カラー超伝導 における 非アーベルボーテックス のフェルミオン構造
DESCRIPTION
カラー超伝導 における 非アーベルボーテックス のフェルミオン構造. Phys. Rev. D81, 105003 (2010). 安井 繁宏 (KEK) in collaboration with 板倉数記 (KEK) and 新田宗土 ( 慶應大学 ). 08 Jun. 2010 @ 東京 大学松井研究室. Contents. Introduction Bogoliubov-de Gennes equation Single Flavor c ase CFL c ase Effective Theory in 1+1 dimension - PowerPoint PPT PresentationTRANSCRIPT
カラー超伝導における非アーベルボーテックスのフェルミオン構造安井繁宏 (KEK)
in collaboration with
板倉数記 (KEK) and 新田宗土 ( 慶應大学 )
08 Jun. 2010@ 東京大学松井研究室
Phys. Rev. D81, 105003 (2010)
1. Introduction2. Bogoliubov-de Gennes equation
A. Single Flavor caseB. CFL case
3. Effective Theory in 1+1 dimension4. Summary
Contents
Introduction
Vortex Δ(r,θ)=|Δ(r)|einθ・ Abrikosov lattice・ 4He (3He) superfluidity・ BEC-BCS・ quantum turbulance・ nuclear superfluidity・ color superconductivity・ cosmic strings
symmetry breaking G→H π1(G/H) π≅ 0(H)≠0
winding number n
Topologically Stableθ=0 → θ=2π
Ginzburg-Landau theoryis effective for r >> ξ.
ξ
Topologically Stable
Vortex Δ(r,θ)=|Δ(r)|einθ
ξ
・ Abrikosov lattice・ 4He (3He) superfluidity・ BEC-BCS・ quantum turbulance・ nuclear superfluidity・ color superconductivity・ cosmic strings
θ=0 → θ=2π
symmetry breaking G→H π1(G/H) π≅ 0(H)≠0
Ginzburg-Landau theoryis effective for r >> ξ.
winding number n
Fermions
Topologically Stable
ξ
Vortex
・ Abrikosov lattice・ 4He (3He) superfluidity・ BEC-BCS・ quantum turbulance・ nuclear superfluidity・ color superconductivity・ cosmic strings
θ=0 → θ=2π
Ginzburg-Landau theoryis effective for r >> ξ.
symmetry breaking G→H π1(G/H) π≅ 0(H)≠0
Fermions
Ginzburg-Landau theoryis effective for r >> ξ.
ξ
Fermions
ξ
Fermions
ξ
Fermions
Fermions appear at short distance.
ξ
Fermions
Fermions appear at short distance.
ξ
FermionsFermions in Topological Objects ・ Soliton (kink, Skyrmion) ・ Quantum Hall Effect ・ Bulk-Edge correspondence ・ Domain Wall Fermion
IntroductionBogoliubov-de Gennes (BdG) equation
de Gennes, ``Superconductivity of Metals and Alloys“, (Benjamin New York, 1966)F. Gygi and M. Shluter, Phy. Rev. B43, 7069 (1991)P. Pieri and G. C. Strinati, Phy. Rev. Lett. 91, 030401 (2003)
Gap profiling function
Hamiltonian of fermions
Solve self-consistently
Gap profiling function Δ(r) is obtained from fermion dynamics.
nkz
E
particle hole
Gap profiling function Δ(r) is obtained from fermion dynamics.
IntroductionBogoliubov-de Gennes (BdG) equation
de Gennes, ``Superconductivity of Metals and Alloys“, (Benjamin New York, 1966)F. Gygi and M. Shluter, Phy. Rev. B43, 7069 (1991)P. Pieri and G. C. Strinati, Phy. Rev. Lett. 91, 030401 (2003)
Gap profiling function
Hamiltonian of fermions
Solve self-consistentlyn
kz
E
vortex Δ(r,θ)=|Δ(r)|eiθ
bound states
Gap profiling function Δ(r) is obtained from fermion dynamics.
