Transcript Presentazione di PowerPoint - Istituto Nazionale di Fisica

Anisotropic color
superconductivity
Roberto Casalbuoni
Department of Physics and INFN - Florence
&
CERN TH Division - Geneva
Nagoya, December 10-13, 2002
Summary
 Introduction
 Anisotropic phase (LOFF). Critical points
 Crystalline structures in LOFF
 Phonons
 Conclusions
Introduction
Study of CS back to 1977 (Barrois 1977, Frautschi 1978,
Bailin and Love 1984) based on Cooper instability:
At T ~ 0 a degenerate fermion gas is unstable
Any weak attractive interaction leads to
Cooper pair formation
 Hard for electrons (Coulomb vs. phonons)
 Easy in QCD for di-quark formation (attractive
channel 3 )
(3  3  3  6)
Good news!!! CS easy for large m due to
asymptotic freedom
At high m, ms, md, mu ~ 0, 3 colors and 3 flavors
Possible pairings:
0 iaa bjb 0
 Antisymmetry in color (a, b) for attraction
 Antisymmetry in spin (a,b) for better use of the
Fermi surface
 Antisymmetry in flavor (i, j) for Pauli principle
 
s s

p
Only possible pairings
LL and RR

p
Favorite state CFL (color-flavor locking)
(Alford, Rajagopal & Wilczek 1999)
a
aL
b
bL
a
aR
b
bR
0  0  0 
0  
abC
abC
Symmetry breaking pattern
SU(3)c  SU(3)L  SU(3)R  SU(3)cLR
What happens going down with m? If m << ms we get
3 colors and 2 flavors (2SC)
a
aL
b
bL
0   0  
ab3
ab
SU(3)c  SU(2)L  SU(2)R  SU(2)c  SU(2)L  SU(2)R
In this situation strange quark decouples. But what
happens in the intermediate region of m? The interesting
region is for
m ~ ms2/
Consider 2 fermions with m1 = M, m2 = 0 at the same
chemical potential m. The Fermi momenta are
pF2  m
pF1  m2  M2
To form a BCS condensate one needs common momenta
of the pair pFcomm
pFcomm
2
1
M
 ( m2  M2  m)  m 
2
4m
With energy cost of ~ M2/4m for bringing the fermions
at the same pFcomm
To have a stable pair the energy cost must be less than the
energy for breaking a pair ~ 
2
M

4m
The problem may be simulated using massless fermions
with different chemical potentials (Alford, Bowers &
Rajagopal 2000)
Analogous problem studied by Larkin &
Ovchinnikov, Fulde & Ferrel 1964. Proposal of
a new way of pairing. LOFF phase
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. Very
difficult experimentally but claims of
observations in heavy fermion superconductors
(Gloos & al 1993) and in quasi-two dimensional
layered organic superconductors (Nam & al.
1999, Manalo & Klein 2000)
LOFF phase
The LOFF pairing breaks translational and rotational
invariance
LOFF (x)(x) LOFF  e

2iqx
 

p1  p2  2q

|q|
fixed variationally
 
q/ | q |
chosen
spontaneously
Strategy of calculations at large m
LQCD
Microscopic description
Quasi-particles (dressed fermions
as electrons in metals). Decoupling
of antiparticles (Hong 2000)
Decoupling of gapped quasiparticles. Only light modes as
Goldstones, etc. (R.C. & Gatto;
Hong, Rho & Zahed 1999)
LHDET
p – pF >> 
LGold
p – pF << 
LHDET may be used for evaluating the
gap and for matching the parameters
of LGold
Gap equation for BCS
Interactions gap the fermions
Quasi-particles

2
2
(p, )    BCS


 
  
E(p)
  E(p)  m  
 (p  pF )  vF  (p  pF )
p p pF
Fermi velocity
residual momentum
Start from euclidean gap equation for 4-fermion
interaction
4
dp
1
1   g
 2 2
4
2
(2) (p4  im)  | p |  BCS
d3p 
1
1  gT 
 2
3 
2
(2) n (( 2n  1)T)  (p, )
nu  nd 
1
e

( p, ) / T
g d3p 1  nu  nd
1 

3
2 (2) (p, )
1
For T T 0
g d3p
1 
2 (2)3
1

2
 (p)  BCS
2
At weak coupling
g p
2
1  2 log
2 vF
BCS
2
F
BCS  2e
(   cutoff )
2 / g
2
F
p
density of states
 2
 vF
Anisotropic superconductivity
m1  m2 or paramagnetic impurities (dm ~ H) give
rise to an energy additive term
HI  dm3
According LOFF this favours pair formation with momenta
  
p1  k  q
 

p2  k  q
Simplest case (single plane wave)
More generally
(x)(x)  e
(x )(x )    cme
m

