Roberto Casalbuoni Department of Physics and INFN - Florence

1. QCD at very high density Roberto Casalbuoni Department of Physics and INFN - Florence http://theory.fi.infn.it/casalbuoni/barcellona.pdf casalbuoni@fi.infn.it 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 • QCDsimplifies 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. q q R Free quarks Asymptotic freedom: When nB >> 1 fm-3 free quarks expected

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

9. High density means high 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 (creatinga hole)

11. 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) SCmuch 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 temperatureTc ~ 4-10 0K. Lower bound 105 ys. in decay time of sc currents. In a superconductor resistivity < 10-23 ohm cm

13. 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). Isotope effect (Maxwell & Reynolds), M the isotopic mass of the material • 1954 - Discontinuity in the specific heat (Corak) Excitation energy ~ 1.5 Tc

15. Implication is that thereisa 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. 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. 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. For most superconductors 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/9210046) 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 aroundeF:

27. Scaling: requiring the action S to be invariant

28. 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. or: Fermi field decomposition

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

36. The following property is useful:

37. 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 BCSinstability 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. Decompose: Mean field theory assumes Hres = 0

44. Minimize w.r.t.q From the expression forG:

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. Gap equation As for the Cooper case choose: