Bose-Einstein Condensation in a
Constant Magneti Field
H. Perez Rojas 1;2;3
and L.Villegas-Lelovski 2
1
HighEnergy Division,DepartmentofPhysis, UniversityofHelsinki,
Siltavuorenpenger 20C,FIN-00014,UniversityofHelsinki,Finland
2
Departamentode Fsia, Centrode Investigaionyde EstudiosAvanzados delIPN,
Apartado Postal14-740,07000Mexio, D.F.,Mexio
3
Grupo deFsiaTeoriaICIMAF,
CalleENo. 309,Vedado, LaHabana4,Cuba.
Reeived22November,1999
WedisusstheourreneofBose-Einsteinondensationinsystemsofnoninteratingharged
par-tilesinthreeandonedimensionsandinpreseneofanexternalmagnetield. Intheone
dimen-sional, as wellas inthemagnetield ases,althoughnotaritial temperature,aharateristi
temperatureanbefound,orrespondingtothease inwhihthegroundstatedensitybeomesa
marosopi frationofthe totaldensity. Theaseof relativistihargedsalar andvetor
parti-les isstudied. Theresultsgivesupporttotheexisteneofsuperondutivityinextremelystrong
magnetields,andleadtothepreditionofsuperondutive-ferromagnetibehaviorinthevetor
eld ase, whihmight beofinterestinondensedmatteras wellasinosmology. Somefeatures
ofthemagnetizationintheearlyuniverseareonjetured.
I Introdution
Condensationinthegroundstateseemstobeageneral
property whenever the onditions of quantum
degen-erayof theBose-Einsteingas aresatised. Quantum
degenerayis usuallyunderstood to beahievedwhen
theDeBrogliethermalwavelengthisgreaterthatthe
mean interpartile separationN 1=3
. Theremarkable
disovery made by Einstein on the Bose distribution
wasthatondensationmayourstartingatsome
rit-ialtemperatureT
dierentfromzero,whihisusually
referredasBose-Einsteinondensation(BEC).
Conden-sation of harged partiles requires a bakground of
harge ofopposite signto sreenthe mutualrepulsion
amongpartiles. Wewillassumethatsuhbakground
exists.
TheonditionforBECtoourinagasofharged
partilesplaedinamagnetieldisusuallyborrowed
fromtheasewithoutmagnetield as=E
0 (where
E
0
is the ground state energy), whih an be
trans-lated into = p
M 2
eBh for the relativistiase
( 0
= M
2
eBh=M inthenonrelativistilimit).
We must ite rst Shafroth [2℄ who proved that for
a non-relativisti boson gas, Bose-Einstein
ondensa-tion (BEC) does not our in presene of a onstant
magneti eld. It was impliit in his paper that he
onsideredonlytheaseofnotveryintenseelds. The
that aphasetransitionlikethat in thefreegasours
for D 5, and later by Daii et al [4℄, [5℄, who
ex-tendedthe onsiderationsmadeby[3℄ tothe
relativis-tihightemperaturease. Later Toms[6℄provedthat
BECinpresene ofaonstantmagnetield doesnot
ourinanynumberofspatialdimensions,andElmfors
etal[7℄whostatedthatinthe3Dase,althoughatrue
ondensateisnotformed,theLandaugroundstatean
beoupiedby alarge hargedensity. Asimple
rite-rionforBECtoour,wasgivenbyTomsandKirsten
[6℄, whoonludedthat usualBEC anouronlyfor
d3.
Therearetwowaysofharaterizingtheourrene
of BEC,whih are: 1) Theexisteneofaritial
tem-peratureT
>0suh that(T
)=E
0
, (Thisondition
isusually takenasaneessaryandsuÆientondition
for ondensation; see [9℄). Then, for T T
, some
signiantamountofpartilesstartstoondenseinthe
groundstate. 2)Theexisteneofanitefrationofthe
totalpartiledensityinthegroundstateandin states
in its neighborhood at some temperature T > 0. We
referto1)asthestrongandto2)astheweakriterion
forBEC.
A magnetield expliitly breaks thespatial
sym-metry. It isreetedin thewavefuntion ofaharged
partile moving in it, and also in its spetrum. The
on-metry in momentum spae: the momentum
ompo-nentsperpendiulartotheeldollapseinasetof
dis-reteLandauquantum states,theirenergyeigenvalues
havinginnite degeneray. A gas ofharged partiles
(eitherBosonsorFermions)inpreseneofveryintense
magneti elds populate mainly the Landau ground
state n = 0 and behaves as a one-dimensional gas in
theaxisp
3
parallel totheeld. ForBosons,itleadsto
a statistial distribution for xed temperature T and
magnetield B,whih dependsonlyonp
3 .
