- Type Phase Transition for a Weakly
Interating Bose Gas
I. Cabrera a;b
,D. Oliva a;b
and H. Perez Rojas a;b;
a
InternationalCentreforTheoretialPhysis,
P.O.Box58634100Trieste,Italy
b
GrupodeFisiaTeoria, ICIMAF,
CalleENo. 309, Vedado,LaHabana4,Cuba
HelsinkiInstitute ofPhysis,
P.O.Box9(Siltavuorenpenger 20C),FIN-00014,Universityof Helsinki,Finland
Reeivedon22November,1999
It is suggestedageneral mehanism throughwhiha-typebehavior is produed inthe spei
heatof aBose gas nearthe ritial temperature T
. It isessentialthat the quasipartile spetra
haveagapproportionaltotheondensate. Itworksforagenerallassofquasipartilespetra,and
inpartiular,for theweaklyinteratingBose gas. Theintrodution ofahemialpotentialinthe
theoryisbrieydisussed.
I Introdution
InhisbookonStatistialMehanis,[1℄Feynmanrefers
tothepointbehaviorandexpresseshisviewthat
per-hapspartofthe explanationofthe transitioninvolves
Bose ondensation. Oneinterestingmodelofthe
be-havioris the one presented by Ceperley [2℄, but in it
thebosonsystemissimulatedbymeansofMonteCarlo
tehniquesusingpathintegrals.However,althoughthe
analytibehaviorofthespeiheatinaneighborhood
of theritialtemperatureisprovidedby
renormaliza-tion group methods, whih gives an aurate
desrip-tion of the behavior(see i.e. [3℄), we are notaware
of anymodelexhibitingthemehanismthroughwhih
the ondensate indues the divergene of the spei
heat
v atT
. Therearetwophenomenawhihare
ex-peted to be explainedby a satisfatorymodel of the
weaklyinteratingbosegas: 1) theexisteneofa
gap-lessmode,whihismoremanifestatT =0,and2)the
behavior,whihwouldappearatnitetemperatures.
Inouropinionbothphenomenamustbemanifest even
at the one-loop level, as it is in the relativisti
mod-elswementionbelow,but whihis nottheasein the
usualnon-relativistimodel,i.e.,thebehaviorannot
befoundbyusingtheBogoliubov'sspetrum[4℄forthe
weaklyinteratingBosegasnearT =0bytakingitas
valid attemperaturesneartheritialpoint.
Our aim is to show that if the bosoni
quasiparti-le spetrahasagapproportionaltotheondensate,a
divergenein appearsatthetransitiontemperature,
andthatthisistheasefortheweaklyinteratingBose
gasnear theritialtemperature.
Firstofall,wewish toremindthesimplemodel of
thetemperatureself-interating salarone-omponent
eldwithZ
2
symmetrybreaking =hiin
relativis-tiquantumstatistis[5℄. Herethesymmetrybreaking
parameterplaystherole ofaeld ondensate. The
ef-fetive potential is V(;T) =
2
1
4
4 a
2
2
2 +V
1 (T;),
where 2
1
istheouplingonstant, a 2
is thenegative
masstermandV
1
(T;)isthesumofT-dependent
tad-pole diagrams. On themassshell V(
;T)=is the
thermodynamialpotential,
being theextremum of
V(;T), where the spetrum is (p) = p
p 2
+M 2
2
,
M=2
1
.
In the high temperature limit one has V
1
(T;) =
2
1 T
2
2
=8, the extremum of V(;T) leading for
T < T
to a temperature-dependent mass M(T) =
1 p
T 2
T
2
, T
= 4a
2
= 2
1
being the ritial
temper-ature for symmetry restoration. The infrared
ontri-bution to the thermodynamial potential in the limit
T M(T) is '
3
M 3
T
12~ 3
. Oneobtainsthen a
spe-iheat
v = T
2
T 2
' 3
3
1 T
4
2h 3
p
T 2
T
2
whih
di-vergesast 1=2
,t =j1 T=T
j,for T !T
, showinga
-typebehavior. Thisexpressionshowsthatthe
diver-geneappearsalreadyattheone-loopapproximationof
v
,althoughitisquantitativelysatisfatoryfort
1
[6℄, sine in the region loser to T
, the ontribution
renormalizationgrouptehniquesgivesanuniversal
ex-pressionforthedivergeneast
,where<1=2.
