Squeezing Squeezing hadronichadronic mattermatter

Roberto CasalbuoniRoberto Casalbuoni

Department of Physics and INFN Department of Physics and INFN -- Florence Florence

http://http://arturo.fi.infn.it/casalbuoniarturo.fi.infn.it/casalbuoni//

casalbuoni@fi.infn.itcasalbuoni@fi.infn.it

1TorinoTorino, December 1, December 1--5, 20035, 2003

SummarySummaryIntroduction and Introduction and basics in Superconductivitybasics in Superconductivity

Effective theoryEffective theory

BCS theoryBCS theory

Color Superconductivity: CFL and 2SC phasesColor Superconductivity: CFL and 2SC phases

Effective theories and Effective theories and perturbativeperturbativecalculationscalculations

LOFF phaseLOFF phase

PhenomenologyPhenomenology2

IntroductionIntroduction

MotivationsMotivations

Basics facts in superconductivityBasics facts in superconductivity

Cooper pairsCooper pairs

3

MotivationsMotivationsImportant to explore the entire QCD Important to explore the entire QCD

phase diagram: Understanding of phase diagram: Understanding of

4

Hadrons Hadrons QCDQCD--vacuumvacuum

Understanding of its modificationsUnderstanding of its modifications

Extreme Conditions in the Universe: Extreme Conditions in the Universe: Neutron Stars, Big BangNeutron Stars, Big Bang

QCDQCD simplifies in extreme conditions:simplifies in extreme conditions:

Study QCD when quarks and gluons are the Study QCD when quarks and gluons are the relevant degrees of freedomrelevant degrees of freedom

Studying the QCD vacuum under different and Studying the QCD vacuum under different and extreme conditions may help our understandingextreme conditions may help our understanding

5Neutron starNeutron star Heavy ion collisionHeavy ion collision Big BangBig Bang

6

( )R 0ρ → ∞ →Limiting case

RRqq qq

AsymptoticAsymptotic freedomfreedom::

FreeFree quarksquarks

fm1<<

When When nnBB >> 1 fm>> 1 fm--33

free quarks expectedfree quarks expected

7

Free Free FermiFermi gasgas andand BCSBCS((highhigh--density density QCD)QCD)

T 0 β = /k 1 T→ → ∞( )For

8

β(E µ) β

1f(E) θ(µ E)e 1− →∞= ⎯⎯⎯→ −

+f(E)f(E)

EE

FE µ≡

ppFF High density means high High density means high ppFF

Typical scattering at Typical scattering at momentamomenta of order of of order of ppFF

F QCDp >> ΛFor

No No chiralchiral breakingbreaking

No confinement No confinement

No generation of masses

Trivial Trivial theory ?theory ?

No generation of masses

9

Grand potential unchanged:Grand potential unchanged:

•• Adding a particle to theAdding a particle to the FermiFermi surfacesurface•• Taking out a particle Taking out a particle (crea(creatingting a holea hole))

µN)E(F −=

F1)µ(N)E(EF F =±−±→

10

For an arbitrary attractive interaction it is For an arbitrary attractive interaction it is convenient to convenient to formform pairs particlepairs particle--particle or particle or

holehole--holehole ((CooperCooper pairspairs))

F B BE + (±2E - E ) - µ(N ± 2) = F - E

In matter In matter SC SC only under particular conditionsonly under particular conditions ((phonon phonon interaction should overcome the Coulomb forceinteraction should overcome the Coulomb force))

430 54

0c 1010

K1010

K 101E(electr.)

(electr.)T −− ÷≈÷

÷≈

≈ ≈cT (quarks) 50 MeV1

E(quarks) 100 MeVIn QCDIn QCD attractive interactionattractive interaction

((antitripletantitriplet channel)channel)

SCSC much more efficient imuch more efficient in QCDn QCD 11

Basics facts in superconductivityBasics facts in superconductivity

1911 1911 –– Resistance experiments in mercury lead Resistance experiments in mercury lead and thin by and thin by KamerlinghKamerlingh OnnesOnnes in in LeidenLeiden: : existence of a critical temperatureexistence of a critical temperature TTcc ~ 4~ 4--10 10 00KK

In a superconductor In a superconductor resistivityresistivity < 10< 10--23 23 ohm cmohm cm

12

1933 1933 –– MeissnerMeissner and and OchsenfeldOchsenfeld discover discover perfect diamagnetismperfect diamagnetism. Exclusion of B except . Exclusion of B except for a penetration depth of ~ 500 Angstrom.for a penetration depth of ~ 500 Angstrom.

