Lattice bosons in quartic confinement

0.22 MB·English

by  R. Ramakumar

EPJ manuscript No.
(will be inserted by the editor)
Lattice bosons in quartic confinement
8
0
0 R. Ramakumar1a and A. N. Das2
2

n
a 1 Department of Physics and Astrophysics,University of Delhi, Delhi-110007, India
J
8 2 TheoreticalCondensedMatterPhysicsDivision,SahaInstituteofNuclearPhysics,1/AFBidhannagar,Kolkata-700064,India
2

] 12 January 2008
r
e
h
t
o Abstract. We present a theoretical study of bose condensation of non-interacting bosons in finite lattices
.
t
a in quartic potentials in one, two, and three dimensions. We investigate dimensionality effects and quartic
m
potential effects on single boson density of energy states, condensation temperature, condensate fraction,
-
d
n and specific heat. The results obtained are compared with corresponding results for lattice bosons in
o
c harmonic traps.
[

2
v
7 PACS. 0 3.75.Lm, 03.75.Nt, 03.75.Hh
9
2
1 1 Introduction of the periodic lattice potential and an overall confining
.
6
0 harmonic potential. Many theoretical studies[9,10,11,12,
7
Overlastfewyears,bosonsandfermionsinopticallattices
0 13,14,15,16,17,18,19,20] of such lattice bosons in har-
:
v haveemergedasimportantcontrollablesystemsforinves-
i monic confinement have appeared in recent years. Unlike
X
tigationsintoseveralpropertiesofquantummany-particle
r the case of lattice bosons in harmonic confinement, the
a
systems[1]. These are clean systems in which experimen-
effect of anharmonic potentials on the properties of lat-
talists have achieved great control over a wide range
tice bosons have begun to be explored only recently[21].
of particle numbers, particle hopping, and strength and
Inthat workGygiandcollaboratorsstudied zerotemper-
sign of inter-particle interactions. Experimental groups
ature properties of strongly interacting lattice bosons in
have conducted extensive studies [2,3,4,5,6,7,8] of sev-
a quartic trap. It has been suggested[21] that it is experi-
eral properties of many-boson systems in one, two, and
mentally possible to create an optical lattice in a quartic
three dimensional optical lattices. The many-boson sys-
trap employing a combination of red-detuned and blue-
tem in these experiments is under the combined influence
detuned Gaussian laser beams. Finite temperature prop-
a e-mail: [email protected]
2 R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement
erties of lattice bosons in a quartic trapis of considerable tonian of the many-boson system we consider is
interest in this context.
H = t c†c +c†c + V(i)n µ n , (1)
In this paper, we present a theoretical study of lattice − X (cid:16) i j j i(cid:17) X i− X i
<ij> i i
bosons in a quartic potential in one, two, and three di-
where t is the kinetic energy gainwhen a boson hop from
mensions (1d, 2d, and 3d). We consider non-interacting
siteitoitsnearestneighborsitej intheopticallattice,c†
i
bosons in a periodic lattice in 1d in a 1d quartic poten-
is the boson creation operator, V(i) is the quartic poten-
tial,asquarelattice ina2dquarticpotential,andacubic
tial at site i, n =c†c the boson number operator, and µ
i i i
lattice in a 3d quartic potential. We study the effects of
thechemicalpotential.Theformsofthequarticpotentials
thepotentialonone-bosondensityofenergystates(DOS)
usedare:V(i)=Qx4 in1d,V(i)=Q(x4+y4)in2d,and
i i i
and the temperature dependence of ground state occu-
V(i) = Q(x4+y4+z4) in 3d. We first obtain the matrix
i i i
pancy andspecific-heat.We comparethe resultsobtained
representation of the system Hamiltonian in a site basis.
for lattice bosons in quartic traps to the corresponding
We numerically diagonalize it to obtain energy levels of
resultsforlatticebosonsinharmonictraps[20].Thiswork
a lattice boson. We have used open boundary conditions.
is presented in the Sections 2 and 3, and conclusions are
We have chosen lattice sizes large enough so that finite
given in Section 4. In the work presented in the following
size effects are absent in the results presented in the next
sections,wehavenotincludedtheeffectofboson-bosonin-
section. The lattice sizes were fixed by finding the lattice
teraction (U). Our results would approximately also hold
size beyond which results remain unchanged with further
in the weak interaction regime (U << t, where t is the
increase of lattice size. This depends on the magnitude of
boson hopping energy) where the interaction induced de-
the quartic potential strength since it decides the spread
pletioneffectsarenotsignificant[22].Aweaklyinteracting
of the boson distribution in the lattice for a given value
regime may be achievedby adjusting the lattice potential
of t and temperature. The energy levels (E ) obtained for
i
depth to a low value as has already been done for lattice
a boson in these large lattices are used in calculations of
bosons in quadratic traps[2,4].
the DOS, ground state occupancy, and the specific heat.
The chemicalpotential and bosonpopulations in the var-
ious energy levels are calculated using the boson num-
2 Model and method
m
ber equation: N = N(E ), where E and E are
Pi=0 i 0 m
the lowest and the highest single boson energy levels and
InthisSection,wegiveabriefpresentationofthemodelof N(E )=1/[exp[β(E µ)] 1]inwhichβ = 1/k T with
i i B
− −
the systemandthe methodfollowedinthe calculationsof k the Boltzmann constant and T the temperature. The
B
various properties presented in later sections.The Hamil- specificheatiscalculatedfromthetemperaturederivative
R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement 3
of total energy (E = m N(E )E ). All energies are a 2d square lattice in a 2d quartic potential (Fig. 2(a))
tot Pi=0 i i
measured in units of t. and the corresponding results for the harmonic case(Fig.
2(b)). In the quartic potential case, we find that the van-
Hovesingularityisstronglysuppressedbytheconfinement
3 Results and Discussion
potential. In comparison, the Van Hove singularity is de-
3.1 One-boson density of energy states stroyed by the harmonic potential[23]. In both cases the
confiningpotentialsspreadtheDOScomparedtothepure
In this section, in Figs. 1-4, we present our results on
latticecase.Fig.3showstheDOSofabosoninacubiclat-
the one-boson density of energy states (DOS) for a lat-
ticeinaquarticpotential.Increasingthequarticpotential
tice bosons in a quartic trap and compare with the corre-
strength clearly has a strong effect on the DOS. Similar
sponding DOS for a lattice boson in a harmonic trap for
results are obtained for the harmonic potential case as
which: V(i)=K(x2+y2+z2). In Fig. 1, we have exhib-
i i i
well,asshowninFig.4.Thedottedlines inFig.4 arethe
ited the DOS for bosons in optical lattices with harmonic
singleparticleDOSforaninfinitedimensionalhypercubic
and quartic confinement potentials for several values of
lattice whose DOS is[24]: ρ(E) =exp[ E2/(2t2)]/√2πt2.
q = Qa4 and k = Ka2, where a is lattice constant. We −
Oncomparingthechangesbroughtaboutbytheharmonic
find that the confining potential has a significant effect
potential in 1d, 2d, and 3d, we notice an approximate di-
on the DOS in both cases. In the case of quartic confin-
mensionality crossover in the DOS for small k. For a 1d
ing potential (Fig. 1(a)), the divergence in DOS at the
latticewithharmonicpotential,theDOShasafinitevalue
band edges is found to be suppressed and eventually de-
at the lower band edge and a weak singularity well inside
stroyedwithincreasingstrengthofthepotential.Further,
theband,whicharecharacteristicsoftheDOSofa2dlat-
the quartic potential spreads the DOS over a wide en-
tice in the absence of confining potential. For a 2d lattice
ergy scale. It is also to be noted that for small potential
with harmonic potential, the DOS almost vanishes at the
strengths shown in the figure, the low energy part of the
lower band edge and has a flat region in the middle part
DOS continues to show similarity to the case of the DOS
of the band, which are characteristics of the DOS of a 3d
of lattice bosons without confinement. In comparison, in
lattice in the absence of any confining potential. Finally,
the case of harmonic potential (Fig. 1(b)), divergence of
the DOS of a 3d lattice with harmonic potential is found
the DOS at the lower edge of the band is destroyedwhile
to be close to that of an infinite dimensional hypercubic
the one at higher edge is significantly suppressed even for
lattice. Hence in DOS, the dimensionality crossoversseen
a weak harmonic potential, and with increasing strength
are:1d(k =0) 2d(k =0),2d(k =0) 3d(k =0),and
6 → 6 →
the DOS is flattened to a wide energy scale[23]. In Fig.
3d(k=0) d(k =0).Incomparison,suchdimension-
6 →∞
2, we have shown our results on the DOS of a boson in
4 R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement
ality crossoversarenot foundin the case oflattice bosons 1(a). At low temperatures the bosons are in low energy
in quartic traps. states.The DOSplotsimplies thatchangeinthe shapeof
the band bottom is not significant for small q. Hence for
3.2 Ground state occupancy and specific heat smallq, the C curves for different q values are very close
v
to each other in the low temperature range.
In this section we present our results on condensate frac-
tion, condensation temperature, and specific heat of lat- InFigs.8-10,we havepresentedour resultsfor bosons
ticebosonsinquarticpotentials.Acomparisonofthetem- in 2d square lattices in a 2d quartic potential. Compared
perature dependence of the fractional ground state occu- to the 1d case, the condensate fraction is found to in-
pancyfor1dlatticebosonsina1dharmonictrapandina creasefasterwithdecreasingtemperatureinthetempera-
1dquadratictrapisshowninFig.5wherewehaveplotted ture range below T . The temperature variation of N /N
0 0
the variation of N /N with T/T . Here N is the boson is seen to be close to the case of pure lattice bosons com-
0 0 0
populationinthelowestenergylevelandT isdetermined pared to lattice bosons in a harmonic trap. This feature
0
by setting N = 0 and µ = E in the number equation is also found in the 3d case discussedlater. The results of
0 0
(i.e., by solving N = im 1/[exp(E E )/k T ) 1]). the quartic confinement case is closer to the pure lattice
Pi=1 i− 0 B 0 −
We note that the dependence of fractional ground state case since, in the central region of the lattice, the quartic
occupancy on scaled temperature (T/T ) is nearly inde- potentialisshallowercomparedtotheharmonicconfining
0
pendentofthestrengthsofthepotentialswhenlatticesare potential. In the low energy range, the DOS of a lattice
largeenoughthatfinite sizeeffectsareabsent.Thelattice with quartic potential is closer to that of a pure lattice
sizes and the strengths of potentials we have used satisfy compared to the DOS of a lattice with harmonic confin-
thiscondition.ThedependenceofT onthestrengthofthe ing potential. Increase in dimensionality leads to smaller
0
quartic potentialpresentedin Fig. 6 shows a fastincrease number of bosons in the ground state for T T . Fig.
0
≥
for small values of q and a monotonic increase for larger 9 shows the dependence of condensation temperature on
q.The bosonnumber dependence ofT is found to be lin- the strength of the quartic potential. While the k depen-
0
ear (not shown) similar to lattice bosons in a harmonic dence is similar to that found in 1d, the magnitude of
potential[20]. In Fig. 7, we have shownthe dependence of T is much lower in 2d compared to that in 1d for the
0
thespecificheatonscaledtemperature.Increasingthethe same value of q. Fig. 10 shows the dependence of specific
strengthofquarticpotentialsuppressC exceptinthelow heat on temperature for various strengths of the quartic
v
temperatureregime.Inthelowtemperaturerange,theC trap potential. Unlike in 1d, the C is found to have a
v v
is nearly independent of the potential strength. This can peak near the condensation temperature. In the interme-
be qualitatively understood if we regard the DOS in Fig. diatetohightemperaturerange,thespecificheatisfound
R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement 5
to be suppressed with increasing potential strength. Low the overall confining potential, it would be better to use
temperatureC showsa slightenhancementwith increas- a quartic confinement rather than a harmonic one. Fig.
v
ing q. When compared with the low temperature C of 13 contains our results on quartic potential strength de-
v
lattice bosons in harmonic traps which has a (T/T )2 de- pendence onthe condensationtemperature.These results
0
pendence, we find that this low temperature part for the are qualitatively similar to 1d and 2d, and quantitatively
quartictraphasapproximatelya(T/T )1.7 dependenceas similarto 2dthan1d.The temperaturedependence ofC
0 v
shown in Fig. 11. This can be understood when we con- is shown in Fig. 14 for various strengths of the quartic
sider the DOS of a lattice boson in quartic and harmonic potential.Increaseddimensionalitymakesthe peakatthe
traps. We first recall that the temperature dependence of condensation temperature sharper. Similar to 1d and 2d,
the specific heat of bosons goes as (T/T )α for a E(α−1) increasing the strength of the potential leads to suppres-
0
dependenceoftheDOS[25].Now,wefindthatthelowen- sion of C except in the low temperature region. The low
v
ergy part of the DOS of lattice bosons in a quartic trap temperature specific heat has a (T/T )2.65 dependence as
0
can be fitted with a value of α 0.8 0.9, as a conse- showninFig.11.Forlatticebosonsinaharmonictrap,the
≈ −
quence of which the specific heat exponent is less than low temperature C has a (T/T )3 dependence. This dif-
v 0
2. For 2d bosons in a 2d harmonic trap, low energy part ferenceoriginatesfromthedifferenceintheDOSforthese
of DOS has a linear E dependence leading to a (T/T )2 two cases which, in the low temperature range, shows a
0
dependence of C . E(α−1) dependence with α = 3 for lattice bosons in a
v
harmonic trap and α 2.5 2.6 for the quartic trap.
≈ −
Now we discuss our results on 3d lattice bosons in a
3d quartic trap. The temperature dependence of the con-
4 Conclusions
densate fraction, shown in Fig. 12, shows that increased
dimensionalitymakesthecondensationsharper.TheBose In this paper, we presented results of our calculations of
condensationis favoredbysmallvaluesofthe DOSatthe singlebosondensityofenergystates,condensatefraction,
bottom of the energy band. With increasing dimension- condensation temperature, and specific heat of bosons in
ality, the DOS decreases at and near the bottom of the someone,two,andthreedimensionalperiodiclatticesina
band and this leads to the sharpness of the condensation. quarticpotential. Wherever possible,the results obtained
Similar to 1d and 2d results, the growth of the conden- are compared with the corresponding results for lattice
sate fraction with temperature is found to lie intermedi- bosons in harmonic traps. In one dimension, we find that
ate between the lattice bosons in a harmonic trap and theDOSofalatticebosoninaquarticpotentialcontinues
pure lattice bosons.Clearly,if one is lookingfor phasesof toretainitspurelatticeformforsmallpotentialstrengths
bosonsinopticallatticeswithminimuminterventionfrom unlike the case of harmonic trap. In two dimensions, the
6 R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement
Van Hovesingularityis suppressedbut not eliminated for 5. B. Paredes, A. Widera, V. Murg, O. Mandel, S. F¨olling, I.
small quartic potential strengths in contrast to the effect Cirac, G. V. Shlyapnikov, T. W. H¨ansch, I. Bloch, Nature
of a harmonic potential. In three dimensions, the quartic 429, 277 (2004).
andharmonicpotentialarefoundtodestroyflatregionsof 6. M. K¨ohl, H. Moritz, T. St¨oferle, C. Schori, T. Esslinger, J.
theDOSofapurelattice.Inaddition,forlatticebosonsin Low Temp. Phys.138, 635 (2005).
aharmonicpotential,onefindsadimensionalitycrossover 7. G. K.Campbell, J. Mun, M. Boyd,P. Medley, A.E. Lean-
intheDOS,whichisnotfoundinthecaseofaquarticpo- hardt,L.G.Maracassa,D.E.Pritchard,W.Ketterle,Science
313, 649 (2006).
tential.The temperaturevariationofcondensatefraction,
8. S.F¨olling,A.Widera,T.Mu¨ller,F.Gerbier,I.Bloch,Phys.
condensationtemperature,andspecificheatisfoundtobe
Rev. Lett.97, 060403 (2006).
intermediatebetweenthoseoflatticebosonsinaharmonic
9. D.Jaksch,C.Bruder,J.I.Cirac,C.W.Gardiner,P.Zoller,
potential and pure lattice bosons.
Phys. Rev.Lett. 81, 3108 (1998).
10. W. Zwerger, J. Opt. B: Quantum Semiclass. Opt. 5, S9
Acknowledgments
(2003).
RRK thanks Professor Bikash Sinha, Director, SINP 11. A. M. Rey, K. Burnett, R. Roth, M. Edwards, C. J.
Williams, C. W. Clark, J. Phys. B: At.Mol. Opt.Phys. 36,
and Professor Bikas Chakrabarti, Head, TCMP Division,
825 (2003).
