Critial Behavior of High Temperature Superondutors
Claude de Calan
Centrede PhysiqueTheorique,
EolePolytehnique,91128PALAISEAUCEDEX,Frane
E-mail: alanpht.polytehnique.fr
Reeivedon5January,2001
Weonsiderthesalingbehaviorintheritialdomainofsuperondutorsatzeroexternalmagneti
eld. TheJosephson'srelationforahargedsuperuidisprovedwithoutassumingthehypersaling
relation. OntheotherhandwedisussthedualGinzburg-Landaumodel. Inthisdualmodel,due
to thepreseneoftwomasssales, aontinuousfamilyofnonequivalentsalings anbedened.
Therelevantritialregimesareidentied,andtheorrespondingritialexponentsarepredited.
I Introdution
The study of high temperature superondutors
(HTSC's) isnot aneasy one,from atheoretial point
ofviewaswellasfrom anexperimentalone.
Up to now, there is no theory of the mirosopi
mehanism generating the pairing of eletrons whih
leads to superondutivity, for these HTSC's.
There-forethetheoretialstudyanbeperformedonlyinthe
frame of phenomenologial models, the most popular
ofwhihistheGinzburg-Landau(GL)model. Butthe
questionraises: isthisGLmodelrelevantforHTSC's,
at leastforwhat onernstheritial behavior, inthe
neighborhood of the superonduting transition ? In
otherwords,dotheHTSC'sandtheGLmodelbelong
tothesameuniversalitylass? Asmanypeoplebelieve,
welaimthatitisthease,butthisanbehekedonly
byomparisonwithexperiments.
Now the experimental situation is also a bit
onfusing[1℄. Dierent experiments,performed on
dif-ferent materials, give dierent results. Many eets
haveto betakenintoaount:
i)nitesizeeets(inpartiularinoneofthethree
spaedimensions,forthin lmssamples);
ii) anisotropy eets, sine the HTSC's generally
present a rystalline struture made of bidimensional
layers;
iii)estimateoftheritialregionprobed,depending
onhowmuh theritialtemperatureisapproahed.
Weshallbeonernedherewiththeritial
behav-iorof HTSC's at zeroexternal magneti eld, and we
aremainlyinterestedin thevaluesoftheritial
expo-nents(theexponentfortheinverseof,whereisthe
0
of,whereisthepenetrationdepth). The
experimen-talresults,forapproximatelyisotropi,extremetypeII
superondutorsareessentiallythefollowingones:
Almostallexperimentalistsagreeonthevalueof,
approximately 0.67, but some nd 0
= =2 ' 0:33.
This orresponds to an unharged three-dimensional
XYuniversalitylass,whihmustberelevantfora
rit-ial region near, but not very near, from the ritial
temperature ; a region where the gauge eld
utua-tionsanbenegleted.
Some other experimentalists, working with thin
lms, nd ' 0:67 and 0
= 1=2. In this ase, the
valueof 0
orrespondsto amean-eld-likebehavior.
Westressthatthehargedritialregion,verynear
of theritialtemperature,is averysmall region, not
yetaessibletotheexperimentalprobes.Andweshall
see that in the harged regime the ritial exponents
must takethevalues = 0
'0:67. Thispreditionis
onrmedbyareentnumerialstudyonthelattie[2℄.
InsetionII,wesetupthemainfeatureswegetfrom
therenormalizationgroupstudyoftheGLmodel.
Se-tionIII givesexatnonperturbativerelationsbetween
ritialexponents,inludingaproof ofJosephson's
re-lation and of hypersaling. Insetion IV,weonsider
the dual GL model and we eluidate the ontroversy
whihhappenedamongtheoretiians,dependingonthe
denition ofthesaling.
II The Ginzburg-Landau model
The ation for the GL model is built from the
self-ouplingofthesalareld:
S= Z
d 3
xf 1
2 (
~
r
^ ~
A
0 )
2
+j( ~
r ie
0 ~
A
0 )
0 j
2
+m 2
0 j
0 j
2
+ u
0
2 j
0 j
4
g
~
A
0
istheeletromagneti eld
0
isthebareorderparameter(relatedtoeletron's
pairs)
m
0
,thebaremassof
0
,isrelatedtothe
tempera-tureby
m 2
0
=T T
=t
The renormalized quantities ~
A, , m, u,e are
de-nedinthestandardway:
~
A
0 =Z
1
2
A ~
A
0 =Z
1
2
m 2
0 =Z
(2)
Z
1
m
2
u
0 =Z
u Z
2
u
e
0 =Z
1
2
A e
Fromgaugeinvariane, thesamefator Z
A
enters the
denition of ~
Aande. Therenormalizedmassmisthe
inverse ofthe orrelation length : m = 1
and the
ritialexponent isdenedbymt
whent!0.
In dimension 3, we dene the dimensionless
ou-plings f andvby
f =e 2
=m and v=u=m
Therenormalizationgroupequationsareestablishedby
hoosingmasthesalingparameteranddierentiating
withrespettolnm,atxedu
0 ande
0 .