tB

c1E

∂∂

−=∧∇Surprising since from Surprising since from Maxwell, Maxwell, for for E = 0, B E = 0, B frozenfrozen

Destruction of Destruction of superconductivity for superconductivity for H = H = HHcc )T(f

8)T(H)T(f n

2c

s =π

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−≈

2

ccc T

T1)0(H)T(HEmpirically:Empirically:

13

1950 1950 –– Role of the phonons (Role of the phonons (FrolichFrolich). ). Isotope effect Isotope effect (Maxwell & Reynolds)(Maxwell & Reynolds)

5.045.0,M1)0(HT cc −≈α≈≈ α

14

1954 1954 -- Discontinuity in the specific heat (Discontinuity in the specific heat (CorakCorak))

Excitation energy ~ Excitation energy ~ 1.5 1.5 TTcc

T/bTcs

ceTac −γ≈Tcn γ≈

Implication is that thereImplication is that there isis a gap in the a gap in the spectrum. spectrum. This was measured by Glover and This was measured by Glover and

TinkhamTinkham in 1956in 1956

15

Two fluid modelsTwo fluid models: phenomenological expressions : phenomenological expressions for the free energy in the normal and in the for the free energy in the normal and in the sueprconductingsueprconducting state (state (GorterGorter and and CasimirCasimir 1934)1934)

London & London theory, 1935London & London theory, 1935: still a two: still a two--fluid fluid models based onmodels based on

Newton equationNewton equation

+ Maxwell+ Maxwell

16

)venJ(,Emen

tJ

sss

2ss −==

∂∂

)venJ(,EJ nnnnn −=σ=

Bmc

enJ2

ss −=∧∇ sJ

c4B π

=∧∇

B1Bmc

en4B 2L

2

2s2

λ=

π=∇ L/xe)0(B)x(B λ−=

1950 1950 -- GinzburgGinzburg--Landau theoryLandau theory. In the . In the context of Landau theory of second order context of Landau theory of second order transitions, valid only around transitions, valid only around TTcc , not appreciated , not appreciated at that time. Recognized of paramount at that time. Recognized of paramount importance after BCS. Based on the construction importance after BCS. Based on the construction of an of an effective theory effective theory (modern terms)(modern terms)

2s |)r(|n ψ=

∫ ⎟⎠⎞

⎜⎝⎛ ψβ+ψα+ψ+∇ψ−=

=−

422**3

ns

|)r(|)T(21|)r(|)T()r(|Aie|)r(

m21rd

)T(F)T(F

17

Cooper pairsCooper pairs

1956 1956 –– Cooper proved that two fermions may Cooper proved that two fermions may form a bound state for an arbitrary form a bound state for an arbitrary attractive interaction in a simple modelattractive interaction in a simple modelOnly two particle interactions considered. Interactions with Only two particle interactions considered. Interactions with

the sea neglected but from Fermi statisticsthe sea neglected but from Fermi statistics

Assume for the Assume for the ground state:ground state:

spinspin zero total momentumzero total momentum

18

))rr(kcos(g)()rr( 21k

k1221210 −⋅βα−βα=−ψ ∑

)rr(E)rr()rr(V)(m21

2102102121

21 −ψ=−ψ⎥⎦

⎤⎢⎣⎡ −+∇+∇−

m2|k|,gVg)2E(2

k'kk'k

'k,kkkF

=ε=ε− ∑>

rde)r(VL1V 3r)'kk(i

3'k,k ∫ ⋅−=

Cooper assumed that only interactions close to Cooper assumed that only interactions close to Fermi surface are relevant (see later)Fermi surface are relevant (see later)

F ck,k '

G, k k kV 0, otherwise− ≤ ≤=

cutoff:cutoff: δ+=ε Fk Ec

∑>

−=ε−Fk'k

'kkk gGg)2E(

∑> −ε

=Fkk k E2

1G1

Summing over k:Summing over k:19

c F

F F

3k E 23 3k E

k k

1 d k 1 d dk dkG (2 ) 2 E (2 ) d 2 E

+δ Ω ε= =

π ε − π ε ε −∫ ∫

F

2F F3

k k

d dk(k ) 2 k(2 ) d

Ωρ = ρ =

π ε∫Defining the density of Defining the density of the states at the Fermi the states at the Fermi

surface:surface:

F2

2F

vk

π=ρFor a sphere:For a sphere:

