Table Of ContentThree-body interactions on a triangular lattice
Xue-Feng Zhang,1,2,∗ Yu-Chuan Wen,3 and Yue Yu1
1Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
2Physics Department and Research Center OPTIMAS,
University of Kaiserslautern, 67663 Kaiserslautern, Germany
3Center of Theoretical Physics, Department of Physics,
Capital Normal University, Beijing 100048, China
(Dated:)
1
We analyze the hard-core Bose-Hubbard model with both the three-body and nearest neighbor
1
0 repulsions on the triangular lattice. The phase diagram is achieved by means of the semi-classical
2 approximation and the quantum Monte Carlo simulation. For a system with only the three-body
interactions,boththesupersolidphaseandonethirdsoliddisappearwhilethetwothirdssolidstably
n
exists. As the thermal behavior of the bosons with nearest neighbor repulsion, the solid and the
a
superfluidundergothe3-statePottsandtheKosterlitz-Thoulesstypephasetransitions,respectively.
J
In a system with both the frustrated nearest neighbor two-body and three-body interactions, the
3
supersolid and one third solid revive. By tuning the strength of the three-body interactions, the
phasediagram is distorted, because theone-third solid and thesupersolid are suppressed.
]
s
a PACSnumbers: 05.30.Jp,03.75.Hh,03.75.Lm,75.40.Mg,75.10.Jm
g
-
t
n I. INTRODUCTION
a
u
While it is an adjustable quantum simulator for solv-
q
. ing some difficult quantum problems, such as the high-
t
a Tc superconductivity and the fractional quantum Hall
m effects, the system of the ultracold molecules trapped in
- the optical lattice also provides an ideal toolbox to an-
d alyze the general properties of the quantum many-body
n systems1–3. In the real materials, comparing with the
o
dominant role that the two-body interactions play, the
c
[ multi-body interactions are usually taken as the high-
order perturbation. On the other hand, the man-made
1
Hamiltonian with leading multi-body interactions ex-
v
hibits many distinctive phenomenons, such as the non-
5
abelian topological phases4 and several novel phases5–7 FIG. 1: (Color online) The hard-core Bose Hubbard Model
6
on the triangular lattice. The particles can hop on the bond
4 originatedfromtheringexchangeinteractions. Recently,
0 because of the engineering development of the ultracold with amplitude t, and the nearest neighbor repulsion V2 ex-
. polarmoleculesconfinedinthe opticallattice,the multi- ists between them. The three-body interactions V3 affect the
1 particles on the triangle marked by red thick lines. In the
0 body interactions can be experimentally realized, espe- 2/3(1/3) solid, the particles(holes) will occupy two sublat-
1 cially,thethree-bodyinteractionscanbevariedinawide tices (black thick line) and form the honeycomb backbone.
1 range with the nearest neighbor interactions from nega-
: tive to positive8.
v
i While the dominant three-body interactions result in
X many exotic phases9–12 on one-dimensional chains and hard-core bosons with nearest neighbor two-body and
r two-dimensionalbipartite lattices, the interplay between three-body repulsions on the triangular lattice. The
a
Hamiltonian in the grand canonical ensemble is shown
the three-body interactions and the geometry frustra-
in Fig.1 and given
tion is still unclear. The frustrated lattices (such as
the triangularandKagomelattices) manifestthemselves
H = t (b†b +b†b )+V n n µ n
by enhancing the quantum fluctuation and often accom- − i j j i 2 i j − i
X X X
panying with the highly degenerate ground state. In hiji hiji i
the magnetic materials with the triangular structure, +V n n n , (1)
3 i j k
the spin liquid which breaks no symmetry has been X
hi,j,k∈△i
observed13. Meanwhile, the triangular optical lattice, in
which the existence of the supersolid has been numeri- where b† (b ) is the creation (annihilation) operator of
i i
cally proved14–21, has been realized by using triple laser bosons; t is the hopping parameter; µ is the chemical
beams22,23. potential; and V and V are strengthes of the nearest
2 3
The simplest model to reflect such an interplay is the neighbor two-body and three-body repulsions, respec-
2
tively, i.e., i,j represents i and j are nearest neighbor
sitesand i,hj,ki meansi,j andk connectwitheach 8 Mott-1 SSE
h ∈△i
other two by two and form a regular triangle. After the SCA
6
Holstein-Primakofftransformationb† S†, b S and
ninito→spSiinZ-1+/21/X2,XtZhemboodseolnic Hamiltoin→iani(1)iis→maipped V3 4 2/3 Solid
µ/ 2
H = −t (Si†Sj +Sj†Si)+(V2+V3) SiZSjZ 0 Superfluid
X X
hiji hiji
Mott−0
−2
−B SiZ +V3 SiZSjZSkZ, (2) 0 0.1 0.2 0.3 0.4 0.5
X X
i hi,j,k∈△i t/V
3
whereB =µ 3V 3V /2,andthe lasttermbreaksthe
2 3
particle-hole s−ymme−try. FIG. 2: (Color online) The phase diagram with V2 =0. The
red line comes from the semi-classical approximation (SCA)
In this work, we used both the semi-classical
and the dot blue line from quantum Monte Carlo with a
approximation14 and quantum Monte Carlo
stochastic series expansion (SSE) arithmetic.
