QCD at very high density
Roberto Casalbuoni
Department of Physics and INFN - Florence
http://theory.fi.infn.it/casalbuoni/barcellona.pdf
[email protected]
Perugia, January 22-23, 2007
1
Summary
 Introduction and basics in Superconductivity
 Effective theory
 BCS theory
 Color Superconductivity: CFL and 2SC phases
 Effective theories and perturbative
calculations
 LOFF phase
 Phenomenology
2
Introduction
 Motivations
 Basics facts in superconductivity
 Cooper pairs
3
Motivations
 Important to explore the entire QCD
phase diagram: Understanding of
Hadrons
QCD-vacuum
Understanding of its modifications
 Extreme Conditions in the Universe:
Neutron Stars, Big Bang
 QCD simplifies in extreme conditions:
Study QCD when quarks and gluons are the
relevant degrees of freedom
4
Studying the QCD vacuum under different and
extreme conditions may help our understanding
Neutron star
Heavy ion collision
Big Bang
5
6
Limiting case   
q
q
 R  0
Free quarks
R
Asymptotic freedom:
When nB >> 1 fm-3
free quarks expected
 1 fm
7
Free Fermi gas and BCS
(high-density QCD)
For T  0 (β = 1/kT  )
f(E)
E
EF  μ
8
pF
 High density means high pF
 Typical scattering at
momenta of order of pF
For p F   QCD
 No chiral breaking
 No confinement
 No generation of masses
Trivial
theory ?
9
Grand potential unchanged:
(F  E  μN)
• Adding a particle to the Fermi surface
 Taking out a particle (creating a hole)
10
For an arbitrary attractive interaction it is
convenient to form pairs particle-particle or
hole-hole (Cooper pairs)
E + (±2E F - E B ) - μ(N ± 2) = F - E B
In matter SC only under particular conditions (phonon
interaction should overcome the Coulomb force)
Tc (electr.)
1  10 0K
3
4


10

10
E(electr.) 104  105 0 K
In QCD attractive interaction
(antitriplet channel)
SC much more efficient in QCD
11
Basics facts in superconductivity
 1911 – Resistance experiments in mercury, lead
and thin by Kamerlingh Onnes in Leiden:
existence of a critical temperature Tc ~ 4-10 0K.
Lower bound 105 ys. in decay time of sc currents.
In a superconductor
resistivity < 10-23 ohm cm
12
 1933 – Meissner and Ochsenfeld discover
perfect diamagnetism. Exclusion of B except
for a penetration depth of ~ 500 Angstrom.
Surprising since from
Maxwell, for E = 0, B frozen

 
1 B
E  
c t
Destruction of
H c2 (T )
superconductivity for H = Hc fs (T ) 
 f n (T )
8
Empirically:
  T 2 
H c (T )  H c (0) 1    
  Tc  
13
 1950 – Role of the phonons (Frolich).
Isotope effect (Maxwell & Reynolds), M the
isotopic mass of the material
1
Tc  H c (0)   ,
M
  0.45  0.5
 1954 - Discontinuity in the specific heat (Corak)
cs  aTce
cn  T
 bTc / T
Excitation energy ~ 1.5 Tc
14
Implication is that there is a gap in the
spectrum. This was measured by Glover and
Tinkham in 1956
15
 Two fluid models: phenomenological expressions
for the free energy in the normal and in the
superconducting state (Gorter and Casimir 1934)
 London & London theory, 1935: still a two-fluid
models based on

J s n se 2 

E,
t
m


J n  n E,


( Js  en s v s )


( J n  en n v n )
 
nse2  + Maxwell
  Js  
B
mc
2 

4

n
e
1 
2
s
 B
B 2 B
2
mc
L
Newton equation
  4 
B
Js
c
B( x )  B(0)e
 x / L
16
 1950 - Ginzburg-Landau theory. In the
context of Landau theory of second order
transitions, valid only around Tc , not appreciated
at that time. Recognized of paramount
importance after BCS. Based on the construction
of an effective theory (modern terms)
 2
ns | ( r ) |
Fs (T )  Fn (T ) 
 2 
 2 1
 4
1 *  
3 
*
  d r 
 ( r ) |   ie A | ( r )  (T ) | ( r ) |  (T ) | ( r ) | 
2
 2m

