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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
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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.
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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
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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
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(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
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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
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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
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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
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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
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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 ?
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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
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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:
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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
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A=B
1st order transition
B-A=2
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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
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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
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Sarma=BCS=Mixed
Sarmamixed
Sarma=normal
EMIXED < ESARMA
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•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 !