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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. 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

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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

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  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 collaborationwith 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

  2. Contents Introduction Bogoliubov-deGennesequation Single Flavorcase CFL case EffectiveTheory in 1+1 dimension Summary

  3. Introduction ・Abrikosovlattice ・4He (3He) superfluidity ・BEC-BCS ・quantumturbulance ・nuclearsuperfluidity ・colorsuperconductivity ・cosmicstrings VortexΔ(r,θ)=|Δ(r)|einθ winding number n θ=0 → θ=2π TopologicallyStable symmetrybreaking G→H π1(G/H)≅π0(H)≠0 ξ Ginzburg-Landautheory iseffectivefor r >> ξ.

  4. ・Abrikosovlattice ・4He (3He) superfluidity ・BEC-BCS ・quantumturbulance ・nuclearsuperfluidity ・colorsuperconductivity ・cosmicstrings VortexΔ(r,θ)=|Δ(r)|einθ winding number n θ=0 → θ=2π Fermions TopologicallyStable symmetrybreaking G→H π1(G/H)≅π0(H)≠0 ξ Ginzburg-Landautheory iseffectivefor r >> ξ.

  5. ・Abrikosovlattice ・4He (3He) superfluidity ・BEC-BCS ・quantumturbulance ・nuclearsuperfluidity ・colorsuperconductivity ・cosmicstrings Vortex θ=0 → θ=2π Fermions TopologicallyStable symmetrybreaking G→H π1(G/H)≅π0(H)≠0 ξ Ginzburg-Landautheory iseffectivefor r >> ξ.

  6. Fermions ξ Ginzburg-Landautheory iseffectivefor r >> ξ.

  7. Fermions ξ

  8. Fermions ξ

  9. Fermions ξ Fermionsappear at short distance.

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

  11. Introduction E Bogoliubov-deGennes (BdG) equation Gap profilingfunction kz particle hole n Solveself-consistently Hamiltonian of fermions Gap profilingfunctionΔ(r)isobtainedfromfermiondynamics. 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)

  12. Introduction E Bogoliubov-deGennes (BdG) equation Gap profilingfunction kz n Solveself-consistently vortexΔ(r,θ)=|Δ(r)|eiθ Hamiltonian of fermions boundstates Gap profilingfunctionΔ(r)isobtainedfromfermiondynamics. 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)

  13. Introduction E Bogoliubov-deGennes (BdG) equation Gap profilingfunction kz n Solveself-consistently vortexΔ(r,θ)=|Δ(r)|eiθ Hamiltonian of fermions boundstates zero mode Gap profilingfunctionΔ(r)isobtainedfromfermiondynamics. 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)

  14. Introduction E Bogoliubov-deGennes (BdG) equation Gap profilingfunction kz n Solveself-consistently vortexΔ(r,θ)=|Δ(r)|eiθ Hamiltonian of fermions bounsdstate dominance boundstates zero mode Gap profilingfunctionΔ(r)isobtainedfromfermiondynamics. 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)

  15. Introduction E Bogoliubov-deGennes (BdG) equation Gap profilingfunction kz r n Solveself-consistently Hamiltonian of fermions boundstates zero mode Gap profilingfunctionΔ(r)isobtainedfromfermiondynamics. 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)

  16. Introduction Density of states in vortex Density of states in vortex iside of vortex outside of vortex Fermisurface B. Sacepe et al. Phys. Rev. Lett. 96, 097006 (2006) I. Guillamon et al. Phys. Rev. Lett. 101, 166407 (2008) non-Abelianstatistics BEC-BCS crossoverwithvortex gapless zero mode BEC ← → BCS D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001) K. Mizushima, M. Ichioka and K. Machida, Phys. Rev. Lett.101, 150409 (2008)

  17. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? QCD lagrangian From confinement phase to deconfinement phase J. C. Collins and M. J. Perry, PRL34, 1353 (1975) quark and gluon (asymptotic free?) QGP = Quark Gluon Plasma baryon and meson

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

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

  20. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? QCD lagrangian From confinement phase to deconfinement phase Early Universe Heavy Ion Collisions Compact Stars RHIC, LHC, GSI RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658

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

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

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

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

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

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

  27. Introduction What‘saboutCOLOR 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 Early Universe Heavy Ion Collisions Compact Stars RHIC, LHC, GSI RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658

  28. Introduction What‘saboutCOLOR 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 Early Universe Heavy Ion Collisions Compact Stars RHIC, LHC, GSI RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658

