1. QCD at very high density
Roberto Casalbuoni
Department of Physics and INFN - Florence
Perugia, January 22-23, 2007

2. Summary Introduction and basics in Superconductivity Effective theory
BCS theory
Color Superconductivity: CFL and 2SC phases
Effective theories and perturbative calculations
LOFF phase
Phenomenology

3. Introduction Motivations Basics facts in superconductivity
Cooper pairs

4. 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

5. Studying the QCD vacuum under different and extreme conditions may help our understanding
Neutron star
Heavy ion collision
Big Bang

7. Free quarks R Asymptotic freedom: q When nB >> 1 fm-3
free quarks expected

8. Free Fermi gas and BCS
(high-density QCD)
f(E) E

9. Trivial theory ? High density means high pF pF
Typical scattering at momenta of order of pF pF
No chiral breaking
No confinement
No generation of masses
Trivial theory ?

10. Grand potential unchanged:
Adding a particle to the Fermi surface
Taking out a particle (creating a hole)

11. In QCD attractive interaction (antitriplet channel)
For an arbitrary attractive interaction it is convenient to form pairs particle-particle or hole-hole (Cooper pairs)
In matter SC only under particular conditions (phonon interaction should overcome the Coulomb force)
In QCD attractive interaction (antitriplet channel)
SC much more efficient in QCD

12. Basics facts in superconductivity
1911 – Resistance experiments in mercury, lead and thin by Kamerlingh Onnes in Leiden: existence of a critical temperature Tc ~ K. Lower bound 105 ys. in decay time of sc currents.
In a superconductor resistivity < ohm cm

13. Surprising since from Maxwell, for E = 0, B frozen
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
Destruction of superconductivity for H = Hc
Empirically:

14. 1950 – Role of the phonons (Frolich)
1950 – Role of the phonons (Frolich). Isotope effect (Maxwell & Reynolds), M the isotopic mass of the material
Discontinuity in the specific heat (Corak)
Excitation energy ~ 1.5 Tc

15. Implication is that there is a gap in the spectrum
Implication is that there is a gap in the spectrum. This was measured by Glover and Tinkham in 1956

16. 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
Newton equation
+ Maxwell

17. 1950 - Ginzburg-Landau theory
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)

18. Assume for the ground state:
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:
spin
zero total momentum

19. Cooper assumed that only interactions close to Fermi surface are relevant (see later)
cutoff:
Summing over k:

20. Defining the density of the states at the Fermi surface:
For a sphere:

21. EB For most superconductors Very important: result not analytic in G
Weak coupling approximation: EB
Very important: result not analytic in G
Close to the Fermi surface

22. Wave function maximum in momentum space close to
Paired electrons within EB from EF:
Only d.o.f. close to EF relevant!!

23. 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

24. Effective theory Field theory at the Fermi surface
The free fermion gas
One-loop corrections

25. Field theory at the Fermi surface
(Polchinski, TASI 1992, hep-th/ )
Renormalization group analysis a la Wilson
How do fields behave scaling down the energies toward eF by a factor s<1?
Scaling:

26. 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)
Expanding around eF:

27. requiring the action S to be invariant
Scaling:
requiring the action S to be invariant

28. corresponding to a Cooper-like interaction
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:
corresponding to a Cooper-like interaction

29. s-1+4
Quartic
s-4x1/2
sd ??
Scales as s1+d

30. Scattering:

31. irrelevant
marginal s0
s-1

32. Higher order interactions irrelevant
Free theory BUT check quantum corrections to the marginal interactions among the Cooper pairs

33. The free fermion gas
Eq. of motion:
Propagator:
Using:

34. Fermi field decomposition
or:
Fermi field decomposition

35. The following representation holds:
with:
The following representation holds:
In fact, using

36. The following property is useful:

37. Closing in the upper plane we get
One-loop corrections
Closing in the upper plane we get

38. d, UV cutoff
From RG equations:

39. BCS instability
Attractive, stronger for

40. BCS theory A toy model BCS theory Functional approach
The critical temperature
The relevance of gauge invariance

41. A toy model
Solution to BCS instability
Formation of condensates
Studied with variational methods, Schwinger-Dyson, CJT, etc.

