr-
x y X/Y-AFM Spiral mion X/Y-stripe
contrast, in spiral-3 phase, Spx=Spy, both are nonzero. 1.0
Spiral-3phaseisalsoobservedintheclassicalmodel,and rs
e
compared to the classical model, the spiral-3 phase re- et 0.8
m p=12/5 p=3 p=4
gion is greatly suppressed in the quantum model. In the a
r
skyrmion phase, the structure factor exhibits strongest Pa 0.6 Sx/y( , )
peaks atk=(±k0,0)and(0,±k0). Furthermorethe non- der Sx( ,0)/Sy( )
conplaner of spin alinement induce a finite srkymion or- Or 0.4 Spx+Spy
der Sk. The skyrmion phase is Neel type[62] and has a Sk
0.2
period3×3(lightpurpleregioninFig. 1)orlarger(dark
purple region in Fig. 1), which is consistent with the
0.0
numerical results for the classic spin model [41].
0.0 0.1 0.2 0.3 0.4 0.5
To understand the quantum effects in a more quanti-
tative way, we plot the groundstate energies per site for
λ=1.3, 0.8,0.3 in Fig. 3 a - c respectively obtained from FIG. 4. (Color online). Order parameters for the effective
classical simulations (E ) and full quantum mechanical spin model derived from Fermi-Hubbardmodel.
c
TNS simulations (E ). In the inserts, we also show the
q
energy differences
find X/Y-AFM phase when α/π <0.08 and X/Y-Stripe
δEfluc =Ec−Eq. (6) phasewhenα/π ∈[0.34,0.5](themirrorsymmetryabout
α = π is assumed). As before the classical model pre-
Obviously Ec ≥ Eq, thus δEfluc ≥ 0. From Fig. 3a. 2
dicts a rotational invariant AFM and vortex phases in
we find that when α = 0, and λ=1.3, E = −0.61537,
c
the x-y plane, which reduce to the X/Y-AFM and X/Y-
and E = −0.61538 in the 12×12 lattice, while the ex-
q
stripe phase due to order-by-order effect. Between the
act classical energy in a infinite size system is E∞ =
2λ−1 =−0.61538. Thisagreementcanbeunderstoocdus- AFM and stripe phases, there are spiral phases and one
2λ skyrmion phase. The spiral phase may also be distin-
ing the Holstein-Primarkoff (HP) transformation to the
guished by the period p = 12,3,4 which can be accom-
XXZ model (see [54]) due to the disappearance of pair- 5
modated by our simulation sizes. In these phases the
ing(orcondensate)term,thusδE =0exactly. Infact
fluc
skyrmion order almost equal to zero and the spiral or-
theXXZmodelcanbeusedasabenchmarkfortheTNS
der dominates. However, when α/π ∈ [0.29,0.34] , the
method,whichshowsgreataccuracyinthisproblem. As
skyrmionorderbecomeimportantalthoughthespiralor-
shown in Fig. 3a, δE ∼ 0 in the whole Z-FM and
fluc
der is still nonzero, similar to that in Fig. 2b.
Z-AFM phase regimes, even when α 6=0. In the spiral
The quantum fluctuation energy is much more pro-
phase, δE ∼ 0.01 - 0.02 is more significant.
fluc
nounced in the FH model than in the BH model for
Thefluctuationenergyincreaseswiththedecreasingof
all phases, as depicted in Fig.3d. For α= 0, we find
λ. Atλ=0.8,δE ∼0.01intheX/Y-FMandX/Ystrip
fluc
E = −0.5, and E = −0.6579, thus δE = 0.1579. In
phases,whichisabout4%ofthetotalenergies. However, c q fluc
theAFMandstripphases,δE isabout30%ofthe to-
even though this energy difference seems not very large, fluc
tal energy. The large quantum fluctuation energy in the
thegroundstatespredictedbyclassicalmodelandquan-
AFM state is due to that there are vast Hilbert spaces
tummodelaretotallydifferent. Fullquantumtreatments
near the S=0 that are energetically close to the ground
are therefore required to capture the correct physics in
state. The δE is slightlysmallinthe spiralphaseand
these phases. δE is also different for different phases, fluc
fluc
skyrmion phase, but still significant.
which is most significant in the skyrmion phase, where
δE ∼0.02. When λ further decrease to 0.3, δE in- It is very interesting to note that the Z-AFM state in
fluc fluc
creasesdramatically. Itisabout0.1-0.3intheX/Y-FM BH model however has very small δEfluc, in sharp con-
and X/Y strip phases, which counts almost 10% - 20% trast with the AFM state derived from the FH model.
of the total energies. The strong quantum fluctuation To understand this difference, we note that the Z-AFM
at small λ suppresses the spiral-3 and skyrmion phases state in BH model can be mapped to the Z-FM state
comparedtothe the classicalphasediagram(seeFig.1). viasymmetrytransformations,whichhassmallquantum
Phase Diagrams for Fermion. Forthespinmodelfrom fluctuationenergy. Therefore,eventhoughthetwoAFM
FH model, we have J = 4t2/U >0, and λ=1. There- states appear very similar to each other at the classical
fore α serves as the only adjustable parameter in this level,theirphysicsareentirelydifferent. Morefundamen-
model. The calculated phase diagram and the order pa- tally, this difference is rooted from the different statistic
rameters from the TNS method are presented in Fig. 4. properties between bosons and fermions.
Similar to the phase diagrams in the bosonic system, we Conclusion. We address the role of quantum fluctua-
5
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