F

F

1 1 2E E 2logG 4 2E E

− + δ≈ ρ

− G/4

G/4

F e1e2E2E ρ−

ρ−

−δ−=

20

3.0G <ρFor most superconductorsFor most superconductors

Weak coupling Weak coupling approximation:

G/4F e2E2E ρ−δ−= EEBBapproximation:

Very important:Very important: result not analytic in Gresult not analytic in G

Close to the Fermi surfaceClose to the Fermi surface

⋅+µ=−⋅∂ε∂

+µ≈µ−ε+µ=ε=

)k(v)kk(k

)( FFkk

kkk

F

21

∑> −ε

⋅=ψ

Fkk k0 E2

)rkcos(N)r(∑>

=Fkk

kgN

∑> +ξ

⋅=ψ

Fkk Bk0 E2

)rkcos(N)r(Fkk E−ε=ξ

Wave function maximum in Wave function maximum in momentum space close to 0k =ξmomentum space close to

22

Paired electrons withinPaired electrons within EEB B fromfrom EEFF:: δ<<BE

Only Only d.o.fd.o.f. close to E. close to EFF relevant!!relevant!!

Effective theoryEffective theory

Field theory at the Fermi surfaceField theory at the Fermi surface

The free The free fermionfermion gasgas

OneOne--loop correctionsloop corrections

23

Field theory at the Fermi surfaceField theory at the Fermi surface((PolchinskiPolchinski, TASI 1992, hep, TASI 1992, hep--th/9210046)th/9210046)

Renormalization group analysis a la WilsonRenormalization group analysis a la WilsonHow do fields behave scaling down the energies How do fields behave scaling down the energies

toward toward εεFF by a factorby a factor s<1?s<1?

+= kp

ksEE ⇒s⇒kk ⇒

Scaling:Scaling:

24

Using the invariance under phase transformations, Using the invariance under phase transformations, construction of the most general action for the construction of the most general action for the effective degrees of freedom: effective degrees of freedom: particles and holes particles and holes close to the Fermi surfaceclose to the Fermi surface (non(non--relativistic relativistic description)description)

( ) ( ) ( )( ) ( ) ( )3 † †σ t σ F σ σdtd p iψ p ψ p ε p ε ψ p ψ p⎡ ⎤∂ − −⎣ ⎦∫

Expanding aroundExpanding around εεFF::

( ) ( ) ( )2F F

0

ε pε p ε v

p=

∂− = ⋅ + ≡ +

∂O

25

( ) ( ) ( ) ( )2 † †σ t σ F σ σS dtd kd iψ p ψ p v ψ p ψ p⎡ ⎤= ∂ −⎣ ⎦∫

SsS 12dψ+→Scaling:

tt

1

ssdd

kdkd

dtsdts

∂→∂→

→

→

→−

1/2ψ s ψ−→

requiring the requiring the action action SS to be to be

invariantinvariant

26

The result of the analysis is that all possible The result of the analysis is that all possible interaction terms are irrelevant (go to zero going interaction terms are irrelevant (go to zero going

toward the Fermi surface) except a toward the Fermi surface) except a marginalmarginal(independent on s) (independent on s) quarticquartic interaction of the interaction of the

form:form:

3 3 † †1 2 σ 1 σ 2 σ' 1 σ' 2

, '

V dtd p d p ψ (p )ψ (p )ψ (-p )ψ (-p )σ σ∑∫

corresponding to a Coopercorresponding to a Cooper--like interactionlike interaction

1p

1p−

2p

2p−27

Higher order interactions Higher order interactions irrelevantirrelevant

Free theory Free theory BUTBUT check quantum corrections check quantum corrections to the marginal interactions among the to the marginal interactions among the

Cooper pairsCooper pairs28

The free The free fermionfermion gasgas

t F(i v ) (p, t) 0σ∂ − ψ =EqEq. of motion:. of motion:

)t()t,p(G)vi( ',',Ft δδ=−∂ σσσσPropagator:Propagator:

[ ] tvi',

',',

Fe)()t()()t(i

)t,p(G)t,p(G−

σσ

σσσσ

−θ−θ−θθδ−=

=δ=

i ti e(t) d2 i

− ω

θ = ωπ ω + ε∫Using:Using:

29

i t

0F F

1 ( ) ( )G(p, t) lim d e2 v i v i+

− ω

ε→

⎡ ⎤θ θ −= ω +⎢ ⎥π ω − + ε ω − − ε⎣ ⎦

∫

)p,p(Gedp21)t,p(G 0

tip0

0−∫π≡

F0 vp)i1(1)p(G

−ε+=

or:or:

ipx

p

xpi

p

e)p(be)t,p(b)x( −σ

⋅σσ ∑∑ ==ψ

)p,v(p),x,t(x F== µµ

Fermi field decompositionFermi field decomposition

30

F

†F

b (p) 0 0 for | p | p

b (p) 0 0 for | p | pσ

σ

= >

= <†

p,p ' , '[b (p),b (p)]σ σ + σ σ= δ δ

† 3, '[ (x, t), (x ', t)] (x x ')σ σ + σ σψ ψ = δ δ −

with:with:

† ip x, ' , ' , '

p p

G (x) i 0 T(b (p, t)b (p,0) 0 e G(p, t)⋅σ σ σ σ σ σ σ σ= − δ = δ∑ ∑

†F

†F F

0 b (p)b (p) 0 (p p) ( )

0 b (p)b (p) 0 1 (p p) (p p ) ( )σ σ

σ σ

= θ − = θ −

= − θ − = θ − = θ

The following representation holds:The following representation holds:

In fact, usingIn fact, using

⎩⎨⎧ >θ−

<−θ=−

−0t,e)(i0t,e)(i)t,p(G

tvi

tvi

F

F 31

32

†,0 0

†F

lim G (0, ) i lim 0 T( (0, ) (0) 0

i 0 0 i

+ +σ σ σ σδ→ δ→

σ σ

−δ = − ψ −δ ψ =

= ψ ψ ≡ ρ

0

4ip

F , 40 00 F

d p 12i lim G (0, ) 2i lim e(2 ) (1 i )p v+ +

δσ σ

δ→ δ→σ

ρ = − −δ = −π + ε −∑ ∫

The following property is useful:The following property is useful:

0>0<

2

3F

F3

3

3

3

F 3p)pp(

)2(pd2)(

)2(pd2

π=−θ

π=−θ

π=ρ ∫∫

OneOne--loop correctionsloop corrections

)v)i1)('EE)((v)i1)('EE((1

)2(dkd'dEGG)E(G

FF4

22

−ε+−−ε++π−= ∫

0 F

1(1 i )p v+ ε −

Closing in the Closing in the upper plane upper plane

we getwe get33

)O(G/E)log(G21GG(E) 32 +δρ−=

( ) )k(v1

2kd2

F3

2

∫ π=ρ δδ, , UV cutoffUV cutoff

From RG equations:From RG equations:2dG(E) 1 G(E)

dE 2E= ρ

34

35

0E →

0E →

BCS BCS instabilityinstability

Attractive, stronger Attractive, stronger forfor

/E)log(2G1

GG(E)δρ+

ρ=ρ

BCS theoryBCS theory

A toy modelA toy model

BCS theoryBCS theory

Functional approachFunctional approach

The critical temperatureThe critical temperature

The relevance of gauge invarianceThe relevance of gauge invariance

36

A toy modelA toy model

Solution to BCSSolution to BCS instabilityinstability

Formation of condensatesFormation of condensates

Studied with Studied with variationalvariational methods, methods, SchwingerSchwinger--Dyson, CJT, etc.Dyson, CJT, etc.

37

Idea of quasiIdea of quasi--particles through a toy model (Hubbard particles through a toy model (Hubbard toytoy--model)model)

2 2 Fermi oscillatorsFermi oscillators::

( )† † † †1 1 2 2 1 2 1 2H ε a a a a Ga a a a= + +

Trial wave functionTrial wave function::

† †1 2trial

Ψ (cosθ sinθ a a ) 0= +

cosθ sinθΨaaΨΓ 21 −== 38

Decompose:Decompose:

res0 HHH +=

( ) ( )† † † † 20 1 1 2 2 1 2 1 2H ε a a a a GΓ a a a a GΓ= + − − +

( )( ) ( )

† † † † 2res 1 2 1 2 1 2 1 2

† †1 2 1 2

H Ga a a a GΓ a a a a GΓ

G a a Γ a a Γ

= + − − =

= + −

Mean field theory assumes Mean field theory assumes HHresres = 0= 039

220 GΓθsin 2εΨHΨ −=

Minimize w.r.t.Minimize w.r.t. θθ

εGΓtan2θ0cos2θ 2GΓsin2θ 2ε −=⇒=+

From the expression forFrom the expression for ΓΓ::

222 ΓGεGΓ

21sin2θ

21Γ

+=−=

40

Gap equationGap equation

22 ∆εG

211

+= GΓ∆ =

Is the fundamental state in the Is the fundamental state in the broken phase where the condensate broken phase where the condensate ΓΓis formedtrial

Ψis formed

41

In fact, via In fact, via BogolubovBogolubov transformationtransformation

†1 1 2

†2 1 2

A a cosθ a sinθ

A a sinθ a cosθ

= −

= +one gets:one gets:

42

0ΨAtrial1,2 =

( ) ( )2 2 2 2 † †0 1 1 2 2H ε ε ∆ ε ∆ A A A A= − + + + +

Energy of quasiEnergy of quasi--particles (created byparticles (created by AA++1,21,2))

22 ∆εE +=

BCS theoryBCS theory† † †

k kq 1 2 2 1k kq

H H N b (k)b (k) V b (k)b ( k)b ( q)b (q)σ σσ

= − µ = ξ + − −∑ ∑

µ−ε=−ε=ξ kFkk E

res0 HHH~ +=

43

† † † * *0 k kq 1 2 q 2 1 k q k

k kq

H b (k)b (k) V b (k)b ( k) b ( q)b (q)σ σσ

⎡ ⎤= ξ + − Γ + − Γ − Γ Γ⎣ ⎦∑ ∑

2211

( ) ( )† † *res kq 1 2 k 2 1 q

kq

H V b (k)b ( k) b ( q)b (q)= − − Γ − − Γ∑

k 2 1b ( k)b (k)Γ = −

† † † * *0 k k 1 2 k 2 1 k k

k k

H b (k)b (k) b (k)b ( k) b ( k)b (k)σ σσ

⎡ ⎤= ξ + ∆ − + ∆ − − ∆ Γ⎣ ⎦∑ ∑

∑ Γ−=∆q

qkqk V

BogolubovBogolubov--ValatinValatin transformation:transformation:

* †1 k 1 k 2

* †2 k 1 k 2

b (k) u A (k) v A (k),

b ( k) v A (k) u A (k)

= ++ − = − +

1vu 2k

2k =+

To bring To bring HH00 in canonical form we choosein canonical form we choose44

⎟⎟⎠

⎞⎜⎜⎝

⎛ ξ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ξ+=

k

k2k

k

k2k E

121v,

E1

21u

2k

2kkE ∆+ξ=

†0 k 0

k

H E A (k)A (k) Hσ σσ

= +∑

† †k k 1 2BCS

k

0 (u v b (k)b ( k)) 0= + −∏

00)k(A0)k(ABCS2BCS1 ==

45

* † †k 2 1 k k 1 1 2 2

* kk k

k

b ( k)b (k) u v (1 A (k)A (k) A (k)A (k)