SINP for hospitality at SINP. RRK also thanks Professor
12. G. Pupillo, E. Tiesinga, C. J. Williams, Phys.Rev. A 68,
Helmut Katzgraber, ETH, Zurich for drawing his atten-
063604 (2003).
tiontobosonsinquartictraps.WethankDr.S.Sil,Visva
13. S.Wessel,F.Alet,M.Troyer,G.G.Batrouni, Phys.Rev.
Bharati, Santiniketan for help in computation.
A 70, 053615 (2004).
14. L.Pollet,S.Rombouts,K.Heyde,J.Dukelsky,Phys.Rev.
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Bloch, Nature415, 39 (2002). Phys. Rev.A 71, 063601 (2005).
3. T. St¨oferle, H. Moritz, C. Schori, M. K¨ohl, T. Esslinger, 17. G. Pupillo, A. M. Rey,G. G. Batrouni, Phys. Rev. A 74,
Phys.Rev.Lett. 92, 130403 (2004). 013601 (2006).
4. C. Schori, T. St¨oferle, M. Henning, M. K¨ohl, T. Esslinger, 18. B.G.Wild,P.B.Blakie,D.A.W.Hutchinson,Phys.Rev.
Phys.Rev.Lett. 93, 240402 (2004). A 73, 023604 (2006).
R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement 7
19. V. Murg, F. Verstraete, J. I. Cirac, Phys. Rev. A 75,
033605 (2007). 3333
20. R.Ramakumar,A.N.Das,S.Sil,Eur.Phys.J.D42,309 2222....5555 bbbb
2222
(2007). SSSS
OOOO 1111....5555
DDDD
21. O. Gygi, H. G. Katzgraber, M. Troyer, S. Wessel, G. G.
1111
Batrouni, Phys. Rev.A 73, 063606 (2006). 0000....5555
0000
22. R. Ramakumar and A. N. Das, Phys. Rev. B 72, 094301
----11110000 0000 11110000 22220000 33330000 44440000 55550000 66660000
(2005). EEEE
444
23. C.HooleyandJ.Quintanilla,Phys.Rev.Lett.93,080404 333...555
333 aaa
(2004). 222...555
SSS
OOO 222
24. W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 DDD
111...555
111
(1989).
000...555
25. C.J.PethickandH.Smith,Bose-Einsteincondensationin 000
---555 000 555 111000 111555 222000
dilute gases (Cambridge University Press, Cambridge, Eng-
EEE
land, 2002).
Fig. 1. Density Of States (DOS) of a boson in a one-
dimensional periodic lattice of size 1000 in quartic (a) and
harmonic(b)potentials.Inbottompanel(a):q = 0.25×10−6
(dots),0.25×10−5 (squares),and0.25×10−4 (crosses).Intop
panel (b): k = 0 (dots), 0.0001 (solid), 0.0002 (dashes), and
0.0003 (dash-dot). All energies are measured in unitsof t.
8 R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement
000...000222 000...111666
qqq === 000
000...111444
bbb
000...000111555 000...111222 aaa
qqq === 000...000000000000111
000...111
SSS SSS
OOO 000...000111 OOO 000...000888
DDD DDD
000...000666
000...000000555 000...000444
qqq === 000...000000000111
000...000222
000 000
000 555000 111000000 111555000 222000000 222555000 333000000 ---555 000 555 111000 111555 222000
EEE EEE
000...000888 00..0044
000...000777 aaa 00..003355
000...000666 00..0033 bb
000...000555 00..002255
SSS SS
OOO 000...000444 OO 00..0022 qq == 00..00000011
DDD DD
000...000333 00..001155
000...000222 00..0011
000...000111 00..000055 qq == 00..000011
000 00
000 111000 222000 333000 444000 555000 666000 777000 00 1100 2200 3300 4400 5500 6600 7700 8800
EEE EE
Fig. 2. The DOS of a boson in a two-dimensional square Fig. 3. The effect of quartic potential strength on the DOS
lattice of size 150 × 150 in a quartic (a) and harmonic (b) of a boson in a three-dimensional cubiclattice of size 50 × 50
potentials. In the bottom panel (a): q = 0.25×10−5 (solid), × 50.
