Study of the LOFF phase diagram in a Ginzburg-Landau approach

G. Tonini, University of Florence, Florence, ItalyR. Casalbuoni,INFN & University of Florence, Florence, Italy

We explained the two tricritical points behavior in the two plane waves case.We provided a general Ginzburg-Landau approach which can be applied to more general casesDifferent crystalline structures have been analyzed by other authors (Bowers-Rajagopal, Combescot-Mora). The scenario coming from all these studies, is a cascade of different structures with the temperature. In particular the two plane waves case seems to be the most

probable near the tricritical point while the face centered cube near T=0. But more work is needed.

CONCLUSIONS

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.

Cooper pairs with non zero total momentum p -p+2q

Fixed |q|~pF arbitrary directionsRotational and traslational symmetry breaking

CrystallineCrystalline structurestructure

Two flavors problem:

critical value

Paring not allowed everywhere:→ blocking and pairing regions

xq2)()( i

qqexx

d u22

222

02

21

normBCS

012

1

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.

2q 2q

-p+2q

-p'+2q

-p-p'

p p'

u d

Applications

Compact starsExplication 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 crustVortices in the nodes of the LOFF crystalExistence of strange stars

Ultracold Fermi gases

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

Single plane wave case

Gap equation:

Integrating for T=0:

blocking region for

Minimizing the granpotential respect to q

),(

1

)2(21

3

3

p

nnpdg du 1

1/)),((,

Tpdu en

))()(1(),(

1

)2(21

3

3

p

pdg

),( p

22.1 Fqv

BCS 754.02BCS 707.01

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 T0.320

Free energies for normal, BCS and LOFF Phase. The LOFF interval is [1, 2].

THE PHASE DIAGRAM IN THE TWO PLANE WAVE CASEFirst order transition near T=Ttric Second order near T=0 one more tricritical point!one more tricritical point!

Ginzburg-Landau expansion

Gap equation with propagator expansion

= + + Introduction of the Matsubara frequencies

Expansion around T=0

...)hexag(3

1

)rhomb(2

1})({

654321

4321

0

***

**

||,

*

qqqqqesag

q

qqqquad

qqqqq

qq

K

J

All the possible vectors configurations

From symmetry considerations

n

TdE

i2 TniE n )12(

),,,(0 aaaa qqqqJJ ),,,,,(0 aaaaaa qqqqqqKK

),,,(1 bbaa qqqqJJ

),,,,,(1 bbaaaa qqqqqqKK

),,,,,(2 bbbbaa qqqqqqKK

Study of the minima of the granpotential

three dimensional space!From the type of solution we find the nature (symmetric or broken) of the phase in every octant

)42(2

10 JJg

)6122(2

210 KKKg

r)q2cos(2)r(

8642

8

1

6

1

4

1)(

Suppose and discuss the results in function of .

20

60 2

0

2

x 020 32

xcxbxaSum over all the vectors configurations

From the study of the second derivative and the equation =0 we find the first order and second order surfaces

SECOND ORDER SURFACESECOND ORDER SURFACE: a=0 between octants 1-2 and, in part, 4-3

FIRST ORDER SURFACEFIRST ORDER SURFACE: D=0, where D is the discriminant of the cubic equation =0Two tricritical lines:Two tricritical lines:

1.

2.

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

0,0

0| 0aD0,0,09

2 2 cbcbTemperature, momentum and chemical potential of the tricritical point respect to

core

superfluid neutrons

crust

10.6 Km

What is LOFF phase?

hep-ph/0310128 to be published on Phys. Rev. B