1u v2 E

Γ = − = − − =

∆= =

k

kk E2

1 ∆=Γ

∑∆

−=∆q q

qkqk E

V21 Gap Gap

equationequation

∑ Γ−=∆q

qkqk V

As for the Cooper case choose:As for the Cooper case choose:46

⎩⎨⎧ δ<ξξ−= |||,|,G

otherwise,0V kk'k,k ∆≈∆k

GE2H

2

kk k

2k

k0F

∆−⎟⎟

⎠

⎞⎜⎜⎝

⎛ ξ−ξ= ∑

>

47

Kinetic Kinetic energyenergy

Interaction Interaction termterm

2 2

0 2 20

2 2 22 2 2 2

H dG

logG

δ ⎛ ⎞ξ ∆= ρ ξ ξ − − =⎜ ⎟

⎜ ⎟ξ + ∆⎝ ⎠

⎡ ⎤δ + δ + ∆ ∆= ρ δ − δ δ + ∆ + ∆ −⎢ ⎥

∆⎢ ⎥⎣ ⎦

∫2Gρ

48

∆∆+δ+δ

∆ρ=∆+ξ

∆ξρ=∆ ∫

δ 22

220

logG21dG

21

[ ] 2222

22222

0

41

2

GG2

2H

∆ρ−≈∆+δδ−δρ

=

=∆

−⎥⎦

⎤⎢⎣

⎡ρ

∆+∆+δδ−δ

ρ=

20 4

1H ∆ρ−≈

δ<<∆<<ρ or,1G

G/2e2 ρ−δ≈∆Pair Pair

condensation condensation energyenergy

0T ≠ H / T

H / TT

Tr e OO

Tr e

−

−

⎡ ⎤⎣ ⎦=⎡ ⎤⎣ ⎦

†H Eb b=For a single Fermi oscillatorFor a single Fermi oscillator

†Eb b / T E / TTr[e ] 1 e− −= +†† Eb b / T E / TTr[b be ] e− −=

†E / TT

1b b f (E)e 1

= =+

Fermi Fermi distributiondistribution

49

* † † *k k k 1 1 2 2 k k kT

u v (1 A (k)A (k) A (k)A (k) u v (1 2f (E ))Γ = − − = −

T2E

tanhE2

V))E(f21(vuV q

q q

qkq

qqq

*qkqk ∑∑

∆−=−−=∆

22E,T2Etanh

EdG

411 ∆+ξ=

ξρ= ∫

δ+

δ−

50

Functional approachFunctional approach

† 4 † † 2t

GS , d x (i (| |) ) ( )2

⎡ ⎤⎡ ⎤ψ ψ = ψ ∂ − ε ∇ + µ ψ + ψ ψ⎣ ⎦ ⎢ ⎥⎣ ⎦∫

51

FierzingFierzing (C = i(C = iσσ22))† † † †a a b b a b a b

† †c d † * Tab ab c d

1 1 C C4 2

ψ ψ ψ ψ = −ψ ψ ψ ψ =

= − ε ε ψ ψ ψ ψ = − ψ ψ ψ ψ

† 4 † † * Tt

GS , d x (i (| |) ) ( C )( C )4

⎡ ⎤⎡ ⎤ψ ψ = ψ ∂ − ε ∇ + µ ψ − ψ ψ ψ ψ⎣ ⎦ ⎢ ⎥⎣ ⎦∫†iS ,†Z D( , )e

⎡ ⎤ψ ψ⎣ ⎦= ψ ψ∫Quantum theoryQuantum theory

4 T * † *i G Gd x ( C ) ( C )* G 2 2const. D( , )e

⎡ ⎤⎡ ⎤− ∆− ψ ψ ∆ + ψ ψ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦∫= ∆ ∆∫2

† 4 † * * To

| | 1 1iS [ , ] i d x ( C ) ( C )G 2 2† *

0 0

Z 1 D( , )D( , )eZ Z

⎡ ⎤∆ψ ψ + − − ∆ ψ ψ + ∆ ψ ψ⎢ ⎥

⎣ ⎦∫

= ψ ψ ∆ ∆∫

⎟⎟⎠

⎞⎜⎜⎝

⎛ψψ

=χ *C21

24 † 1

0| |S d x S

G−⎛ ⎞∆

+ ⋅ ⋅ ⋅ = χ χ −⎜ ⎟⎝ ⎠

∫

⎥⎦

⎤⎢⎣

⎡ξ+∆−

∆−ξ−=−

p0*

p01

pp

)p(S52

Since Since ψψ∗ ∗ appears already in appears already in χ χ we are doublewe are double--counting. Solution:counting. Solution: integrate over the integrate over the

fermions with the fermions with the “replica trick”“replica trick”::

[ ] eff

24

iSG||xdi2/11

00

ee)Sdet(Z1

ZZ

≡∫=∆

−−

[ ] ∫∆

−−=∆∆ −

G||xd)SSlog(Tr

2i),(S

241

0*

eff

Evaluating the saddle point:Evaluating the saddle point:

22p

3

3

||)2(pd

2G

∆+ξ

∆π

=∆ ∫22p

20

4

4

||p)2(pdiG

∆−ξ−∆

π=∆ ∫

53

At T not 0, introducing the Matsubara frequenciesAt T not 0, introducing the Matsubara frequencies

T)1n2(n π+=ω

∑ ∫+∞

−∞= ∆+ξ+ω∆

π=∆

n22

p2n

3

3

||)2(pdGT

and usingand using

))E(f21(TE2

1||

1p

pn22

p2n

−=∆+ξ+ω∑

+∞

−∞=

)T2/Etanh(||)2(

pd2G

p22p

3

3

∆+ξ

∆π

=∆ ∫54

By saddle point:By saddle point:2

† 4 † * * To

| | 1 1iS [ , ] i d x ( C ) ( C )G 2 2† *

0 0

Z 1 D( , )D( , )eZ Z

⎡ ⎤∆ψ ψ + − − ∆ ψ ψ + ∆ ψ ψ⎢ ⎥

⎣ ⎦∫

= ψ ψ ∆ ∆∫

ψψ=∆ C2G T

Introducing the Introducing the emem interaction in Sinteraction in S0 0 we see we see thatthat Z is gauge invariant underZ is gauge invariant under

