Download PowerPoint ********* - Fermion structure in non-Abelian

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
1. Introduction
2. Bogoliubov-de Gennes equation
A. Single Flavor case
B. CFL case
3. Effective Theory in 1+1 dimension
4. Summary
Introduction
・ Abrikosov lattice
・ 4He (3He) superfluidity
・ BEC-BCS
・ quantum turbulance
・ nuclear superfluidity
・ color superconductivity
・ cosmic strings
Vortex Δ(r,θ)=|Δ(r)|einθ
winding number n
θ=0 → θ=2π
Topologically Stable
symmetry breaking G→H
π1(G/H)≅π0(H)≠0
ξ
Ginzburg-Landau theory
is effective for r >> ξ.
lattice
superfluidity
Vortex Δ(r,θ)=|
winding
turbulance
uperfluidity
erconductivity
rings
θ=0 → θ=2π
Topologically
ξ
symmetry brea
π1(G/H)≅π
Ginzburg-Land
is effective for
dity
Vortex
ce
ity
tivity
θ=0 →
Topo
ξ
symm
π1
Ginzbu
ξ
ξ
Fermions appear at short distance.
Fermions in Topological Objects
・ Soliton (kink, Skyrmion)
・ Quantum Hall Effect
・ Bulk-Edge correspondence
・ Domain Wall Fermion
Fermions appear at short distance.
Introduction
E
Bogoliubov-de Gennes (BdG) equation
Gap profiling function
particle hole
n
kz
Solve self-consistently
Hamiltonian of fermions
Gap profiling function Δ(r) is obtained from fermion dynamics.
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)
Introduction
E
Bogoliubov-de Gennes (BdG) equation
Gap profiling function
n
kz
Solve self-consistently
vortex Δ(r,θ)=|Δ(r)|eiθ
Hamiltonian of fermions
Gap profiling function Δ(r) is obtained from fermion dynamics.
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)
Introduction
E
Bogoliubov-de Gennes (BdG) equation
Gap profiling function
n
kz
Solve self-consistently
vortex Δ(r,θ)=|Δ(r)|eiθ
Hamiltonian of fermions
zero mode
Gap profiling function Δ(r) is obtained from fermion dynamics.
E=0
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)
Introduction
E
Bogoliubov-de Gennes (BdG) equation
Gap profiling function
n
kz
Solve self-consistently
vortex Δ(r,θ)=|Δ(r)|eiθ
Hamiltonian of fermions
bounsd state
dominance
zero mode
Gap profiling function Δ(r) is obtained from fermion dynamics.
E=0
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)
Introduction
E
Bogoliubov-de Gennes (BdG) equation
Gap profiling function
r
n
kz
Solve self-consistently
Hamiltonian of fermions
zero mode
Gap profiling function Δ(r) is obtained from fermion dynamics.
E=0
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)
Introduction
Density of states in vortex
Density of states in vortex
iside of vortex
outside of vortex
Fermi surface
B. Sacepe et al. Phys. Rev. Lett. 96, 097006 (2006)
non-Abelian statistics
I. Guillamon et al. Phys. Rev. Lett. 101, 166407 (2008)
BEC-BCS crossover with vortex
gapless
zero mode
D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001)
BEC ←
→ BCS
K. Mizushima, M. Ichioka and K. Machida,
Phys. Rev. Lett.101, 150409 (2008)
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
J. C. Collins and M. J. Perry, PRL34, 1353 (1975)
quark and gluon
(asymptotic free?)
baryon and meson
QGP
= Quark Gluon Plasma
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
From confinement phase to deconfinement phase
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
CFL (Color-Flavor Locking) phase
・ pairing gap
From confinement phase to deconfinement phase
・ symmetry breaking
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
CFL (Color-Flavor Locking) phase
・ pairing gap
From confinement phase to deconfinement phase
・ symmetry breaking
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
vortex structure inside the star
What‘s about COLOR
QCD lagrangian
・ nuclear clust → glitch (star quake)
・ neutron matter → p-wave
SUPERCONDUCTIVITY?
・ CFL phase → non-Aelian vortex ?