IntroductionBogoliubov-de Gennes (BdG) equation
de Gennes, ``Superconductivity of Metals and Alloys“, (Benjamin New York, 1966)F. Gygi and M. Shluter, Phy. Rev. B43, 7069 (1991)P. Pieri and G. C. Strinati, Phy. Rev. Lett. 91, 030401 (2003)
Gap profiling function
Hamiltonian of fermions
Solve self-consistentlyn
kz
E
zero mode
E=0
vortex Δ(r,θ)=|Δ(r)|eiθ
bound states
Gap profiling function Δ(r) is obtained from fermion dynamics.
IntroductionBogoliubov-de Gennes (BdG) equation
de Gennes, ``Superconductivity of Metals and Alloys“, (Benjamin New York, 1966)F. Gygi and M. Shluter, Phy. Rev. B43, 7069 (1991)P. Pieri and G. C. Strinati, Phy. Rev. Lett. 91, 030401 (2003)
Gap profiling function
Hamiltonian of fermions
Solve self-consistentlyn
kz
E
zero mode
E=0
vortex Δ(r,θ)=|Δ(r)|eiθ
bound statesbounsd state
dominance
Gap profiling function Δ(r) is obtained from fermion dynamics.
IntroductionBogoliubov-de Gennes (BdG) equation
de Gennes, ``Superconductivity of Metals and Alloys“, (Benjamin New York, 1966)F. Gygi and M. Shluter, Phy. Rev. B43, 7069 (1991)P. Pieri and G. C. Strinati, Phy. Rev. Lett. 91, 030401 (2003)
Gap profiling function
Hamiltonian of fermions
Solve self-consistentlyn
kz
E
zero mode
E=0
bound states
r
IntroductionDensity of states in vortex
non-Abelian statistics
D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001)
B. Sacepe et al. Phys. Rev. Lett. 96, 097006 (2006)
Density of states in vortex
BEC-BCS crossover with vortex
K. Mizushima, M. Ichioka and K. Machida,Phys. Rev. Lett.101, 150409 (2008)
→ BCSBEC ←
zero mode
gapless
I. Guillamon et al. Phys. Rev. Lett. 101, 166407 (2008)
outside of vortex
iside of vortex
Fermi surface
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
From confinement phase to deconfinement phase
baryon and meson
QGP = Quark Gluon Plasma
quark and gluon(asymptotic free?)
QCD lagrangian
J. C. Collins and M. J. Perry, PRL34, 1353 (1975)
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Early UniverseCompact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Early Universe
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Early Universe
What‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Introduction
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
CFL (Color-Flavor Locking) phase
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R
・ pairing gap
・ symmetry breaking
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
CFL (Color-Flavor Locking) phase
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R
・ pairing gap
・ symmetry breaking
Early Universe
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658
RHIC, LHC, GSI
QCD lagrangian
From confinement phase to deconfinement phase
CFL (Color-Flavor Locking) phase
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R
・ pairing gap
・ symmetry breaking
vortex structure inside the star
・ nuclear clust → glitch (star quake)・ neutron matter → p-wave・ CFL phase → non-Aelian vortex ?
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
CFL gap
Δiα =
SU(3)c+F
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Abelian vortex ?・ M. M. Forbes and A. R. Zhitntsky, Phy. Rev. D65, 085009 (2002)・ K. Ida and G. Baym, Phys. Rev. D66, 014015 (2002)・ K. Iida, Phys. Rev. D71, 054011 (2005)
s u dCFL gap
Δiα =
SU(3)c+F
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Abelian vortex ?・ M. M. Forbes and A. R. Zhitntsky, Phy. Rev. D65, 085009 (2002)・ K. Ida and G. Baym, Phys. Rev. D66, 014015 (2002)・ K. Iida, Phys. Rev. D71, 054011 (2005)
s u dsCFL gap SU(3)c+F → SU(2)c+F x U(1)c+F
・ A. P. Balachandran, S. Digal, T. Matsuura, Phys. Rev. D73, 074009 (2006)
non-Abelian vortex !!
Δiα =
SU(3)c+F
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Abelian vortex ?・ M. M. Forbes and A. R. Zhitntsky, Phy. Rev. D65, 085009 (2002)・ K. Ida and G. Baym, Phys. Rev. D66, 014015 (2002)・ K. Iida, Phys. Rev. D71, 054011 (2005)
s u ds
u
CFL gap SU(3)c+F SU(3)c+F → SU(2)c+F x U(1)c+F
・ A. P. Balachandran, S. Digal, T. Matsuura, Phys. Rev. D73, 074009 (2006)
non-Abelian vortex !!
Δiα =
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
Abelian vortex ?・ M. M. Forbes and A. R. Zhitntsky, Phy. Rev. D65, 085009 (2002)・ K. Ida and G. Baym, Phys. Rev. D66, 014015 (2002)・ K. Iida, Phys. Rev. D71, 054011 (2005)
s u ds
u
d
CFL gap SU(3)c+F → SU(2)c+F x U(1)c+F
・ A. P. Balachandran, S. Digal, T. Matsuura, Phys. Rev. D73, 074009 (2006)
non-Abelian vortex !!
Δiα =
SU(3)c+F
Introduction
repulsive force
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
NG boson
・ E. Nakano, M. Nitta, T. Matsuura, Phys. Lett. B672, 61 (2009), ibid Phys. Rev. D78, 045002 (2008)・ M. Eto and M. Nitta, arXiv:0907.1278 [hep-ph], 0908.4470 [hep-ph]
What‘s about COLOR SUPERCONDUCTIVITY?
attractive force repulsive force
vortex-vortex vortex-antivortex vortex-vortex
Introduction
repulsive force
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
NG boson
・ E. Nakano, M. Nitta, T. Matsuura, Phys. Lett. B672, 61 (2009), ibid Phys. Rev. D78, 045002 (2008)・ M. Eto and M. Nitta, arXiv:0907.1278 [hep-ph], 0908.4470 [hep-ph]
What‘s about COLOR SUPERCONDUCTIVITY?
attractive force
vortex-vortex vortex-antivortex vortex-vortex
→ But Ginzburg-Landau theory is effective only at large length scale.
repulsive force
ξ
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
We will study the vortex for any length scale.
non-Abelian vortex
ξ
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
We will study the vortex for any length scale.
What‘s fermion modes?
non-Abelian vortex
ξ
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
We will study the vortex for any length scale.
Bogoliubov-de Gennes (BdG) equation !!
non-Abelian vortex
What‘s fermion modes?
Single FlavorSingle flavor fermion with Abelian vortex
nkz
E
For vacuum (μ=0), see R. Jackiw and P. Rossi,Nucl. Phys. B190, 681 (1981).
Bogoliubov-de Gennes (BdG) equation
Single FlavorSingle flavor fermion with Abelian vortex
nkz
E
Solution with E=0 (n=0, kz=0)
For vacuum (μ=0), see R. Jackiw and P. Rossi,Nucl. Phys. B190, 681 (1981).
Bogoliubov-de Gennes (BdG) equation
Single FlavorSingle flavor fermion with Abelian vortex
Right solutionn
kz
E
Solution with E=0 (n=0, kz=0)Fermion Zero mode (E=0)
For vacuum (μ=0), see R. Jackiw and P. Rossi,Nucl. Phys. B190, 681 (1981).
vortex configuration |Δ(r)|eiθ
as background field
Bogoliubov-de Gennes (BdG) equation
|Δ(r)| → 0 for r → 0|Δ(r)| → |Δ| for r → ∞・ Localization with e-|Δ|r
・ Oscillation with J0(μr), J1(μr)
Left solution is similar.
CFLBogoliubov-de Gennes equation with non-Abelian vortex
snon-Abelian vortex
Bogoliubov-de Gennes equation
nkz
E
CFLBogoliubov-de Gennes equation with non-Abelian vortex
nkz
E
tripletsinglet
SU(3)c+F → SU(2)c+F x U(1)c+F
From CFL basis to SU(3) basis
doublet (no zero mode)
CFLBogoliubov-de Gennes equation with non-Abelian vortex
nkz
E
tripletsinglet
SU(3)c+F → SU(2)c+F x U(1)c+F
From CFL basis to SU(3) basis
doublet (no zero mode)
CFLBogoliubov-de Gennes equation with non-Abelian vortex
nkz
E
tripletsinglet
SU(3)c+F → SU(2)c+F x U(1)c+F
From CFL basis to SU(3) basis
doublet (no zero mode)
CFLBogoliubov-de Gennes equation with non-Abelian vortex
nkz
E
tripletsinglet
SU(3)c+F → SU(2)c+F x U(1)c+F
From CFL basis to SU(3) basis
doublet (no zero mode)
CFLBogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)triplet
nkz
E
Right solution
CFLBogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)singlet
nkz
E
Right solution
CFLBogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)singlet
nkz
E
Right solution
CFLBogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)
nkz
E
multiplet most stable mode radius tripletsinglet
doublet
zero modezero mode
non-zero mode
1/|Δ|2/|Δ|
---
SU(2)c+F x U(1)c+F
CFLBogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)
CFLSU(3)c+F
VortexSU(2)c+FxU(1)c+F
non-Abelian vortex
tripletsinglet
CFLSU(3)c+F
VortexSU(2)c+FxU(1)c+F
Fermion zero modes (E=0)
non-Abelian vortex
Effective Theory in 1+1 dimension
Fermion zero modes (E=0)
What is effective theoryof fermion zero modesin 1+1 dim. along z axis?
