L - Florence Theory Group - Infn

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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


In this situation strange quark decouples. But what

happens in the intermediate region of µ? The interesting

region is for (see later)

µ ∼ m 2 /∆ ∆

s

Possible new inhomogeneous phase of QCD


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


2

BCS

p

v

2

F

F


ξ

log

2δe

1

( p)

+




BCS

2



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


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



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


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


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


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



=

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



=

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



∆e

iΦ(

x)

ψ

T


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





iεiΦ

e

i=

1,

2,

3;

ε = ±

i

(

i )

( x)

( i)

ψ

T

|

2


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?

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