Wemustbearin mind at thispointthat the usual
propertiesofseond orderphasetransitions annotbe
validinpreseneofexternalelds(seeLandau-Lifshitz,
[12℄). Theexternaleldintroduesin theHamiltonian
a perturbing operator whih is linear in the external
eld strength, in our ase B, and in the order
opera-tor , in our ase ^
M, the magnetization. As a result,
at any value of B, M beomes dierent from zero at
any temperature. Thus, B redues the symmetry of
the usual moresymmetrial phase, and the dierene
betweenthetwophasesoftheusualtheorydissapears;
the disretetransition pointdissapears; thetransition
is\smoothedout".
Thus,asBECin 3Dhasthepropertiesofaseond
order phasetransition,in studying it in presene of a
magnetield, thestrongriteriumannotbeapplied;
no ritial temperature separating phases of dierent
symmetry exists, and that was onrmed in many of
the resultsof referenes [2℄- [6℄. Butan order
param-eter exists at any temperature, the magnetizationM,
whih for large elds and densities, and temperatures
small enough(see below)beomesproportionalto the
ground state density. Atually, the ground state
den-sity an be onsidered also as another order
param-eter losely related to M and may beome a
maro-sopifrationofthetotaldensity,notatadeniteT
,
but in someintervalof valuesofthetemperature. We
may think in the ase in whih one starts from a gas
of hargedpartiles,withoutmagnetield,belowthe
ritial temperature. If a suÆiently strong magneti
eld is applied, thepartilesin theondensateremain
ondensed,andaritialtemperatureisnotpossibleto
be dened. (It is interestingto mention herethat, as
shownin ref. [8℄, there is noritial temperaturealso
in theaseofagasonnedto aparabolipotential.)
Thispaperisamoredetailed andompleteversion
of some of the resultsof twoprevious ones,[13℄, [14℄,
in whih it was pointed out that Bose-Einstein
on-densation,in thesenseof havingalargepopulationin
thegroundstateinaontinuousrangeoftemperatures
(i.e.,noritialtemperatureexists),mayourin
pres-ene of a magneti eld B, for very dense systemsat
verylowtemperatures. Theexistene ofdisrete
Lan-dau quantum states allows this "weak" BEC. In this
ase,thegroundstatepopulationisexpliitlyinluded
in theBose-Einsteindistribution.
Theourreneof Bose-Einstein ondensationin a
strong magneti eld gives theoretial support to the
phenomenon of re-appearane of superondutivity in
elds strongenough, and even leadsto the predition
of superondutive-ferromagneti behavior in the
ve-toreldase,aphenomenonwhihwouldbeofespeial
interestin ondensedmatterandin osmology.
Thestrutureofthepresentpaperisasfollows. In
setionIIweshalldisusssomefeaturesoftheusual3D
Bose-Einsteinondensation,whih givesabasisto
un-derstandits"weak"ourreneintheone-dimensional
ase(setionIII)andinthemagnetieldase(setion
IV).InsetionVitisexpliitlyshowntheferromagneti
behaviorofthemagnetizationofthevetoreldase. It
isalsodisussedtheonnetionwithsuperondutivity,
theexisteneoflow-eldandstrong-eldondensation
andtheorrespondingsuperondutivebehavior,
sepa-ratedbyagap,andtheferromagneti-superondutive
behaviorappearingin thevetoreld ase. setionVI
isdevotedtosomeosmologialonsiderations,and
se-tionVII dealswiththeonlusions.
II The usual Bose-Einstein
Condensation
Inordertobeself-ontained,wereallsomeresultsand
formulaefrom [13℄, [14℄. onerning the standard 3D
BEC theory of the free Bose gas. We must
empha-size that our onsiderations in the present and next
setionare essentiallyonerned with thease of
sys-temsfar from thethermodynami limit. Weshall use
throughout the paper T in energy units, i.e., as the
produt of theBoltzmann onstant k by the absolute
temperature; thus, T = 1
. The hemial potential
=f(N;T)<0is a dereasingfuntion of
tempera-tureatxeddensityN,andfor=0onegetsan
equa-tion dening T
= f
(N). Fortemperatures T < T
,
as = 0, the expression for the density gives values
N 0
(T) < N, and the dierene N N 0
= N
0 is
in-terpretedasthedensityof partilesintheondensate.