42. Idea of quasi-particles through a toy model (Hubbard toy-model)
2 Fermi oscillators:
Trial wave function:

43. Mean field theory assumes Hres = 0
Decompose:
Mean field theory assumes Hres = 0

44. From the expression for G:
Minimize w.r.t. q
From the expression for G:

45. Gap equation
Is the fundamental state in the broken phase where the condensate G is formed

46. In fact, via Bogolubov transformation
one gets:
Energy of quasi-particles (created by A+1,2)

47. BCS theory 1 2

48. Bogolubov-Valatin transformation:
To bring H0 in canonical form we choose

50. As for the Cooper case choose:
Gap equation
As for the Cooper case choose:

51. Kinetic energy
Interaction term

52. Pair condensation energy

53. For a single Fermi oscillator
Fermi distribution

55. Functional approach
Fierzing (C = is2)
Quantum theory

57. Evaluating the saddle point:
Since y* appears already in c we are double-counting. Solution: integrate over the fermions with the “replica trick”:
Evaluating the saddle point:

58. At T not 0, introducing the Matsubara frequencies
and using

59. By saddle point:
Introducing the em interaction in S0 we see that Z is gauge invariant under
Therefore also Seff must be gauge invariant and it will depend on the space-time derivatives of D through

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

61. got the result (with a convenient renormalization of the fields):
charge of the pair
This result gave full justification to the Landau treatment of superconductivity

62. The critical temperature
By definition at Tc the gap vanishes. One can perform a GL expansion of the grand potential
with extrema:
a and b from the expansion of the gap equation up to normalization

63. To get the normalization remember (in the weak coupling and relatively to the normal state):
Starting from the gap equation:
Integrating over D and using the gap equation one finds:
Rule to get the effective potential from the gap equation: Integrate the gap equation over D and multiply by 2/G

64. Integrating over x and summing over n up to N
Expanding the gap equation in D:
One gets:
Integrating over x and summing over n up to N

65. Requiring a(Tc) = 0
Also
and, from the gap equation

66. 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.
Interaction:
may give attraction
Coulomb interaction screened by electrons and ions

67. The relevance of gauge invariance
(See Weinberg (1990))
In the BCS ground state:
The U(1)em is broken since Qem(O) = - 2e.
Introduce an order parameter F transforming as the operator O:

68. r(x) is gauge invariant, whereas
As usual the phase of O is the Goldstone field associated to the breaking of the global U(1). Decompose:
Goldstone
Order parameter
r(x) is gauge invariant, whereas
f dependence through
U(1) broken to Z2

69. Gauge invariant Fermi field
Effective theory in terms of
From gauge invariance only combinations
Eqs. of motion for f:

70. Assume that Ls gives a stable state in absence of A and f
Assume that Ls gives a stable state in absence of A and f. This implies that
is a local minimum and that
Well inside the superconductor we will be at the minimum. The em field is a pure gauge and
Meissner effect

71. L3 = volume, l some typical length where the field is not a pure gauge
Close the minimum:
L3 = volume, l some typical length where the field is not a pure gauge

72. Cost of expelling B
Convenience in expelling B if
Since
the current flows at the surface in a region of thickness l

73. In stationary conditions the voltage V(x) = 0, with J not zero
Superconductivity
Current density conjugated to f:
Hamilton equation:
In stationary conditions the voltage V(x) = 0, with J not zero

74. and expand Ls for small F
Close to the phase transition the Goldstone field f is not the only long wave-length mode Consider again
and expand Ls for small F

75. Looking at the fluctuations:
Coherence length:
Notice that in the SM: