L - Florence Theory Group - Infn

QCD@Work 2003
International Workshop on
Quantum Chromodynamics
Theory and Experiment
Conversano (Bari, Italy)
June 14-18 2003
Inhomogeneous color
superconductivity
Roberto Casalbuoni
Department of Physics and INFN – Florence
&
CERN TH Division - Geneva

Introduction
Effective theory of CS
Gap equation
The The inhomogeneous phase (LOFF): phase
diagram and crystalline structure
Phonons
LOFF phase in compact stellar objects
Outlook
Summary

Introduction
mu, , md , ms s

Effective theory of
Color
Superconductivity

Relevant scales in CS
δ (cutoff)
∆ (gap)
Fermi momentum defined by
pF
r
E( pF
)
=
µ
The cutoff is of order ωD in
superconductivity and > ΛQCD QCD
in QCD
∆

Hierarchies of effective lagrangians
Microscopic description LQCD QCD
Quasi-particles Quasi particles (dressed fermions
as electrons in metals). Decoupling
of antiparticles (Hong ( Hong 2000) 2000
Decoupling of gapped quasi-
particles. Only light modes as
Goldstones, etc. (R.C. ( R.C. & Gatto
Gatto; ; Hong,
Rho & Zahed 1999)
1999
p – pF >> δ
LHDET HDET
δ >> p – pF >> ∆
p – pF

Physics near the Fermi surface
(see:
( ∆

SM M gives rise di-fermion di fermion condensation producing a
Majorana mass term. Work in the Nambu-Gorkov
Nambu Gorkov basis:
r
1 ⎛ ψ ( p)
⎞
χ = ⎜ r ⎟
⎜ ⎟
⎝Cψ
∗
2 ( −p)
⎠
Near the Fermi surface
ξr
p ≡
r
r ∂E(
p)
( p)
− µ ≈ r
∂p
r r
r r r r r
⋅(
p − pF
) ≡ vF
⋅(
p − p
E F
r
p
F
r
= µ v l r
F
r
p
r
µ v
= F
p=
p
F
+
r
l
)

S
=
S
E
−1
=
Dispersion relation
⎡E
− ξ
⎢ *
⎣ − ∆
1
r
p
E
−
+
∆
ξ
r
p
⎤
⎥
⎦
⎡E
+ ξr
p
2 ⎢ *
∆ ⎣ ∆
∆
− ξ
r
ε(
p)
= ±
2 2
ξr
+ ∆
2
− ξr
p −
E
At fixed vF only energy and
momentum along vF are relevant
v1 Infinite copies of 2-d 2 d physics
p
r
p
⎤
⎥
⎦
v 2

Gap equation
Gap equation
2
BCS
2
2
4
4
4
|
p
|
p
1
)
2
(
p
d
G
1
∆
+
+
π
= ∫ r
∑
∫
+∞
−∞
=
∆
ε
+
π
+
π
=
n
2
2
3
3
)
,
p
(
)
T
)
1
n
2
((
1
)
2
(
p
d
GT
1 r
)
,
p
(
n
n
1
)
2
(
p
d
2
G
1
d
u
3
3
∆
ε
−
−
π
= ∫ r
1
e
1
n
n T
/
)
,
p
(
d
u
+
=
= ∆
ε r

For T T 0
1
G
2
At weak coupling
d p
( 2π)
3
= ∫ 3 2 r
1 ≅
∆
G
2π
2
BCS
p
v
2
F
F
≈
ξ
log
2δe
1
( p)
+
2δ
∆
∆
BCS
2
−
Gρ
2
BCS
( δ =
ρ
cutoff
=
π
p
2
2
F
v
)
F
density of states

With G fixed by χSB SB at T = 0, requiring
Mconst const ~ 400 MeV
and for typical values of µ ~ 400 – 500 MeV one gets
∆
≈
10 ÷ 100
MeV
Evaluation from QCD first principles at asymptotic µ
∆
≈
(Son Son 1999) 1999
bµ
g
5
se
Notice the behavior exp(-c/g) exp( c/g) and not exp(-c/g exp( c/g2 ) as one
would expect from four-fermi four fermi interaction
For µ ~ 400 MeV one finds again ∆
≈10
÷ 100 MeV
−
3π
2
2g
s

The inhomogeneous
phase (LOFF)
In many different situations the “would be” pairing fermions
belong to Fermi surfaces with different radii:
• Quarks with different masses
• Requiring electrical neutrality and/or weak equilibrium

Consider 2 fermions with m 1 = M, m 2 = 0 at the same
chemical potential µ. . The Fermi momenta are
2
pF1 = µ −
M
2
pF2
To form a BCS condensate one needs common momenta
of the pair pF comm
p
comm
F
Grand potential at T = 0
for a single fermion
=
µ
−
2
M
4µ
pF
3
d p r
Ω = 2∫
) 3
( 2π)
0
=
µ
( ε(
p − µ )

2
2 comm
comm
∆Ω ≈ 2∑ µ ( pF
− pFi)(
εi
( pF
) − µ ) ≈
i=
1
Pairing energy
Pairing possible if
≈
−µ
2
∆
2
2
M
The problem may be simulated using massless fermions with
different chemical potentials (Alford, ( Alford, Bowers & Rajagopal 2000) 2000
Analogous problem studied by Larkin &
Ovchinnikov,
Ovchinnikov,
Fulde & Ferrel 1964. 1964.
Proposal of
a new way of pairing. LOFF phase
µ
≤
∆
M
4

LOFF: ferromagnetic alloy with paramagnetic
impurities.
The impurities produce a constant exchange
field acting upon the electron spins giving rise to
an effective difference in the chemical potentials
of the opposite spins. spins.
Very difficult experimentally but claims of
observations in heavy fermion superconductors
(Gloos Gloos & al 1993) 1993 and in quasi-two quasi two dimensional layered
organic superconductors (Nam ( Nam & al. 1999, Manalo & Klein
2000)
2000

µ ≠ µ or paramagnetic impurities (
1
I
2
H = −δµσ
Ω
or paramagnetic impurities (δµ δµ ∼ H) give
rise to an energy additive term
BCS
−
3
4
d p 1
1 = G∫
4
2 r
( 2π)
( p + iδµ
) + | p
Ω
normal
4
=
ρ
− ( ∆
4
Gap equation
2
BCS
|
2
+ ∆
Solution as for BCS ∆ = ∆ BCS, BCS,
up to (for T = 0)
δµ
1
=
∆
BCS
2
≈
0.
707
∆
BCS
2
− 2δµ
2
)

According LOFF, close to first order line, possible
condensation with non zero total momentum
r
p 1
r r r r r
= k + q = −k
+ q
p 2
More generally
| q
r
r
q
/
|
ψ(
x)
ψ(
x)
ψ(
x)
ψ(
x)
|
r r
p + p 1 2
r
q
|
=
=
=
∆
∆e
∑
m
r
2q
r r
2iq⋅x
c
m
e
r r
2iq
⋅x
fixed variationally
chosen
spontaneously
m

Simple plane wave: energy shift
r r r
E(
p)
− µ → E(
± k + q)
− µ m δµ ≈ ξ m µ
r r
µ = δµ − ⋅q
Gap equation:
n ≠
u
n
d
For T T 0
3
g d p 1
= ∫ r
2 ( 2π)
ε(
p,
∆)
1 3
v F
3
g d p 1−
n u − n
1 = ∫ 3 r
2 ( 2π)
ε(
p,
∆)
( 1
e
d
1
n = u,
d ( ε( p,
∆)
± µ ) / T
r
blocking region
− θ(
−ε
− µ ) − θ(
−ε
+ µ ))
+ 1
ε
< | µ
|

The blocking region reduces the gap:
∆
LOFF

Ω
=
α∆
2
+
β
2
∆
4
+
γ
3
∆
6
+
⋅⋅⋅
(for regular crystalline structures all the ∆q are equal)
The coefficients can be determined microscopically for
the different structures (Bowers ( Bowers and Rajagopal (2002))
(2002)

Gap equation
Propagator expansion
Insert in the gap equation

We get the equation
α∆
+
β∆
3
+
γ∆
5
+ ⋅⋅⋅
∂Ω
Which is the same as = 0
∂∆
α∆
β∆ 3
γ∆ 5
=
=
=
=
0
with
The first coefficient has
universal structure,
independent on the crystal.
From its analysis one draws
the following results

δµ
δµ
1
2
Ω
Ω
=
BCS
LOFF
∆
− Ω
∆
BCS
− Ω
LOFF
/
normal
≈ 0.
754∆
normal
ρ
= − ( ∆
4
2
BCS
− 2δµ
= −0.
44ρ(
δµ − δµ
≈1.
15 ( δµ 2 − δµ )
2
BCS
Small window. Opens up in QCD?
(Leibovich Leibovich, , Rajagopal & Shuster 2001;
Giannakis, Giannakis,
Liu & Ren 2002
2
2002)
2
)
)
2

Results of Leibovich, Rajagopal &
µ(MeV)
LOFF
400
1000
Shuster (2001)
δµ 2/ /∆ BCS
0.754
1.24
3.63
(δµ 2 −δµ 1 )/∆ BCS
0.047
0.53
2.92

Along the critical line
( at T = 0,
q = 1.
2δµ
2
)
Single plane wave
∂Ω
∂∆
Critical line from
=
∂Ω
0 , =
∂q
0

General
analysis
(Bowers Bowers and
(2002))
Rajagopal (2002)
Preferred
structure:
face-centered
face centered
cube