Working at xed dimensiond = 3in theone-loop
approximation,onendstheowdiagram[3℄:
O
A
C
B
f
v
Figure1
InFig.1,thearrowsgivetheevolutionof(v;f)when
mgoesto zero. Fourxedpointsappear:
OisthetrivialGaussianxedpoint,infrared
repul-sive.
A istheneutral(e=0) XYpoint,infrared
attra-tiveonlyat zeroharge(i.e. inthe 4
model).
Bis a triritialpoint. Theline from A to B
sep-arates two basins : starting on the left of this line,
v goes to negativevalues and there is no xed point,
whih orrespondsto arstorder transition; starting
on the right, (v;f) goes to the attrative, IR stable,
harged xed point C, whih orresponds to aseond
ordertransition.
Itmustbenotiedthatwhenworkingwiththe
tra-ditionalexpansion,where=4 d,thehargedxed
points B and C are not found[4℄, unless is an
N-omponent eld with a very large number of
ompo-nents (N>365). The absene of harged xed points
orrespondsto arst-order transition, appropriatefor
thedesriptionofsuperondutorsinthetypeIregime,
butnotforthetypeIIregime.
The 4 method provides a good ontrol of the
renormalizationgroup,asfar asone isinterested in a
dimension equal to (or near from) 4. But for = 1
(in dimension 3), this ontrol is usually lost. On the
other hand, the ontrol in our method (xed d = 3)
dependsonthesmallnessoftheouplingonstants. At
theharged xedpoint,these onstantsarenot really
small. Howeveronemayhopethattheone-loop
ontri-butionsremaindominant,asitgenerallyhappens.
Fur-thermore,weshallseeinsetion4below,fromthedual
GLmodel,aqualitativeonrmationoftheresults.
III Exat non perturbative
re-sults
Inthissetionwetakeanarbitrary (yet xed)
dimen-siond, 2<d4,andwedeneorrespondingly:
f =e 2
m d 4
v=um d 4
We shall prove some exat results (not depending on
the one-loop approximation), with only one
assump-tion,theexisteneofthestableinfraredxedpoint[5℄.
The -funtions for the renormalization onstants
aredenedas:
A =m
m lnZ
A
=m
lnZ
(2) =m m lnZ (2) Z A
is dened from the two-pointsfuntion for the ~
A
eld. At the infrared xed point,
A
!
A when
m ! 0,
A
being the anomalous dimension of the ~
A
eld.
Similarly,Z
isdenedfromthetwo-pointsfuntion
of the eld, and Z (2)
from the two-points funtion
of with massinsertion. At the infrared xed point,
!
=, beingtheanomalousdimensionof .
Fromthedenition off,its-funtionis :
m
m f =(
A
+d 4)f
Assumingf !f
6=0whenm!0,sinem
m f must
vanishatthexedpoint,wend:
A
=4 d
Now in the normal phase (m 2
0
> 0;T > T
) the
photonis massless. But in the Meissnerphase(m 2
0 <
0;T <T
), itiswellknown that thesymmetryis
bro-ken, thelassial potential beomes adouble-well one
andthephotonaquiresamassm
A
. Theritial
expo-nentofm
A
islabeledas 0 : m A t 0
anditsinverse=m 1
A
isthepenetrationdepth.
Themassm
A
isgivenby
m 2 A =e 2 s
wherethesuperuiddensity
s =<jj 2 >satises s = m 2 u
TheGinzburgparameterisdenedastheratioof
thetwomasses. Thuswehave:
= m m A = u e 2 1 2 = v f 1 2
Fromthedenitionof m
A
,the-funtionform 2 A is: m m m 2 A =( A
+d 2 1 v m m v)m 2 A
Attheinfraredxedpoint,v!v
6=0; m m vanishes and A ! A
=4 d. Therefore
m m m 2 !2m 2 A
Thismeansthatm
A
behaveslikem, andwend:
= 0 Rememberingm 2 A =e 2 s
andusing = 0
,wendfor
theritialbehaviorof
s : s t (d 2)
whihisnothingbuttheJosephson'srelation[6℄. In
on-trastwith whatis usuallydone,weprovethis relation
withoutusingthehypersalingrelation.
Ontheotherhand
s =<jj 2 >=Z 1 <j 0 j 2 >
Calling the ritial exponent of
0 and remember-ing Z m
,weanwritethe ritialexponentof
s
as 2 . But therelation 2 =(d 2) has
beenshown[7℄tohold onlyifthehypersalingrelation
d=2 holds( beingtheritialexponentforthe
speiheat. Thereforewealsoprovethehypersaling
relationfortheGLmodel.
Finallywemaynotiethatifthegaugeeld
utua-tionswerenegleted,i.e. atzeroharge,wewouldhave
A
=0(insteadof4 d)andonsequentlym 2
A m
d 2
(instead ofm 2
). Thereforeinthisase
0 = (d 2) 2 and 0 = 2
for d = 3. We reover the XY
behav-iorwhihorrespondstoapproahingtheneutralxed
pointA(seeg. 1).