simulation24–26 to study the model described by
the Hamiltonian Eq. (2). We first assume V = 0 and
2
merely consider the three-body interactions. We depict
is taken the semi-classical approximation. The ground
the phase diagram and find the coexistence of both
state is determined by minimizing the energy per site.
the charge-density wave order and the bond-ordered
Becauseofthe3-foldrotationalandthetransitionalsym-
wave9. Furthermore, at the finite temperature, the
metries, the lattice canbe divided into three sublattices,
Kosterlitz-Thouless and the 3-state Potts type phase
and the spins are equivalent in each one.
transitions are found in the superfluid phase and the
The superfluid is classically identified by checking
solid phase, respectively. After the antiferromagnetic
whether all the spins point at the same direction. By
nearest neighbor interaction is switched on, the phase
using the semi-classical approximation, we find that the
diagramisderivedinthewholeparameterspace. Forthe
second order Mott-0(Mott-1) insulator-superfluid phase
large three-body interactions, we observe the separation
transitionath= 2∆(h=2(2+∆)),where∆=2t/V
3
oftheminimaofthesuperfluiddensityandthestructure −
and h = 2µ/3V . Meanwhile, the phase transition be-
3
factor.
tween solid and superfluid state is the first order, and
the critical lines can be analytically expressed with
II. THREE-BODY INTERACTIONS 16 3∆2 c1/3cos((θ 2π)/3)
h= − + − (3)
12 6
InRefs. [14,15],theauthorsshowedthatthefrustrated
nearest neighbor interaction leads to the charge density and
wave (CDW) order. In this section, we show that the
16 3∆2 c1/3cos(θ/3)
three-bodyinteractionsplaythesameroleasthenearest h= − + , (4)
neighbor interaction does. 12 6
When the hopping is forbidden, i.e., t=0, the ground
where
state at the absolute zero temperature is exactly known.
All the sites are empty (Mott-0 Insulator)when µ is less a = 4096 2304∆2 5760∆3 2160∆4
then 0. As the chemical potential increases, a √3 √3 − − −
× 216∆5 27∆6
solid phase appears in order to maximizes the chemical − −
b = c2 a2 =48 6∆3(8 9∆)(8+4∆+∆2)
potential part without costing energy on the three-body
repulsions. When µ > 6V3, the system is fully occupied c = (p256−96∆2+p48∆3+9−∆4)3/2
−
(Mott-1Insulator),becausetheenergypaidinthethree-
θ = arccos(a/c).
body interactions is less than its gain from the chemical
potential. For a finite t there may be a superfluid phase.
Andthesummitofthelobeisat∆=8/9andh=64/27,
Then, the phase diagram is extended to the finite t and
whichisgivenbyequalizingEq.(3)andEq.(4)atθ =π.
shown in Fig.2.
The peak and the shape of the lobe in the Fig.2 also
reflect the breaking of the particle-hole symmetry. No-
tice that the semi-classicalapproximationis exactin the
A. Semi-classical approximation large-S limit and the result from this approximation is
onlyqualitativelycorrect. Todepicttheprecisephasedi-
By approximating the quantum spin in Eq. (2) as a agram,we use a stochastic series expansion (SSE) arith-
classicalunit vector with the magnitude 1/2,the system metic in quantum Monte Carlo simulation.
3
−3
x 10
0.25 8
ρ L=24
s
0.2 Sρ( QL)= L48= 24 5x 10−3 ρ−2/3 6
0.15 s )
S(Q) L=48 0 (Q 4
b
0.1 −5 S
L=24
−10 L=48 2
0.05 −105.2 5 0.3 0.35 0.4 L=24
t/V L=48
00. 25 0.3 0.35 0.4 0.435 0.5 00.2 5 0.3 0.35 0.4 0.45 0.5
t/V t/V
3 3
FIG.3: (Coloronline)Thestructurefactorandthesuperfluid FIG. 4: (Color online)The Sb(Q) at µ/V3 = 3.5 and T =
density (Inset: The density per site) at µ/V3 = 3.5 and T = 0.01V3 vs t/V3. Inset: the distribution of Ki for L = 12 in
0.01V3 vst/V3. L is thelinear size of thelattice. the solid phase. The thick line is 0.24 and the thin line is
0.07.
B. Quantum Monte Carlo
Fig.4 gives a picture of such an order, in which the local
The cluster SSE24–26 is taken because of its high ac- kineticenergyinunitoftismuchhigheronthehalffilled
curacy and efficiency on simulating the system with bonds than the fully filled ones.
the multi-body interactions. The sufficiently low tem-
The phase diagrams (Fig.2), derived from both
perature and large system size ensure the thermody-
the semi-classical approximation and SSE calculations,
namic limit to be achieved , and the number of the
matchingwell,provesthevalidityofthesemi-classicalap-
quantum Monte Carlo steps are up to one million. A
proximation for such a model. However, unlike a model
solid phase may be described by a charge density wave
with the nearest neighbor repulsive bosons, neither the
(CDW) order parameter, the structure factor S(Q) =
N n eiQ·rk 2 /N2 where N is the number of sites 1/3 filling solid nor the supersolid is found. The super-
h| k=1 k | i solid is in fact the superfluid of the hole (particle) exci-
anPd Q = (4/3π,0). Meanwhile, the long range off-
tationsonthe backboneconstructedbyparticles(holes).
diagonal order is reflected by the finite superfluid den-
Becausethe bosonsareofthe hardcore,the excitedpar-
sity ρ = W2/4βt ,where W isthe winding number. In
s ticle can not hop to the nearest site to have the long
h i
the 2/3 filling CDW state, the bosons fully occupy two
range off-diagonal order. Meanwhile, due to the domain
sublattices, so that the particle and hole on same bond wall formation15, the solid order can be destructed by
can partly form a singlet state due to the second order
the infinitesimal density but infinite number of hole ex-
hopping process. Thus, the bond order wave appears. It
citations, and the phase transition from the solid to the
is described by the non-zero bond order structure factor
superfluid is allowed.
S (Q) = Nb K eiQ·rl 2 /N2, where N is the num-
b h| l=1 l | i b
ber of bondPs, K = b†b +b†b and i, j are two ends of
l i j j i
the bond l. 0.04
In the inset of the Fig.3, the density plateau indicates 0.03
0.00318
the incompressible state. And in the Fig.3, the finite
0.02
valuesofthe S(Q)andzerosuperfluiddensity inthe left
0.01
part(small t/V ) supportthe existence of the solid state (a)
3 Q) 0
(ρ = 2/3) in this region. Meanwhile, the finite jump of
( 0.02
b
the superfluid density and zero S(Q) in the right part S
indicate the first order superfluid-solid phase transition 0.01 −3.29×10−6
atthecriticalpointt/V =0.36(1). TheS (Q)alsodrops
b 0
down at the same critical point, as shown in the Fig.4.
(b)
However, the dependence on the lattice size requires the −0.01
0 0.005 2 0.01 0.014
finite size scaling analysis, which is shown in Fig.5. We 1/L
see that the system has a finite bond order wave order
in the solid phase and not in the superfluid. Thus, we
FIG. 5: (Color online)The finite size scaling of Sb(Q). (a) In
confirmed that the bond order wave and CDW coexist the solid phase at µ/V3 = 3.5 and t/V3 = 0.36. (b) In the
in the solid and do not separate. However, it is not a superfluid phaseat µ/V3 =3.5 and t/V3 =1.
novel phase because it can be easily understood by the
localvibrationsoftheparticlesorholes. Theinsetofthe
4
is the integral function
00..1155
0.4
R1 dt
0.3
0.1 2/9Q)L0.2 4ln(L2/L1)=ZR2 t2(ln(t)−κ)+t (5)
Q) S(0.1 andR=3πρs/2tT. WesetdataofL=24asR1 andthe
(
S 0 other sizes as R2, and then plot the κ in the inset of the
0.05 LL==2448 −50 δL06/5 50 Fig.7. From the Fig.7, we observe that the critical point
is around T 0.4. And the inset shows the Kosterlitz-
L=72 c ≈
Thouless transition happens at T =0.3342.
L=96 c
0
0.4 0.5 0.6 0.7 0.8 0.9 1
T
FIG.6: (Coloronline)Thestructurefactorvstemperaturefor
different sizes at t/V3 =0.1 and µ/V3 =3. Inset: the critical
behavior of the 3-states Potts Model universality class and
δ=(T −Tc)/t with Tc =0.65.
