COLOR SUPERCONDUCTIVITY
Massimo Mannarelli INFN-LNGS
GGI-Firenze Sept. 2012 venerdì 21 settembre 12 “Compact Stars in the QCD Phase Diagram”, Copenhagen August 2001
venerdì 21 settembre 12 Outline
• Motivations • Superconductors • Color Superconductors • Low energy degrees of freedom • Crystalline color superconductors
Reviews: hep-ph/0011333, hep-ph/0202037, 0709.4635 Lecture notes by Casalbuoni http://theory.fi.infn.it/casalbuoni/barcellona.pdf
venerdì 21 settembre 12 MOTIVATIONS
venerdì 21 settembre 12 QCD phase diagram
Hadronic Color ? Superconductor (CSC) µ
venerdì 21 settembre 12 QCD phase diagram
T Quark Gluon Plasma (QGP) Tc
Hadronic Color ? Superconductor (CSC) µ
venerdì 21 settembre 12 QCD phase diagram
T Quark Gluon Plasma (QGP) Tc
Hadronic Color ? Superconductor (CSC) µ
venerdì 21 settembre 12 QCD phase diagram
T Quark Gluon Plasma (QGP) Tc
Hadronic Color ? Superconductor (CSC) µ Compact stars
venerdì 21 settembre 12 QCD phase diagram
T Quark Gluon Plasma (QGP) Tc
Hadronic Color ? Superconductor (CSC) µ Compact stars
Warning: QCD is perturbative only at asymptotic energy scales
venerdì 21 settembre 12 QCD phase diagram
T Quark Gluon Plasma (QGP) Tc
Hadronic Color ? Superconductor (CSC) µ Compact stars
Warning: QCD is perturbative only at asymptotic energy scales
HOT MATTER ENERGY-SCAN EMULATION RHIC RHIC Ultracold fermionic EXPERIMENTS LHC NA61/SHINE@CERN-SPS atoms CBM@FAIR/GSI MPD@NICA/JINR venerdì 21 settembre 12 Compact stars
quark⌧hybrid traditional neutron star star N+e N+e+n
n,p,e, µ n superfluid hyperon n d u c o c t neutron star with star e r i n p g pion condensate u n, s p,e p , µ r u,d,s o quarks t o , 2SC n , CFL s , H crust Fe color−superconducting 6 3 strange quark matter K 10 g/cm (u,d,s quarks) 11 10 g/cm 3 2SC CFL 14 3 CSL CFL−K + 10 g/cm gCFL 0 CFL−K LOFF 0 Hydrogen/He CFL− atmosphere strange star nucleon star
R ~ 10 km F. Weber, Prog.Part.Nucl.Phys. 54 (2005) 193
venerdì 21 settembre 12 Compact stars
quark⌧hybrid traditional neutron star star N+e “Probes” N+e+n
n,p,e, µ n superfluid cooling hyperon n d u c o c t neutron star with star e r i n glitches p g pion condensate u n, s p,e p , µ r instabilities u,d,s o quarks t o
n mass-radius , 2SC , CFL s , H crust magnetic field Fe color−superconducting 6 3 strange quark matter K 10 g/cm GW (u,d,s quarks) 11 10 g/cm 3 2SC ...... CFL 14 3 CSL CFL−K + 10 g/cm gCFL 0 CFL−K LOFF 0 Hydrogen/He CFL− atmosphere strange star nucleon star
R ~ 10 km F. Weber, Prog.Part.Nucl.Phys. 54 (2005) 193
venerdì 21 settembre 12 Compact stars
quark⌧hybrid traditional neutron star star N+e “Probes” N+e+n
n,p,e, µ n superfluid cooling hyperon n d u c o c t neutron star with star e r i n glitches p g pion condensate u n, s p,e p , µ r instabilities u,d,s o quarks t o
n mass-radius , 2SC , CFL s , H crust magnetic field Fe color−superconducting 6 3 strange quark matter K 10 g/cm GW (u,d,s quarks) 11 10 g/cm 3 2SC ...... CFL 14 3 CSL CFL−K + 10 g/cm gCFL 0 CFL−K LOFF 0 Hydrogen/He CFL− atmosphere strange star nucleon star
R ~ 10 km F. Weber, Prog.Part.Nucl.Phys. 54 (2005) 193
Example PSR J1614-2230 mass M ~ 2 M⊙ Demorest et al Nature 467, (2010) 1081 hard to explain with quark matter models Bombaci et al. Phys. Rev. C 85, (2012) 55807
venerdì 21 settembre 12 SUPERCONDUCTORS
In 1911, H.K. Onnes, cooling mercury, found almost no resistivity at T = 4.2 K.
arbitrary units
-D T CV ~ e CV ~T ê
R ~ T3
T
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Tc
venerdì 21 settembre 12 Superconductivity is a quantum phenomenon at the macroscopic scale
venerdì 21 settembre 12 Superconductivity is a quantum phenomenon at the macroscopic scale
T=0
BOSONS
Bosons occupy the same quantum state: They “like” to move together, no dissipation
4He becomes superfluid at T ≃ 2.17 K, Kapitsa et al (1938) BEC
venerdì 21 settembre 12 Superconductivity is a quantum phenomenon at the macroscopic scale
T=0
BOSONS FERMIONS
Bosons occupy the same quantum Fermions cannot occupy the same state: They “like” to move together, quantum state. A different theory of no dissipation superfluidity
4He becomes superfluid at 3He becomes superfluid at T ≃ 2.17 K, Kapitsa et al (1938) T ≃ 0.0025 K, Osheroff (1971) BEC BCS
venerdì 21 settembre 12 Superconductivity is a quantum phenomenon at the macroscopic scale
T=0
BOSONS FERMIONS
Bosons occupy the same quantum Fermions cannot occupy the same state: They “like” to move together, quantum state. A different theory of no dissipation superfluidity
4He becomes superfluid at 3He becomes superfluid at T ≃ 2.17 K, Kapitsa et al (1938) T ≃ 0.0025 K, Osheroff (1971) BEC BCS
venerdì 21 settembre 12 Superconductivity is a quantum phenomenon at the macroscopic scale
T=0
BOSONS FERMIONS
Bosons occupy the same quantum Fermions cannot occupy the same state: They “like” to move together, quantum state. A different theory of no dissipation superfluidity
4He becomes superfluid at 3He becomes superfluid at T ≃ 2.17 K, Kapitsa et al (1938) T ≃ 0.0025 K, Osheroff (1971) BEC ? BCS venerdì 21 settembre 12 BCS Theory
Bardeen-Cooper-Schrieffer (BCS) in 1957 proposed a microscopic theory of fermionic superfluidity
“active” Fermi sphere T=0 fermions PF
“frozen” fermions
venerdì 21 settembre 12 BCS Theory
Bardeen-Cooper-Schrieffer (BCS) in 1957 proposed a microscopic theory of fermionic superfluidity
“active” Fermi sphere T=0 fermions PF
“frozen” fermions
Cooper pairing : Any attractive interaction produces correlated pairs of “active” fermions Cooper pairs effectively behave as bosons and condense
venerdì 21 settembre 12 BCS Theory
Bardeen-Cooper-Schrieffer (BCS) in 1957 proposed a microscopic theory of fermionic superfluidity
“active” Fermi sphere T=0 fermions PF