17
Cooper pairs
1956 – Cooper proved that two fermions may
form a bound state for an arbitrary
attractive interaction in a simple model
Only two particle interactions considered. Interactions with
the sea neglected but from Fermi statistics
Assume for the
ground state:
  
 
0 ( r1  r2 )  (12  21 ) g k cos( k  (r1  r2 ))
k
spin
zero total momentum
    
 
 1
2
2

(



)

V
(
r

r
)

(
r

r
)

E

(
r

r
)
1
1
1
2
0
1
2
0
1
2
 2m

 2
|k|
( E  2k )g k   Vk ,k 'g k ' , k 
18
2m
k ' k F
 i ( k k ')r 3
1
Vk ,k '  3  V( r )e
dr
L
Cooper assumed that only interactions close to
Fermi surface are relevant (see later)

 G, k F  k  k c
Vk,k ' 
0,
otherwise
cutoff:
Summing over k:
( E  2 k ) g k   G  g k '
g
g k = -G
kc  E F  
k ' k F
k'
k'>k F
E - 2ε k
1
1

G k  k F 2 k  E
19
3
kc d k
E F  d
1
1
d
2 dk


k
3
3
k
E
F (2 )
F
G
2 k  E
(2)
d k 2  E
Defining the density of
the states at the Fermi
surface:
For a sphere:
d 2 dk
  (k F )  2 
k
3 F
(2)
d k
kF
k 2F
 2
 vF
1 1
2E F  E  2
  log
G 4
2E F  E
e 4 / G
E  2 E F  2
4 / G
1 e
20
G  0.3
For most superconductors
Weak coupling
approximation:
E  2E F  2e
4 / G
EB
Very important: result not analytic in G
Close to the Fermi surface
k
 k    (  k  )    
k
 
  
 ( k  k F )    v F ( k)  
kk F
21
N = G  gk
k>k F
k  k  EF
 

cos( k  r )
0 ( r )  N 
k  k F 2 k  E
 

cos( k  r )
0 ( r )  N 
k  k F 2 k  E B
Wave function maximum in
momentum space close to
Paired electrons within EB from EF:
k  0
E B  
Only d.o.f. close to EF relevant!!
22
R
2

=
2
2
ψ 0 (r) r d r

2
3
ψ 0 (r) d r
3
2
4 vF
=
2
3 EB
Assuming EB of the order of the critical
temperature, 10 K and vF ~ 108 cm/s we get
that the typical size of a Fermi Cooper is
about 10-4 cm ~ 104 A. In the corresponding
volume about 1011 electrons (one electron
occupies roughly a volume of about (2 A)3 ).
In ordinarity SC the attractive interaction
is given by the electron-phonon interaction
that in some case can overcome the Coulomb
interaction
23
Effective theory
 Field theory at the Fermi surface
 The free fermion gas
 One-loop corrections
24
Field theory at the Fermi surface
(Polchinski, TASI 1992, hep-th/9210046)
Renormalization group analysis a la Wilson
How do fields behave scaling down the energies
toward F by a factor s<1?



k
  
pk
Scaling:
E

sE


 s
kk
25
Using the invariance under phase transformations,
construction of the most general action for the
effective degrees of freedom: particles and holes
close to the Fermi surface (non-relativistic
description)
†
†


dtd
p
iψ
p

ψ
p

ε
p

ε
ψ
p
ψ
p












σ
t
σ
F
σ
σ



3
Expanding around F:
ε  p 
ε  p  εF 
p
 O
2
v
F

0
26
S   dtd kd iψ  p   t ψσ  p   v Fψ  p  ψ σ  p 
2dψ 1
2
†
σ
Scaling:
  s
1
dt  s dt


dk  dk


d   sd 
 t  s t
†
σ
Ss
S
requiring the
action S to be
invariant
ψs
1/2
ψ
27
The result of the analysis is that all possible
interaction terms are irrelevant (go to zero going
toward the Fermi surface) except a marginal
(independent on s) quartic interaction of the
form:
V  dtd p1d p2 ψ (p1 )ψσ (p2 )ψ (-p1 )ψσ' (-p2 )
3
3
†
σ
 ,'
†
σ'
corresponding to a Cooper-like interaction
p1
p2
 p1
 p2
28
s-1+4
Quartic








2
2
2
2
 dtd k1d 1d k 2d 2d k 3d 3d k 4d 4
   
  

 
V( k1 , k 2 , k 3 , k 4 )ψ σ (p1 )ψ σ (p3 )ψ σ' (p2 )ψ σ' (p4 )
  
3 
δ (p1  p2  p3  p4 )
s ??
s-4x1/2
Scales as s1+
29
Scattering:
 
 
p1  p2  p3  p4


 
p3  p1  δk 3  δ 3




p 4  p 2  δk 4  δ  4





3
δ δk3  δk4  δ 3  δ 4

30
irrelevant
marginal

 


3

 4 p1
  p

2

 4 p1
δk 3
δk 3


δk 4

 3
δk 4
0


s
p2  p1






3
δ δk3  δk4  δ 3  δ 4




2
-1
δ (δk3  δk 4 )(δ 3  δ 4 )
s
31
Higher order interactions
irrelevant
Free theory BUT check quantum corrections
to the marginal interactions among the
Cooper pairs
32
The free fermion gas
Eq. of motion:
Propagator:
(i t  v F ) (p, t)  0

(i t  v F )G ,' ( p, t )  ,'( t )


G , ' ( p, t )  , 'G( p, t ) 
 i, ' ( t )()  (  t )( )e iv F t
Using:
i
e  it
(t) 
d

2
  i
33
1
( )
(  ) 
 it 
G(p, t)  lim
de 



 0 2


v

i



v

i

F
F


or:


1
ip0 t
G( p, t ) 
dp0 e G( p0 , p)

2
1
G ( p) 
(1  i)p0  v F
0
Fermi field decomposition
0


  ipx
ipx
  ( x )   b ( p, t )e   b ( p)e

p

p



x  ( t, x ), p  (v F , p)

34
with:
b (p) 0  0
for
| p | pF
b (p) 0  0 for
| p | p F
†

[b (p), b† (p)]  p,p', '
[ (x, t),  (x ', t)]  , ' (x  x ')
†

3
The following representation holds:
G , ' (x)  i, '  0 T(b (p, t)b† (p,0) 0 e ipx  , '  G(p, t)
In fact, using
p
p
0 b† (p)b (p) 0  (p F  p)  (  )
0 b (p)b (p) 0  1  (p F  p)  (p  p F )  ( )
†

 iv F t


i

(

)
e
, t0

G( p, t )  
iv F t
i

(


)
e
, t0

35
The following property is useful:
lim G , (0, )  i lim 0 T( (0, )† (0) 0 
0
0
 i 0   0  i F
†

0
0
d 4 p ip0
1
 F  2i lim  G , (0, )  2i lim 
e
4
0
0
(2)
(1  i)p0  v F


3
dp
dp
p3F
F  2
( )  2
( p F  p)  2
3
3
( 2 )
( 2 )
3
3
36
One-loop corrections
1
(1  i)p0  v F
2
dE'd
kdl
1
2
iG(E) = iG - G 
(2π)4 ((E + E')(1+ iε) - v Fl)((E - E')(1 + iε) - v Fl)
Closing in the
upper plane
we get
37
1 2
3
G(E)  G  G log( /E)  O(G )
2