Themeaninterpartile separationisthenl=N 1=3
.
Inouronsiderationswewilluseintegralswhih,as
it is usually done, must be interpreted as
approxima-tionsofsums overdisrete quantum states, whih
im-plynottobeworkingatthethermodynamilimit. For
usual marosopisystems, as the separationbetween
quantum statesisp=h=V 1=3
, theapproximationof
thesumbytheintegralisquitewelljustied.
Now, above the ritial temperature for
ondensa-tion
N =4 1=2
3
Z
1
0 x
2
d
e x
2
+
1 =
3
g
3=2
(z); (1)
where = =T(> 0), x = p=p
T
is the relative
mo-mentum, p
T =
p
2mT the harateristi thermal
mo-mentum,and=h=(2mT) 1=2
thedeBrogliethermal
Pathria [10℄), asa funtion of the fugaity z =e =T
,.
AtT =T
,wehaveg
n
(1)=(n),andg
3=2
(1)=(3=2),
and thedensityis N
=(3=2)= 3
,orin other words,
N 3
=(3=2)'2:612. Herewemustpointoutthatif
thepartile densityis written as asumoverquantum
states
N = X
i (z
1
e E
i
T
1) 1
; (2)
for stritly z = 1( = 0), the ground state term
di-verges. (In eld theoretial language, it is due to an
infrared divergene of the one-partileGreen funtion
at = 0). This divergene usually serves as an
ar-gument[10℄toindiatetheourreneofBose-Einstein
ondensation,andatuallywemusttake=0
+ .
How-ever,inonsideringtemperaturesbelowT
,itisavery
good approximationto take=0[10℄, [12℄, andthus
thegroundstatedensityisdesribedbytheexpression
N
0
=N[1 ( T
T
)
3=2
℄ (3)
It leads to onlude that for T = T
, N
0
= 0. This
seemstoberatherinonsistentwithourprevious
anal-ysis aboutthe limit of (2) for ! 0,and oneexpet
that (3) isnotexat, butmust ontaintermsof order
smaller than N, aounting for the groundstate
den-sity at T = T
. One an proeed in a dierent way,
by keepingin mind the temperature Green's funtion
formalism[15℄,whihinthenon-relativistiinnite
vol-ume limit leads to N = R
d 3
pG(p). By onsidering a
nite volume, N results as asum overdegenerate
en-ergy states, the degeneray fator being proportional
to p 2
p=h 3
. Thisleads to anite ontributionin the
limitp=0,sinetheinfrareddivergene oftheGreen
funtionatthepole E
0
isaneledbythefatorp 2
,
leadingto a non-zerobut nite density at theground
stateatT =T
, aswillbeseenbelow.
Beforedoingthat,itis interestingto investigatein
detailthepartiledensityinrelativemomentum spae
f
3
(x; ) = x
2
e x
2
+
1 :
Byalulating the rst and seond derivatives of this
funtion, we ndthat for6=0 ithas aminimum at
x=0andamaximumat x=x
wherex
is the
solu-tionofe x
2
+
=1=(1 x 2
).
Inthissensef(x;) hasasimilarbehaviorthanthe
Maxwell-Boltzmann distributionof lassial statistis.
Butas !0, alsox
!0, and themaximum ofthe
density, for stritly = 0, is loated at x = 0. The
onvergenetothelimitx=0isnotuniform. Anite
fration of the total density falls in a viinity of the
Figure1. ThedensityinmomentumspaeoftheBosegas
for T >T has aminimumat p3 =0and a maximumat
some p3 6=0. Astemperature is dereased, the maximum
approahes to the value p3 =0and at T =T itis
rigor-ouslyonit,theminimumdisappears. Theurvesa,b,,d
orrespondrespetivelyto=1,0.1,0.001and0.
If we go bak and substitute the original integral
overmomentumbyasumovershellsofquantumstates
ofmomentum(energystates),weanwriteforthe
rit-ialvalue=0,
N
=
4
h 3
1
X
i=0 p
2
i p
e p
2
i =2MT
1
: (4)
Taking p ' h=V 1=3
, where V is the volume of the
vessel ontainingthe gas, the ontribution of the rst
termp
0
=0ofthegroundstatedensityisN
0 =
4
V 1=3
2
.
Wehavethusafrationof
N
0 =N
=4(3=2)=V 1=3
=4(3=2) 2=3
=N 1=3
(5)
partiles in the groundstate, (where N = NV is the
totalnumberofpartiles)whihisthemostpopulated,
asdesribedby thestatistialdistribution, atT =T
.
