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

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



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



Fermions


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






Solve self-consistentlyn

kz

E

vortex Δ(r,θ)=|Δ(r)|eiθ

bound states







kz

E

zero mode

E=0


bound states







kz

E

zero mode

E=0


bound statesbounsd state

dominance







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)


Early UniverseCompact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658

RHIC, LHC, GSI

QCD lagrangian



Compact StarsHeavy Ion CollisionsRX J1856,5-37544U 1728-34SAXJ1808.4-3658

RHIC, LHC, GSI

QCD lagrangian


Early Universe

Early Universe



RHIC, LHC, GSI

QCD lagrangian


Early Universe



RHIC, LHC, GSI

QCD lagrangian


Early Universe

What‘s about COLOR SUPERCONDUCTIVITY?


RHIC, LHC, GSI

QCD lagrangian


Introduction

Early Universe



RHIC, LHC, GSI

QCD lagrangian


Early Universe



RHIC, LHC, GSI

QCD lagrangian


Early Universe



RHIC, LHC, GSI

QCD lagrangian


Early Universe



RHIC, LHC, GSI

QCD lagrangian


Early Universe



RHIC, LHC, GSI

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

Early Universe



RHIC, LHC, GSI

QCD lagrangian




・ pairing gap


Early Universe



RHIC, LHC, GSI

QCD lagrangian




・ pairing gap


vortex structure inside the star

・ nuclear clust → glitch (star quake)・ neutron matter → p-wave・ CFL phase → non-Aelian vortex ?


CFL gap

Δiα =

SU(3)c+F


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



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



s u ds

u

CFL gap SU(3)c+F SU(3)c+F → SU(2)c+F x U(1)c+F



Δiα =



s u ds

u

d

CFL gap SU(3)c+F → SU(2)c+F x U(1)c+F



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]


attractive force repulsive force

vortex-vortex vortex-antivortex vortex-vortex

Introduction

repulsive force


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]


attractive force

vortex-vortex vortex-antivortex vortex-vortex

→ But Ginzburg-Landau theory is effective only at large length scale.

repulsive force

ξ


We will study the vortex for any length scale.

non-Abelian vortex

ξ



What‘s fermion modes?

non-Abelian vortex

ξ



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


nkz

E

Solution with E=0 (n=0, kz=0)




Right solutionn

kz

E

Solution with E=0 (n=0, kz=0)Fermion Zero mode (E=0)


vortex configuration |Δ(r)|eiθ

as background field


|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


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)


nkz

E

tripletsinglet





nkz

E

tripletsinglet





nkz

E

tripletsinglet





Fermion zero modes (E=0)triplet

nkz

E

Right solution


Fermion zero modes (E=0)singlet

nkz

E

Right solution


Fermion zero modes (E=0)singlet

nkz

E

Right solution


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



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


non-Abelian vortex

Effective Theory in 1+1 dimension


What is effective theoryof fermion zero modesin 1+1 dim. along z axis?

z


zSeparate (r,θ) and (t,z).

Integrate out (r, θ).

Effective Theory in 1+1 dim.

original equation of motion

Single flavor case


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


z


Plane wave solution

Dispersion relation


v 0.027 for ≅ μ=1000 MeV, |Δ|=100 MeV

n kz

E

Right


z


Plane wave solution

Dispersion relation


v 0.027 for ≅ μ=1000 MeV, |Δ|=100 MeV

n kz

E

Right


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:


z

equation of motion




Single flavor case

Left:


z

equation of motion




Single flavor case

Left:

Right

Left


z

equation of motion




Single flavor case

Left:

Right

Left

E

kz

light

Right

Left


equation of motion




CFL case

ztriplet

singlet

t : triplets : singlet

i =

Right


equation of motion




CFL case

ztriplet

singlet


i =

E

kz

lighttripletsinglet

Right

n kz

E


equation of motion




CFL case

ztriplet

singlet


i =

Right

n kz

E

triplet

singlet


equation of motion




CFL case

ztriplet

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


NG boson


repulsive force (?)repulsive force (?)

repulsive force (?)

Introduction

non-Abelian Abrikosov lattice (?) Nitta-san‘s seminar in KEK 2009


NG boson


We need to study structure of non-Abelian vortex from micro- to macroscopic scale.

repulsive force (?)repulsive force (?)

repulsive force (?)


QCD lagrangian


QCD lagrangian



・ pairing gap


Documents

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

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