CFL (Color-Flavor Locking) phase
・ pairing gap
From confinement phase to deconfinement phase
・ symmetry breaking
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R
Heavy Ion Collisions
RHIC, LHC, GSI
Compact Stars
RX J1856,5-3754
4U 1728-34
SAXJ1808.4-3658
Early Universe
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
CFL gap
Δiα =
SU(3)c+F
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
CFL gap
SU(3)c+F
s
u
d
Δiα =
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)
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
CFL gap
SU(3)c+F
SU(3)c+F → SU(2)c+F x U(1)c+F s
s
u
d
Δiα =
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)
non-Abelian vortex !!
・ A. P. Balachandran, S. Digal, T. Matsuura,
Phys. Rev. D73, 074009 (2006)
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
CFL gap
SU(3)c+F
SU(3)c+F → SU(2)c+F x U(1)c+F s
s
u
d
u
Δiα =
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)
non-Abelian vortex !!
・ A. P. Balachandran, S. Digal, T. Matsuura,
Phys. Rev. D73, 074009 (2006)
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
CFL gap
SU(3)c+F
SU(3)c+F → SU(2)c+F x U(1)c+F s
s
u
d
u
Δiα =
d
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)
non-Abelian vortex !!
・ A. P. Balachandran, S. Digal, T. Matsuura,
Phys. Rev. D73, 074009 (2006)
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
NG boson
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
vortex-vortex
vortex-antivortex
repulsive force
attractive force
vortex-vortex
repulsive force
・ 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]
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
NG boson
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
vortex-vortex
vortex-antivortex
repulsive force
attractive force
vortex-vortex
repulsive force
→ But Ginzburg-Landau theory is effective only at large length scale.
・ 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]
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
non-Abelian vortex
We will study the vortex
for any length scale.
ξ
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
non-Abelian vortex
We will study the vortex
for any length scale.
What‘s fermion modes?
ξ
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
non-Abelian vortex
We will study the vortex
for any length scale.
What‘s fermion modes?
Bogoliubov-de Gennes
ξ (BdG) equation !!
Single Flavor
Single flavor fermion with Abelian vortex
E
Bogoliubov-de Gennes (BdG) equation
n
kz
For vacuum (μ=0), see R. Jackiw and P. Rossi,
Nucl. Phys. B190, 681 (1981).
Single Flavor
Single flavor fermion with Abelian vortex
E
Bogoliubov-de Gennes (BdG) equation
Solution with E=0
(n=0, kz=0)
n
kz
For vacuum (μ=0), see R. Jackiw and P. Rossi,
Nucl. Phys. B190, 681 (1981).
Single Flavor
Single flavor fermion with Abelian vortex
E
Bogoliubov-de Gennes (BdG) equation
Fermion Zero mode (E=0)
Solution with E=0
(n=0, kz=0)
n
kz
Right solution
・ Localization with e-|Δ|r
・ Oscillation with J0(μr), J1(μr)
Left solution is similar.
vortex configuration |Δ(r)|eiθ
as background field
|Δ(r)| → 0 for r → 0
|Δ(r)| → |Δ| for r → ∞
For vacuum (μ=0), see R. Jackiw and P. Rossi,
Nucl. Phys. B190, 681 (1981).
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
non-Abelian vortex
s
Bogoliubov-de Gennes equation
E
kz
n
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
E
From CFL basis to SU(3) basis
SU(3)c+F → SU(2)c+F x U(1)c+F
triplet
singlet
doublet (no zero mode)
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
E
From CFL basis to SU(3) basis
SU(3)c+F → SU(2)c+F x U(1)c+F
triplet
singlet
doublet (no zero mode)
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
E
From CFL basis to SU(3) basis
SU(3)c+F → SU(2)c+F x U(1)c+F
triplet
singlet
doublet (no zero mode)
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
E
From CFL basis to SU(3) basis
SU(3)c+F → SU(2)c+F x U(1)c+F
triplet
singlet
doublet (no zero mode)
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)
triplet
Right solution
E
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)
singlet
Right solution
E
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)
singlet
Right solution
E
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)
E
SU(2)c+F x U(1)c+F
multiplet most stable mode radius
triplet
singlet
doublet
zero mode
zero mode
non-zero mode
1/|Δ|
2/|Δ|
---
n
kz
CFL
Bogoliubov-de Gennes equation with non-Abelian vortex
Fermion zero modes (E=0)
non-Abelian vortex
CFL
SU(3)c+F
Vortex
SU(2)c+FxU(1)c+F
ro modes (E=0)
non-Abelian
triplet
singlet
CFL
SU(3)c+F
Vortex
SU(2)c+FxU(1)c+F
Effective Theory in 1+1 dimension
z
Fermion zero modes (E=0)
What is effective theory
of fermion zero modes
in 1+1 dim. along z axis?
Effective Theory in 1+1 dimension
Single flavor case
original equation of motion
Separate (r,θ) and (t,z).
Integrate out (r, θ).
Effective Theory in 1+1 dim.
z
Effective Theory in 1+1 dimension
Single flavor case
Effective Theory in 1+1 dim.
z
If |Δ(r)| is a constant |Δ|, ...
E
Plane wave solution
light
Right
kz
Dispersion relation
v ≅ 0.027 for μ=1000 MeV, |Δ|=100 MeV
Effective Theory in 1+1 dimension
Single flavor case
Effective Theory in 1+1 dim.
z
If |Δ(r)| is a constant |Δ|, ...
E
Plane wave solution
Right
Dispersion relation
n
kz
v ≅ 0.027 for μ=1000 MeV, |Δ|=100 MeV
Effective Theory in 1+1 dimension
Single flavor case
Effective Theory in 1+1 dim.
z
If |Δ(r)| is a constant |Δ|, ...
E
Plane wave solution
Right
Dispersion relation
n
kz
v ≅ 0.027 for μ=1000 MeV, |Δ|=100 MeV
Effective Theory in 1+1 dimension
Single flavor case
Dirac operator in 1+1 dim.
equation of motion
solution corresponding to a(t,z)
Right:
Spinor form of fermion zero mode
z
Effective Theory in 1+1 dimension
Single flavor case
Dirac operator in 1+1 dim.
equation of motion
solution corresponding to a(t,z)
Left:
Spinor form of fermion zero mode
z
Effective Theory in 1+1 dimension
Single flavor case
Dirac operator in 1+1 dim.
z
Right
equation of motion
solution corresponding to a(t,z)
Left
Left:
Spinor form of fermion zero mode
Effective Theory in 1+1 dimension
Single flavor case
Dirac operator in 1+1 dim.
z
E
light
Right
Right
equation of motion
solution corresponding to a(t,z)
kz
Left
Left
Left:
Spinor form of fermion zero mode
Effective Theory in 1+1 dimension
CFL case
i=
Dirac operator in 1+1 dim.
t : triplet
s : singlet
triplet
singlet
equation of motion
solution corresponding to a(t,z)
Spinor form of fermion zero mode
Right
z
Effective Theory in 1+1 dimension
CFL case
i=
Dirac operator in 1+1 dim.
t : triplet
s : singlet
triplet
singlet
E
equation of motion
light
triplet
singlet
kz
solution corresponding to a(t,z)
Spinor form of fermion zero mode
Right
z
Effective Theory in 1+1 dimension
CFL case
t : triplet
s : singlet
i=
Dirac operator in 1+1 dim.
triplet
singlet
E
equation of motion
solution corresponding to a(t,z)
n
Spinor form of fermion zero mode
kz
Right
z
Effective Theory in 1+1 dimension
CFL case
t : triplet
s : singlet
i=
Dirac operator in 1+1 dim.
triplet
singlet
E
triplet
equation of motion
solution corresponding to a(t,z)
singlet
n
Spinor form of fermion zero mode
kz
Right
z
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
What‘s about COLOR SUPERCONDUCTIVITY?
NG boson
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
repulsive force (?)
repulsive force (?)
repulsive force (?)
non-Abelian Abrikosov lattice (?)
Nitta-san‘s seminar in KEK 2009
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
NG boson
CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F
repulsive force (?)
repulsive force (?)
repulsive force (?)
We need to study structure of
non-Abelian vortex from microto macroscopic scale.
non-Abelian Abrikosov lattice (?)
Nitta-san‘s seminar in KEK 2009
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
Introduction
What‘s about COLOR SUPERCONDUCTIVITY?
QCD lagrangian
CFL (Color-Flavor Locking) phase
・ pairing gap
・ symmetry breaking
SU(3)c x SU(3)L x SU(3)R → SU(3)c+L+R