Phase Diagram of Bosons in a 2D Optical Lattice with infinite-range Cavity-mediated Interactions

0.73 MB·

by  T. Flottat

Phase Diagram of Bosons in a 2D Optical Lattice with infinite-range Cavity-mediated
Interactions
T. Flottat,1 L. de Forges de Parny,2,3 F. H´ebert,4,∗ V.G. Rousseau,5 and G.G. Batrouni4,6,7
1Universit´e Coˆte d’Azur, CNRS, INLN, Valbonne, 06560, France
2Laboratoire de Physique, CNRS UMR 5672, E´cole Normale Sup´erieure de Lyon,
Universit´e de Lyon, 46 All´ee d’Italie, Lyon, F-69364, France
3Physikalisches Institut, Albert-Ludwigs Universit¨at Freiburg,
Hermann-Herder Straße 3, D-79104, Freiburg, Germany
4Universit´e Coˆte d’Azur, CNRS, INLN, France
7 5Physics Department, Loyola University New Orleans, 6363 Saint Charles Ave., LA 70118, USA
1 6MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit UMI 3654, Singapore
0 7Centre for Quantum Technologies, National University of Singapore; 2 Science Drive 3 Singapore 117542
2
High-finesse optical cavity allows the establishment of long-range interactions between bosons

r in an optical lattice when most cold atoms experiments are restricted to short-range interactions.
p
Supersolid phases have recently been experimentally observed in such systems. Using both exact
A
quantumMonteCarlosimulationsandGutzwillerapproximation, westudythegroundstatephase
diagramsofatwo-dimensionalBose-Hubbardmodelwithinfinite-rangeinteractionswhichdescribes
4
such experiments. In addition to superfluid and insulating Mott phases, the infinite-range checker-
] boardinteractionsintroducechargedensitywavesandsupersolidphases. Westudyherethesystem
s atvariousparticledensities,elucidatethenatureofthephasesandquantumphasetransitions,and
a
discuss the stability of the phases with respect to phase separation. In particular we confirm the
g
existence and stability of a supersolid phase detected experimentally.
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t
n
PACSnumbers: 03.75.Hh,05.30.Jp,64.60.F-
a
u
q
I. INTRODUCTION tional Bose-Hubbard model with an additional infinite
.
t rangeinteraction12,whichtakesintoaccounttheeffectof
a
m the cavity field. This model and similar ones have been
In the past fifteen years, cold atoms and Bose Ein-
mostly studied within the (static and dynamic) mean-
- stein condensates have proven invaluable tools to study
d the physics of interacting quantum systems1. Optical field theory16–20; only one study has employed an exact
n quantum Monte Carlo method in the hard-core limit21.
lattices1,2 and Feshbach resonances3 have been used to
o Inthecurrentstateoftheliterature, exactstudiesinthe
drivethesesystemsintostronglyinteractingregimes. Re-
c
strong (but finite) interacting regime are still missing.
[ cently, it has been shown that coupling between cold
In this work, we use exact quantum Monte Carlo sim-
atoms and an electromagnetic field, for example by
3 ulations to determine the phase diagram of this model
putting a condensate in a cavity, leads to interesting col-
v
4 lective phases of light and matter4. For example, the for several particle fillings, and elucidate the nature of
2 Dicke transition5, superradiant Mott phases6, self struc- the observed quantum phase transitions. We also com-
6 turation of atoms and light7, and crystallization8 have pare our results with mean-field results, obtained with
1 been observed in such systems. Many theoretical pre- an approach based on the Gutzwiller ansatz and classi-
0 dictions have also been made such as the possibility to cal Monte Carlo simulations22,23.
1. observe Bose glasses9, localization10, or synchronization The paper is organized as follows: The Hamiltonian
0 of quantum dipoles11. andthemethodsusedarepresentedinSec.II.SectionIII
7 is devoted to the discussion of the mean field phase di-
Here we are especially interested in a recent experi-
1 agrams whereas the exact ground-state phase diagrams,
ment by the ETH-Zurich group12 where a cloud of cold
: obtainedbyusingquantumMonteCarlosimulations,are
v atomsisplacedinanopticallatticeandinsideanoptical
Xi cavity. The field of the cavity mediates an effective infi- discussedinSec.IV.Finally,conclusionsandoutlookare
provided in Sec. V.
nite range interaction between the atoms which favours
r
a adensitydifferencebetweenneighbouringsitesoftheop-
tical lattice. This experiment attracted a strong interest
as it provides one of the first observations of the elusive II. HAMILTONIAN AND METHODS
supersolid phase13,14. This exotic phase is characterized
bybothlongrangephasecoherenceandspatialordering, A. Bose-Hubbard model with infinite-range
i.e. simultaneousdiagonalandoff-diagonallongrangeor- checkerboard interactions
ders. In a recent experiment, the same group observed a
supersolid phase with a symmetry breaking of a contin- We consider spinless bosons in a two-dimensional
uous space invariance15. square optical lattice inside a high-finesse optical cav-
Theexperimentalsystemiswelldescribedbyaconven- ity. The particles can hop between nearest neighbouring
2
sites of the lattice and interact repulsively on site. An quantumMonteCarlo(QMC)techniquethatallowssim-
additionaleffectduetotheexternalcavityfieldgenerates ulationsinthecanonical(CE)orgrandcanonical(GCE)
aneffectiveinfinite-rangeinteractionbetweenparticles12. ensembles of the system at finite temperatures, as well
Integrating out the effect of the field, the system is then as measurements of many-particle Green functions.
shown to be governed by a Bose-Hubbard model with We treat L×L lattices with sizes up to L = 14 and
Hamiltonian12: fix t = 1 to set the energy scale. Large enough inverse
temperatures allow to eliminate thermal effects from the
Hˆ =−t(cid:88)(cid:0)b†b +H.c.(cid:1)+U (cid:88) nr(nr−1) QMC and GMC results (we used inverse temperatures
r s s 2 βt=2LfortheQMCsimulationsanduptoβt=104 for
(cid:104)r,s(cid:105) r∈e,o
the GMC simulations).
(cid:32) (cid:33)2
−LU2l (cid:88)nr−(cid:88)nr . (1) deInnsitpyarρtic=ul(cid:80)ar w(cid:104)ne (cid:105)fo/cLu2s. mNain=lyρLon2 sisimtuhleattiootnasl antumfixbeedr
r∈e r∈o r r
of particles. The phase coherence is captured by the one
The bosonic operator b† (b ) creates (annihilates) an body Green function,
r r
atom at site r and n = b†b is the corresponding num-
r r r 1 (cid:88)
ber operator. The indices e and o denote respectively G(R)= (cid:104)b†b +H.c.(cid:105) (3)
2L2 r r+R
even and odd lattice sites. The first term of the Hamil- r
tonianisthekinetictermdescribingtunnellingwitham-
and its Fourier transform n(k) is the density of parti-
plitudetbetweennearestneighboursitesrandsdefined
cles occupying the wave vector k. The condensate frac-
on a square lattice of L×L sites with periodic bound-
tion, i.e. the fraction of particles occupying the k = 0
ary conditions. The second term represents the on-site (cid:80)
mode, is given by n(k = 0) = G(R)/N. We also
repulsive interactions between the atoms with strength R
calculate the superfluid density ρ given, in the QMC
U > 0. The third term describes the infinite-range in- s
s algorithm, by fluctuations of the winding number27 W,
teraction with amplitude U >0 and favours imbalanced
l ρ =(cid:104)W2(cid:105)/(4tβ). Finally, we also calculate the density-
populations between even and odd sites. µ will denote s
density correlation
the chemical potential for simulations performed in the
grand canonical ensemble (GCE). 1 (cid:88)
TheHamiltonianhasaU(1)×Z2symmetry,associated D(R)= L2 (cid:104)nrnr+R(cid:105) (4)
with the mass conservation (U(1) symmetry), times the r
Ising Z2 symmetry between the even and odd checker- and its Fourier transform, the structure factor S(k) =
board sublattices. (cid:80) eik.RD(R)/L2. We particularly focus on S(π,π) as
R
we expect checkerboard phases to appear.
B. Methods
III. GUTZWILLER PHASE DIAGRAMS
The approximate Gutzwiller Monte Carlo (GMC)22,23
approachisanumericalmethodbuiltonthecombination Since competing terms are involved in the Hamilto-
of both the Gutzwiller ansatz and the classical Monte nian,Eq.(1),weexpectfourdifferentphasesatzerotem-
Carlo method with Metropolis algorithm24. This results perature. For U = 0, i.e. the standard Bose-Hubbard
l
inasemi-classicallatticefieldtheorywhichpreservesthe model,itiswellknownthatthecompetitionbetweenthe
U(1)symmetry,whichisanadvantagecomparedtosome kinetic and interacting terms leads to two phases. Most
of the mean-field approaches conventionally used. This ofthephasediagramconsistsofaBosecondensed(BEC)
methodalsoallowsthereconstructionofcorrelationfunc- superfluidphase(SF)whichexhibitsphasecoherencein-
tionsonafinitelatticecluster. TheGutzwillermean-field dicated by n(k=0)(cid:54)=0 and ρ (cid:54)=0. For integer particle
s
state takes the form densities,ρ,andstrongrepulsion,U ,therearealsoMott
s
insulating (MI) phases with n(k = 0) = 0 and ρ = 0.
|Ψ(f)(cid:105)=(cid:79)L2 |ψ (cid:105)=(cid:79)L2 (cid:32)n(cid:88)maxf(r)|n (cid:105)(cid:33) , (2) Adding the checkerboard interaction Ul (cid:54)= 0 offesrs the
r nr r possibility to stabilize spatial ordering, i.e. oscillations
r=1 r=1 nr=0 in the density signalled by S(π,π) (cid:54)= 0. In addition to
the SF and MI phases – for which S(π,π) = 0 as the
where |n (cid:105) is the state with n particles on site r and
r r populationsarebalancedinthesephases–thereis,there-
whereweintroducedacut-offn onthenumberofpar-
max fore,thepossibilityoftwootherphases: achargedensity
ticles per site. The ensemble f = {fn(rr)} of the complex wave (CDW) solid with vanishing coherence (for inte-
f(r) coefficients is then sampled with the Monte Carlo ger or half-integer densities) and a supersolid (SS) phase
nr
method22,23 which is especially useful at finite tempera- exhibiting both spatial ordering S(π,π) (cid:54)= 0 and phase
turebutcanalsobeusedinthelowtemperatureregime. coherence. In Fig. 1, we show ρ, n(k = 0), and S(π,π)
The Hamiltonian is also simulated by using the asfunctionsofthechemicalpotential,µ. Weobservethe
stochasticGreenfunctionalgorithm(SGF)25,26,anexact four previously mentioned phases. The truncation n
max
3
we systematically observe SS phases at the tip of the
2
CDW lobes. Note that this phase diagram is in a good
0 SF SF SF agreementwithpreviousmeanfieldstudies17,18. TheSS-
= CDW MI, ρ=1 SS
ρ SF phase transition at the tip of the lobes is found to be
1.5 pty, ρ=1/2 L=10, continuous, as observed in Ref.17, and not first-order as
m
E observed in Ref.18.
U/U=0.4,
l s As reported in Refs.17,18, beyond U = U /2, the na-
l s
1 t/Us=0.04 ture of the phases change, as phases that show a density
modulation are favoured (Fig. 3). The Mott phases are
thus replaced with CDW phases. In the small t limit,
0.5 the Mott phase at ρ=1 is replaced with a phase where,
ρ depending on the way the symmetry breaks, even (odd)
2n(k=0)
sites are occupied by 2 bosons while odd (even) sites are
2S(π,π)
empty. This is what we call a CDW (2,0) phase. The
0
0 0.5 1 1.5 CDWphasesathalf-integerfillingsarealsoaffected. For
µ/U example, the CDW (2,1) phase, i.e. having alternately
s
doubly and singly occupied sites, observed at ρ = 3/2
below U /U = 1/2 is replaced with a (3,0) phase. The
FIG. 1. (Color online) The density, ρ, condensate fraction, l s
supersolidregionisalsomuchlargerasitsurroundscom-
n(k = 0), and structure factor, S(π,π), as functions of the
pletely the CDW lobes for ρ > 1/2. The bosons are
chemical potential, µ, obtained with the GMC method at
low temperature. We observe four phases: solid with charge expected to collapse for Ul >Us in t=0 limit18.
density wave (CDW), superfluid (SF), Mott insulator (MI), We now focus on the phase diagrams in the
and supersolid (SS). (U /t,U /U ) plane maintaining the density fixed by ad-
s l s
justing the chemical potential. The GMC mean field
phase diagrams for densities ρ=0.5, ρ=1, and ρ=1.5
2
CDW ρ=2.5 SS areplottedinFig.3. Ourresultsaresimilartothephase
Mean Field GMC U l/U s =0.4 diagramsofRef.17. Weobservethatthesupersolidphase
1.5 MIρ=2 surrounds CDW phases, which means that it only ap-
pears for U /U > 1/2 for integer densities (as ρ = 1)
SF l s
whileitispresentforsmallerU athalf-integerdensities.
Us 1 CDW (2,1) ρ=1.5 SS l
Atρ=1/2,whatappearsatfirstsighttobeasupersolid
/
µ phase, where n(k = 0) and S(π,π) are both non zero,
0.5 MIρ=1 is not a stable phase but a region of the phase diagram
where we observe a phase separation between a super-
fluid with ρ<1/2 and a supersolid with ρ>1/2 (SF-SS
0 CDW (1,0) ρ=0.5 SS
PS region). We will discuss this point later when we will
vacuum
study the stability of the different phases (Sec. IVC).
0 0.02 0.04 0.06 0.08 0.1
t/U
s
IV. QUANTUM MONTE CARLO PHASE
FIG.2. (Coloronline)Ground-statemeanfieldphasediagram
DIAGRAMS
intheplane(t/U ,µ/U )obtainedwiththeGMCmethodfor
s s
U/U =0.4. Fourphasesareobserved: superfluid(SF),Mott
l s
insulator (MI), charge density wave (CDW) and supersolid Using QMC simulations in the canonical ensemble, we
(SS) phases. determine exactly the properties of the ground state.
We will detail our observations in the following but we
first compare the phase diagrams at fixed fillings that
(up to nmax = 8) is chosen large enough for the results we obtained with QMC (Fig. 4) and GMC (Fig. 3).
not to depend on it and the temperature is chosen low We observe the same features for all phase diagrams,
enough to be in the ground state limit. with mostly quantitative differences between the QMC
Using cuts such as Fig. 1, we determined the phase and the GMC predictions. The main difference is that
diagrams in the (t/U ,µ/U ) plane for given values of the GMC technique generally overestimates the size of
s s
U /U (see Fig. 2 for U /U =0.4). The four phases, SF, supersolid or phase separation regions. For example,
l s l s
MI,CDW,andSS,areclearlyseeninthisphasediagram. at ρ = 1/2 the PS region is difficult to observe below
In the zero hopping limit t=0, the CDW phases appear U /U = 0.6 as the transition lines marking the onset
l s
between the Mott insulator phases, reducing the energy of density ordering and the disappearance of condensa-
gap of the MI phases to ∆=U −U for 0<U <U /2. tionappearsuperimposedinthelimitsofoursimulations
s l l s
At large hopping amplitude t the system is in the SF (Fig. 4 (a)). For ρ = 1, the GMC and QMC phase di-
phase. Intheintermediateregime,formoderatehopping, agrams are essentially the same but we observe a region
4
1 1 1
(a)
Mean Field GMC ρ=0.5
0.8 0.8 SS Mean Field GMC ρ=1 0.8 Mean Field GMC ρ=1.5
(b) CDW (2,0) CDW (3,0)
0.6 CDW (1,0) 0.6 0.6
Us SF-SS PS (c)
U/l0.4 0.4 0.4 SS CDW (2,1)
SF MI SF
0.2 SF 0.2 0.2
0 0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 10 20 30 40 50
U/t U/t U/t
s s s
FIG. 3. (Color online) Ground-state mean field phase diagrams obtained with the GMC method for three fillings (a) ρ=1/2,
(b)ρ=1and(c)ρ=3/2. Fourstablephasesareobserved: superfluid(SF),Mottinsulator(MI),chargedensitywave(CDW)
and supersolid (SS) phases. The dashed lines indicate first-order transitions, otherwise the transitions are second order. The
greenlinein (c)corresponds tothecut through theSSphaseused inFig. 10(b). InSec. IVC,we willdiscussthestabilityof
these phases and show that there is a region of phase separation between a superfluid and a supersolid (SF-SS PS) at ρ=1/2
instead of a SS phase.
1 1 1
(a)
0.8 SF-SS PS QMC ρ=0.5 0.8 SS QMC ρ=1 0.8 QMC ρ=1.5
SS
(b)
CDW (2,0) CDW (3,0)
0.6 CDW (1,0) 0.6 0.6
Us
U/l (c) SS
0.4 0.4 0.4 CDW (2,1)
SF SF MI SF
0.2 0.2 0.2
0 0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 10 20 30 40 50
U/t U/t U/t
s s s
FIG.4. (Coloronline)Ground-statephasediagramsobtainedwithQMCsimulationsinthecanonicalensembleforthreefillings
(a) ρ=1/2, (b) ρ=1 and (c) ρ=3/2. Four phases are observed: superfluid (SF), Mott insulator (MI), charge density wave
(CDW) and supersolid (SS) phases. Dashed lines indicate first-order transition. The black line indicates a 3D XY transition
associated with the appearance of a condensed superfluid state. The red line indicates the 3D Ising transition associated with
theappearanceofdensitywave. Atρ=1,thereisasmallregionwherethereisadirecttransitionbetweentheSFandtheCDW
(2,0) phases where both the Ising Z and XY U(1) symmetries are broken simultaneously. The green line in (c) corresponds
2
to the cut through the SS phase used in Fig. 10 (a). In Sec. IVC, we will discuss the stability of these phases and show that,
while the ρ = 1 and ρ = 3/2 SS phases are stable, at ρ = 1/2 there is, instead, phase separation between a superfluid and a
supersolid (SF-SS PS).
wherethereseemstobeadirecttransition,withoutpass- which is not equal to one because of quantum depletion
ing via an intermediate SS phase, between the SF and due to the interaction. On the contrary, there is no di-
CDW (2,0) region (Fig. 4 (b)). Finally, for ρ = 3/2, agonal order as the density correlation D(R) tends to a
we observe that the SS phase does not surround the two plateau with a value equal to the square of the density,
CDW phases completely but exists for U /U (cid:38)0.3 (Fig. ρ2 = 1 in that case. The CDW phase shows strong os-
l s
4 (c)). cillation around ρ2 at long distance in D(R) while G(R)
decays exponentially to zero, which is characteristic of a
solid phase. Finally, the supersolid phase shows simulta-
neouslyoscillationsaroundplateauxinbothD(R)andin
A. Phases
G(R), which shows that both diagonal and off-diagonal
long range orders are present at the same time. In the
We first analyse the different properties of the phases Mottphase(notshownhere)G(R)decaysexponentially
usingthebehaviourofthedensity-densitycorrelationand tozerowhileD(R)simplytendstoaplateauwithavalue
Green functions (Fig. 5) in the case where ρ=1. In the ρ2.
