Two talks for the price of Two talks for the price of one:one:

Cooling by angulon Cooling by angulon annihilation annihilation

andand Asymmetrical fermion Asymmetrical fermion

superfluidssuperfluids

P. Bedaque (Berkeley Lab)

G. Rupak, M. Savage, H. Caldas

Cooling of neutron stars by "angulon" annihilationCooling of neutron stars by "angulon" annihilationLight element

envelope

Heavy element envelope

GemingaVela

• Neutron stars start their lives hot and cool down by neutrino and photon emission

• After a minute, degenerate

• Cooling curves depend on emissivity, specific heat, thermal conduction, etc …

Determined by low-lying excitations

Unique laboratory to learn about dense matter

Page et al.

Neutron pairing in the Neutron pairing in the 33PP22 channel for channel for

densities higher than densities higher than nuclear saturationnuclear saturation

most attractive channel

2T

i j ijn n �

traceless symmetric matrix

2T

ij i jn n �

ij real (favored close to Tc)

determined by an orthonormal frame + 2 eigenvalues

2

30

2

3

1 0 0

0 0

0 0

i

i

e

e

eigenvalues are roots of 1

1

0 2

1 2

0 0

0 0

0 0

(2) (3) (1)S LSU SO U (3) (1)JSO U

Symmetry (breaking) patternSymmetry (breaking) pattern

tensor/spin-orbit

discrete discretetensor/spin-orbit

0nn 0nn

approximate exact

4 exactly massless Goldstone bosons4 exactly massless Goldstone bosons

Ripples in the sea of frames =

angulonsangulons

• Contrary to other excitations, not exponentially suppressed by Boltzman factors, dominate low energy/temperature properties

• Couples to neutrino through neutral currents

0 † iA iC Z n n

• Dimensional analysis:

2 9 3,F VG T c T

Effective theory

For E << 0 :i T

S LU e U R R Δ

phasespin

rotationorbital

rotation

2† 2 † 2 †

0 02

00

0

† †0

8 i i i ik j kj

iiV

fv w

H

L Tr Tr

i Z Tr J

Matching to a simple "microscopic" theory

22

2 2 2

20

31

5v

4

F

F

VV

Mk

v

f

f

w

CH

Rate for angulon+angulon

• No rotation symmetry: big complicated mess

• = v k

enhancement on the number of angulons ~ (T/v)3

suppression due to E ~ v

suppression due to kinematical constraint (no decay with v=0)

suppression on the coupling ~ v39

2 17 9 393

3 316 3 3 0

V 9

0.1510 /

v v

0.154 10 /

v v

F

TG T erg cm s

Tc T erg cm K

Does it matter for cooling ?Does it matter for cooling ?

Above Tc no: (modified) Urca N+N N+N+l+

Below Tc maybe: e+e e+e+

But pair formation around TBut pair formation around Tc c on a shellon a shell

~ 1020 T97

A more detailed study is A more detailed study is probably necessaryprobably necessary

Pairing in asymmetrical superfluidsPairing in asymmetrical superfluids

Two fermion species: A and B

up and down quarks, two

different atoms or hyperfine states in a trap,

…

A Bn ndifferent Fermi surfaces,

pairing ?

Relevance for neutron (quark) stars, atomic traps renewed the interest on the problem: some new,

interesting suggestions for the ground state

Variational arguments can rigorously show what is NOT a

ground state

To make it concrete:

3 3 3

† † † †3 3 3

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )(2 ) (2 ) (2 )

A Bi i k A A k B B A B B A

d k d k d pH N k k k k g p p k k

2

2ik i

i

k

M

mean-field, low density, /<<1

2 3 3

† †3 3

( ) ( ) ( ) ( )(2 ) (2 )

Bi i k k k k

d k d kH N E E k k E k k

g

3

3( ) ( )

(2 ) B A

d kg k k

2

2 2

A B A Bi k k k kkE

Bogoliubov transformation:

2 3 3

† †3 3

( ) ( ) ( ) ( )(2 ) (2 )

Bi i k k k k

d k d kH N E E k k E k k

g

slightly different problem: A and B fixed: minimize H-N

Fill states with negative Ek

† † † †

0 0 0

| ( ) | 0 ( ( ) ( )) ( ) | 0k k k

k k B A BE E E

S k u v k k k

BCS filled with B

A=B

1st order transition

B-A=2

In the real problem nA and nB are fixed: local maximum

(Sarma state) suggested as the ground state (Liu &Wilczek,

gapless superconductor)

† † †

0 0

| ( ( ) ( )) ( ) | 0k k

k k B A BE E

S u v k k k

Mixed state:

(1 )

(1 )A A

B B

n xn x n

n xn x n

,( , ) min ( , ) (1 ) ( )MIXED A B N A B BCS

x nE n n x E n n x E n

minimize

Which is smaller: EMIXED or ESARMA ?

normal fraction

Sarma=BCS=Mixed

Sarmamixed

Sarma=normal

EMIXED < ESARMA

•In atomic traps, bubbles can be imaged. Long sought evidence for superfluidity in fermionic atomic traps ?

• Different dispersion relations (relativity) don’t change anything

•Three flavors is (may be) different !