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QCD at very high density
Roberto Casalbuoni Department of Physics and INFN
- Florence
http//theory.fi.infn.it/casalbuoni/barcellona.pdf
casalbuoni_at_fi.infn.it
Perugia, January 22-23, 2007
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Summary

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

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Introduction

• Motivations
• Basics facts in superconductivity
• Cooper pairs

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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
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Studying the QCD vacuum under different and
extreme conditions may help our understanding
Neutron star
Heavy ion collision
Big Bang
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Free quarks
Asymptotic freedom
When nB gtgt 1 fm-3 free quarks expected
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Free Fermi gas and BCS
(high-density QCD)
f(E)
E
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• 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 ?
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Grand potential unchanged

• Adding a particle to the Fermi surface

• Taking out a particle (creating a hole)

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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
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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 lt 10-23 ohm cm
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• 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
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• 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
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Implication is that there is a gap in the
spectrum. This was measured by Glover and Tinkham
in 1956
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• 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
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• 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)

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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
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Cooper assumed that only interactions close to
Fermi surface are relevant (see later)
cutoff
Summing over k
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Defining the density of the states at the Fermi
surface
For a sphere
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For most superconductors
Weak coupling approximation
EB
Very important result not analytic in G
Close to the Fermi surface
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Wave function maximum in momentum space close to
Paired electrons within EB from EF
Only d.o.f. close to EF relevant!!
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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
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Effective theory

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

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Field theory at the Fermi surface
(Polchinski, TASI 1992, hep-th/9210046) Renormaliz
ation group analysis a la Wilson
How do fields behave scaling down the energies
toward eF by a factor slt1?
Scaling
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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
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Scaling
requiring the action S to be invariant
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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
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s-14
Quartic
s-4x1/2
sd ??
Scales as s1d
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Scattering
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irrelevant
marginal
s0
s-1
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Free theory BUT check quantum corrections to the
marginal interactions among the Cooper pairs
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The free fermion gas
Eq. of motion
Propagator
Using
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or
Fermi field decomposition
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with
The following representation holds
In fact, using
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The following property is useful
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One-loop corrections
Closing in the upper plane we get
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d, UV cutoff
From RG equations
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BCS instability
Attractive, stronger for
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BCS theory

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

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A toy model
Solution to BCS instability
Formation of condensates
Studied with variational methods,
Schwinger-Dyson, CJT, etc.
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Idea of quasi-particles through a toy model
(Hubbard toy-model)
2 Fermi oscillators
Trial wave function
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Decompose
Mean field theory assumes Hres 0
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Minimize w.r.t. q
From the expression for G
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Gap equation
Is the fundamental state in the broken phase
where the condensate G is formed
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In fact, via Bogolubov transformation
one gets
Energy of quasi-particles (created by A1,2)
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BCS theory
2
1
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Bogolubov-Valatin transformation
To bring H0 in canonical form we choose
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Gap equation
As for the Cooper case choose
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Kinetic energy
Interaction term
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Pair condensation energy
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For a single Fermi oscillator
Fermi distribution
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Functional approach
Fierzing (C is2)
Quantum theory
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Since y appears already in c we are
double-counting. Solution integrate over the
fermions with the replica trick
Evaluating the saddle point
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At T not 0, introducing the Matsubara frequencies
and using
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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
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In fact, evaluating the diagrams (Gorkov 1959)
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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
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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
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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
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Expanding the gap equation in D
One gets
Integrating over x and summing over n up to N
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Requiring a(Tc) 0
Also
and, from the gap equation
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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
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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
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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

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• Gauge invariant Fermi field
• Effective theory in terms of
• From gauge invariance only combinations

Eqs. of motion for f
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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
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Close the minimum
L3 volume, l some typical length where the
field is not a pure gauge
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Cost of expelling B
Convenience in expelling B if
Since
the current flows at the surface in a region of
thickness l
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Superconductivity
Current density conjugated to f
Hamilton equation
In stationary conditions the voltage V(x) 0,
with J not zero
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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
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Looking at the fluctuations
Coherence length
Notice that in the SM