Phonons
In the LOFF phase translations and rotations are broken
phonons
Phonon field through the phase of the condensate (R.C., ( R.C.,
Gatto, Gatto,
Mannarelli & Nardulli 2002): 2002):
ψ(
x)
ψ(
x)
Introduce:
=
∆e
r r
2iq⋅x
→
∆e
iΦ(
x)
Φ(
x)
1 r r
φ(
x)
= Φ(
x)
− 2q
⋅ x
f
=
r r
2q
⋅ x

L
phonon
2 2
2
⎡1
⎛ ∂ φ ∂ φ ⎞ ∂ φ⎤
= ⎢ φ&
2 2
2
− v ⎜ + ⎟
⊥ − v
2 2 || 2 ⎥
⎣2
⎝ ∂x
∂y
⎠ ∂z
⎦
Coupling phonons to fermions (quasi-particles) (quasi particles) trough
the gap term
∆(
x)
ψ
T
Cψ
→
∆e
iΦ(
x)
ψ
T
Cψ
It is possible to evaluate the parameters of Lphonon phonon
v
2
⊥
=
1
2
(R.C., R.C., Gatto, Gatto,
Mannarelli & Nardulli 2002) 2002
⎛
⎜ ⎛ δµ ⎞
1−
⎜ r
⎟
⎜ | q |
⎝ ⎝ ⎠
2
⎞
⎟
⎟
⎠
≈
+
0.
153
v
2
||
⎛ δµ ⎞
= ⎜ r ⎟
⎝ | q | ⎠
2
≈
0.
694

Cubic structure
Cubic structure
∑
∑
∑
±
=
ε
=
Φ
ε
±
=
ε
=
ε
=
⋅
∆
⇒
∆
=
∆
=
∆
i
)
i
(
i
i
i
i
k
;
3
,
2
,
1
i
)
x
(
i
;
3
,
2
,
1
i
x
|
q
|
i
2
8
1
k
x
q
i
2
e
e
e
)
x
(
r
r
r
i
)
i
(
x
|
q
|
2
)
x
(
r
=
Φ
i
)
i
(
)
i
(
x
|
q
|
2
)
x
(
)
x
(
f
1 r
−
Φ
=
ϕ

Using the symmetry group of the cube one gets:
L
−
phonon
b
2
∑
=
i=
1,
2,
3
1
2
∑
i=
1,
2,
3
( ⎛ ∂φ
⎜
⎝ ∂t
i)
−
∑
i=
1,
2,
3
( ( i)
)
2
∑(
( i)
( j)
∂ φ − ∂ φ ∂ φ )
i c i j
⎞
⎟
⎠
2
i<
j=
1,
2,
3
a
2
r
| ∇φ
Coupling phonons to fermions (quasi-particles) (quasi particles) trough
the gap term
∆(
x)
ψ
T
Cψ
→
∆
∑
iεiΦ
e
i=
1,
2,
3;
ε = ±
i
(
i )
( x)
( i)
ψ
T
|
2
Cψ

we get for the coefficients
1
a = b = 0
12
c
1 ⎛ ⎞
⎜ ⎛ δµ ⎞
= 3 − ⎟
⎜ ⎜⎜ r ⎟⎟ 1
12
⎟
⎝ ⎝ | q | ⎠ ⎠
One can also evaluate the effective lagrangian for the
gluons in the anisotropic medium. For the cube one finds
Isotropic propagation
This because the second order invariant for the cube
and for the rotation group are the same!
2

LOFF phase in CSO
Why the interest
in the LOFF
phase in QCD?

In neutron stars CS can be studied at T = 0
T
∆
ns
BCS
≈10
−6
÷ 10
−7
( 1MeV
20
≤
≈10
10
∆
BCS
K)
For LOFF state from δpF ∼ 0.75 ∆ BCS
14 ≤ δµ ( MeV)
≤
( MeV)
70
≤100
Orders of magnitude from a crude model: 3 free quarks
M =
M = 0,
M
u d
s
≠
0

ρ n.m.
.m.is is the saturation nuclear density ~ .15x10 15
At the core of the neutron star ρB ~ 10 15
Choosing µ ~ 400 MeV
M s = 200
M s = 300
δp F = 25
δp F = 50
15 g/cm 3
ρB
≈ 5 ÷
ρ
n.
m.
15 g/cm 3
6
Right ballpark
(14 - 70 MeV)
MeV

Glitches: discontinuity in the period of the pulsars.
Standard explanations require: metallic crust +
superfluide inside (neutrons)
LOFF region inside the star might provide a
crystalline structure + superfluid CFL phase
New possibilities for strange stars
( ∆Ω/Ω
≈10
−6
)

Outlook
Theoretical problems: problems Is the cube the optimal
structure at T=0? Which is the size of the LOFF
window?
Phenomenological problems: problems:
Better discussion
of the glitches (treatment of the vortex lines)
New possibilities:
possibilities:
Recent achieving of degenerate
ultracold Fermi gases opens up new fascinating
possibilities of reaching the onset of Cooper pairing of
hyperfine doublets. Possibility of observing the LOFF
crystal?