“frozen” fermions
Cooper pairing : Any attractive interaction produces correlated pairs of “active” fermions Cooper pairs effectively behave as bosons and condense
It costs energy to break a Cooper pair E(p)= (✏(p) µ)2 + (p, T )2 quasiparticle dispersion law: p
venerdì 21 settembre 12 BCS Theory
Bardeen-Cooper-Schrieffer (BCS) in 1957 proposed a microscopic theory of fermionic superfluidity
“active” Fermi sphere T=0 fermions PF
“frozen” fermions
Cooper pairing : Any attractive interaction produces correlated pairs of “active” fermions Cooper pairs effectively behave as bosons and condense
It costs energy to break a Cooper pair E(p)= (✏(p) µ)2 + (p, T )2 quasiparticle dispersion law: p
Increasing the temperature the coherence is lost at T 0.3 c ' 0 venerdì 21 settembre 12 Superfluid vs Superconductors
Definitions Superfluid: frictionless fluid with potential flow v = �ϕ. Irrotational: �× v = 0
Superconductor: perfect diamagnet (Meissner effect)
Cooper pairing is at the basis of both phenomena (for fermions)
venerdì 21 settembre 12 Superfluid vs Superconductors
Definitions Superfluid: frictionless fluid with potential flow v = �ϕ. Irrotational: �× v = 0
Superconductor: perfect diamagnet (Meissner effect)
Cooper pairing is at the basis of both phenomena (for fermions)
Superfluid Broken global symmetry Transport of the quantum numbers Goldstone boson ϕ of the broken group with (basically) no dissipation v = �ϕ
venerdì 21 settembre 12 Superfluid vs Superconductors
Definitions Superfluid: frictionless fluid with potential flow v = �ϕ. Irrotational: �× v = 0
Superconductor: perfect diamagnet (Meissner effect)
Cooper pairing is at the basis of both phenomena (for fermions)
Superfluid Broken global symmetry Transport of the quantum numbers Goldstone boson ϕ of the broken group with (basically) no dissipation v = �ϕ
Superconductor Broken gauge symmetry
Broken gauge fields with mass, M, Higgs mechanism penetrates for a length 1/M / venerdì 21 settembre 12 BCS-BEC crossover v correlation length ⇠ F ⇠ vs 1/3 average distance n up down
1/3 ⇠ n BCS fermi surface phenomenon
venerdì 21 settembre 12 BCS-BEC crossover v correlation length ⇠ F ⇠ vs 1/3 average distance n up down g weak 1/3 ⇠ n BCS fermi surface phenomenon strong
venerdì 21 settembre 12 BCS-BEC crossover v correlation length ⇠ F ⇠ vs 1/3 average distance n up down g weak 1/3 ⇠ n BCS fermi surface phenomenon
1/3 BCS-BEC crossover ⇠ n depleting the Fermi sphere ⇠ strong
venerdì 21 settembre 12 BCS-BEC crossover v correlation length ⇠ F ⇠ vs 1/3 average distance n up down g weak 1/3 ⇠ n BCS fermi surface phenomenon
1/3 BCS-BEC crossover ⇠ n depleting the Fermi sphere ⇠ strong BEC 1/3 4 ⇠ n equivalent to He ⌧
venerdì 21 settembre 12 COLOR SUPERCONDUCTIVITY
venerdì 21 settembre 12 A bit of history
• Quark matter inside compact stars, Ivanenko and Kurdgelaidze (1965), Paccini (1966) ...
• Quark Cooper pairing was proposed by Ivanenko and Kurdgelaidze (1969)
• With asymptotic freedom (1973) more robust results by Collins and Perry (1975), Baym and Chin (1976)
• Classification of some color superconducting phases: Bailin and Love (1984)
venerdì 21 settembre 12 A bit of history
• Quark matter inside compact stars, Ivanenko and Kurdgelaidze (1965), Paccini (1966) ...
• Quark Cooper pairing was proposed by Ivanenko and Kurdgelaidze (1969)
• With asymptotic freedom (1973) more robust results by Collins and Perry (1975), Baym and Chin (1976)
• Classification of some color superconducting phases: Bailin and Love (1984)
Interesting studies but predicted small energy gaps ~ 10 ÷100 keV negligible phenomenological impact for compact stars
venerdì 21 settembre 12 A bit of history
• Quark matter inside compact stars, Ivanenko and Kurdgelaidze (1965), Paccini (1966) ...
• Quark Cooper pairing was proposed by Ivanenko and Kurdgelaidze (1969)
• With asymptotic freedom (1973) more robust results by Collins and Perry (1975), Baym and Chin (1976)
• Classification of some color superconducting phases: Bailin and Love (1984)
Interesting studies but predicted small energy gaps ~ 10 ÷100 keV negligible phenomenological impact for compact stars
• A large gap with instanton models by Alford et al. (1998) and by Rapp et al. (1998)
• The color flavor locked (CFL) phase was proposed by Alford et al. (1999)
venerdì 21 settembre 12 The idea with a cartoon
“particle” quark baryon diquark
“size” point-like ~1 fm ~10 fm
venerdì 21 settembre 12 The idea with a cartoon
“particle” quark baryon diquark
“size” point-like ~1 fm ~10 fm
High density
Liquid of neutrons
venerdì 21 settembre 12 The idea with a cartoon
“particle” quark baryon diquark
“size” point-like ~1 fm ~10 fm
High density Very high density
Liquid of neutrons Liquid of quarks with correlated diquarks
venerdì 21 settembre 12 The idea with a cartoon
“particle” quark baryon diquark
“size” point-like ~1 fm ~10 fm
High density Very high density
Liquid of neutrons Liquid of quarks with correlated diquarks
Models for the lowest-lying baryon excited states with diquarks Anselmino et al. Rev Mod Phys 65, 1199 (1993)
venerdì 21 settembre 12 Do we have the ingredients?
Recipe for superconductivity
• Degenerate system of fermions • Attractive interaction (in some channel) Le chef BCS • T < Tc
venerdì 21 settembre 12 Do we have the ingredients?
Recipe for superconductivity
• Degenerate system of fermions • Attractive interaction (in some channel) Le chef BCS • T < Tc
Color superconductivity
• At large µ, degenerate system of quarks • Attractive interaction between quarks in 3 color channel
• We expect Tc ~ (10 - 100) MeV >> Tneutron star ~ 10 ÷100 keV
venerdì 21 settembre 12 Do we have the ingredients?