  29. Introduction vortex structure inside the star ・ nuclear clust → glitch (star quake) What‘saboutCOLOR SUPERCONDUCTIVITY? ・ neutron matter → p-wave ・ CFL phase → non-Aelian vortex ? 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 Early Universe Heavy Ion Collisions Compact Stars RHIC, LHC, GSI RX J1856,5-3754 4U 1728-34 SAXJ1808.4-3658

  30. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? CFL gap SU(3)c+F Δiα =

  31. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? CFL gap SU(3)c+F d u s 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)

  32. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? SU(3)c+F → SU(2)c+F x U(1)c+F CFL gap SU(3)c+F d u s s 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-Abelianvortex !! ・A. P. Balachandran, S. Digal, T. Matsuura, Phys. Rev. D73, 074009 (2006)

  33. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? SU(3)c+F → SU(2)c+F x U(1)c+F CFL gap SU(3)c+F d u u s s 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-Abelianvortex !! ・A. P. Balachandran, S. Digal, T. Matsuura, Phys. Rev. D73, 074009 (2006)

  34. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? SU(3)c+F → SU(2)c+F x U(1)c+F CFL gap SU(3)c+F d d u u s s 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-Abelianvortex !! ・A. P. Balachandran, S. Digal, T. Matsuura, Phys. Rev. D73, 074009 (2006)

  35. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? NG boson CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F vortex-antivortex vortex-vortex vortex-vortex repulsive force attractive force 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]

  36. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? NG boson CP2 = SU(3)c+F / SU(2)c+F x U(1)c+F vortex-antivortex vortex-vortex vortex-vortex repulsive force attractive force repulsive force → ButGinzburg-Landautheoryiseffectiveonly at large lengthscale. ・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]

  37. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? non-Abelianvortex We will studythevortex foranylengthscale. ξ

  38. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? non-Abelianvortex We will studythevortex foranylengthscale. What‘sfermionmodes? ξ

  39. Introduction What‘saboutCOLOR SUPERCONDUCTIVITY? non-Abelianvortex We will studythevortex foranylengthscale. What‘sfermionmodes? Bogoliubov-deGennes (BdG) equation !! ξ

  40. Single Flavor E Single flavorfermionwith Abelian vortex Bogoliubov-deGennes (BdG) equation kz n For vacuum (μ=0), see R. Jackiw and P. Rossi, Nucl. Phys. B190, 681 (1981).

  41. Single Flavor E Single flavorfermionwith Abelian vortex Bogoliubov-deGennes (BdG) equation kz n Solution with E=0 (n=0, kz=0) For vacuum (μ=0), see R. Jackiw and P. Rossi, Nucl. Phys. B190, 681 (1981).

  42. Single Flavor E Single flavorfermionwith Abelian vortex Bogoliubov-deGennes (BdG) equation kz n Solution with E=0 (n=0, kz=0) Fermion Zero mode (E=0) Right solution vortexconfiguration |Δ(r)|eiθ as backgroundfield |Δ(r)| → 0 for r → 0 |Δ(r)| → |Δ| for r → ∞ ・ Localization with e-|Δ|r ・ Oscillation with J0(μr), J1(μr) For vacuum (μ=0), see R. Jackiw and P. Rossi, Nucl. Phys. B190, 681 (1981). Leftsolutionissimilar.

  43. CFL Bogoliubov-deGennesequationwithnon-Abelianvortex non-Abelianvortex s Bogoliubov-deGennesequation E kz n

  44. CFL E Bogoliubov-deGennesequationwithnon-Abelianvortex From CFL basis to SU(3) basis kz n SU(3)c+F → SU(2)c+F x U(1)c+F triplet singlet doublet (no zero mode)

  45. CFL E Bogoliubov-deGennesequationwithnon-Abelianvortex From CFL basis to SU(3) basis kz n SU(3)c+F → SU(2)c+F x U(1)c+F triplet singlet doublet (no zero mode)

  46. CFL E Bogoliubov-deGennesequationwithnon-Abelianvortex From CFL basis to SU(3) basis kz n SU(3)c+F → SU(2)c+F x U(1)c+F triplet singlet doublet (no zero mode)

  47. CFL E Bogoliubov-deGennesequationwithnon-Abelianvortex From CFL basis to SU(3) basis kz n SU(3)c+F → SU(2)c+F x U(1)c+F triplet singlet doublet (no zero mode)

  48. CFL E Bogoliubov-deGennesequationwithnon-Abelianvortex Fermionzeromodes (E=0) triplet kz n Right solution

  49. CFL E Bogoliubov-deGennesequationwithnon-Abelianvortex Fermionzeromodes (E=0) singlet kz n Right solution

  50. CFL E Bogoliubov-deGennesequationwithnon-Abelianvortex Fermionzeromodes (E=0) singlet kz n Right solution

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