0.25×10−4 (dashes),and0.10×10−3 (dash-dot).Intoppanel
(b): k = 0.01 (solid line), 0.02 (dashed line), and 0.03 (dash-
dot).
R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement 9
40
00000.....1111144444
kkk === 000 35
00000.....1111122222
30
00000.....11111
25
SSSSS
OOOOO 00000.....0000088888 kkk === 000...000111 T0 20
DDDDD
00000.....0000066666
15
00000.....0000044444 10
kkk === 000...000222
00000.....0000022222 5
00000 0
-----55555 00000 55555 1111100000 1111155555 2222200000 2222255555 3333300000 0 5e-05 0.0001 0.00015 0.0002 0.00025
EEEEE q
Fig. 4. The effect of the strength of harmonic potential on Fig.6. Quarticpotentialstrengthdependenceofcondensation
DOS of a boson in a three-dimensional cubic lattice of size 70 temperaturefor600bosonsina1dperiodiclatticeofsize1000.
×70×70forthevaluesofkshownonthecurves.Thedotted Thelowest value of q in this figureis 0.25×10−6.
linesareDOS’sforaninfinitedimensionallatticeintheabsence
000...333
of any confining potential, as discussed in thetext.
000...222555
000...222
11 BBB
kkk
NNN 000...111555
///
00..88 CCCvvv
000...111
NN 00..66
//
00 000...000555
NN
00..44
000
000 000...555 111 111...555 222
00..22 TTT///TTT
000
00 Fig. 7. Temperature dependence of the specific heat of 600
00 00..55 11 11..55 22
TT//TT
00 bosons in a 1d lattice of size 1000 in a quartic potential of
Fig. 5. Temperature dependence of condensate fraction for stregnthsq = 0.25×10−6 (solid line),0.25×10−5 (dots),and
bosons in 1d optical lattices with confining potentials: lattice 0.25×10−4 (dash-dot).ThevaluesofkBT0 forthesecases are
bosons in a quartic trap of strength q = 0.25×10−6 (dots) 3.23, 6.91, and 14.98, respectively.
and lattice bosons in a harmonic trap of strength k = 0.001
(solid).Thelatticesizeusedis1000andthenumberofbosons
is 600. The values of kBT0 for these cases are 3.23, and 6.773,
respectively.
10 R. Ramakumarand A.N. Das: Lattice bosons in quartic confinement
111
222...555
000...888
222
000...666 BBB
NNN kkk 111...555
/// NNN
000
NNN ///vvv
000...444 CCC
111
000...222 000...555
000 000
000 000...555 111 111...555 222 000 000...555 111 111...555 222
TTT///TTT TTT///TTT
000 000
Fig. 8. Comparison of the growth of condensate fraction of Fig. 10. Temeperature dependence of specific heat of 600
lattice bosons in 2d: bosons in a 2d square lattice of size 100 bosons in a 2d lattice of size 100 × 100 in a quartic potential
× 100 (dash-dot), lattice bosons in a quartic trap of strength of stregnths q = 5×10−5 (solid line), 0.10×10−4 (dotted),
q = 0.25×10−5 (dotted), and lattice bosons in a harmonic and0.25×10−4 (dash-dot).ThevaluesofkBT0 forthesecases
trap of strength k = 0.01 (solid). The lattice sizes used are are 1.187, 1.817, and 2.421, respectivley.
150 × 150 (harmonic trap case) and 100 × 100 (quartic trap
77
case), and number of bosons is 600. The values of kBT0 for 66
55
BB
kk 44
these cases are 0.18, 1.187, and 3.19, respectively. C/NC/Nvv 33
22
11
6 00
00 00..22 00..44 00..66 00..88 11 11..22
TT//TT00
44
5 33..55
33
kkBB 22..55
4 NN 22
C/C/vv 11..55
11
T0 3 00 ..0055
00 00..22 00..44 00..66 00..88 11 11..22
2 TT//TT00
1 Fig. 11. Fits to low temperature specific heats. 2d (bottom
0 panel): the dotted line is a plot of 2.6×(T/T0)1.7. 3d (top
0 5e-05 0.0001 0.00015 0.0002 0.00025
q panel): thedotted line is a plot of 7.5×(T/T0)2.65
Fig.9. Quarticpotentialstrengthdependenceofcondensation
temperature for 600 bosons in a 2d square lattice of size 100
× 100.