Recipe for superconductivity
• Degenerate system of fermions • Attractive interaction (in some channel) Le chef BCS • T < Tc
Color superconductivity
• At large µ, degenerate system of quarks • Attractive interaction between quarks in 3 color channel
• We expect Tc ~ (10 - 100) MeV >> Tneutron star ~ 10 ÷100 keV
N.b. Quarks have color, flavor as well as spin degrees of freedom: complicated dishes. A long menu of colored dishes.
venerdì 21 settembre 12 Two good dishes ...
C ✏ ✏ h ↵i 5 ji⇠ I↵ Iij I
venerdì 21 settembre 12 Two good dishes ...
C ✏ ✏ h ↵i 5 ji⇠ I↵ Iij I
CFL Color superconductor Baryonic superfluid 3 = 2 = 1 > 0 “e.m.” insulator
SU(3) SU(3) SU(3) U(1) SU(3) Z c ⇥ L ⇥ R ⇥ B ! c+L+R ⇥ 2 { { U(1)Q U(1) ˜ Q
venerdì 21 settembre 12 Two good dishes ...
C ✏ ✏ h ↵i 5 ji⇠ I↵ Iij I
CFL Color superconductor Baryonic superfluid 3 = 2 = 1 > 0 “e.m.” insulator
SU(3) SU(3) SU(3) U(1) SU(3) Z c ⇥ L ⇥ R ⇥ B ! c+L+R ⇥ 2 { { U(1)Q U(1) ˜ Q
2SC Color superconductor “e.m.” conductor 3 > 0 , 2 = 1 =0
SU(3) SU(2) SU(2) U(1) U(1) SU(2) SU(2) SU(2) U(1) ˜ U(1)
c ⇥ L ⇥ R ⇥ B ⇥ S ! c ⇥ L ⇥ R ⇥ B ⇥ S
{ {
U(1)Q U(1) ˜ Q venerdì 21 settembre 12 The main course: Color-flavor locked phase
Condensate Using instantons or NJL models (Alford, Rajagopal, Wilczek hep-ph/9804403) (10 100) MeV CFL ' ↵iC 5 j CFL ✏I↵ ✏Iij h i⇠ µ 400 MeV n1/3 µ ' / 1/3 ⇠ & n in between BCS and BEC
venerdì 21 settembre 12 The main course: Color-flavor locked phase
Condensate Using instantons or NJL models (Alford, Rajagopal, Wilczek hep-ph/9804403) (10 100) MeV CFL ' ↵iC 5 j CFL ✏I↵ ✏Iij h i⇠ µ 400 MeV n1/3 µ ' / 1/3 ⇠ & n in between BCS and BEC
Symmetry breaking
SU(3) SU(3) SU(3) U(1) SU(3) Z c ⇥ L ⇥ R ⇥ B ! c+L+R ⇥ 2 { { U(1) U(1) ˜ Q Q
• Higgs mechanism: All gluons acquire “magnetic” mass • χSB: 8 (pseudo) Nambu-Goldstone bosons (NGBs)
• U(1)B breaking: 1 NGB g • “Rotated” electromagnetism mixing angle cos ✓ = (analog of the Weinberg angle) g2 +4e2/3 p venerdì 21 settembre 12 Quark-hadron complementarity
Mapping of the NGBs of the hadronic phase with the NGBs of the CFL phase
Lagrangian Casalbuoni and Gatto, Phys. Lett. B 464, (1999) 111
2 f⇡ 2 = Tr[@ ⌃@ ⌃† v @ ⌃@ ⌃†] Le↵ 4 0 0 ⇡ i i
i a /f a 0 0 0 where ⌃=e a ⇡ describes the octet (⇡±,⇡ ,K±,K , K¯ ,⌘)
21 8 log 2 µ2 1 f 2 = v2 = ⇡ 18 2⇡2 ⇡ 3
venerdì 21 settembre 12 Quark-hadron complementarity
Mapping of the NGBs of the hadronic phase with the NGBs of the CFL phase
Lagrangian Casalbuoni and Gatto, Phys. Lett. B 464, (1999) 111
2 f⇡ 2 = Tr[@ ⌃@ ⌃† v @ ⌃@ ⌃†] Le↵ 4 0 0 ⇡ i i
i a /f a 0 0 0 where ⌃=e a ⇡ describes the octet (⇡±,⇡ ,K±,K , K¯ ,⌘)
21 8 log 2 µ2 1 f 2 = v2 = ⇡ 18 2⇡2 ⇡ 3
Masses Son and Sthephanov, Phys. Rev. D 61, (2000) 74012
2 m = A (mu + md)ms + ⇡± ⇡ (d¯s¯)(us) kaons are lighter than mesons! ⇠ 2 mK = A (mu + ms)md ± K+ (d¯s¯)(ud) m2 = A (m + m )m ⇠ K0,K¯ 0 d s u 3 2 A = 2 2 ⇡ f⇡
venerdì 21 settembre 12 “Phonons”
There is an additional massless NGB, ϕ, associated to U(1)B breaking to Z2 Quantum numbers ϕ ~ <Λ Λ > like the H-dibaryon of Jaffe, Phys. Rev. Lett. 38, 195 (1977)
Effective Lagrangian up to quartic terms Son, hep-ph/0204199
3 2 (')= (µ @ ')2 (@ ')2 Le↵ 4⇡2 0 i ⇥ ⇤
venerdì 21 settembre 12 “Phonons”
There is an additional massless NGB, ϕ, associated to U(1)B breaking to Z2 Quantum numbers ϕ ~ <Λ Λ > like the H-dibaryon of Jaffe, Phys. Rev. Lett. 38, 195 (1977)
Effective Lagrangian up to quartic terms bulk “sound” or phonon Son, hep-ph/0204199 '(x)=' ¯(x)+ (x) 3 2 2 2 e↵ (')= 2 (µ @0') (@i') L 4⇡ classical field long-wavelength ⇥ ⇤ fluctuations
venerdì 21 settembre 12 “Phonons”
There is an additional massless NGB, ϕ, associated to U(1)B breaking to Z2 Quantum numbers ϕ ~ <Λ Λ > like the H-dibaryon of Jaffe, Phys. Rev. Lett. 38, 195 (1977)
Effective Lagrangian up to quartic terms bulk “sound” or phonon Son, hep-ph/0204199 '(x)=' ¯(x)+ (x) 3 2 2 2 e↵ (')= 2 (µ @0') (@i') L 4⇡ classical field long-wavelength ⇥ ⇤ fluctuations
Phenomenology MM et al., Phys. Rev. Lett. 101, 241101 (2008)
Dissipative processes due to vortex-phonon interaction damp r-mode oscillation for CFL stars rotating at frequencies < 1 Hz
venerdì 21 settembre 12 Mismatched Fermi spheres (3 flavor quark matter)
venerdì 21 settembre 12 More realistic conditions
sizable strange quark mass mismatch of the Fermi + momenta around weak equilibrium + µ + µ + µ µ = u d s electric neutrality 3 d u s
Fermi spheres of u,d, s quarks
venerdì 21 settembre 12 More realistic conditions
sizable strange quark mass mismatch of the Fermi + momenta around weak equilibrium + µ + µ + µ µ = u d s electric neutrality 3 d No pairing case u Fermi momenta pF = µ pF = µ pF = µ2 m2 s u u d d s s s p
Fermi spheres of u,d, s quarks
venerdì 21 settembre 12 More realistic conditions
sizable strange quark mass mismatch of the Fermi + momenta around weak equilibrium + µ + µ + µ µ = u d s electric neutrality 3 d No pairing case u Fermi momenta pF = µ pF = µ pF = µ2 m2 s u u d d s s s p
weak decays µu = µd µe u d +¯e + ⌫e u ! s +¯e + ⌫e µd = µs u !+ d u + s $ 2 1 1 Fermi spheres of electric neutrality N N N N =0 3 u 3 d 3 s e u,d, s quarks
venerdì 21 settembre 12 More realistic conditions
sizable strange quark mass mismatch of the Fermi + momenta around weak equilibrium + µ + µ + µ µ = u d s electric neutrality 3 d No pairing case u Fermi momenta pF = µ pF = µ pF = µ2 m2 s u u d d s s s p
weak decays µu = µd µe u d +¯e + ⌫e u ! s +¯e + ⌫e µd = µs u !+ d u + s $ 2 1 1 Fermi spheres of electric neutrality N N N N =0 3 u 3 d 3 s e u,d, s quarks
2 ms F 1 F 2 F 5 µe p = µ + µ p = µ µ p µ µ ' 4µ d 3 e u 3 e s ' 3 e
Alford, Rajagopal, JHEP 0206 (2002) 031 venerdì 21 settembre 12 Mismatch vs Pairing
d u • Energy gained in pairing 2 CFL s ⇠ m2 Energy cost of pairing µ s • ⇠ ⇠ µ
m2 The CFL phase is favored for s 2 µ . CFL
venerdì 21 settembre 12 Mismatch vs Pairing
d u • Energy gained in pairing 2 CFL s ⇠ m2 Energy cost of pairing µ s • ⇠ ⇠ µ
m2 The CFL phase is favored for s 2 µ . CFL
Casalbuoni, MM et al. Phys.Lett. B605 (2005) 362
Forcing the superconductor to a homogenous gapless phase E(p)= µ + (p µ)2 + 2 2 p Leads to the “chromomagnetic instability” Mgluon < 0
venerdì 21 settembre 12 Mismatch vs Pairing
d u • Energy gained in pairing 2 CFL s ⇠ m2 Energy cost of pairing µ s • ⇠ ⇠ µ
m2 The CFL phase is favored for s 2 µ . CFL
Casalbuoni, MM et al. Phys.Lett. B605 (2005) 362
Forcing the superconductor to a homogenous gapless phase E(p)= µ + (p µ)2 + 2 2 p Leads to the “chromomagnetic instability” Mgluon < 0
2 ms For 2 CFL some less symmetric CSC phase should be realized µ &
venerdì 21 settembre 12 LOFF-phase
For µ 1 < µ< µ 2 the superconducting phase named LOFF is favored with Cooper pairs of non-zero total momentum
For two flavors
LOFF: Larkin-Ovchinnikov and Fulde-Ferrel 0 µ1 µ 0.75 ' p2 2 ' 0
• In momentum space P1 2 q < (p ) (p ) > (p + p 2q) 1 2 ⇠ 1 2
P2 • In coordinate space i2q x < (x) (x) > e · ⇠
venerdì 21 settembre 12 LOFF-phase
For µ 1 < µ< µ 2 the superconducting phase named LOFF is favored with Cooper pairs of non-zero total momentum
For two flavors
LOFF: Larkin-Ovchinnikov and Fulde-Ferrel 0 µ1 µ 0.75 ' p2 2 ' 0
• In momentum space P1 2 q < (p ) (p ) > (p + p 2q) 1 2 ⇠ 1 2
P2 • In coordinate space i2q x < (x) (x) > e · ⇠
The LOFF phase corresponds to a non-homogeneous superconductor, with a spatially modulated condensate in the spin 0 channel
venerdì 21 settembre 12 LOFF-phase
For µ 1 < µ< µ 2 the superconducting phase named LOFF is favored with Cooper pairs of non-zero total momentum
For two flavors
LOFF: Larkin-Ovchinnikov and Fulde-Ferrel 0 µ1 µ 0.75 ' p2 2 ' 0
• In momentum space P1 2 q < (p ) (p ) > (p + p 2q) 1 2 ⇠ 1 2
P2 • In coordinate space i2q x < (x) (x) > e · ⇠
The LOFF phase corresponds to a non-homogeneous superconductor, with a spatially modulated condensate in the spin 0 channel The dispersion law of quasiparticles is gapless in some specific directions. No chromomagnetic instability.
venerdì 21 settembre 12 Crystalline structures
venerdì 21 settembre 12 Crystalline structures
• Structures combining more plane waves
venerdì 21 settembre 12 Crystalline structures
• Structures combining more plane waves
venerdì 21 settembre 12 Crystalline structures
• Structures combining more plane waves
venerdì 21 settembre 12 Crystalline structures
• Structures combining more plane waves
venerdì 21 settembre 12 Crystalline structures
• Structures combining more plane waves
• From GL studies: “no-overlap” condition between ribbons
venerdì 21 settembre 12 Crystalline structures
• Structures combining more plane waves
• From GL studies: “no-overlap” condition between ribbons
CX 2cube45z
Z Z
Three flavors
2iqa r < ↵iC 5 j > I e I · ✏I↵ ✏Iij ⇠ Y I=2,3 qa qa Y X I X2{ I } X X
Rajagopal and Sharma Phys.Rev. D74 (2006) 094019
venerdì 21 settembre 12 Crystalline structures
• Structures combining more plane waves
• From GL studies: “no-overlap” condition between ribbons
CX 2cube45z
Z Z
Three flavors
2iqa r < ↵iC 5 j > I e I · ✏I↵ ✏Iij ⇠ Y I=2,3 qa qa Y X I X2{ I } X X
Rajagopal and Sharma Phys.Rev. D74 (2006) 094019
• Crystal oscillations Casalbuoni, MM et al. Phys.Rev. D66 (2002) 094006 MM, Rajagopal and Sharma Phys.Rev. D76 (2007) 074026
venerdì 21 settembre 12 Shear modulus
z The shear modulus describes the response of a crystal to a shear stress y x yz ij yy ij for i = j ⌫ = ij yx 2s 6
ij stress tensor acting on the crystal sij strain (deformation) matrix of the crystal
venerdì 21 settembre 12 Shear modulus
z The shear modulus describes the response of a crystal to a shear stress y x yz ij yy ij for i = j ⌫ = ij yx 2s 6
ij stress tensor acting on the crystal sij strain (deformation) matrix of the crystal
• Crystalline structure given by the spatial modulation of the gap parameter • It is this pattern of modulation that is rigid (and oscillates)
venerdì 21 settembre 12 Shear modulus
z The shear modulus describes the response of a crystal to a shear stress y x yz ij yy ij for i = j ⌫ = ij yx 2s 6
ij stress tensor acting on the crystal sij strain (deformation) matrix of the crystal
• Crystalline structure given by the spatial modulation of the gap parameter • It is this pattern of modulation that is rigid (and oscillates)
MeV 2 µ 2 More rigid than diamond!! ⌫ =2.47 fm3 10MeV 400MeV 20 to 1000 times more rigid than the crust of neutron star ✓ ◆ ⇣ ⌘
MM, Rajagopal and Sharma Phys.Rev. D76 (2007) 074026
venerdì 21 settembre 12 Gravitational waves from “mountains”
z If the star has a non-axial symmetric deformation (mountain) it can emit gravitational waves y ellipticity GW amplitude
2 2 x Ixx Iyy 16⇡ G ✏Izz⌫ ✏ = h = 4 Izz c r
venerdì 21 settembre 12 Gravitational waves from “mountains”
z If the star has a non-axial symmetric deformation (mountain) it can emit gravitational waves y ellipticity GW amplitude
2 2 x Ixx Iyy 16⇡ G ✏Izz⌫ ✏ = h = 4 Izz c r
venerdì 21 settembre 12 Gravitational waves from “mountains”
z If the star has a non-axial symmetric deformation (mountain) it can emit gravitational waves y ellipticity GW amplitude
2 2 x Ixx Iyy 16⇡ G ✏Izz⌫ ✏ = h = 4 Izz c r
• The deformation can arise in the crust or in the core
• Deformation depends on the breaking strain and the shear modulus
venerdì 21 settembre 12 Gravitational waves from “mountains”
z If the star has a non-axial symmetric deformation (mountain) it can emit gravitational waves y ellipticity GW amplitude
2 2 x Ixx Iyy 16⇡ G ✏Izz⌫ ✏ = h = 4 Izz c r
• The deformation can arise in the crust or in the core
• Deformation depends on the breaking strain and the shear modulus
• Large shear modulus To have a “large” GW amplitude • Large breaking strain
venerdì 21 settembre 12 Gravitational waves Using the non-observation of GW from the Crab by the LIGO experiment 4
3
1e-07 Full star 30 Naked core assuming a value of Izz. However, the moment of inertia Core 1e-08 is very sensitive to the poorly known dense matter equa- 25 tion of state (EOS). It can change by a factor of seve1e-09n 20 depending on the stiffness of the EOS [40]. Alternatively, with Eq. (2), one can use Eq. (1) to set a limit on tεh1e-10e 15 pulsar’s quadrupole moment without assuming a value -3 1e-11 (MeV) =10 σmax of I [41]. The limit can in turn set a constraint on the Δ zz 10 shear modulus of crystalline color superconducting quar1e-12k
-2 matter by Eq. (4). In particular, with the expression (5) 5 σ =10 1e-13 max for ν, we can define an exclusion region in the ∆−µ plane by the the following constraint: 0 1e-14 0 200 400 600 800 3 3 4 5 6 7 8 µ (MeV) 10 km 1 Hz ρ / ρn ∆µ < 7.3 × 104 MeV2 ∼Andersson et al. Phys.Rev.R Lett.99.f Lin, Phys.Rev. D76 (2007) 081502 ! " ! " FIG. 1: The areas above the solid lines are excFluIdGed. 2b:yIn the left panel, the ellipticities obtained from ou1r/2analysis are shown as a function of the core transition 231101 (2007) −3 the direct upper limit for the Crab pulsar obtainalloweded from t hregionse h˜0 M 10 r −3 −2 density ρc/ρn, where ρn is the nuclear saturation density. We consider two breaking strains, a maximum value S3/S4 runs for the cases σmax = 10 and 10 . The dotted −2 × −24 −5 ,(7) 10 1.4M" σmax 1 kpc (dashed) line is the constraint set by the spin-downσ¯lbimr i=t f1or0 and a m'(inimum)o!f σ¯br = 1"0! . Fo"r e!ach of "th*ese values we examine the region between the maximum −3 −2 and the minimum shear modulus µ, obtained for the range of parameters in [10], cf. (2). The shaded region σmax = 10 (10 ). The rectangular box is the theoretically ˜ allowed region of µ and ∆. indicates thwe hreergeiohn0 pisertmheittoebdsebrvyatLioIGnaOl uopbpseerrvlaimtiiotnosnanh0d ftohreahorizontal line is the current LIGO upper limit of # = 7.1×10−gi7vefonrpPuSlsRar.J2124-3358 [5]. If the breaking strain is close to the maximum the observations are already constraining tUhendtehreothrey.asIsfuimt pistiocnlostehattotthheepmulsinarimisuman, ihsoowlaetevder, there are no significant constraints yet. In the Based on the extrapolation of terrestrial materials, val- rigid body and that the observed spin-down of the pulsar 2 right panel we compare the ellipticity obtained by considering the full star, only the core of the full star and just ues as high as 10− has been suggested for neutron star is due to the loss of rotational kinetic energy as gravita- −5 the naked core, with no fluid around it. We take the breaking strain to be σ¯br = 10 . As one can see the results tional radiation, one can also obtain the so-called spin- crusts [36, 37], which may also be favored by tfhoer etnhoer-two cores do not differ significantly, while the ellipticity of the core plus fluid star is smaller, more so as mous energy (∼ 1046 erg) liberated in the 2004 December down limit on the gravitational-wave amplitude hsd = venerdì 21 settembre 12 the core size decrea˙ses a3t2hig1h/2er transit˙ion densities. 27 giant flare of SGR 1806-20 according to the magnetar (5GIzz|f|/2c r f) , where f is the time derivative of model [38, 39]. In this work, we consider the effects of the pulsar’s spin frequency [30]. As it is expected that −3 −2 < σmax in the range 10 to 10 , values that have been the strain amplitude satisfies h0 ∼ hsd in general, we used in the study of compact stars by othdeerrauttohomrsa(kee.g.r,eal cimanpdreorviveemaecnotnss,troaninet wonouthlde p(raogdauicnt)∆wµabnatsed onnuthme bers PP/E001025/1 and PP/C505791/1. [18, 19, 27, 38]). to use a realisticspeqinu-daotwionnlimofits: tate. This would require Using the shear modulus given in Eq. (5), the max- the calculation to be carried out within Gene−r3al 1R/2elativ- 3/4 imum quadrupole moment for a solid quark star in the < 5 2 10 M ity. Then the det(e∆rmµ)isnd a∼tio2n.1o×f 1t0heMbeaVckground solution crystalline color superconducting phase can be estimated σmax 1.4M" [1] G. Ushomirsky, C. Cutler, L. Bildsten, MNRAS 319 902 ! " ! 1 4 " by Eq. (4). However, the moment of ineritsiasItzrzaiisgnhetefdoerdward, but the calculatio5n2 of the mountain/ −5 4 10 km / |f˙| f (20/00) to calculate the corresponding maximumseiqzueawtooriualldellriepq- uire implemen×ting the General Relativistic 373 R 10−10 Hz s−1 1 H[2z] B. Haskell, D.I. Jones, N. Andersson, MNRAS 1423 ticity. Using the empirical formula for strange stars given ! " ( ) ! " theory of elasticity, see for example [15]. To date, there1 4 (2006) by Bejger and Haensel (see Eq. (10) of [40]), Eq. (2) gives M 10 km / the maximum equatorial ellipticity as have been no such calculat×ion1s+. 0W.14ork in this direction , [3] B(.8)Haskell, L. Samuelsson, K. Glampedakis, N. An- should clearly be encouraged. 1.4M" R dersson, Modelling magnetically deformed neutron stars, 2 % ! " ! "& −4 ν σmax 1.4M" preprint arXiv:0705.1780 "max = 2.6 × 10 Finally, we neewdhteore iwmephraovveeuoseudr Euqn.d(e1r0s)toafn[d4i0n] gforofthtehemoment of MeV/fm3 10−3 M [4] T. Akgun, I. Wasserman, Toroidal Magnetic Fields ! " brea!king st"rain. Tinheretiarafnorgsetroafngvealsutaerss.that we have used, 4 # $−5 −1 −2 in Type II Superconducting Neutron Stars, preprint R M 1100km σ¯br 10 I,n i[s30r]eilteivsarneptofroterdathcartutshte gcroanvsitiastiionngalo-wfave strain × 1 + 0.14 . (6) −24 ≤ ≤ upper limit for the Crab pulsar (h˜0 = 3.1 × 10 ) is thearXiv:0705.2195 10 km 1.4M" noRrmal matter. The upper limit represents the limit on ! " % ! " ! "& closest to the spin-down limit (at a ratio of 2.2). For[t5h]eB. Abbott et al, Upper limits on gravitational wave emis- We use the values M = 1.4M", R C=ou10lomkmb, foanrcde doomthienraptueldsamrs,ictrhoe cdriryescttaolsbswerhvailteiotnhael ulopwpeerrlimits aresion from 78 radio pulsars, preprint arXiv:gr-qc/0702039 −2 σmax = 10 in order to compare withlim[1i9t], iisn awhpicehssimtiyspticcalelystaitmleaatset oonfe hlaunrgderedsctaimleesblarregaekr itnhgan. the sp[6in]-B.J. Owen, Phys. Rev. Lett. 95 211101 (2005) −4 Owen obtained "max ∼ 2 × 10 for solHidoswtreavnegre, stthares phydsoiwcns loimf itths.e Fcoorrepuilsarvseriny gdloiffbuelraernctlufsrtoerms, in wh[7ic]hP. Haensel, pp 129-149 in Relativistic Gravitation and (with the quarks clustered in groups oftahbaotutof18t)h.eFcorrust.caTsehsetrhee sispin-odorwenasmoenastuorebmeelnietviseobthscautretdhbey the clus-Gravitational Radiation, Ed. J-A. Marck and J-P. Lasota the estimated range of the shear modulus given above, ter’s dynamics, the gravitational-wave observations pro-(Cambridge Univ Press, Cambridge 1997) estimates−2on the breaking strain for the crust should be we find that "max could be as large as ∼ 5 × 10 for vide the only direct upper limits. In the following,[8w]eK. Rajagopal, R. Sharma, Phys. Rev. D. 74 094019 solid quark stars in a crystalline color sauppperliccoanbdulecttinogtheshcaolrlef.ocWuseonustehde tcohnessteraeinsttsimseattbeys sthime poblyservational(2006) phase. This relatively large value of "mbaxecisauabseount ofodurata rdealteavfaonrtthfeorCroaubrpsutlusadry. is available. The [9] K. Rajagopal, R. Sharma, J. Phys. G. 32 S483 (2006) orders of magnitude larger than the tighrteestpuopnpseerolifmtiht e crysTthaelliSn3e/Sq4uraerskultms afotrtetrhetoCrlabrgpeuslstarresasresplotte[1d0i]nM. Mannarelli, K. Rajagopal, R. Sharma, The rigidity of obtained by the combined S3/S4 result for the pulsar Fig. 1. In the figure, the solid lines represent Eq. (7) when is also uncertain. Normal matter will be predominantly −3 crystalline color superconducting quark matter, preprint PSR J2124-3358 [30]. the equality holds for the breaking strain σmax = 10 −2 arXiv:hep-ph/0702021 Constraints set by the Crab pulsar. Ebqri.t(t1le) saungdgebstrseak ainntdo1p0iec,eassswuhmeinngtthheatteMmp=er1a.4tMur"e aisndsuRf-= 10 km. 305 that the observational upper limits on hfi0 coibetnatilnyedfafrrombelowThtehpeumlsaerl’tsinspgintefrmeqpueenracytuisref a=nd29.w8iHllzraen-d the[1d1i]s-E. Kim, M.H.W. Chan, Science 1951 (2004) known isolated pulsars can be used to sspetoandlimbiyt ponla"stic fltaonwce(iuspr t=o s2omkpec.limFoitr)cootmhpearrwisiosne,. tHheowdotted[1a2n]dA.T. Dorsey, P.M. Goldbart, J. Toner, Phys. Rev. Lett. 96 055301 (2006) an elastic quark core will respond is completely unknown. [13] C. Josserand, T. Pomeau, S. Rica, Phys. Rev. Lett. 98 Yet, for our purposes it may not matter which scenario 195301 (2007) is realised as long as the timescale for plastic flow at a [14] M. Alford, S. Reddy, Compact stars with color supercon- given strain is longer than the observation time. ducting quark matter, preprint nucl-th/0211046 We thank Krishna Rajagopal for useful discussions. [15] M. Karlovini, L. Samuelsson, Class. Quantum Grav. 20 This work was supported by PPARC/STFC via grant 3613 (2003) Gravitational waves Using the non-observation of GW from the Crab by the LIGO experiment 4
3
1e-07 Full star 30 Naked core assuming a value of Izz. However, the moment of inertia Core 1e-08 is very sensitive to the poorly known dense matter equa- 25 tion of state (EOS). It can change by a factor of seve1e-09n 20 depending on the stiffness of the EOS [40]. Alternatively, with Eq. (2), one can use Eq. (1) to set a limit on tεh1e-10e 15 pulsar’s quadrupole moment without assuming a value -3 1e-11 (MeV) =10 σmax of I [41]. The limit can in turn set a constraint on the Δ zz 10 shear modulus of crystalline color superconducting quar1e-12k
-2 matter by Eq. (4). In particular, with the expression (5) 5 σ =10 1e-13 max for ν, we can define an exclusion region in the ∆−µ plane by the the following constraint: 0 1e-14 0 200 400 600 800 3 3 4 5 6 7 8 µ (MeV) 10 km 1 Hz ρ / ρn ∆µ < 7.3 × 104 MeV2 ∼Andersson et al. Phys.Rev.R Lett.99.f Lin, Phys.Rev. D76 (2007) 081502 ! " ! " FIG. 1: The areas above the solid lines are excFluIdGed. 2b:yIn the left panel, the ellipticities obtained from ou1r/2analysis are shown as a function of the core transition 231101 (2007) −3 the direct upper limit for the Crab pulsar obtainalloweded from t hregionse h˜0 M 10 r −3 −2 density ρc/ρn, where ρn is the nuclear saturation density. We consider two breaking strains, a maximum value S3/S4 runs for the cases σmax = 10 and 10 . The dotted −2 × −24 −5 ,(7) 10 1.4M" σmax 1 kpc (dashed) line is the constraint set by the spin-downσ¯lbimr i=t f1or0 and a m'(inimum)o!f σ¯br = 1"0! . Fo"r e!ach of "th*ese values we examine the region between the maximum −3 −2 and the minimum shear modulus µ, obtained for the range of parameters in [10], cf. (2). The shaded region σmax = 10 (10 ). The rectangular box is the theoretically ˜ a...wellowed canregio nrestrictof µ and the∆. parameter space!indicates thwe hreergeiohn0 pisertmheittoebdsebrvyatLioIGnaOl uopbpseerrvlaimtiiotnosnanh0d ftohreahorizontal line is the current LIGO upper limit of # = 7.1×10−gi7vefonrpPuSlsRar.J2124-3358 [5]. If the breaking strain is close to the maximum the observations are already constraining tUhendtehreothrey.asIsfuimt pistiocnlostehattotthheepmulsinarimisuman, ihsoowlaetevder, there are no significant constraints yet. In the Based on the extrapolation of terrestrial materials, val- rigid body and that the observed spin-down of the pulsar 2 right panel we compare the ellipticity obtained by considering the full star, only the core of the full star and just ues as high as 10− has been suggested for neutron star is due to the loss of rotational kinetic energy as gravita- −5 the naked core, with no fluid around it. We take the breaking strain to be σ¯br = 10 . As one can see the results tional radiation, one can also obtain the so-called spin- crusts [36, 37], which may also be favored by tfhoer etnhoer-two cores do not differ significantly, while the ellipticity of the core plus fluid star is smaller, more so as mous energy (∼ 1046 erg) liberated in the 2004 December down limit on the gravitational-wave amplitude hsd = venerdì 21 settembre 12 the core size decrea˙ses a3t2hig1h/2er transit˙ion densities. 27 giant flare of SGR 1806-20 according to the magnetar (5GIzz|f|/2c r f) , where f is the time derivative of model [38, 39]. In this work, we consider the effects of the pulsar’s spin frequency [30]. As it is expected that −3 −2 < σmax in the range 10 to 10 , values that have been the strain amplitude satisfies h0 ∼ hsd in general, we used in the study of compact stars by othdeerrauttohomrsa(kee.g.r,eal cimanpdreorviveemaecnotnss,troaninet wonouthlde p(raogdauicnt)∆wµabnatsed onnuthme bers PP/E001025/1 and PP/C505791/1. [18, 19, 27, 38]). to use a realisticspeqinu-daotwionnlimofits: tate. This would require Using the shear modulus given in Eq. (5), the max- the calculation to be carried out within Gene−r3al 1R/2elativ- 3/4 imum quadrupole moment for a solid quark star in the < 5 2 10 M ity. Then the det(e∆rmµ)isnd a∼tio2n.1o×f 1t0heMbeaVckground solution crystalline color superconducting phase can be estimated σmax 1.4M" [1] G. Ushomirsky, C. Cutler, L. Bildsten, MNRAS 319 902 ! " ! 1 4 " by Eq. (4). However, the moment of ineritsiasItzrzaiisgnhetefdoerdward, but the calculatio5n2 of the mountain/ −5 4 10 km / |f˙| f (20/00) to calculate the corresponding maximumseiqzueawtooriualldellriepq- uire implemen×ting the General Relativistic 373 R 10−10 Hz s−1 1 H[2z] B. Haskell, D.I. Jones, N. Andersson, MNRAS 1423 ticity. Using the empirical formula for strange stars given ! " ( ) ! " theory of elasticity, see for example [15]. To date, there1 4 (2006) by Bejger and Haensel (see Eq. (10) of [40]), Eq. (2) gives M 10 km / the maximum equatorial ellipticity as have been no such calculat×ion1s+. 0W.14ork in this direction , [3] B(.8)Haskell, L. Samuelsson, K. Glampedakis, N. An- should clearly be encouraged. 