IV The dual Ginzburg-Landau
model
ThedualGL modelhasbeenproposed usingplausible
argumentsonthedynamisofavortexgas[8℄. In
prin-iple, itistheontinuumlimitofthegeometrialdual
(onthelattie) ofthelattieversionforthediretGL
model. Lattie duality in Abelian gaugemodels have
beenusedto preditthataseond orderphase
transi-tionshouldtakeplae,atleastinthetypeIIregime[9℄.
Buttheontinuouslimitgivingtheontinuum
ver-sionofthedual GLmodel isnoteasytoperform,and
impliesseveralapproximations. Wedonotrepeathere
theargumentsleadingtothisdualmodel,whihareset
upinourpaper[5℄. Theresultisthefollowingation:
S= Z d 3 xf 1 2 ( ~ r ^ ~ h 0 ) 2 + 1 2 M 2 0 ~ h 0 2 +j( ~ r iM 0 q 0 ~ h 0 ) 0 j 2 + 2 0 j 0 j 2 + w 0 2 j 0 j 4 g
ThedualGLmodelisthus analogousto thediretGL
model,butthebaregaugeeld ~
h
0
(relatedtothe
This is similar to what happens in the broken
sym-metry region of the diret model. Thus the duality
exhanges theordered and disorderedphases,and the
hargedsalareld
0
issometimesalled a"disorder
parameter". On theother hand,thebare harge q
0 is
relatedtothebarehargee
0
ofthediretmodelbythe
Dira'srelation
q
0 =
2
e
0
Thusthedualityalsoexhangesthelargeandsmall
ou-pling situations, giving a better support to the small
oupling expansion implied in our (xed dimension)
method.
Therenormalized quantities in the dualmodel are
dened inawayquiteanalogoustothepreviousone:
~
h
0 =Z
1
2
h ~
h
0 =Z
1
2
2
0 =Z
(2)
Z 1
2
M
0 q
0 =Z
1
2
h Mq
w
0 =Z
w Z
2
w
After renormalization, onehas to identify the ritial
behavior of and M with the ritial behavior of m
and m
A
inthediret model,retaining thesame
riti-al exponents and 0
t
; M t
0
Inpartiular,thedualGinzburgparameter
d =
M
has thesamexed point valueasthe diret Ginzburg
parameter
= m
m
A
Nowaproblemomes fromthefat that,from the
beginning, two masses M
0 and
0
enter the ationof
the dual model : taking as the saling parameter,
what is the saling for M
0
? In priniple, thesaling
lawforM
0
mustbededuedfromtherelationbetween
thediretandthedualGLmodels. Butsinethesteps
leadingto thedualmodelarenotsimple,this relation
is pratially lost. By working just in the dual model
(forgettingwhereitomesfrom),onehastoassumethe
M
0
salingbehavior. Indeed,variousauthorsmade
dif-ferentassumptions,whihledtosomeontroversy. Let
us generallydenetheM
0
salingbehaviorby
M 2
t
and look at the results of the renormalization group
study.
i)Ifistakentobezero,itamountstotakethelimit
!0at axedvalueofM
0
. Theresultisthen[10℄
0
=
2
with a vanishing limit for the Ginzburg parameter .
This is harateristi of a neutral XY behavior (the
only one presently seen by the experimentalists) and
isrelatedto theapproahof theneutralxedpointA
inthediretmodel.
ii) If istaken to be 1,it amounts to identify M
0
withthebaremassm
A
0
ofthegaugeeldinthebroken
symmetryphase of the diret model. The resulthere
beomes[11℄
0
= 1
2
amean-eld valuefortheexponentofthe penetration
depth. As wesawin theintrodution,this ouldbea
relevantbehaviorforthesuperondutingtransitionof
thinlms.
iii) If and only if wetake = 2, whih amounts
toidentify thesalingbehaviorofM
0
withthesaling
behavioroftherenormalizedmassm
A
,wend[5℄
= 0
withanon-vanishingxed-pointvalueoftheGinzburg
parameter. Thishoienoworrespondstotheharged
regime,approahingtheinfraredstablexedpointCof
thediret model.
Attheendofthisshortreview,letusmentionthat
it would be veryuseful to ask for the samequestions
inpreseneofanexternalmagnetield. Manyreent
experimentsare onerned with suh a situation, and
indiatethatseveraltransitionsour,betweenvarious
phases. Startingfromthenormalmetalliphase,when
thetemperatureisdereasing(dependingonthevalue
oftheexternaleld),oneprobablymeetsaphasewith
auid of vorties, then the Abrikosov's phase with a
regularlattieofvorties,beforereahingtheMeissner
superondutingphase.
Butfromatheoretialpointofviewtheproblemis
extremelyhard. Wearepresentlyworkingonthis
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