2
dk 1

  2
3
2  v F (k)
, UV cutoff
on v F
From RG equations:
dG(E) 1

G(E) 2
dE
2E
38
G
G(E) 
G
1
log( /E)
2
E0
BCS
instability
Attractive, stronger
for E  0
39
BCS theory
 A toy model
 BCS theory
 Functional approach
The critical temperature
 The relevance of gauge invariance
40
A toy model
Solution to BCS instability
Formation of condensates
Studied with variational methods,
Schwinger-Dyson, CJT, etc.
41
Idea of quasi-particles through a toy model (Hubbard
toy-model)
2 Fermi oscillators:
H  ε  a a  a a   Ga a a a
†
1 1
†
2 2
† †
1 2 1 2
Trial wave function:
Ψ
trial
 (cosθ  sinθ a a ) 0
† †
1 2
Γ  Ψ a1a 2 Ψ  sinθ cosθ
42
Decompose:
H  H 0  H res
H 0  ε  a a  a a   GΓ  a1a 2  a a   GΓ
†
1 1
†
2 2
† †
1 2
H res  Ga a a a  GΓ  a1 a 2  a a
† †
1 2 1 2
† †
1 2
  GΓ
 G  a a  Γ  a1 a 2  Γ 
† †
1 2
Mean field theory assumes Hres = 0
43
2
2

Ψ H0 Ψ  2ε sin θ  GΓ
2
Minimize w.r.t.
2

GΓ
2ε sin2 θ  2GΓ cos2θ  0  tan2θ  
ε
From the expression for
G:
1
1
GΓ
Γ   sin2 θ 
2
2 ε 2  G 2 Γ2
44
Gap equation
1
G
1
2 ε 2  Δ2
Ψ
Δ  GΓ
Is the fundamental state in the
broken phase where the condensate G
trial
is formed
45
In fact, via Bogolubov transformation
A1  a1cosθ  a sinθ
†
2
A 2  a sinθ  a 2cosθ
†
1
one gets:


0
trial
A1,2 Ψ
H0  ε  ε  Δ
2
2

ε  Δ  A A1  A A 2 
2
2
†
1
†
2
Energy of quasi-particles (created by A+1,2)
E  ε Δ
2
2
46
BCS theory
H  H  N   k b† (k)b (k)   Vkq b1† (k)b†2 ( k)b 2 ( q)b1(q)
k
kq
k  k  EF  k  
~
H  H 0  H res
1 2
H 0   k b† (k)b (k)   Vkq b1† (k)b†2 ( k) Gq  b2 ( q)b1(q) G*k  Gq G*k 
k
kq
H res   Vkq  b1† (k)b†2 ( k)  G*k   b2 ( q)b1(q)  Gq 
kq
Gk  b2 (k)b1 (k)
47
H 0   k b† (k)b (k)     k b1† (k)b†2 ( k)  *k b 2 ( k)b1(k)   k G*k 
k
k
 k    VkqGq
q
Bogolubov-Valatin transformation:
b1 (k)  u*k A1 (k)  v k A†2 (k),

*
†
b2 (  k)   v k A1 (k)  u k A 2 (k)
uk  vk  1
2
2
To bring H0 in canonical form we choose
48
uk
2
1
k 
 1  ,
2  Ek 
vk
2
1
k 
 1  
2  Ek 
Ek    k
2
k
2
H 0   E k A† (k)A  (k)  H 0
k
0
BCS
  (u k  v b (k)b ( k)) 0
†
k 1
†
2
k
A1 (k) 0
BCS
 A2 ( k ) 0
BCS
0
49
G k  b2 (  k)b1 (k)  u*k v k (1  A1† (k)A1 (k)  A †2 (k)A 2 (k) 
1 k
 u vk 
2 Ek
*
k
 k    VkqGq
q
1 k
Gk 
2 Ek
q
1
 k    Vkq
2 q
Eq
As for the Cooper case choose:
Gap
equation
50
 G, | k |, | k |  