AnumerialestimationforonelitreofHegasleadsto
N
0
=N '10 6
. Inquantum statesin asmall
neighbor-hood of the ground state, the momentum density has
slightlylowervalues. Thus, attheritialtemperature
for BEC, there is a set of states lose to the ground
state,havingrelativelargedensities. Althoughthis
re-sultwasobtainedusingexatly=0,whihaording
to (2) implies an innite ground state population, it
representsanimprovementomparedwith N
0
=0for
T =T
.
Eq. (5) indiates an interesting relation: the
largertheseparationbetweenquantumstates,(i.e.,the
smaller the volume) the larger the population of the
ground stateat the ritial temperature. In the
ther-modynami limit the quantum states form a
ontin-uum,and (5)hasnomeaning. However,mostsystems
astro-We onlude that for valuesof T < T
, the urve
desribing thedensityin momentum spae attens on
the p (or x) axis, and its maximum dereases also.
As onservation of partiles is assumed, we get the
ground state density by adding to f(x) the quantity
2N[1 (T=T
)
3=2
℄(T
T)
3
Æ(x)asanadditional
den-sity. As T ! 0, the density in the ground state
in-reases at the expense of non-zero momentum states.
ThisleadstotheusualBose-Einsteinondensation. We
must remarkthat in that ase,forvaluesof T smaller
but lose to T
, boththe weak andthe strongriteria
aresatised.
III The one-dimensional ase
The ondensation in 1D is interesting in itself, sine
there is an inresinginterestto study ondensationin
systemsoflowerdimensionalitythan3D(wedisussed
the ideal 2D gas ase in [14℄), taking into aount its
possibleexperimental realization(even theproblemof
superuidityinquasi1Dsystemsisurrentlydisussed
in theliterature, see [11℄). Here wewill be interested
onitthinkinginitsonnetionwiththemagnetield
ase. For D = 1,there is reduedsymmetry with
re-gard to the 3D ase, and no ritial temperature is
expeted to our. In this ase, the mean
interpar-tile separation is l = N 1
. The density in
momen-tum spae oinides with the Bose-Einstein
distribu-tion,f
1
(p;T;)=(e p
2
=2MT =T
1) 1
. Thisfuntion
has only one extremum, a maximum, at p = 0. We
havetheexpressionforthedensityofpartilesas
N =2 1
1=2
Z
1
0
dx
e x
2
+
1 =
1
g
1=2
(z): (6)
We have thus that N = g
1=2
(z). The fat that
g
1=2
(z)divergesasz!1indiatesanenhanementof
the quantum degeneray regime. But this fat
atu-ally meansthat is adereasingfuntion of T for N
onstant,and forverysmall oneanwrite,
approxi-mately
N '2 1
1=2
Z
x0
0 dx
x 2
+ '
1=2
1=2
; (7)
where x
0 =p
0
=MT,p
0
being someharateristi
mo-mentum p
0 p
T
. Thus, doesnot vanishat T 6=0,
and for small T it is approximately given by =
=N 2
2
. Thenx
0
impliesp
0
2
2
M 2
T 2
=N 2
h 3
,
anditisfullled ifT=N!0.
Bysubstitutingthelastexpressionforbakin(7),
onehas
N '2 1
1=2
Z
x0
x0 dx
x 2
+ 2
; (8)
where = 1=2
=N. Due to the properties of the
Cauhy distribution,onean nd that onehalf of the
densityisin theinterval[0;℄,
1
2
N =2 1
1=2
Z
0 dx
x 2
+ 2
:
Weanalsond
f
1
(p;T;N)
!0 '
2N
1=2
Æ(p):
Thus,forsmallone-halfofthedensityisonentrated
intheintervalofmomentum[ ;℄. Topreisegures,
we shalldene 0
=p
T
, where p
T =
p
2MT is the
deBrogliemomentum, to givetheproperdimensions,
and ompare it with the ground state quantum ell,
2h=L,where L is theonedimensional volumeof our
system. For2h=L 0
, mostofthesystemisin the
groundstate. Thismeansthat theadimensionalphase
spaedensityismuhgreaterthanthe"volume"ofthe
systemmeasuredinunits,NL=.
For 2h=L ' 0
, N L=, and we an dene
a temperature T
d
= Nh 2
=2 3=2
mL whih, although
notbeingaritialtemperatureforaphasetransition,
establishes the order of magnitude of T in whih the
ground state density is a marosopi fration of the
totaldensity N. Thus, in the present ase we havea
weakondensation.
IV The magneti eld ase
By assuming as in [7℄ a onstant mirosopi
mag-neti eld B along the p
3
axis (the external eld is
H ext
=B 4M(B),where M(B)is the
magnetiza-tion),foreBh=mT,allthedensityanbetakenas
onentratedintheLandaustaten=0,themaximum
ofthedensityin momentum spaeis just at thepoint
x=0even for 0
6=0;thus,weonludethat without
anyritialtemperatureinthataseanitefrationof
thedensityisfoundinthegroundstate.
Theenergyofahargedpartileinamagnetield
inthenon-relativistiaseisp 2
3
=2M+eBh(n+ 1
2 )=M.
Here we name
1
= eBh=2M as the eetive
hemialpotential. Bydening theelementaryellas
v=h=eB weanwrite
N =
1
v Z
1
0
dx
e x
2
+
1
1
(9)
1
v g
1=2 (z)=
1
v 1
X
m=1 e
1m
m 1=2
(10)
Byintroduingtheontinuousvariablex=Tmand
bywritingT P
1
n=1 !
R
1
0
weget
N =
1=2
v 1=2
andinasimilarwayasbeforeoneangetforthe
rela-tivepopulationin thegroundstate
N 0 N = L 1=2 1 : (12)
Here L is the harateristi dimension of the system
(along the magneti eld). We observe that N
0 =N
inreases with dereasing
1
, i.e., as temperature
de-reases. Thelimit, whih isnottransparentfrom (12)
mustbeunity,asweshallseebelow. Weonludethat
the density in momentum spae in the magneti eld
aseonentratesina(dereasingwithT)narrowpeak
aroundthegroundstatep
3
=0(Fig. 2).
Weturntotherelativistiase. Thethermodynamipotentialforagasofhargedsalarpartilesplaedinthe
magnetieldis,
s = eB 4 2 h 2 1 X n=0 Z 1 1 dp 3 h ln(1 e ( q ) )(1 e ( q +) )+ q i (13) where q = q p 2 3 2 +M 2 4
+2eBh(n+ 1
2
) , the last term in (1) aounts for the vauum energy and is the
hemialpotential. For avetoreldtheone-loopthermodynami potentialis
v = eB 4 2 h 2 Z 1 1 dp 3 h ln(1 e (0q ) )(1 e (0q+) )+ q i + eB 4 2 h 2 1 X n=0 n Z 1 1 dp 3 h ln(1 e ( q ) )(1 e ( q +) )+ q i (14) where n
=3 Æ
0n , 0q = p p 2 3 2 +M 2 4 eBh, q = q p 2 3 2 +M 2 4
+2eBh(n+ 1
2 ) .
d
Figure 2. In the magneti eld ase, for eBh=mT 1,
thesystemisonnedtothen=0Landauquantumstate.
Themaximumofthedensityinthemomentumomponent
along the magneti eld is loated at zero momentum at
anytemperature,andforverylowT,ithasapeakedÆ-like
form. Hereurvesd,e,forrespondto=1,0.1and0.001.
The mean density of partiles minus antipartiles
(average harge divided by e) is given by N
s;v =
s;v =.
Wehaveexpliitly
N s = eB 4 2 h 2 1 X Z 1 1 dp 3 (n + p n p ) : (15) wheren p =[exp( q ) 1℄ 1 .
Wetaketheexpressionfrothevetoreldasefrom
theWeinberg-SalamLagrangianinamediumina
mag-neti eld[16℄. Wehave,
N v = eB 4 2 h 2 Z 1 1 dp 3 (n + 0p n 0p ) + eB 4 2 h 2 1 X 0 n Z 1 1 dp 3 (n + p n p ) (16) where n 0p = [exp( 0q ) 1℄ 1 , 0q = p p 2 3 2 +M 2 4
eBh,and
0n
=3 Æ
0n .
For T ! 0, ! M
2
(we named M
= p M 2 eBh= 3
), the population in Landau quantum
states other than n = 0 vanishes (this was shown in
[1℄)and the density forthe n=0stateis infrared
di-vergent. Weexpetthen mostofthepopulationto be
in the ground state, sine for small temperaturesn
0p
is vanishing smalland n +
0p
is abell-shaped urvewith
its maximum at p
3
= 0. We will proeed as in [16℄
and all p
0 (
p
2M 0
) some harateristi
momen-tum. Wehavethen,byassuming 0
T,forthenet
densityinasmallneighborhoodofp
N 0s;v = eBT 2 2 h 2 Z p0 0 dp 3 p p 2 3 2 +M 2 4 eBh Z p0 0 dp 3 p p 2 3 2 +M 2 4
eBh+
' eBT 2 2 h 2 2 Z p0 0 (M 2 +)dp 3 p 2 3 2 +M 2 4 2 Z p0 0 (M 2 )dp 3 p 2 3 2 +M 2 4 2 = eBT 4h 2 [ s M 2 + M 2 s M 2 M 2 + ℄ eBT 4h 2 s 2M 0 (17) d whereN s;v =N 0s;v +ÆN s;v andÆN s;v
isthedensityin
the interval[p
0
;1℄. AtuallyÆN
s;v
isnegligibly small
and N 0
s;v 'N
s;v
. Weobservethat theontributionof
antipartilesanbenegleted as!M
. Thus (17)
leadsto 0 = e 2 B 2 T 2 M 8 2 N 2 0s;v h 4 2 : (18)
We observe that 0
is a dereasingfuntion of T and
vanishes for T = 0, where the "ritial" ondition
=M
2
is reahed. As shown below, in that limit
the Bose-Einstein distribution degenerates in a Dira
Æ funtion, whih means to haveallthesystem in the
ground statep
3
=0. Tosee this, weshallrewrite the
momentumdensityofpartilesaroundthegroundstate
p
3
=0,n=0approximatelyas
n 0 (p 3 ) = T p 2 3 2M + e 2 B 2 T 2 M 8 2 h 4 2 N 2 0s;v = 4h 2 N 0s;v eB p 2 3 + 2 (19) where = eBTM 2h 2 N 0s;v = p T vN 0s;v = p 2M 0 where p T = p 2M
T is the thermal momentum,
v = h=eB the elementary volume ell, = h=p
T
beingtheDeBrogliethermalwavelength(observethat
v dereaseswithinreasingB). Wehaveapproximated
the Bose-Einstein distribution by one proportional to
aCauhydistribution,havingitsmaximum 1
for
p
3
=0. Wehavethat !0forT !0,but for small
xedT,alsodereasesasvN
0s;v
inreases. Weremind
that in thezeroeld ase, theondensation ondition
demands N 3 >2:612. approximately, 1 2 N 0s;v = eB 4 2 h 2 Z n 0 (p 3 )dp 3 : (20)
Thus,approximatelyonehalfofthetotaldensityis
on-entrated in the narrow strip of width 2 around the
p
3
= 0 momentum. It results that for densities and
magnetieldslargeenough,ifwehooseanarbitrary
smallneighborhoodofthegroundstate,ofmomentum
width 2p
30
, oneanalways nda temperature T >0
smallenoughsuhthat p
30
and(17)and(20)are
satised.Theondensateappearsanditisdesribedby
thestatistial distribution. Graphsof n
0 (p
3
)forsome
valuesofareshownin Fig. 2. Wehavealso
lim !0 n 0 (p 3 )=4
2 h 2 N s;v eB Æ(p 3 ): (21) Orequivalently, N s;v = lim !0 eB 4 2 h 2 Z 1 1 n 0 (p 3 )dp 3 : (22)
Thus, if T ! 0, all the density N
s;v
falls in the
on-densate, as our in the zero eld ase, but here the
total density is desribed expliitly by the integralin
momentumspae.
Oneannotxany(small)valueforatwhihthe
distribution starts to havea manifest Æ(p
3
) behavior;
and there is no denite value of ritial temperature
forondensation to startto be onstrainedessentially
inthegroundstate. Buttheonnementin then=0
LandaustateoursiftheratioeB=T islargeenoughto
maketheaverage oupationnumbersin exited
Lan-daustatesnegligiblysmallasomparedto theground
state. Theonnementtothep
3
=0momentumstate
alongtheeld,foragivenB,isdeterminedbytheratio
T=N. If itissmall enoughto make 2h=L where
L is the harateristi length of the system along the
in non-relativistisystemse. g.,formasses10 27
g,
elds of order 10 6
Gauss,T =10 2Æ
Kand L 1m;
for Kaons in a neutron star, assuming T = 10 8Æ
K,
N = 10 39
, B = 10 10
Gauss, L 10 Km, it results
' 10 29
and 2h=L '10 33
. Onehalf of the
total density an be assumed then to be distributed
among10 4
states;i.e.,thegroundstatedensityis10 35
,
whih is a marosopi number ), The harateristi
order of magnitude of the temperature for having a
marosopifrationofthetotaldensityintheground
state,ifListhedimensionofthesystemparalleltothe
eld, isgivenbyL='vN,or
T
d '
4 2
h 3
N
eBM
L
: (23)
Weobservethat forgiven N=T, thequantity
in-reases with B, that is, the inreasing eld tends to
spreadthedistributionalongp
3
, orin otherwords,to
enlarge the density in states loseto the groudstate,
and ifT 'T
d
, ondensation in statesneighborto the
groundstatebeome alsosigniant. ButforT T
d ,
we have almost all the partiles in the ground state
n=0,p
3
=0,andatrueondensateexists.