2iqx
 
2iqm x
Simple plane wave: energy shift
 

E(p)  m  E(k  q)  m  dm    m
 
m  dm  vF  q
g d p 1  nu  nd
1 

3
2 (2) (p, )
3
Gap equation:
nu  nd
For T T 0
nu,d 
1
e

( ( p, ) m ) / T
1
blocking region
g d3p
1
1 
 (1  (  m )  (  m ))
3
2 (2) (p, )
 | m |
The blocking region reduces the gap:
LOFF  BCS
Possibility of a crystalline structure (Larkin &
Ovchinnikov 1964, Bowers & Rajagopal 2002)
(x)(x ) 
e

 qi

|qi |1.2 dm
 
2iqi x
see later
The qi’s define the crystal pointing at its vertices.
The LOFF phase is studied (except for the single plane
wave) via a Ginzburg-Landau expansion of the grand
potential
b 4  6
  a        
2
3
2
(for regular crystalline structures all the q are equal)
The coefficients can be determined microscopically for
the different structures. The first coefficient has
universal structure, independent on the crystal. From its
analysis one draws the following results
Two critical values in dm:
1
dm  (m1  m 2 )
2
BCS  normal
 2
  ( BCS  2dm2 )
4
LOFF  normal  .44(dm  dm2 )2
LOFF  1.15 (dm2  dm)
dm1  BCS / 2
dm2  0.754BCS
Small window. Opens
up in QCD? (Leibovich,
Rajagopal & Shuster
2001; Giannakis, Liu &
Ren 2002)
The LOFF gap equation around zero LOFF gap gives
BCS 1
log
 f z
2dm 2
1
f (z)   log(1  uz )du
1
dm
z 
|q|
For dm > dm2, f(z) must reach a minimum
dm
z  coth( z)    1.2  dm2  0.754BCS
|q|
The expansion
and the results as
given by Bowers
& Rajagopal 2002
Preferred
structure:
face-centered
cube
Phonons
In the LOFF phase translations and rotations are broken
phonons
Phonon field through the phase of the condensate (R.C.,
Gatto, Mannarelli & Nardulli 2002):
 
(x)(x)  e
 e
(x)  2q  x
 
1
introducing
(x )  (x )  2q  x
f

2iqx
i ( x )

q  0,0,1
2
2
2
1 2








2
2
phonon     v   2  2   v ll 2 
y 
z 
 x
2
Coupling phonons to fermions (quasi-particles) trough
the gap term
i ( x )
(x) C  e
T
 C
T
It is possible to evaluate the parameters of Lphonon
(R.C., Gatto, Mannarelli & Nardulli 2002)
+
2




1
dm
v 2  1       0.153
2  | q | 


2
 dm 
v      0.694
| q |
2
||
Cubic structure
8
(x )    e
 
2iqk x

k 1
e

2i|q|i xi
i1,2,3;i  
3 scalar fields (i)(x)

 (x)  2 | q | xi
(i )

1 (i )
(i )
 (x )   (x )  2 | q | xi
f

e
ii( i ) ( x )
i1,2,3;i  

(x)  0

(x)  4

(x)  4
(i)(x) transforms under the group Oh of the cube.
Its e.v. ~ xi breaks O(3)xOh T Ohdiag. Therefore we get
phonon
2
 
1
  
2 i1,2,3  t
 (i) 2
 a
   |  |
 2 i1,2,3


(i)

b
(i ) 2
  i  c  i(i) j( j)
2 i1,2,3
i j1,2,3

Coupling phonons to fermions (quasi-particles) trough
the gap term
(x ) C  
T
e
ii ( i ) ( x )
i1,2,3;i  
 C
T
we get for the coefficients

1   dm 
c
3    1

12   | q | 


2
1
a
12
b0
One can evaluate the effective lagrangian for the gluons in
tha 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!
Outlook
Why the interest
in the LOFF
phase in QCD?
Neutron stars
Glitches: discontinuity in the period of the pulsars.
(Ω/Ω  106 )
Possible explanation: LOFF region inside the star
Recent achieving of degenerate ultracold Fermi
gases opens up new fascinating possibilities of
reaching the onset of Cooper pairing of hyperfine
doublets. However reaching equal populations is
a big technical problem. (Combescot 2001)
New possibility for the LOFF state?
Normal
LOFF
weak coupling
BCS
strong coupling
dm2  0.75BCS

| q | 0.9BCS

|q|
 1.2
dm
dm
cos q  
|q|
QCD@Work 2003
International Workshop on
Quantum Chromodynamics
Theory and Experiment
Conversano (Bari, Italy)
June 14-18 2003
Anisotropic color
superconductivity
Roberto Casalbuoni
Department of Physics and INFN – Florence
&
CERN TH Division - Geneva