Vk ,k '  
otherwise
 0,
k  

2k  2
H 0  2   k   
Ek  G
kk F 
2


   d   
2
2




0


H0
Kinetic Interaction
energy
term
 2


 G

2
2 
2
 2






      2   2   2 log



 G
2
G
51

1

1
  2  2
   G  d
  G log
2
2
2
2




0
2
2

 2
2


2
2
H 0        



2
G  G
 2
1 2
2
2
         
2
4


 G  1, or   
Pair
condensation
energy
1 2
H 0   
4
  2e
2 /  G
52
T0
O
T

Tr e
H /T
Tr e
O 
H /T

H  Eb b
For a single Fermi oscillator
Tr[e
 Eb†b / T
†
Tr[b be
†
bb
T
 f (E) 
]  1 e
 Eb†b / T
1
e
E/T
†
1
E / T
]e
E / T
Fermi
distribution
53
Gk  u*k v k (1  A1† (k)A1 (k)  A†2 (k)A2 (k)
T
 u*k v k (1  2f (E k ))
 k   V u v q (1  2f ( Eq ))   Vkq
*
kq q
q
q
q
2 Eq
tanh
Eq
2T

1
d
E
2
2
1   G  tanh
, E  
4
E
2T

54
Functional approach
G † 2

†
S ,     d x  (i t  (|  |)  )  ( ) 
2


†
4
Fierzing (C = i2)
†a a †b b  †a †ba  b 
1
1 † * T
† †c
d
  ababc d    C  C
4
2
G † * T


†
S ,     d x  (i t  (|  |)  )  ( C )( C) 
4


†
4
Quantum theory
Z   D(,  )e
†
iS ,† 


55
const.   D( ,  )e
*

i 4  G T
G


d x   (  C )  *  ( †C* ) 
G
2
 2



Z
1
†
*

D(

,

)D(

,

)e

Z0 Z0
 ||2 1

1
iSo [  , ] i d x  
  ( †C* )  * ( TC ) 
2
 G 2

†

4
1   

 * 
2  C 
2


|

|
4
† 1
S0     d x   S  

G 

 p 0  p
S ( p)  
*



1
 
p0  p 
56
Since * appears already in  we are doublecounting. Solution: integrate over the
fermions with the “replica trick”:
||2
i d x
1/ 2
G
Z
1

*
1



D
(

,

)
det(
S
)
e


Z0 Z0 

4
e
iSeff

i
1
4 ||
Seff ( ,  )   Tr log( S0S )   d x
2
G
*
2
Evaluating the saddle point:
d4p

  iG 
(2)4 p02  2p  |  |2
G d 3p
 
2 ( 2 ) 3

2p  |  |2
57
At T not 0, introducing the Matsubara frequencies
n  (2n  1)T
dp

  GT  
3
2
2
2
n   ( 2 ) n  p  |  |

3
and using

1
1

(1  2f ( E p ))

2
2
2
2 E pT
n   n  p  |  |
G d 3p
 
2 ( 2 ) 3

2p  |  |2
tanh( E p / 2T )
58
By saddle point:
Z
1
†
*

D(

,

)D(

,

)e

Z0 Z0
 ||2 1

1
iSo [  , ]i d x  
  ( †C* )  * ( TC ) 
2
 G 2

†

4
G T

 C
2
Introducing the em interaction in S0 we see
that Z is gauge invariant under
  e
i( x )
,   e
2 i( x )
Therefore also Seff must be gauge invariant
and it will depend on the space-time
derivatives of  through
59
D    2ieA 
In fact, evaluating the diagrams (Gor’kov 1959):
60
got the result (with a convenient
renormalization of the fields):
1
 1 *
2
2
4
H  d r 
 (r) | (  i2eA) | (r)   | (r) |   | (r) | 
2
 4m

3
charge of the pair
This result gave full justification to the
Landau treatment of superconductivity
61
The critical temperature
By definition at Tc the gap vanishes. One
can perform a GL expansion of the grand
potential
1 2 1 4
    
2
4
with extrema:
  3  0
 and  from the

d 3p

expansion of the   GT  
3
2
2
2
(
2

)




|

|
n  
n
p
gap equation up to
normalization
62
To get the normalization remember (in the
weak coupling and relatively to the normal
state):
1 2
H 0     
4
1
2
Starting from the gap equation:    G log
0
2

