Transcript Presentazione di PowerPoint - Istituto Nazionale di Fisica

Study of the LOFF phase diagram in a
Ginzburg-Landau approach
G. Tonini, University of Florence, Florence, Italy
R. Casalbuoni,INFN & University of Florence, Florence, Italy
What is LOFF phase?
Applications
Initially studied as a pairing mechanism for an electron superconductor with Zeeman splitting between spin-up and
spin-down Fermi surfaces. Applied also to dense QCD with three flavors.
2q
2q
-p+2q
Cooper pairs with non zero total momentum
p -p+2q
Fixed |q|~pF arbitrary directions
Rotational and traslational symmetry breaking
Crystalline structure
 ( x) ( x) 
 BCS   norm  
1
2
u    
d    
Two flavors problem:
  
2
0
2
-p'+2q
-p'
2iqx

e
q
Explication of pulsar glitches: jumps of the
rotational frequency due to the angular
momentum stored and then suddenly released
by the superfluid neutrons
→ from the interaction between the rigid crust
and the vortices in the neutron superfluid
→ pinning of the vortices in the
crust
Vortices in the nodes of the
LOFF crystal
Existence of strange stars
-p
u
p
d
p'
q
2

Compact stars
2 2


2
crust
The LOFF pairing geometry for a Cooper pair with
momentum 2q. Green/red sphere=up/down quark
Fermi surface. An up quark with momentum p near its
Fermi surface coupled with a down quark with
momentum -p+2q. Strongest coupling for up and down
quark near the pink rings.
1
0
critical value 1 
2
Paring not allowed everywhere:→ blocking and pairing regions
core
10.6 Km
superfluid neutrons
Single plane wave case
Gap equation:
g d p 1  nu  nd
1 

3
2 (2 )  ( p, )
3
nu ,d 
1
e

( ( p ,  )   ) / T
1
Integrating for T=0:
Ultracold Fermi gases
LOFF phase diagram. The transition
between LOFF and normal phase is
always second order. The transition
between BCS and LOFF is first order.
There is one tricritical point at
T0.320
BEC: cold bosons→ cold fermions
(lithium-6 or potassium-40)
 Feshbach resonance provides an attractive
interaction between two different hyperfine states
 Control the two different atomic densities
 Expansion of the gas when the trap is
switched off →spatial distribution of momenta
 Observation of LOFF phase by the periodic
modulation of the atom densities in the crystalline
superfluid

g d3 p
1
1 
(1  (   )  (   ))

3
2 (2 )  ( p, )
Free energies for normal, BCS and LOFF
Phase. The LOFF interval is [1, 2].
 ( p, )  
blocking region for
Minimizing the granpotential respect to q qvF  1.22
2  0.754 BCS
1  0.707 BCS
THE PHASE DIAGRAM IN THE TWO PLANE WAVE CASE
(r )  2 cos( 2q  r)
First order transition near T=Ttric
Second order near T=0
 one more tricritical point!
All the possible vectors configurations
Ginzburg-Landau expansion
({ q }) 
*


 qq 
q ,|q|  q0
J 0  J (qa , qa , qa , qa ) K 0  K (qa , qa , qa , qa , qa , qa )
1
*
*
J
(
rhomb
)




q1 q2 q3  q4
2 quad
1
  K (hexag )*q1  q2 *q3  q4 *q5  q6  ...
3 esag
Sum over all the vectors
configurations
Gap equation with propagator expansion
=
+
 
+
2
( 2 J 0  4 J1 )
g
K1  K (qa , qa , qa , qa , qb , qb )
K 2  K (qa , qa , qb , qb , qb , qb )
From symmetry considerations
 
1
1
1
()  2  4  6  8
4
6
8
2
(2 K 0  12 K1  6 K 2 )
g

60 20

2
x 2
0

 0  2a  bx  cx 2  x 3  0

three dimensional space!
From the type of solution we find the nature (symmetric or
broken) of the phase in every octant
From the study of the
second derivative and the
equation =0 we find the
first order and second
order surfaces
Suppose and discuss
the results in function
of .
Introduction of the Matsubara frequencies

dE
i
 T 
2
n  
J1  J (qa , qa , qb , qb )
Study of the minima of the granpotential
E  in  (2n  1)T
Expansion around T=0
SECOND ORDER SURFACE: a=0 between octants 1-2 and, in part, 4-3
FIRST ORDER SURFACE: D=0, where D is the discriminant of the
cubic equation =0
Two tricritical lines:
1.
    0,   0
2.
2 2
b  c  0, b  0, c  0
9
D |a  0  0
Temperature, momentum and chemical
potential of the tricritical point respect to 
Second tricritical point: when the second order line on the plane =0 meets the
tricritical line given by D=0 the position of the tricritical point in the phase space is -dependent
CONCLUSIONS
We explained the two tricritical points behavior in the two plane waves case.
hep-ph/0310128 to be published on
We provided a general Ginzburg-Landau approach which can be applied to more general cases
Rev.
B all these studies, is a cascade of differe
Different crystalline structures have been analyzed by other authors (Bowers-Rajagopal, Combescot-Mora). The scenarioPhys.
coming
from
probable near the tricritical point while the face centered cube near T=0. But more work is needed.