In the two omponent salar eld ase, having
U(1) symmetry, two modes appear after the
symme-trybreaking, themassive
1 (p)=
p
p 2
+M 2
2
, M =
2
1
, and the massless
2
(p) = p. The ontribution
to the thermodynami potential infrared term is the
same as in the previous ase (the masslessterm does
notontribute),and
v
divergesast 1=2
.
Atually,bothintherelativistiandnon-relativisti
ases,withspetra
r =
p
p 2
+,and
n =p
2
=2m+
, respetively. By taking some harateristi small
momentum p
0
and introduing x = p=T, alling =
p
0
=T,itiseasytoseethatthereisaommoninfrared
ontribution to AT 5=2
R
0 x
4
dx=[x 2
+T 1
℄
A 0
T 3=2
, where the mass (gap) b[1 (T=T
)
℄,
>1,andA;A 0
;bbeingonstants. Thenoneanstate
asatheoremthat forT !T
,theone-loop
thermody-nami potentialobtainedfrom suhspetraleadsto a
divergent
v
behavioras 1=2
.
Intherelativistilimitthisomesfrom(Aontains
afator~ 3
):
=AT Z
1
0 p
2
dpln(1 e
r
)= A
2
3 Z
1
0 p
4
dp
r (e
r
1)
(1)
Thetheinfraredlimitofthelast expressionis
AT
5=2
3 Z
0 x
4
dx
x 2
+T 1
AT
5=2
(T 1
) 2
3
Z
0
dx
x 2
+T 1
+O(T 1
) (2)
This meansthat nearT
AT 5=2
(T 1
) 3=2
3
Artan
p
T 1
=
AT 3=2
6
: (3)
From
V = T
2
T 2
;wegetthat
V
divergesas 1=2
:
Inthenon-relativistiasewehaveasimilarformula,bytakingx=p= p
2mT,sine
=AT Z
1
0 p
2
dpln(1 e n
)= A
3 Z
1
0 p
4
dp
m(e n
1)
A(2mT) 5=2
3m Z
0 x
4
dx
x 2
+T 1
: (4)
fromwhihwegetthesamebehaviornearT
:
d
It is expeted that any system having a spetrum
of similar infrared properties than the previous ones,
would have also a spei heat having an innite
be-haviorattheritialpoint. Wewouldliketoshowthat
suhistheaseoftheBogoliubovmodeloftheweakly
interatingBosegasneartheritialpoint. Itontains
a gap due to the ondensate, whih leads to a
diver-gene in the spei heat (in the region T=T
< 1),
alreadyattheone-loopapproximation.
II Bogoliubov Hamiltonian near
the ritial point
We will start from the quantized Hamiltonian for a
weakly interating Bosegas expanded in termsof the
elementaryreationandannihilationoperatorsfor
spin-vation in the interations and all U(r) the
repul-sive potential of the two body interation, U p
0
1 ;p
0
2
p1;p2 =
R R
d 3
re ipr
~
U(r). Forp=0(zeromomentum transfer
in the ollision) U
o =
R
d 3
rU(r). In the temperature
intervalweareonsideringthemomentaareverysmall
andweanassumethatthemomentumtransferineah
ollisionisalmostzero,p'0. Forthatreasonitis
pos-sible toexpress approximatelythematrixelementsby
usingU
o
. Thesatteringlengthanbewrittenthenas
a= m
4~ 2
U
0
,anda>0sinethepotentialisrepulsive.
We shall assume the onditions a= 1 and
a 3
1 where is the thermal de Broglie
wave-length and = N=V the partile density. Then, by
starting from the fat that the oupation number of
the ground state n
0
is a large number, we will write
a
o =
p
n
0 e
i
, a +
o =
p
n
0 e
i
. From thiswewouldget
[7℄ (x) !e i
(x) ( (x) =V 1=2 P e ik x a k
), and we
takethephysial representationasthe onewith =0
whihleadsto,
^ H = U o 2V (n 2 o n o )+ X p6=0 p 2 2m a + p a p (5) + U o n o 2V X p6=0 fa + p a + p +a p a p +4a + p a p g + U o 2V X pi6=0;p 0 i 6=0 a + p 0 1 a + p 0 2 a p2 a p1
Weassumeonservationofthetotalnumberofpartiles
N andthen P p a + p a p +n 0
=N. Theusualproedure
forT !0,byfollowingBogoliubov[8℄[9℄(seealso[10℄),
assuming that P a + p a p = P n p
N, is to substitute
in(5)n 2 0 by[N P a + p a p ℄ 2
intherstandn
0
byN in
thethirdterm,whihleadstotheanellationofaterm
U0n0 V P a + p a p
in the last termin urlybrakets of(5).
Thismeanstonegletonetermofseondorderin P
n
p ,
the number of exited partiles. But if P
n
p N,
oneannotneglettheterm( P n p ) 2 = P a + p a p a + p 0 a p 0.
This is the ase for temperaturesnear and below the
transition point, and even for very low temperatures;
e.g., as in the ase n
0
' N=2, P
n
p
' N=2. Thus,
we keep n
0
(T) expliit in our formulae as a quantity
dereasing with inreasing temperature. We onlude
thatBogoliubov'gaplessspetrumisanapproximation
valid atually forT =0 ,for almost allthe systemin
theondensate.
Inourpresentapproahthelasttermin(5)aounts
fortheenergydue totheinterationsofpartiles with
momenta p 6= 0. Sine we are onsidering an
inter-val of temperatures very near (but below) the
transi-tiontemperature,weassumethatsuhtermis
approx-imately onstant and take it as E
N = O(U 0 N 2 =2V).
Then the problem we are left with is the
diagonal-ization of the sum of the seond and third terms in
(5). Our assumptions are valid for a wide range of
values of n
0
(if N ' 10 23
our approximationis good
up to, say, n
0 ' 10
21
). Thus, we keep our
alula-tions in the one-loop approximation and in terms of
n
0
andwhenomparedwiththestandardBogoliubov's
model [8℄ it diers in the oeÆient of the last term
in urly brakets, whih is hanged from 2 to 4. The
next step is to make the usual Bogoliubov's
anoni-al transformation a
p = (b p p b + )= q 1 2 and a + p = (b + p p b p )= q 1 2 p
as aresult of whih we
obtainthediagonalizedHamiltonian,
^
H =E
o +E N + X p6=0 "(p)b + p b p ; (6) where E o = U o 2V [n 2 o n o ℄ U o n o 2V P p6=0 p and p = V Uono [4 Uono 2V + p 2 2m
"(p)℄,andifwedenotebyK= Uon0 2V , then "(p)= r 12K 2 +8K p 2 2m +( p 2 2m ) 2 ; (7)
is the spetrum of the new Bose quasipartiles
repre-sentingtheelementaryexitationsofthesystem,where
b +
p ;b
p
are their reation and annihilation operators.
Thelimit T ! 0 does notlead to Bogoliubov's
spe-trum,inwhihthegapterm2 p
3Kisabsent,sineboth
models orrespond todierentapproximations.
Atu-ally,havingonlyonemode,thepresentmodelbearsin
this respet more resemblane with the Z
2
ase than
withthe U(1)one. Inour model thelongwavelength
limitis obviouslynot linear in p, asit is usually. We
may argue that our limit is not T ! 0and also that
its small value probably makes diÆult to identify it
amongtheexperimentalerrorsforp!0nearT
.
Atu-ally,asKkTisverysmall(forn
0 '10
20
;K'10 12
eV)it an be usually negleted, but it is able to
pro-due the marosopi eet of a typial -type
diver-gent spei heat near T
. As lim
p!0
"(p) = 2 p
3K,
thegapparameterK formallybehavesastheanalogof
arestenergyinrelativistidynamis. Thisgaphasthe
property that it dereases with temperature and goes
tozero,aswellasE
0 ;
p
,forT !T
:
Belowwewill alulate thethermodynami
poten-tialbytakingthehemialpotential=0byassuming
thatthenumberofquasipartilesn
q = P p b + p b p isnot
onserved(See,however,nextsetion).
The term K is proportional to n
0
, the eigenvalue
of a
0
with regard to the oherent ground state. In
ourpresentapproximation, the quasi-partiles are
a-tually massive sine their eetive mass is m = [ 2 "(p)=p 2 ℄ 1 j p=0 = p
3m=4.Anexpliitomputation
of
v
showingthedivergentbehaviorlosetotheritial
temperatureis easilydoneby doingrstthe
tempera-tureexpansionof thethermodynami potentialof the
quasipartilesbytakingthespetrum(7).
Duetotheinterationterm,thegroundstateenergy
E
0
E
o =
2~ 2
an 2
o
mV (
1+ 16
15 r
6a 3
n
o
V "
7E 1; r
2
3 !
2F 1; r
2
3 !#)
; (8)
whereE,F aretheusualelliptiintegrals.
d
III The spei heat
We will alulate the thermodynamial potential of
the quasipartiles . The total energy is then U =
E
0 + E
N
T(=T)+ . We will do also the
asymptoti expansion of lose to T
for T < T
.
By hanging to the variable x = q
2m
p it reads
= 2(2m)
3=2
V
5=2
~ 3
1
R
0 dxx
2
ln[1 e p
12M 2
+8Mx 2
+x 4
℄,where
M =K 1andweobtainanexpansionintermsof
it,inthesamewaydoneintheaseoftheeetive
po-tentialinthetemperaturesalarmodel[5℄. Aftersome
alulations whih we outline in the appendix (these
aregivenindetailin [11℄),the perturbativeexpansion
ofinpowersofKisobtained,andfromitthespei
heatisgivenby
v =
E
0
T T
2
T 2
as,
v
= [n
0 +
8
3 q
2a 3
n 3
0 (7E(1;
r
2
3
) 2F(1; r
2
3 ))℄
K
T +
(2m) 3=2
~ 3
(2) 2
f 5
2 (
5
2 )(
5
2 )k
5=2
T 3=2
3 ( 3
2 )(
3
2 )k
3=2
T 1=2
K (9)
+4kT[2K 1=2
3 ( 3
2 )(
3
2 )k
1=2
T 1=2
℄ K
T 4kT
2
[ ( 3
2 )(
3
2 )k
1=2
T 1=2
K 1=2
℄
2
K
T 2
+2kT 2
K
T
2
K 1=2
)
;
d
Here = ( 5
4 +
p
3
2 )(
p
6 p
2). The last term
diverges at T
. We shall write n
o
= Nf(T) where
f(T) = (1 N
e
=N) and N
e
is the density of
parti-les notin the ground state. By starting from N
p =
n
q (1+
2
p
)=(1 2
p )+
2
p
=1
2
p
, [12℄ theexpression
forN
e =~
3 R
d 3
pN
p
isgivenintheAppendixasan
ex-pansionsimilartotheonefor. Undertheassumption
K1,theleadingtermsare
N
e =N
e0
+CT; (10)
where N
e0 = AT
3=2
and A = (2mk=h 2
) 3=2
(3=2),
C = 1=2
( p
6+ p
2)AK 1=2
=k 1=2
(3=2). This equation
for N
e
isto besolved byiteration, bytaking in M as
arstapproximationf(T)=[1 N
e0
=N℄. Itmust be
observedthatasT dereases,therateofondensation
in (10)is strongerthanfortheidealgas. Wewill take
N =AT 3=2
. Byevaluating(9)withheliumparameters
and takingthesattering lengthasa=10 10
m, the
urvefor
v
(T)denedintheregionT T
isdepited
in Fig. 1. Then an innite -type behavior of
v (T)
similar to the salareld aseresults. Suh a
behav-iorannotbeobtainedbyusingtheusualBogoliubov's
spetrum[4℄.
Themoreexatpartof the
v
urveliesin the
re-gionnearandbelowT
sineweassumedE
N
onstant
and K 1. Butthelast onditionis valid evenfor
verylowT andE
N
dereasestosmallvaluesasT !0.
Thus, the
v
urvedrawn is approximatelyvalid even
inthatregion. Thebehaviorof
v
forT >T
isbeyond
thesopeofthepresentletter. Suhproblem mustbe
investigatedbystartingfromanadequatemodelof
0 2 4 6 8 10 12
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
C
v =Nk
(T=T
)
Figure 1:
v
=Nk-urveobtainedfromourmodel, wherethe shapeisshown. ForT >T
theidealgasurvehas
beendrawnforomparison.
IV The introdution of a
hemi-al potential
Theabseneofaseond(gapless)Goldstonemodedoes
not mean an inompleteness of our model, sine
al-though we are dealing with a two-dimensional
prob-lem, the present representation behaves as similar to
the one-dimensional salar ase seen in the
Introdu-tion,whosespetrumbearsagap. Ontheotherhand,
an investigation of the Bose gasby using theGreen's
funtionmethodmadebyBeliaev[14℄leadstotheusual
Bogoliubovspetrumin termsofn
0
at stritly T =0.
However,thisismadebyintroduingahemial
poten-tial,whihlosesitsusualmeaning,sinethe
alula-tion oftheGreen funtions bystartingfrom adensity
matrix, demands ommutation between the
Hamilto-nianH andthenumberN ofexitedpartiles,whihis
nottheasesinethequasipartilenumberisnot
on-served. If=0in[14℄ourspetrumisreprodued,and
thegapannotbeerasedin higherloopalulations.
Thepapers[14℄werewrittenbeforetheformulation
ofGoldstonetheorem[15℄,whihleadtoabetter
under-standingofthetheoryofspontaneoussymmetry
break-ing. Atually,Beliaevresultsarebasedinthesame
op-erator algebrarepresentationused byBogoliubov,and
desribetheexisteneof onlyonemode,whihis
gap-less. Itmissesaseondmodehavingagap,whihis
ex-petedtoappearinusualU(1)gaugeinvariantmodels
with spontaneous symmetry breaking, as was pointed
out in the Introdution. If the hemial potential is
understood notin theusualsense,but asaparameter
ndsthattherepresentationusedbyBeliaevisnotthe
uniqueone: thereisasetofinniteunequivalentones,
whihistypialofthespontaneouslybrokensymmetry
ase[7℄. One of these is being investigated by two of
thepresentauthors(D.O.andH.P.R.)forthe
tempera-turease. Init,asintheU(1)salarrelativistimodel
onsideredintheintrodution,atwomodespetrumis
found. Oneof them is agapless Goldstone mode and
theother onehasagap,the laterproduingalambda
behaviorin the spei heat. Work in this problem is
inprogress.
V Conlusions
Inrelativistimodelsofasalarbosegaswith
symme-trybreaking,itisfoundadivergentbehaviorofthe
spe-iheatattheonelooplevelofthethermodynami
po-tential, whih omes from thetemperature dependent
mass(gap) in the spetrum, whih is onsistent
qual-itatively with renormalization group preditions. By
analogy,weonsiderthe usualBogoliubovmodelnear
the ritial temperature for the phase transition, and
agap is shown to be present, leading to a divergene
inthespeiheat. Wedisussedbrieythe
introdu-tion of a hemial potential in the model, as done in
Beliaevproedure[14℄,andtheabseneofaGoldstone
mode. Weonludethat themodelwehavedisussed,
although not being perhaps the nal one, is an step
Aknowledgments
TheauthorswouldliketothankProfessorM.Virasoro,
IAEAandUNESCOforhospitalityattheInternational
Centre of Theoretial Physis. All authors thank A.
Amezaga,A.Cabo,S.Fantoni,S.Giovannazi,F.
Hus-sain, G. A. Mezinesu, C. Montonen, K. Narain A.
Polls,A.SmierziandR.Sorkinforvaluableomments
andsuggestions. H.P.R.wouldliketothankhisformer
student R. Torres Rivero for having alled his
atten-tion to the possible relation of the -type spetrum
withsometemperature-dependentmassmodel,andM.
Chaihianforseveralilluminatingdisussionsand
hos-pitality in the High EnergyPhysisDivision,
Depart-ment of Physis, University of Helsinki. The partial
support oftheAademyofFinland under ProjetNo.
163394isgreatlyaknowledged.
VI Appendix
To make an asymptoti expansion of in terms of
M = K, we write = (0)+
M j
M=0
M +R (M)
, where we stop ourexpansion in the rst-order term
andR (M)isertainfuntion ofMwhihwearegoing
tondoutbyusingR (M)=(M) (0)
M j
M=0 M
and R
M =
M
M j
M=0 . Then
M =
(2m) 3=2
V
5=2
~ 3
(2) 2
1
Z
0
dxx 2
[24M+8x 2
℄
p
12M 2
+8Mx 2
+x 4
[exp( p
12M 2
+8Mx 2
+x 4
) 1℄ ;
fromwhih
M j
M=0
=4(2m) 3=2
V (3=2)(3=2)= 5=2
~ 3
(2) 2
)TheR (M)funtionisobtainedbyalulatingR =M
from these expressions,by usingtheMatsubarasum 1
"[exp(") 1℄ =
1
P
1 1
" 2
+4n 2
1
2"
, and performingthe adequate
regularizations,wegetnally(seealso[11℄),
=
1
4 3
1=2
2
3 (
5
2 )(
5
2
)kT+4 ( 3
2 )(
3
2
)K
8
3 (kT)
1=2
K 3=2
(kT) 3=2
4 p
2
15 K
5=2
+( 3
2 )(kT)
1 O(K
3
)
12 p
3
g
(11)
Byasimilarproedure,weget,
N
e =
1
3
f( 3
2
)
1=2
( p
6+ p
2)M 1=2
6
(3=2)M 2
+O(M 3
)g (12)
d
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