)x(i2)x(i e,e αα ∆→∆ψ→ψ

55

Therefore also Therefore also SSeffeff must be gauge invariant must be gauge invariant and it will depend on the spaceand it will depend on the space--time time

derivatives of derivatives of ∆∆ throughthrough

µµµ +∂= ieA2D

In fact, evaluating the diagrams (In fact, evaluating the diagrams (Gor’kovGor’kov 1959):1959):

56

got the result (with a convenient got the result (with a convenient renormalization of the fields):renormalization of the fields):

3 * 2 2 41 1H d r (r) | ( i A) | (r ) | (r ) | | (r ) |4

2em 2

⎛ ⎞= − ψ ∇ + ψ + α ψ + β ψ⎜ ⎟⎝ ⎠∫

charge of the paircharge of the pair

This result gave full justification to the This result gave full justification to the Landau treatment of superconductivityLandau treatment of superconductivity

57

The critical temperatureThe critical temperature

By definition at By definition at TTcc the gap vanishes. One the gap vanishes. One can perform a GL expansion of the grand can perform a GL expansion of the grand potentialpotential

42

41

21

∆β+∆α=Ω

3 0α∆ + β∆ =with with extremaextrema::

∑ ∫+∞

−∞= ∆+ξ+ω∆

π=∆

n22

p2n

3

3

||)2(pdGT

α α andand β β from the from the expansion of the expansion of the gap equation up to gap equation up to normalizationnormalization 58

To get the normalization remember (in the To get the normalization remember (in the weak coupling and relatively to the normal weak coupling and relatively to the normal state):state):

20 4

1H ∆ρ−=Ω=

Starting from the gap equation:Starting from the gap equation: 02logG21

=∆δ

∆ρ−∆

Integrating overIntegrating over ∆ ∆ and using and using the gap equationthe gap equation one findsone finds:

2

8G

∆ρ

−:

Rule: Rule: Integrate the gap equation and multiply by 2/GIntegrate the gap equation and multiply by 2/G

59

60

Expanding the gap equation:Expanding the gap equation: )T)1n2(( n π+=ω

0)()(

dReTG2 222n

3

22n0n 0

=⎥⎦

⎤⎢⎣

⎡⋅⋅⋅+

ξ+ω∆

−ξ+ω

∆ξρ−∆ ∑∫

∞

=

δ

One gets:One gets:

Integrating over Integrating over ξξand summing over and summing over

n up to N⎟⎟⎠

⎞⎜⎜⎝

⎛ξ+ω

ξρ−=α ∑∫

∞

=

δ

0n 022

n )(dReTG21

G2

n up to N

∑∫∞

=

δ

ξ+ωξ

ρ=β0n 0

222n )(dReT4

T2NN π

δ≈⇒δ=ω

0

T(T) log πα = ρ

γ∆

c 0T 0.56693γ= ∆ ≈ ∆

πRequiring Requiring αα(T(Tcc) = 0) = 0

)3(T8

7)T( 2c

2 ζπ

ρ≈βAlsoAlso

c

T(T) 1T

⎛ ⎞α ≈ −ρ −⎜ ⎟

⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

and, from the gap equationand, from the gap equation

61

1/ 2 1/ 22 c

cc c

(T) 2 2 T T T(T) (T) 1 3.06T 1(T) T T7 (3)

⎛ ⎞ ⎛ ⎞α π∆ = − ∆ ≈ − ≈ −⎜ ⎟ ⎜ ⎟β ζ ⎝ ⎠

⇒⎝ ⎠

Origin of the attractive Origin of the attractive interactioninteraction

•• Coulomb force repulsive, need of an attractive Coulomb force repulsive, need of an attractive interaction interaction

•• ElectronElectron--phonon interaction (phonon interaction (FrolichFrolich 1950)1950)

•• Simple description: Simple description: JelliumJellium modelmodel (Pines et al. 1958): (Pines et al. 1958): electrons + ions treated as a fluid.electrons + ions treated as a fluid.

•• Interaction:Interaction:22 2q

q s2 2 2 2 2 2s s q

4 e 4 e , v qq k q k

ωπ π+ ω ≈

+ + ω − ω2

2s

F

6 nekEπ

= may give attractionmay give attraction

s(1/ k 1A)≈

62Coulomb interaction screened by electrons and ionsCoulomb interaction screened by electrons and ions

The relevance of gauge invarianceThe relevance of gauge invariance(See Weinberg (1990))(See Weinberg (1990))

In the BCS ground state:In the BCS ground state:

0O ≠ψψε= βααβ

The The U(1)U(1)emem is broken since is broken since QQemem(O(O) = ) = -- 2e.2e.

Introduce an order parameter Introduce an order parameter ΦΦ transforming as transforming as the operator the operator OO::

Φ→Φψ→ψΛ∂+→ ΛΛµµµ

ie2ie e,e,AA

63

As usual the phase of O is the Goldstone field As usual the phase of O is the Goldstone field associated to the breaking of the global associated to the breaking of the global U(1). U(1). Decompose:Decompose:

)x(ie2e)x()x( φρ=Φ

GoldstoneGoldstoneOrder parameterOrder parameter

ρρ(x(x) ) is gauge invariant, whereasis gauge invariant, whereas

)x()x()x( Λ+φ→φ

•• φ φ dependence throughdependence through

•• U(1)U(1) broken to broken to ZZ2 2

φ∂µ

⎟⎠⎞

⎜⎝⎛ π

=Λ=Λe

and,064

•• Gauge invariant Fermi fieldGauge invariant Fermi field

•• Effective theory in terms ofEffective theory in terms of

•• From gauge invariance only combinations

ψ=ψ φ−iee~φ∂ψ µµ ,A,~

From gauge invariance only combinationsF A A , Aµν µ ν ν µ µ µ= ∂ − ∂ − ∂ φ

3s

1L d x F F L (A )4

µνµν µ µ= − + − ∂ φ∫

65

s s JL L0Aµ µ

µ

µ

µµ

δ δ= ∂ = −∂ =

δ−∂

δ∂ φEqsEqs. of motion for. of motion for φ:φ:

Assume that LAssume that Lss gives a stable state in absence of gives a stable state in absence of AA and and φ. φ. This implies thatThis implies that

Aµ µ= ∂ φ

is a local minimum and thatis a local minimum and that

2s

2A

L 0(A )

µ µµ µ =∂ φ

δ≠

δ − ∂ φ

Well inside the superconductor we will be at the Well inside the superconductor we will be at the minimum. minimum. The The emem field is a pure gaugefield is a pure gauge andand

F 0 B 0µν = ⇒ = MeissnerMeissner effecteffect 66

Close the minimum:Close the minimum:

22s

s s 2A

1 LL (A ) L (0) (A )2 (A )

µ µ

µ µ µ µµ µ =∂ φ

δ− ∂ φ ≈ + − ∂ φ

δ − ∂ φ

2dim E E L−= × =

3 2

s 2LL A≈ − ∇φλ

LL33 = volume, = volume, λ λ some typical length where the some typical length where the field is not a pure gaugefield is not a pure gauge

67

A BL− ∇φ ≈2 5

s 2B LL ≈λ

2 3B LCost of expelling Cost of expelling BB2 5

2 32

B L B Lλ

Convenience in expelling Convenience in expelling BB ifif

L λ

J B∝ ∇ ∧

68

Since Since the current flows at the current flows at the surface in a the surface in a region of thickness region of thickness λλ

69

SuperconductivitySuperconductivity

Current density conjugated to Current density conjugated to φ:φ: s s0

0

L L JA

δ δ= − = −

δδφ

s

0

H(x) V(x)( J (x))

δφ = = −

δ −Hamilton equation:Hamilton equation:

In stationary conditions In stationary conditions the voltage the voltage V(xV(x) = 0) = 0, with , with JJnot zeronot zero

Close to the phase transition Close to the phase transition the Goldstone field the Goldstone field φ φ is not the only long waveis not the only long wave--length modelength mode. . Consider againConsider again

)x(ie2e)x()x( φρ=Φ

and expand and expand LLs s for smallfor small ΦΦ

2 2 43 *s

1 1 1L d x ( 2ieA)2 2 4

⎡ ⎤≈ − Φ ∇ − Φ − α Φ − β Φ⎢ ⎥⎣ ⎦∫

70

3 2 2 2 2 2 4s

1 1 1L d x 2e ( eA) ( )2 2 4

⎡ ⎤≈ − ρ ∇φ − − ∇ρ − αρ − βρ⎢ ⎥⎣ ⎦∫

2 2

1

4eλ =

ρ2 α

ρ = −β

'ρ = ρ + ρLooking at the fluctuations:Looking at the fluctuations:

2 ' 2 '∇ ρ = − αρ

12

ξ =− α

Coherence length:Coherence length:

2 2V H

1 1,M M

λ = ξ =Notice that in the SM:Notice that in the SM:

71