BEC or superfluid phase, we observe long range coher-
ence of the phase as evidenced by the saturation at long Toconfirmthepresenceofthesephases, weperformed
distance in the Green function G(R) value. The satura- finitesizescalinganalysesofthestructurefactor,S(π,π),
tionvalueofG(R)correspondstothecondensatefraction and of the condensate fraction, n(k = 0). In Fig. 6, we
5
show that both quantities extrapolate to non zero values 0.7
inthethermodynamiclimitforvaluesofparametersthat
are chosen in the supersolid region at ρ=1.
0)0.6
=
2.5 BEC (Us = 4.95) L = 14, Ul/Us = 0.7, (a) k( Ul/Us = 0.7, Us=6.95,
Supersolid (U = 6.95) n
CDW (Us = 9.5s5) ρ = 1, βt=2L nd 0.5 ρ = 1, βt=2L
2 a

)
π
π, n(k=0)
R)1.5 S(0.4 S(π,π)
( linear fits
D
1
0.3
0 0.005 0.01 0.015 0.02 0.025 0.03
2
0.5 1/ L
FIG. 6. (Color online) The structure factor S(π,π) and the
0
0 1 2 3 4 5 6 7 condensate fraction n(k=0) as a function of 1/L2 for ρ=1
R
inthesupersolidphase. Bothquantitiesextrapolatetoanon
zero value for large sizes.
1
(b)
0.8 For 1.25 (cid:46) ρ < 1.5, we once again have a supersolid
and a CDW (2,1) phase at ρ=1.5. On the contrary for
U = 0.8U (Fig. 7(b)), we observe that, for ρ ≥ 1/2,
l s
0.6 there are only CDW or supersolid phases, as S(π,π) is
)
R always non zero. There is no Mott phase and the super-
G( BEC (Us = 4.95) fluid phase is limited to the region where ρ(cid:46)0.25.
0.4 Supersolid (Us = 6.95)
CDW (Us = 9.55) L = 14, Ul/Us = 0.7,
11
ρ = 1, βt=2L
0.2 L = 6 (open symbols) and L=8 (filled symbols), n(k=0)
00..88
S(π,π)
U/U = 0.45, U = 25, βt=2L
00..66 l s s
0
0 1 2 3 4 5 6 7 00..44
R (a)
00..22
00
FIG. 5. (Color online) The density-density correlation func- 33 00..55 11 11..55
tion D(R) (a) and the Green function G(R) (b) as functions L = 6 (open symbols) and L=8 (filled symbols),
ofdistanceRindifferentphasesatρ=1andforagivenratio
22 U/U = 0.8, U = 30, βt=2L
ofinteractionUl/Us. Dexhibitslongrangeoscillationsinthe l s s
CDW and supersolid phases. G is non zero at long distances 3n(k=0)
in the supersolid and superfluid phases. 11 S(π,π)
(b)
Asmentionedearlier,thephasesobservedchangedras-
00
00 00..55 11 11..55
ticallydependingonwhetherUl issmallerorlargerthan ρ
U /2. In Fig. 7 (a), we show a cut in the phase dia-
s
gram, varying ρ for U = 0.45U where we observe all
l s FIG. 7. (Color online) Cuts in the phase diagram as func-
four phases. For small ρ, the system is superfluid; for
tions of ρ for U/U =0.45 (a) and U/U =0.8 (b) and two
0.25 (cid:46) ρ (cid:46) 0.75, the system shows non zero S(π,π), l s l s
differentsystemsizesL=6andL=8. In(a)weseethefour
which means that it is supersolid (since the condensate phases SF, CDW, MI and SS (see text). In (b) we see that
is still non zero) except at ρ = 0.5 where we find a the larger value of U forbids the existence of homogeneous
l
CDW (1,0) phase. This is somewhat surprising because phasesexceptforlowdensitiesρ(cid:46)0.25. Onthecontrary,we
supersolid phases generally do not appear in large re- observe MI and SF phases in (a), i.e. regions where S(π,π)
gions of parameter space for ρ < 1/228. This effect is is zero.
due to the infinite range interaction in the system. For
0.75 (cid:46) ρ (cid:46) 1.25, S(π,π) is zero and the system is su- We remark that these results have been obtained at
perfluid except at ρ = 1 where it adopts a Mott phase. fixed densities (canonical ensemble). The stability of
6
these phases with regard to density fluctuations, i.e. the 8
behaviourofthesysteminthegrandcanonicalensemble,
will be discussed below (see Sec. IVC). ν
β/ Dashed lines and
2L Full lines open symbols: S
)6 and symbols: n
π
π,
B. Quantum Phase Transitions at fixed densities (
S
d
n4
We show here how the phase diagrams (Fig. 4) were ν a βIsing= 0.326, βXY= 0.348,
obtained. To this end, we analyse the quantum phase 2β/ νIsing= 0.630 νXY= 0.671
transitions between the different phases, focusing on the L
)
ρ=1case,andusingfinitesizescalinganalysis. Wesum- =02
marize in Table. I the different types of phase transition kn( LL == 68 βt = 2L
that we observed. When a spatial modulation develops, L = 10
L = 12 U/U = 0.7
as in the CDW or SS phases, the discrete Z2 transla- 0 l s
5 6 7 8 9 10
tionsymmetryisbroken. Whenthesuperfluidityorcon-
U /t
densateappears, thecontinuousU(1)phasesymmetryis s
broken. For the successive transitions between the su-
perfluid,supersolidandCDWphases,theseZ andU(1) FIG. 8. (Color online) The transitions between the SF and
2
symmetriesarethenbrokenseparatelyorsimultaneously. SS and between the SS and the CDW phases for ρ = 1.
When they are broken together, we expect a first order ThefirsttransitionissignalledbyS(π,π)becomingnonzero
(opensymbols,dashedlines),whilethesecondtransitioncor-
transition. When they are broken separately, we expect
respondston(k=0)becomingzero(fullsymbols,fulllines).
3D Ising and 3D XY universality classes and our QMC
Intheintermediatesupersolidregion,botharenonzero. The
resultsconfirmthis. Werescaledn(k=0)byL2βXY/νXY
data have been rescaled with critical exponents β ,ν
and S(π,π) by L2βIsing/νIsing. βXY and νXY here denote for S(π,π) and β ,ν for n(k=0), correspondiInsigngtoIdsiinfg-
XY XY
thecriticalexponentsofthe3DXYmodelandβIsing and ferentuniversalityclasses(3DXYand3DIsing). Thearrows
νIsingthoseofthe3DIsingmodel. Withthisrescaling,we point to the two transition points where the curves obtained
expect the curves obtained for different sizes to cross at for different sizes cross each other.
asinglepointcorrespondingtothetransition. Increasing
U /tforagivenvalueofU /U ,weobservesuchcrossings
s l s
for the SF-SS transition where S(π,π) becomes non zero ues of S(π,π) while the two initial conditions yield the
and for the SS-CDW transition where n(k=0) becomes same values when we are far away from the transition
zero (see Fig. 8). We do not obtain such crossings if we point and no longer have coexistence of metastable and
use mean field scaling exponents. stable states.
The transitions between different solid phases are Examining the ground state energy, i.e. the free en-
found to be of first order, as suggested by12 for ρ = 1. ergy, measured with both initial states, the transition
To confirm this, we did QMC simulations with different point is located where the two energy curves cross. Be-
starting states (CDW or MI), for a large enough sys- lowthiscrossingpoint,theCDWstateismetastableand
tem. In the region where both the MI and CDW (2,0) theMottstateisstable(Fig. 9(b)), whiletheoppositeis
phases coexist, the system remains in the local energy true above this point.
minimacorrespondingtothephasewithwhichwestarted We observe the same hysteresis effects at ρ=1 in the
the simulation. An hysteresis effect is then observed for small region (13 < Us/t < 17 in Fig. 4) between the SF
S(π,π)asU /U isvaried(Fig. 9(a)). Inthecoexistence and CDW phases. This confirms the presence of a direct
l s
region around the transition we obtain two different val- first order transition from the SF to the CDW (2,0) in
this region and the absence of a supersolid which then
only exists for U /t<13.
s
Quantum Phase symmetry Type AstheinteractionUs/tgrows,forρ=1/2andρ=3/2,
Transitions broken the PS and SS regions becomes narrower and eventually
disappears. There is then a direct transition between SF
MI-SF U(1) 3D XY
and CDW (1,0) at ρ = 1/2 and between SF and CDW
SS-CDW U(1) 3D XY
(2,1) at ρ = 3/2. We were not able to obtain a definite
SF-SS Z2 3D Ising conclusionconcerningthenatureofthesetransitions,due
MI-CDW Z first-order to the difficulty of the QMC simulations in this large
2
SF-CDW (2,0) U(1),Z first-order interaction regime.
2
CDW (2,1)-CDW (3,0) Ø first-order Forρ=3/2,wealsoobserveafirstordertransitionbe-
tween the CDW (2,1) and the CDW (3,0) phases. This
TABLE I. Universality classes for the quantum phase transi- is somewhat surprising since the same spatial symmetry
tionsofthephasediagramsFig.4,determinedusingquantum is broken in both phases and one might have expected a
Monte Carlo simulations. crossover. The proximity of this first order transition to
7
1.5 (a)
11 QMC, L = 6,
n(k=0)
00..88 MCDotWt i ninitiitaial ls statatete 1 ρ = 3/2, βt=2L S(π,π)
)π00..66 )
(π,00..44 (a) L=8, Us/t = 22.5, π,π0.5
S (
ρ = 1, βt = 2L S
00..22
d
00 an 0 8 10 12 14 16 18 20
00..44 00..55 00..66 00..77 00..88 00..99 2
)
0
00 = GMC, L = 6,
k 1.5
--22 n( (b) ρ = 3/2, n =6, βt=103
2L--44 1 max
E/ (b)
--66 0.5
--88
0
00..44 00..55 00..66 00..77 00..88 00..99 10 12 14 16 18 20
U /t
Ul/Us s
FIG. 9. (Color online) The structure factor (a) and ground FIG. 10. (Color online) Cuts along lines inside the ρ = 3/2
state energy (b) obtained by varying U/U around the MI- supersolid phase (see the green lines in Figs. 3(c) and 4(c))
l s
CDW(2,0) transition. The two curves in each panel corre- madewithQMC(a)andGMC(b). Inbothcases,weobserve
spondtodifferentstartingstatesfortheQMCsimulationthat a crossover between two different regimes. As Us/t grows
yield different results in the region where both phases coex- (Ul/Us diminishes) the SS goes from a regime where S(π,π)
ist and the same result outside the coexistence region. The is large, due to the proximity with the CDW (3,0) phase,
transitionpointislocatedatthepointwherethetwoground to a regime where it is much smaller, when the SS is close
state energy curves meet. to the CDW (2,1). As the crossover between these two SS
regimeshappens(aroundU /t=12),thecondensatefraction
s
n(k=0) increases.
