Interaction of non-Abelian vortices with quasiparticles in a color superconductor

1. Interaction of non-Abelian vortices with quasiparticles in a color superconductor Yuji Hirono [The Univ. of Tokyo] Collaborators: T. Kanazawa [Univ. Regensburg], M. Nitta [Keio Univ.]

2. Quantum vortices in a color superconductor • Color-flavor locked phase of QCD • Superconductor &Superfluid color magnetic flux & quantized circulation [Balachandran, Digal and Matsuura (2006)] • The existence of a vortexbreaks the isotropy group H Nambu-Goldstone mode on the vortex “Orientational zero mode” [Eto, Nakano and Nitta (2009)] Arrows denote the value of the orientational mode.

3. Main purpose of this study • We study CFL vortex – quasiparticle interaction • How (or whether) the orientational zero modes couple with quasiparticle? • Method: dual transformation

4. Outline • Introduction • What are quantum vortices? • Quantum vortices in CFL phase • Dual transformation • Vortex-quasiparticle interaction in the CFL phase • Summary

5. Introduction

6. What are quantum vortices?

7. What are quantum vortices ? • Topological defects (string-like in 3+1D) • Field configurations which cannot be continuously deformed into trivial one • Symmetry breaking Topologically stable vortex exists complex scalar Ex) Goldstone model

8. Abelian vortex - “local” and “global” • Global vortex • breaking of global symmetry • Local vortex • breaking of local symmetry Abelian Higgs model Goldstone model Local Global Superfluids Superconductors

9. Abelian vortex - Quantization • Superconductor • Quantized magnetic flux • Superfluid • Quantized circulation

10. Abelian vortex - Force btw vortices • Superfluid • Superconductor Phonon : Repulsive Massive photon : Repulsive Scalar mode : Attractive Scalar mode : Attractive Type II SC Type I SC

11. Abelian vortex – Collective structure • Superconductor • vortex lattice • vortex liquid • vortex glass • Superfluid • vortex lattice • quantum turbulence [http://cua.mit.edu/ketterle_group/Nice_pics.htm] [Hess, et. al, Phys. Rev. Lett. 62 214.] [M. Kobayashi and M. Tsubota, J. Phys. Soc. Jpn. 74, 3248 (2005).] Possible collective structure can be investigated by making use of vortex-vortex interaction

12. Quantum vortices in the color-flavor locked phase

13. Color superconductivity • Order Parameter color index flavor index : charge conjugation matrix Color rep., Flavor rep., diquark condensate • Color-Flavor Locked Phase • Expected to be realized at high baryon density and low T u d s [Alford, Rajagopal and Wilczek (1998)]

14. Symmetry of the Ground States of CFL • Order parameter space Topologically stable vortex exists !

15. Vortex solution [Balachandran, Digal and Matsuura (2006)] , At spatial infinity, This winding is generated by • Global & Local vortex • quantized circulation • color magnetic flux [Eto, Nakano and Nitta (2009)]

16. Orientational zero mode • Existence of a vortex further breaks the symmetry • Low effective theory on the vortex model Nambu-Goldstone mode : complex 3 component [Eto, Nakano and Nitta (2009)]

17. Dual Transformation

18. Dual Transformation • Change of variables • Particles and topological defects interchange their roles Ex) Electric-Magnetic duality magnetic monopole electron Ex) XY spin model in 2+1 dim. Spin wave: Real scalar Vector particle [José, Kadanoff, Kirkpatrick and Nelson (1977)]

19. Dual Transformation (cont’d.) • Advantages of the dual transformation Topological defects are described as particles Ex) Radiation of phonons from quantum vortices in superfluids Radiation of axions from cosmic strings

20. Vortex - quasiparticle interaction in the CFL phase [Y. Hirono, T. Kanazawa and M. Nitta, PRD 83, 085018 (2011) [arxiv:1012.6042] ]

21. Start: Low energy effective Lagrangian for a color superconductor • Massless 3 flavors • Order Parameter • Ginzburg-Landau Effective Lagrangian where • CFL phase

22. Quasiparticles in the CFL phase gluon phonon CFL pion CFL pion scalar modes scalar modes • Consider the interaction of vortices with gluons & phonons

23. Dual transformation of gluons • 1. Consider the partition function massive vector field massive two-form field Gauge fixing condition – imposed on • 2. Introduce by Hubbard-Stratonovich transformation [Seo, Okawa and Sugamoto (1979)] • 3. Integrate out the original gauge fields

24. Dual transformation of gluons (cont’d.) • Dual Lagrangian for gluons mass term Interaction , Non-Abelian vorticity tensor where ,

25. Dual transformation of phonons phonons massless two-form field Interaction btw. phonons & vortices , , Abelian Vorticity tensor for a solution of the form • For a vortex of general shape, : vortex coordinate

26. Non-Abelian vorticity tensor • Express vorticity in terms of orientational zero modes Correction due to the modulation in the space • Localized around the vortex • Gluons interact with orientational zero modes

27. Orientation dependence of interaction of two vortices via gluon exchanges • Interaction energy of two vortices Orientation dependence : Vortex coordinate Attractive Repulsive Orientation of the vortex 1

28. Summary • Interaction of CFL vortices with quasi-particles is investigated in terms of a dual transformation • phonons are described by massless two-form fields, while Gluons are described by massive two-form fields in the dual description. Their interaction with vortex is derived. • We have investigated the orientation dependence of interaction of two vortices via gluon exchanges phonon gluon

29. Outlook • Possible applications • Effect of massive scalar modes • Radiation of phonons / gluons • Reconnection • Low energy effective theory of a vortex • Collective structure of vortices (lattice, turbulence, …) • Interaction with fermionic zero modes • Play some role in glitch phenomena in neutron stars? • Other interesting topics • Fermionic zero modes trapped in the vortices • Novel non-Abelian statistics of fermionic zero modes • Monopoles in the vortices as a kink of (1+1)D effective theory • …

30. Back up slides

31. Correspondence under the dual transformation # of d.o.f 2+1 dim massless scalar massless vector 1 massless 2-form 3+1 dim massless scalar 1 2 massless vector massless vector massive vector massive 2-form 3

32. Orientational zero mode Moduli space translational zero mode orientational zero mode

33. Phase diagram of QCD T Quark-gluon plasma Early universe Heavy ion collisions Color superconductivity Hadron Compact stars 〜 1 GeV baryon chemical potential

34. Other Phases of Color Superconductivity • 2SC

35. Example :global vortex in 3+1D [K. Lee, Phys. Rev. D 48, 2493 (1993)] [H. Kleinert, Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation (World Scientific, 2008).] [M. Hatsuda, S. Yahikozawa, P. Ao, and D. J. Thouless, Phys. Rev. B 49, 15870 (1994).] Vorticity tensor : Vortex coordinate

36. Attractive when the color flux is reduced • Repulsiveif their orientations are close