C. Finite Temperature
In the following, we study the finite temperature be-
haviors in the solid and superfluid phases. In Fig.6, we
showthe thermalmelting processofthe CDW order. As
the hard-core bosons with only nearest neighbor inter- FIG. 8: (Color online)The phase diagram by SCA with V2+
actions on the triangular lattice16, the phase transition V3 = 1. The solid phases are under the color surface, the
supersolid (SS) exists between the black net and the color
is expected to be in the universality class of the 3-state
surface, and the rest part is the superfluid. The black lines
Potts model with the critical exponents ν = 5/6 and
are thetricritical lines.
β = 1/9. The critical behavior of the structure factor is
S(Q) = f(δL1/ν) L−2β, where δ = (T T )/t and T
c c
× −
is the fitting critical temperature. The Fig.6 shows the
phase transition happens in the critical point T = 0.65
c III. INFLUENCE ON THE SYSTEM WITH
and also confirms our expectation by the same function
NEAREST NEIGHBOR INTERACTION
f for different lattice sizes.
In the system with only the nearest neighbor interac-
0.25 tion, the 1/3 filling solid and supersolid phase14–16 can
L=48 stably exist. It is intuitively thought that the additional
L=60
0.2 1.01 L=72 three-body interactions should have more influences on
κ L=84 the2/3CDWorderthan1/3. Byusingthesemi-classical
L=96
0.15 1 fit line approximationandsettingV2+V3 =1,weplotthephase
ρs diagrams for different ratios of V2 and V3 in Fig.8. We
0.1 0.3342 find that the three-body repulsionstrongly enhances the
0.99 2/3 CDW order, and the 1/3 CDW order will disappear
L=24 0.3 0.32 0.34 0.36
0.05 L=48 T when V2 is 0 which indicates that the 1/3 CDW order is
L=72 associated with the nearest neighbor interaction.
L=96
0 The strong-coupling expansion is powerful on solving
0 0.5 1 1.5 2
T thecompressible-incompressiblesecondorderphasetran-
sition. Therefore,we apply it to the supersolid-solidand
FIG. 7: (Color online)The superfluid density vs temperature Mott-superfluid phase transitions. The critical line for
for different sizes at t/V3 = 0.4 and µ/V3 = 3.5. Inset: The Mott-1(Mott-0) insulator-superfluid phase transition is
solution of theEq.(5) and the dash line is κ=1. µ = 6(V + V )+ 6t (µ = 6t), and the 1/3 and 2/3
2 3
−
supersolid-solidphasetransitionboundariesaregivenby
The superfluid to the normal liquid undergoes the
12t2 15t2 3t2
Kosterlitz-Thouless transition. The critical temperature µ =3V 3t
can be determined by the renormalization flow and the 1/3 2− − 2V2+V3 − 2V2 − 3V2+2V3
universal jump of the superfluid density at T (Fig. 7). 60t3 9t3 9t3
c +
The critical point is given by finding κ = 1, where κ(T) −V2(2V2+V3) − V2(3V2+2V3) (3V2+2V3)2
5
00..1122
6
0.1
5 00..77
V=0.1
3
4 2/3 Solid 0.08 ρ V=0.1 0.6 V3=1
s 3 ρ V3=10
µ 3 0.06 S(Q) V3=0.1 0.5
Superfluid ρ V=1
2 SS QMC 0.04 Ss(Q)3 V3=1 0.4
ρ V=10 0.3
1 1/3 Solid SCE 0.02 Ss(Q)3 V=10 2 2.5µ3/V3.5 44
SCA 3
00 0.1 0.2 0.3 0.4 0 2 2.5 3 3.5 4
t µ/V
FIG. 9: (Color online)The phase diagrams at V2 = 0.6 and FIG. 10: (Color online)The superfluid density and the struc-
V3 = 0.4 by using the QMC (red dot), SCA (blue line) and ture factor at T = 0.02, L = 24, V2 = 1 and t = 0.1 for
strong-coupling expansion(SCE) (black dash line). different V3 varyingas µ. Inset: The density per site
36t3 15t3 region. Two possible reasons may be used to interpret
+ + (6)
(2V2+V3)2 4V22 it. (i) The minimum of the S(Q) determines the 1/3-
2/3supersolidphasetransition,because the competition
and between both orders in the critical line may minimize
the S(Q). (ii) Because the second order hopping pro-
12t2 15t2 3t2
µ =3V +3t+ + + cess in the 2/3 supersolid are partly prohibited by the
2/3 2 2V +3V 2V +4V 3V +4V
2 3 2 3 2 3 three-body interactions, the holes moving on the honey-
60t3 9t3 comb backbone constructed by the particles can be ap-
+ +
(V +2V )(2V +3V ) (V +2V )(3V +4V ) proximately treated as, the quasi-particles forming the
2 3 2 3 2 3 2 3
9t3 36t3 15t3 superfluid flux on the honeycomb lattice. For the same
. (7) reasonmentioned in the Ref.27,28, such dip indicates the
−(3V +4V )2 − (2V +3V )2 − 4(V +2V )2
2 3 2 3 2 3 geometric hindrance in the superfluid flow.