the narrowing of the ρ = 3/2 SS phase (see Fig. 4(c))
raises the question of a possible transition between two
We,therefore,analysethestabilityofphasesbystudy-
different SS phases. Cutting along a line through the SS
ing the evolution of the density, ρ, as a function of the
phase (see Fig. 4(c)), we do not find a transition but
chemical potential µ. In the grand canonical ensemble ρ
a crossover between two different regimes (Fig. 10(a)).
is calculated directly as an average for a given value of
Close to the CDW (3,0) phase (U /t < 12), the oscilla-
s µ. Inthecanonicalensemble,ρisfixedandµcanbecal-
tions of density are much larger in the SS phase, which
culated, for kT (cid:39)0, as µ(N)=E(N +1)−E(N) where
is marked by larger value of S(π,π). S(π,π) becomes
E(N) is the ground state energy of the system with N
much smaller close to the CDW (2,1) phase (U /t>12).
s particles. IntheGCE,anunstableregionischaracterized
Such a large change is expected as, in the large interac-
byajumpinthedensityasµisvaried. Theintermediate
tion limit, S(π,π) goes from 9/4 down to 1/4 between
densitiesdonotcorrespondtoastablephase. UsingCE,
the CDW (3,0) and (2,1) phases. While n(k=0) gener-
one can choose a filling corresponding to these interme-
ally decreases with increasing U /t, it increases slightly
s diatedensitiesbutthentheregionissignalledbyacurve
inthecrossoverregion(U /t(cid:39)12). Wealsoobservethis
s ρ(µ) with a negative slope29.
behaviour with the GMC approach (Fig. 10(b)).
We found that some regions of the phase diagrams
are unstable. For example doping around ρ = 0.5 for
U /t = 5.5 and U /U = 1 (Fig. 11(a)), we observe a
s l s
C. Stability of phases large unstable region which encompasses ρ = 0.5 and
leads directly from a superfluid phase for ρ (cid:46) 0.39 to a
So far, we mainly discussed the phases and transitions supersolid phase for ρ (cid:38) 0.60. This is attested by the
thatappearataconstantparticlenumberandmentioned behavior of ρ(µ): a jump is observed in the GCE simu-
that, insomecases, Hubbardsystemscanbeunstableto lations and a corresponding negative slope region is ob-
phase separation as the density is allowed to fluctuate. servedinCEsimulations. Thismeansthatwhatappears
In particular, it was shown that such a phase separation to be a supersolid phase with the canonical simulations
can occur between superfluid and solid phases and be for ρ = 0.5 in the phase diagram Fig. 4(a) is, as men-
mistaken for a stable supersolid phase29. This is espe- tioned earlier, not a stable phase but corresponds to a
ciallyimportantinthepresentcaseaslongrangeattrac- region of separation between SF and SS phases (SF-SS
tiveinteractions,ifstrongenough,tendtodestabilizethe PS region in Fig. 4(a)).
systemandleadtoacollapseoftheparticles(forexample Ontheotherhand,cuttingthroughthephasediagram
for U /U >1 in the MF description) or to the presence at U /t = 6.5 and U /U = 0.8 (Fig. 11(b)), i.e. going
l s s l s
of metastable states30. through the SS regions at ρ = 1 and ρ = 1.5 (see Fig.
8
L=8, U/U =1, U /t=5.5, βt=2L 15
l s s
0.5 QMC
CE and GCE
CDW
(a) 10
00
22 -3.25 --33 -2.75 U/tl
ρ QMC - GCE
n(k=0) (b) 5 SS
MI
11 S(π,π)
L=8, U/U =0.8,
l s SF ρ =1, βt = 2L
U /t=6.5, βt=2L
s
00
--33 --22 --11 00 11 22 00 5 10 15 20 25 30
U /t
1.5 s
GMC - GCE
1 FIG. 12. (Color online) The phase diagram (Fig. 4 (b)) for
L=8, U/U =0.8, ρ=1rescaledforadirectcomparisonwithexperimentaldata
0.5 (c) U /t=6.l5, sβt=104 (figure 2 of extended data12). Compared to the experiment,
s thesuperfluidandsupersolidregionsarefoundtobesmaller.
