1. 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)

2. 1.Introduction 2.Bogoliubov-de Gennes equation A.Single Flavor case B.CFL case 3.Effective Theory in 1+1 dimension 4.Summary Contents

3. Introduction Vortex Δ(r,θ)=|Δ(r)|e inθ ・ Abrikosov lattice ・ 4 He ( 3 He) superfluidity ・ BEC-BCS ・ quantum turbulance ・ nuclear superfluidity ・ color superconductivity ・ cosmic strings symmetry breaking G→H π 1 (G/H) ≅ π 0 (H)≠0 winding number n Topologically Stable θ=0 → θ=2π Ginzburg-Landau theory is effective for r >> ξ. ξ

4. Topologically Stable Vortex Δ(r,θ)=|Δ(r)|e inθ ξ ・ Abrikosov lattice ・ 4 He ( 3 He) superfluidity ・ BEC-BCS ・ quantum turbulance ・ nuclear superfluidity ・ color superconductivity ・ cosmic strings θ=0 → θ=2π symmetry breaking G→H π 1 (G/H) ≅ π 0 (H)≠0 Ginzburg-Landau theory is effective for r >> ξ. winding number n Fermions

5. Topologically Stable ξ Vortex ・ Abrikosov lattice ・ 4 He ( 3 He) superfluidity ・ BEC-BCS ・ quantum turbulance ・ nuclear superfluidity ・ color superconductivity ・ cosmic strings θ=0 → θ=2π Ginzburg-Landau theory is effective for r >> ξ. symmetry breaking G→H π 1 (G/H) ≅ π 0 (H)≠0 Fermions

6. Ginzburg-Landau theory is effective for r >> ξ. ξ Fermions

7. ξ

8. ξ

9. Fermions appear at short distance. ξ Fermions

10. Fermions appear at short distance. ξ Fermions Fermions in Topological Objects ・ Soliton (kink, Skyrmion) ・ Quantum Hall Effect ・ Bulk-Edge correspondence ・ Domain Wall Fermion

11. Introduction Bogoliubov-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. n kzkz E particle hole

12. Gap profiling function Δ(r) is obtained from fermion dynamics. Introduction Bogoliubov-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 n kzkz E vortex Δ(r,θ)=|Δ(r)|e iθ bound states

13. Gap profiling function Δ(r) is obtained from fermion dynamics. Introduction Bogoliubov-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 n kzkz E zero mode E=0 vortex Δ(r,θ)=|Δ(r)|e iθ bound states

14. Gap profiling function Δ(r) is obtained from fermion dynamics. Introduction Bogoliubov-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 n kzkz E zero mode E=0 vortex Δ(r,θ)=|Δ(r)|e iθ bound states bounsd state dominance

15. Gap profiling function Δ(r) is obtained from fermion dynamics. Introduction Bogoliubov-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 n kzkz E zero mode E=0 bound states r

16. Introduction Density 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

17. Introduction What‘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)

18. Introduction What‘s about COLOR SUPERCONDUCTIVITY? Early Universe Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase

19. Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase Early Universe

20. Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase

21. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase

22. Early Universe What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase Introduction

23. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase

24. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase

25. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase

26. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 RHIC, LHC, GSI QCD lagrangian From confinement phase to deconfinement phase

27. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 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

28. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 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

29. Early Universe Introduction What‘s about COLOR SUPERCONDUCTIVITY? Compact StarsHeavy Ion Collisions RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658 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 ?

30. Introduction What‘s about COLOR SUPERCONDUCTIVITY? CFL gap Δ iα = SU(3) c+F

31. Introduction What‘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 d CFL gap Δ iα = SU(3) c+F

32. Introduction What‘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 d s 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 !! Δ iα = SU(3) c+F

33. Introduction What‘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 d s u CFL gapSU(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 !! Δ iα =

34. Introduction What‘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 d s 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 !! Δ iα = SU(3) c+F

35. Introduction repulsive force CP 2 = 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 forcerepulsive force vortex-vortex vortex-antivortex vortex-vortex

36. Introduction repulsive force CP 2 = 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

37. ξ Introduction What‘s about COLOR SUPERCONDUCTIVITY? We will study the vortex for any length scale. non-Abelian vortex

38. ξ Introduction What‘s about COLOR SUPERCONDUCTIVITY? We will study the vortex for any length scale. What‘s fermion modes? non-Abelian vortex

39. ξ Introduction What‘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?