Wereturn to expression (17). It is easy to obtain
theinfraredontributiontothethermodynami
poten-tialas
= eBT
4h 2
q
M 2
2
(24)
andfrom ,wegettheenergyU =T=T ,and
thespeiheat,whihasT!0tendstotheonstant
value,
V =
2e 2
B 2
M
k
2
4 2
h 4
2
N
0s;v
; (25)
wherek istheBoltzmannonstant.
V Magnetization and
superon-dutivity
For thevetoreldase,themagnetizationinthe
on-densationlimitispositivesineallthesystemisinthe
Landaun=0state,and
M =
B
= e
2
B
8 2
Z
1
1 lim
!0 n
0 (p
3 )dp
3
(26)
= eN
v h
2M
: (27)
and we have that the ondensate of vetor partiles
leads to atrue ferromagnetibehavior. In partiular,
if we write H ext
=0, we get B = M(B) as the
on-startedat hightemperatureT, wherethesystemhave
a diamagneti ontribution oming from the
distribu-tion in Landau states, n = 1;2::, and aparamagneti
eetduetothespinoupling,weobservethatby
low-eringT gradually,thesystemends inatrue
ferromag-netistate. Aphasetransitionwithouthavingaritial
temperature,asthe\diuse"ones[17℄,hastakenplae.
Usualsuperondutivitymaybeunderstoodin
prin-ipleasamanifestationofBECofCooperpairs,whih
are spin zeroelds. In onnetion to it Shafroth [2℄
in his study of the analogy between Bose
ondensa-tionof salarharged partilesand superondutivity,
foundaritialappliedeldB
s =eN
s =2M
+
,(where
N
s = N
s
[1 (T=T
)
3=2
℄, N
s
is thetotal density and
T
the ritial temperature for normal ondensation),
suh that the magnetization is equaland opposite to
the applied eld. In this asewhen the ondensateis
ontained in some body limited by asurfae, and the
systemisplaedinanexternalmagnetield,itreats
by reating a surfae magnetization (Meissner eet)
whih leadsto a magneti eld equal and opposite to
the applied one. The harges inside does notfeel the
external eld and remain in theondensate. For eld
intensities B > B
s
the (zero eld) ondensate is
de-stroyed. Obviously,this istheaseofnotveryintense
elds,inwhihthesystemofhargesatthesurfaeare
distributedonalargesetofLandaustatesproduinga
magnetizationwhihisequalandoppositetothe
exter-naleld,keepingthepartilesinsideinfreeeldstates.
Butasthemagnetieldgrowssomeofthepopulation
inside starts to leavethe ground state, and for larger
eld intensities, the ondensate dissapears. However,
if the magneti eld inreases enough, all the system
falls in the n=0Landau groundstate, and ifthe
re-lation N=T islargeenough,theondensationbeomes
manifestagain. ByalulatingM
s
= =B,asthe
densityis(21),wehave
M
s =
eN
s h
2M
+
(28)
Thus,forgrowingeldsstartingfromzero,wehave
usual ondensation(andsuperondutivity),upto the
Shafroth's ritial eld. Then appears a gap where
ondensationandsuperondutivityarenotobservable,
andforeld andN=T largeenoughondensation(and
superondutivity)reappears.
Thephenomenonofondensationofsalarand
ve-tor harged partiles in strong magneti elds may
have, thus, espeial interest in onnetion with
on-densed matter physis and osmology (the early
uni-verse). Inrelationwiththerst, ithasbeenreported
(seei.e[18℄), thattheritialeld ofsomehighT
su-perondutorsdiverges with dereasingT, whih
sug-gests the appearane of re-entering superondutivity
in extremely strong magneti elds. The vetor
pre-superondutive phase in extremely strong magneti
elds wemayexpettheappearaneof
ferromagneti-superondutivepropertiesinthoseasesinwhihthere
isasigniantamountofparallelspinpairedfermions.
VI Magnetization in the early
universe
Wewillonsiderthespontaneousmagnetizationof
ve-torpartilesinonnetionwiththeondensationofW
bosons, aphenomenon whih wassuggested by Linde
[19℄ to ourin superdense matter, and by oneof the
present authors and Kalashnikov [21℄, to our near
theeletroweakphasetransition. Theondensationof
W'smustbeunderstoodassomehemialequilibrium
proess,sinetheW'sinteratingwiththequark
bak-groundwoulddeayin n,por
; 0
pairs.
If we onsider at rst the large density ase, by
taking N
W ' N
v
, where N
v
= M
3
W
3
=6 2
h 3
'
10 45
m 3
, the magnetization would be of order B '
10 19
Gauss. Onemay thinkas themehanismof
aris-ing these huge elds as follows: the mirosopields
inside hadrons are of order 10 15
Gauss. Suh elds
lead the W-ondensate to magnetize, reating a eld
10 4
times larger. Thenthe onditionsfor spontaneous
magnetizationindomainsarise. Obviously,Mbeomes
verylarge(oforderB
)foreldsneartheritialvalue
B
' M
2
W
4
=eh ' 10 24
gauss, sine for B B
the
system beomesunstable leadingto imaginaryenergy
eigenvalues. [20℄. Near thesymmetry restoration
tem-peratureasM=M(T)<M(0)theritialeldours
forsmallervaluesofB.
The ase of temperatures near the symmetry
restoration lead to veryinteresting results. As shown
in [21℄,[1℄,ondensationoftransversevetorW bosons
oursforarbitrarysmalldensities,andtakinginto
a-ountthatthestrongloalmagnetieldswould
mag-netizetheondensate,wemaythinkthat(27)beomes
singular,(forM
v (T)=
p
M 2
W
(T) eBh= 3
!0,
lead-ing toinnitelargemagnetizationofthe medium,due
to the vanishing of the (transverse, sine the
longitu-dinal modes aquire a Debye mass gT) vetor
bo-son mass. Onemay think that this would lead to
in-stabilities also, but we must remember that the zero
mass oftheW
s
for T >T
is onlyanapproximate
re-sultrstbeauseatuallythemassisexpetedtobeof
order g 2
T [22℄, seond,beausetheself-magnetization
B = M(B) that would arise, leads to a mehanism
preventing the divergene to our. In any ase, the
existeneofaondensateinpresene ofstronghadron
(orquark)eldsmayleadtothearisingofa
ferromag-netibehaviorleadingtoextremelylargeelds,oforder
B B
,attemperaturesT T
. Theexisteneofsuh
largeeldsoforderB
,asutuatingeldsonasale
M 1
W
,wasrstsuggestedbyVahaspati[23℄. Thesame
ofsomesortofequipartitionofmagnetiandradiation
energies that may our in a magnetially turbulent
environment in the bubble formation arising from an
eletroweakrstorderphasetransition.
VII Conlusions
We onlude that Bose-Einstein ondensation of
hargedpartiles in astrongmagnetield is possible
and leads to several new and interesting phenomena,
astheourreneofphasetransitionin preseneofan
externalmagnetield,withoutaritialtemperature.
Forloweldintensitywehaveusualondensation,and
for very strong elds, ondensation is manifest again.
The ondensate in the strong magneti eld suggests
theexisteneofsuperondutivityin extremelystrong
magneti elds and the existene of a
ferromagneti-superondutivephase. Thishasinterestinondensed
matterphysis. Inastrophysisandosmologywehave
also interestingonsequenes. It givessupport to the
onjetured existene of superuid and
superondu-tivephasesin neutronstars [25℄. It suggestsalsothat
at the eletroweak phase transition, extremely strong
magnetields mayarise asaonsequeneof
onden-sationandself-magnetizationeetsofthemedium.
The authors would like to thank J. G. Hirsh for
very important and detailed disussions and
sugges-tions. They thank also R. Baquero and G. Gonzalez
for fruitful omments. One of the authors (H.P.R.)
thanks H. Rubinstein and G. Andrei Mezinesu for
important disussions and M. Chaihian for
hospital-ity in the High EnergyDivision of the Department of
Physis,andaknowledgesthenanialsupportofthe
AademyofFinlandunder theProjetNo. 163394.
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