Integrating over  and using
G 2


the gap equation one finds:
8
Rule to get the effective potential from the gap
equation: Integrate the gap equation over  and multiply
63
by 2/G
Expanding the gap equation in :
(n  (2n  1)T)
 



3
  2GT Re   d  2
 2
     0
2
2 2
n 0 0
 (n   ) (n   )

One gets:
 
2
d 

  1  2GT Re   2
2 
G
n 0 0 ( n   ) 
 
d
  4T Re   2
2 2
(



)
n 0 0
n
Integrating over 
and summing over
n up to N

N    N 
2T
64
T
(T)   log
 0
Requiring (Tc) = 0
Also
γ = eC , C = 0.577...

Tc   0  0.56693  0

7ρ
β(T) = 2 2 ζ(3)
8π T
and, from the gap equation


T 
 (T)    1   
 Tc  

1/ 2
(T)
2 2 Tc 
T
 (T)  
 (T) 
1  
(T)
7(3)  Tc 
2
1/ 2

T
 3.06Tc 1  
 Tc 
65
Origin of the attractive
interaction
 Coulomb force repulsive, need of an attractive
interaction
 Electron-phonon interaction (Frolich 1950)
 Simple description: Jellium model (Pines et al. 1958):
electrons + ions treated as a fluid.
q2
4e2
4e2
 2
, q  v sq
 Interaction:
2
2
2
2
2
q  ks q  ks   q
2
6

ne
ks2 
EF
(1/ k s  1A)
may give attraction
Coulomb interaction screened by electrons and ions
66
The relevance of gauge invariance
(See Weinberg (1990))
In the BCS ground state:


O     0
The U(1)em is broken since Qem(O) = - 2e.
Introduce an order parameter F transforming as
the operator O:
ie
A  A   ,   e , F  e
2 ie
F
67
As usual the phase of O is the Goldstone field
associated to the breaking of the global U(1).
Decompose:
F( x )  ( x )e
2 ie( x )
Order parameter
Goldstone
(x) is gauge invariant, whereas
( x )  ( x )  ( x )
  dependence through   


 U(1) broken to Z2    0, and   

e
68
ie
~
e 
 Gauge invariant Fermi field
~
 Effective theory in terms of , A ,  
 From gauge invariance only combinations
F    A     A , A   
1 3

L    d x F F  Ls (A    )
4
Ls
Ls
Eqs. of motion for : 0  
 
 J

A
69
Assume that Ls gives a stable state in absence of
A and . This implies that
A    
is a local minimum and that
 Ls
(A  )2
2
0
A  
Well inside the superconductor we will be at the
minimum. The em field is a pure gauge and
F  0  B  0
Meissner effect
70
Close the minimum:
1
 Ls
Ls (A  )  Ls (0) 
2
2 (A  )
2
(A  )
A  
dim  E  E 2  L
3
2
L
Ls  2 A  

L3 = volume,  some typical length where the
field is not a pure gauge
71
2
2 5
BL
Ls  2

A    BL
2 3
Cost of expelling B
BL
Convenience in expelling B if
L
Since
J B
2 5
BL
2
2 3
BL

the current flows at
the surface in a
region of thickness 
72
Superconductivity
Current density conjugated to :
Hamilton equation:
Ls
Ls

 J 0
A0

H s
(x) 
  V(x)
( J 0 (x))
In stationary conditions the voltage V(x) = 0, with J
not zero
73
Close to the phase transition the Goldstone field
 is not the only long wave-length mode.
Consider again
F( x )  ( x )e
2 ie( x )
and expand Ls for small F
2
1
1
2
4
 1 *
Ls   d x   F (  2ieA) F   F   F 
2
4
 2

3
1
1 2 1 4

2 2
2
2
Ls   d x  2e  (  A)  ()     
2
2
4


3

1
4e 
2
2

2



74
   ' 
Looking at the fluctuations:
  '  2 '
2
Coherence length:
Notice that in the SM:

1
2
1
1
 2 ,  2
MV
MH
75