40. Single Flavor Single flavor fermion with Abelian vortex n kzkz E For vacuum (μ=0), see R. Jackiw and P. Rossi, Nucl. Phys. B190, 681 (1981). Bogoliubov-de Gennes (BdG) equation

41. Single Flavor Single flavor fermion with Abelian vortex n kzkz E Solution with E=0 (n=0, k z =0) For vacuum (μ=0), see R. Jackiw and P. Rossi, Nucl. Phys. B190, 681 (1981). Bogoliubov-de Gennes (BdG) equation

42. Single Flavor Single flavor fermion with Abelian vortex Right solution n kzkz E Solution with E=0 (n=0, k z =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)|e iθ 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 J 0 (μr), J 1 (μr) Left solution is similar.

43. CFL Bogoliubov-de Gennes equation with non-Abelian vortex s non-Abelian vortex Bogoliubov-de Gennes equation n kzkz E

44. CFL Bogoliubov-de Gennes equation with non-Abelian vortex n kzkz E triplet singlet SU(3) c+F → SU(2) c+F x U(1) c+F From CFL basis to SU(3) basis doublet (no zero mode)

45. CFL Bogoliubov-de Gennes equation with non-Abelian vortex n kzkz E triplet singlet SU(3) c+F → SU(2) c+F x U(1) c+F From CFL basis to SU(3) basis doublet (no zero mode)

46. CFL Bogoliubov-de Gennes equation with non-Abelian vortex n kzkz E triplet singlet SU(3) c+F → SU(2) c+F x U(1) c+F From CFL basis to SU(3) basis doublet (no zero mode)

47. CFL Bogoliubov-de Gennes equation with non-Abelian vortex n kzkz E triplet singlet SU(3) c+F → SU(2) c+F x U(1) c+F From CFL basis to SU(3) basis doublet (no zero mode)

48. CFL Bogoliubov-de Gennes equation with non-Abelian vortex Fermion zero modes (E=0) triplet n kzkz E Right solution

49. CFL Bogoliubov-de Gennes equation with non-Abelian vortex Fermion zero modes (E=0) singlet n kzkz E Right solution

50. CFL Bogoliubov-de Gennes equation with non-Abelian vortex Fermion zero modes (E=0) singlet n kzkz E Right solution

51. CFL Bogoliubov-de Gennes equation with non-Abelian vortex Fermion zero modes (E=0) n kzkz E multiplet most stable mode radius triplet singlet doublet zero mode non-zero mode 1/|Δ| 2/|Δ| --- SU(2) c+F x U(1) c+F

52. CFL Bogoliubov-de Gennes equation with non-Abelian vortex Fermion zero modes (E=0) CFL SU(3) c+F Vortex SU(2) c+F xU(1) c+F non-Abelian vortex

53. triplet singlet CFL SU(3) c+F Vortex SU(2) c+F xU(1) c+F Fermion zero modes (E=0) non-Abelian vortex

54. Effective Theory in 1+1 dimension Fermion zero modes (E=0) What is effective theory of fermion zero modes in 1+1 dim. along z axis? z

55. Effective Theory in 1+1 dimension z Separate (r,θ) and (t,z). Integrate out (r, θ). Effective Theory in 1+1 dim. original equation of motion Single flavor case

56. 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 kzkz light Right

57. 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 kzkz E Right

58. 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 kzkz E Right

59. 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:

60. 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:

61. 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

62. 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 kzkz light Right Left

63. 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 z triplet singlet t : triplet s : singlet i = Right

64. 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 z triplet singlet t : triplet s : singlet i = E kzkz light triplet singlet Right

65. n kzkz 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 z triplet singlet t : triplet s : singlet i = Right

66. n kzkz 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 z triplet singlet t : triplet s : singlet i = Right

67. 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.

68. Introduction non-Abelian Abrikosov lattice (?) Nitta-san‘s seminar in KEK 2009 CP 2 = SU(3) c+F / SU(2) c+F x U(1) c+F NG boson What‘s about COLOR SUPERCONDUCTIVITY? repulsive force (?)

69. Introduction non-Abelian Abrikosov lattice (?) Nitta-san‘s seminar in KEK 2009 CP 2 = 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 (?)

70. Introduction What‘s about COLOR SUPERCONDUCTIVITY? QCD lagrangian

71. Introduction What‘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