1.4M" R dersson, Modelling magnetically deformed neutron stars, 2 % ! " ! "& −4 ν σmax 1.4M" preprint arXiv:0705.1780 "max = 2.6 × 10 Finally, we neewdhteore iwmephraovveeuoseudr Euqn.d(e1r0s)toafn[d4i0n] gforofthtehemoment of MeV/fm3 10−3 M [4] T. Akgun, I. Wasserman, Toroidal Magnetic Fields ! " brea!king st"rain. Tinheretiarafnorgsetroafngvealsutaerss.that we have used, 4 # $−5 −1 −2 in Type II Superconducting Neutron Stars, preprint R M 1100km σ¯br 10 I,n i[s30r]eilteivsarneptofroterdathcartutshte gcroanvsitiastiionngalo-wfave strain × 1 + 0.14 . (6) −24 ≤ ≤ upper limit for the Crab pulsar (h˜0 = 3.1 × 10 ) is thearXiv:0705.2195 10 km 1.4M" noRrmal matter. The upper limit represents the limit on ! " % ! " ! "& closest to the spin-down limit (at a ratio of 2.2). For[t5h]eB. Abbott et al, Upper limits on gravitational wave emis- We use the values M = 1.4M", R C=ou10lomkmb, foanrcde doomthienraptueldsamrs,ictrhoe cdriryescttaolsbswerhvailteiotnhael ulopwpeerrlimits aresion from 78 radio pulsars, preprint arXiv:gr-qc/0702039 −2 σmax = 10 in order to compare withlim[1i9t], iisn awhpicehssimtiyspticcalelystaitmleaatset oonfe hlaunrgderedsctaimleesblarregaekr itnhgan. the sp[6in]-B.J. Owen, Phys. Rev. Lett. 95 211101 (2005) −4 Owen obtained "max ∼ 2 × 10 for solHidoswtreavnegre, stthares phydsoiwcns loimf itths.e Fcoorrepuilsarvseriny gdloiffbuelraernctlufsrtoerms, in wh[7ic]hP. Haensel, pp 129-149 in Relativistic Gravitation and (with the quarks clustered in groups oftahbaotutof18t)h.eFcorrust.caTsehsetrhee sispin-odorwenasmoenastuorebmeelnietviseobthscautretdhbey the clus-Gravitational Radiation, Ed. J-A. Marck and J-P. Lasota the estimated range of the shear modulus given above, ter’s dynamics, the gravitational-wave observations pro-(Cambridge Univ Press, Cambridge 1997) estimates−2on the breaking strain for the crust should be we find that "max could be as large as ∼ 5 × 10 for vide the only direct upper limits. In the following,[8w]eK. Rajagopal, R. Sharma, Phys. Rev. D. 74 094019 solid quark stars in a crystalline color sauppperliccoanbdulecttinogtheshcaolrlef.ocWuseonustehde tcohnessteraeinsttsimseattbeys sthime poblyservational(2006) phase. This relatively large value of "mbaxecisauabseount ofodurata rdealteavfaonrtthfeorCroaubrpsutlusadry. is available. The [9] K. Rajagopal, R. Sharma, J. Phys. G. 32 S483 (2006) orders of magnitude larger than the tighrteestpuopnpseerolifmtiht e crysTthaelliSn3e/Sq4uraerskultms afotrtetrhetoCrlabrgpeuslstarresasresplotte[1d0i]nM. Mannarelli, K. Rajagopal, R. Sharma, The rigidity of obtained by the combined S3/S4 result for the pulsar Fig. 1. In the figure, the solid lines represent Eq. (7) when is also uncertain. Normal matter will be predominantly −3 crystalline color superconducting quark matter, preprint PSR J2124-3358 [30]. the equality holds for the breaking strain σmax = 10 −2 arXiv:hep-ph/0702021 Constraints set by the Crab pulsar. Ebqri.t(t1le) saungdgebstrseak ainntdo1p0iec,eassswuhmeinngtthheatteMmp=er1a.4tMur"e aisndsuRf-= 10 km. 305 that the observational upper limits on hfi0 coibetnatilnyedfafrrombelowThtehpeumlsaerl’tsinspgintefrmeqpueenracytuisref a=nd29.w8iHllzraen-d the[1d1i]s-E. Kim, M.H.W. Chan, Science 1951 (2004) known isolated pulsars can be used to sspetoandlimbiyt ponla"stic fltaonwce(iuspr t=o s2omkpec.limFoitr)cootmhpearrwisiosne,. tHheowdotted[1a2n]dA.T. Dorsey, P.M. Goldbart, J. Toner, Phys. Rev. Lett. 96 055301 (2006) an elastic quark core will respond is completely unknown. [13] C. Josserand, T. Pomeau, S. Rica, Phys. Rev. Lett. 98 Yet, for our purposes it may not matter which scenario 195301 (2007) is realised as long as the timescale for plastic flow at a [14] M. Alford, S. Reddy, Compact stars with color supercon- given strain is longer than the observation time. ducting quark matter, preprint nucl-th/0211046 We thank Krishna Rajagopal for useful discussions. [15] M. Karlovini, L. Samuelsson, Class. Quantum Grav. 20 This work was supported by PPARC/STFC via grant 3613 (2003) Summary
• The study of matter in extreme conditions allows to shed light on the basic properties of QCD
• Color superconductivity is a phase of matter predicted by QCD
• At asymptotic densities matter should be color-flavor locked
• In realistic conditions a crystalline rigid color superconducting phase should be favored
venerdì 21 settembre 12 Back-up slides
venerdì 21 settembre 12 Increasing the baryonic density
Density H low He ...... Ni
neutron high drip
deconfinement
very large (stellar core ?)
extreme ⇢ venerdì 21 settembre 12 Increasing the baryonic density
Density ↵ ↵ (µ) s ⌘ s H low He
..... Confining ...... Ni
neutron high drip
deconfinement
very large Strong coupling (stellar core ?)
extreme Weak coupling ⇢ venerdì 21 settembre 12 Increasing the baryonic density
Density ↵ ↵ (µ) Degrees of freedom s ⌘ s H low light nuclei He ...... Confining ...... heavy nuclei Ni
neutrons neutron high and protons drip
deconfinement quarks and gluons very large Strong coupling (stellar core ?) Cooper pairs of quarks? quarkyonic phase?....
extreme Weak coupling Cooper pairs of quarks ⇢ NGBs venerdì 21 settembre 12 Phonons in cold atom experiments Experiments with ultracold fermionic atoms in an optical trap helpful to understand properties of NGBs
Phonons originate from the breaking of particle number At low temperature they should dominate the thermodynamics and the dissipative processes
At very low temperature they are ballistic (but still produce dissipation)
2
1,5
x MM, Manuel, Tolos 1201.4006 s
/ ballistic a=0.3 R 1 3ph
0,5
1 4 0 0,1 0,2 0,3 T/TF
venerdì 21 settembre 12 Pairing
fermions spin up • Cooper pairs: di-fermions with total spin 0 and total momentum 0 spin down momentum
venerdì 21 settembre 12 Pairing
fermions spin up • Cooper pairs: di-fermions with total spin 0 and total momentum 0 spin down momentum
⇠
venerdì 21 settembre 12 Pairing
fermions spin up • Cooper pairs: di-fermions with total spin 0 and total momentum 0 spin down momentum
v ⇠ F ⇠
1/3 ⇠ BCS: loosely bound pairs ⇠ & n
1/3 BEC: tightly bound pairs ⇠ . n
venerdì 21 settembre 12 Pairing
fermions spin up • Cooper pairs: di-fermions with total spin 0 and total momentum 0 spin down momentum
v ⇠ F ⇠
1/3 ⇠ BCS: loosely bound pairs ⇠ & n
1/3 BEC: tightly bound pairs ⇠ . n
Type I (Pippard): ⇠ first order phase transition to the normal phase ⌧ Type II (London): ⇠ second order phase transition to the normal phase
venerdì 21 settembre 12 Chiral symmetry breaking
At low density the χSB is due to the condensate ¯ that locks left-handed and right-handed fields h i
In the CFL phase we can write the condensate as
L L = R R = h ↵i ji h ↵i ji 1 ↵i j 2 ↵j i
Color is locked to both left-handed and right-handed rotations.
F C C F
< L L> < R R> SU(3)L rotation SU(3)c rotation SU(3)R rotation
venerdì 21 settembre 12