When V goes to the infinity, the Mott-1 insulator will
3
disappear and the first order phase transition line given
0.07
by µ disappears. In contrary, the µ is partly af-
2/3 1/3
fected in this limit and the Mott-0 insulator-superfluid
critical line is not changed. Furthermore, these critical 0.06
linesarecomparedwithresultsfromthequantumMonte µ=3.06
Carlo and semi-classical approximation at V2 = 0.6 and 0.05
V =0.4inFig.9. Weseeagainthatthesemi-classicalap- µ=3.08
3
proximation gives a qualitative matched phase diagram
0.04
to that comes from the quantum Monte Carlo. In the ρ
s
region that the strong-coupling expansion is valid, the
S(Q)
results fitting with the data from quantum Monte Carlo 0.03
2.9 2.95 3 3.05 3.1 3.15
are better than those from the semi-classicalapproxima- µ/V
tion.
Inordertodetectthethree-bodyinteractions’impacts
FIG.11: (Color online)The superfluiddensity(blue)andthe
on the different orders, we use quantum Monte Carlo to
structure factor (red) in small region at T = 0.01, L = 24,
simulateseveralvariables. IntermsoftheFig.10,wecan V2=1 and t=0.1 for V3 =10 by varyingthe µ.
find, comparing to the 1/3 solid phase which hardly af-
fected by the three-body repulsion, the 2/3 filling CDW
order are strongly enhanced due to decreasing of the lo-
cal vibrations. In the supersolid phase, the CDW order
IV. CONCLUSION
is also enhanced, but the superfluid order becomes weak
at the same time. The influence on the 2/3 supersolid
is stronger than that on the 1/3 supersolid, because the We studied the hard-core Bose-Hubbard Model with
repulsive effect is larger in former case on the superfluid nearest neighbor and three-body repulsions on the tri-
flux. It is also interesting that the minima of the super- angularlatticebyusingthesemi-classicalapproximation
fluiddensityandthestructurefactorareseparatedinthe and the Quantum Monte Carlo simulation. In the only
large V . In the Fig.11, we can observe it in the small three-body repulsions case, we got the complete phase
3
6
diagram and find no 1/3 solid and supersolid phase ex- can form the superfluid flow without feeling the three-
ist. By comparing with the CDW and BOW order, body repulsions. For the large V , the minima of the
3
we demonstrate they coexist in the solid phase and the superfluidandCDWorderareseparated. Thedipofthe
BOW order trivial results from the local vibrations, so CDWordermayindicatethefirstorder1/3-2/3SSphase
it enlighten us it needs to be more careful to judge the transition16,andthe dipofsuperfluiddensitymayresult
BOW+CDW order. At the finite temperature, the su- from the geometry hindrance.
perfluidphasechangesintothe normalliquidphaseafter
a Kosterlitz-Thouless transition. Meanwhile, the 3-state
Potts Model universality class phase transition emerges
V. ACKNOWLEDGEMENT
whenheatingupthesolidstate. Andthesethermalprop-
ertiesaresameasthesolidphaseinthesystemwithonly
nearest neighbor interaction16. The authors would like to thank for discussions with
After adding the nearest neighbor repulsion, the su- S.Eggert and his suggestions. XFZ thanks the members
persolid and 1/3 solid phase revive. The three-body re- ofSFB/TRR49. ThisworkissupportedbytheNational
pulsions can enhance the 2/3 CDW order and suppress Natural Science Foundation of China, the national pro-
the high order hopping, so they affect the phase with gramfor basic researchof MOST of China, the Key Lab
2/3 CDW order harder than 1/3 CDW order. However, of Frontiers in Theoretical Physics of CAS, the Natural
evenupto infinity, itstill cannot destroythe 2/3super- Science Foundation of Beijing under Grant No.1092009,
solid, because the excited holes in the honeycomb back- the DAAD, and the DFG via the ResearchCenter Tran-
bone constructed by the particles in two sublattices still sregio 49.
∗
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