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

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

Introduction Motivations Basics facts in superconductivity Cooper pairs

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

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

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

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

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 ?

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

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

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

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:

1950 – Role of the phonons (Frolich) 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

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

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

1950 - Ginzburg-Landau theory 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)

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

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

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

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

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

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

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

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:

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:

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

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

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

Scattering:

irrelevant marginal s0 s-1

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

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

Fermi field decomposition or: Fermi field decomposition

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

The following property is useful:

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

d, UV cutoff From RG equations:

BCS instability Attractive, stronger for

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

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

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

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

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

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

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

BCS theory 1 2

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

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

Kinetic energy Interaction term

Pair condensation energy

For a single Fermi oscillator Fermi distribution

Functional approach Fierzing (C = is2) Quantum theory

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:

At T not 0, introducing the Matsubara frequencies and using

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

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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