z
Effective Theory in 1+1 dimension
zSeparate (r,θ) and (t,z).
Integrate out (r, θ).
Effective Theory in 1+1 dim.
original equation of motion
Single flavor case
Effective Theory in 1+1 dimension
z
If |Δ(r)| is a constant |Δ|, ...
Plane wave solution
Dispersion relation
Effective Theory in 1+1 dim.Single flavor case
v 0.027 for ≅ μ=1000 MeV, |Δ|=100 MeV
E
kz
light
Right
Effective Theory in 1+1 dimension
z
If |Δ(r)| is a constant |Δ|, ...
Plane wave solution
Dispersion relation
Effective Theory in 1+1 dim.Single flavor case
v 0.027 for ≅ μ=1000 MeV, |Δ|=100 MeV
n kz
E
Right
Effective Theory in 1+1 dimension
z
If |Δ(r)| is a constant |Δ|, ...
Plane wave solution
Dispersion relation
Effective Theory in 1+1 dim.Single flavor case
v 0.027 for ≅ μ=1000 MeV, |Δ|=100 MeV
n kz
E
Right
Effective Theory in 1+1 dimension
z
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
Single flavor case
Right:
Effective Theory in 1+1 dimension
z
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
Single flavor case
Left:
Effective Theory in 1+1 dimension
z
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
Single flavor case
Left:
Right
Left
Effective Theory in 1+1 dimension
z
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
Single flavor case
Left:
Right
Left
E
kz
light
Right
Left
Effective Theory in 1+1 dimension
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
CFL case
ztriplet
singlet
t : triplets : singlet
i =
Right
Effective Theory in 1+1 dimension
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
CFL case
ztriplet
singlet
t : triplets : singlet
i =
E
kz
lighttripletsinglet
Right
n kz
E
Effective Theory in 1+1 dimension
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
CFL case
ztriplet
singlet
t : triplets : singlet
i =
Right
n kz
E
triplet
singlet
Effective Theory in 1+1 dimension
equation of motion
Spinor form of fermion zero mode
Dirac operator in 1+1 dim.
solution corresponding to a(t,z)
CFL case
ztriplet
singlet
t : triplets : singlet
i =
Right
Summary• Fermion structure in non-Abelian vortex in color superconductivity.
• Bogoliubov-de Gennes (BdG) equation with non-Abelian vortex. - Single flavor: single zero mode (Cf. Y.Nishida, Phys.Rev.D81,074004(2010)) - CFL: triplet and singlet zero modes in SU(2)c+F x U(1)c+F symmetry.
• Effective theory of fermion zero mode in 1+1 dimension.
• Application to neutron (quark, hybrid) stars and experiments of heavy ion collisions will be interesting.
Introduction
non-Abelian Abrikosov lattice (?) Nitta-san‘s seminar in KEK 2009
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
NG boson
What‘s about COLOR SUPERCONDUCTIVITY?
repulsive force (?)repulsive force (?)
repulsive force (?)
Introduction
non-Abelian Abrikosov lattice (?) Nitta-san‘s seminar in KEK 2009
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
NG boson
What‘s about COLOR SUPERCONDUCTIVITY?
We need to study structure of non-Abelian vortex from micro- to macroscopic scale.
repulsive force (?)repulsive force (?)
repulsive force (?)
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
IntroductionWhat‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
CFL (Color-Flavor Locking) phase
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R
・ pairing gap
・ symmetry breaking