0
-4 -2 0 2 µ 4 6 8
FIG. 11. (Color online) ρ, n(k = 0) and S(π,π) vs µ. Dot-
sition around that density. The parameters used for Fig.
tedlinescorrespondtocanonicalsimulationswhilecontinuous
11(b) and (c) are the same but, as the phase diagrams
linescorrespondtograndcanonicalones. (a)UsingQMCCE
Fig. 3 and Fig. 4 are slightly different, the phases that
simulations, we observe a region in ρ(µ) with negative slope
are present in these two cuts are different. For example,
around ρ = 1/2 indicating instability. n(k = 0) is always
at ρ = 1.5, we find a SS phase in the QMC simulations
non zero (not shown to keep the figure uncluttered). There
is, therefore, a discontinuous transition from a superfluid for while we have a solid phase in the GMC simulations.
ρ(cid:46)0.39toasupersolidforρ(cid:38)0.60whichisalsosignalledby
ajumpinρ(µ)whenusingtheGCE.Thereisnostablephase
at ρ=0.5 for these parameters. (b) For a different set of pa-
V. CONCLUSIONS
rameters, we do not observe jumps in the ρ(µ) curve (QMC
GCE simulations) which shows that the supersolid phases at
ρ=1andρ=1.5arestable. n(k=0)isalwaysnonzero(not Using exact quantum Monte Carlo and approximate
shown). (c)SimilarbehaviorisobservedusingtheGutzwiller mean field calculations, we studied the phase diagram of
approximation. Inthecasepresentedhereweobserveadirect abosonicHubbardmodelwithinfiniterangeinteractions
transition from SF at ρ(cid:46)0.44 to SS for ρ(cid:38)0.58 and a dis- whichhasbeenproposedtodescriberecentexperimental
continuous transition from SS for ρ(cid:46)1.29 to solid phase for results of the ETH-Zurich group12. Our results confirm
ρ=1.5. We then have two density regions where the system
thatthemodelcorrectlycapturestheessentialphysicsof
isunstable. In(b)and(c)S(π,π)hasbeenmultipliedby0.5
the experiments. In particular, we confirm the existence
to improve visibility.
of a supersolid phase and also the nature of the phase
transitions, especially the first order transition between
the MI and CDW at ρ = 1. We observe a small region
4(b) and (c)), we do not observe jumps in ρ(µ) using where there is, at ρ=1, a direct first order SF to CDW
GCE simulation (Fig. 11(b)) and conclude that the su- transition that was not experimentally observed.
persolidphasesareinfactstableforthesedensities,with Thereremains,however,quantitativedifferencesinthe
these parameters. Finally, we performed some simula- extent of the different phases. This is easier to discuss
tions (not shown here) for large values of the interac- by comparing the phase diagram Fig. 12, represented in
tions (Us/t = 30,Ul/Us = 0.8) where we observed that the plane (Us/t,Ul/t), with the one provided as Figure
the only stable phases are the solid phases obtained for 2 of extended data in Ref. 12. In the experimental data,
integer or half-integer densities: there is no stable super- the SF phase is observed up to U /t (cid:39) 25 whereas we
s
fluid or supersolid phases for intermediate densities. observe a transition around U /t (cid:39) 17. The SS region
s
This question of stability can also be addressed within is also much larger in the experimental figure extending
the Gutzwiller approximation. In Fig. 11(c), we observe up to U /t(cid:39)30 and U /t(cid:39)25 whereas we observe it in
s l
jumpsinρ(µ)thatmarkthepresenceofunstableregions. the region where U /t and U /t are smaller than 10. We
s l
Inparticular, againobservethatthesystemisnotstable believe these discrepancies to be due to the fact that we
forρ=0.5andthatthere isaSF-SSdiscontinuous tran- performed bulk simulations (i.e. with no trap) whereas
9
theexperimenttakesplaceinaharmonictrap. However, ACKNOWLEDGMENTS
the main point is that the existence of the supersolid
phase observed in the experiment is confirmed. It is a
true thermodynamically stable phase and not a simple
mixture of superfluid and solid phases.
Wealsoextendedthephasediagramtootherdensities
and confirmed predictions that were made using mean- We thank T. Roscilde for constructive discussion on
field calculations17,18. We also discussed the stability of theGMCtechnique. WethankG.MorigiandT.Donner
theobservedphaseswithrespecttophaseseparationand for useful discussions. This work is supported by the
found that the ρ = 1 and ρ = 1.5 supersolid phases are AlexandervonHumboldt-Foundation. Somecalculations
stable whereas there is phase separation between a SF have been performed on the PSMN centre of the ENS-
and a SS for ρ=1/2. Lyon.
∗ Corresponding author: [email protected] 15 J. L´eonard, A. Morales, P. Zupancic, T. Esslinger, and T.
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