1. Lattice studies of topologically nontrivial non-Abelian gauge field configurations in an external magnetic field in an external magnetic field P. V. Buividovich (Regensburg University) Workshop on QCD in strong magnetic fields 12-16 November 2012, Trento, Italy 12-16 November 2012, Trento, Italy

2. Chiral Magnetic Effect: charge separation in an external magnetic field due to chirality fluctuations Chiral Magnetic Effect: charge separation in an external magnetic field due to chirality fluctuations Chirality fluctuations: reflect the fluctuations of the topology of non-Abelian gauge fields Chirality fluctuations: reflect the fluctuations of the topology of non-Abelian gauge fields In real QCD: instanton tunneling (zero temperature) or sphaleron transitions In real QCD: instanton tunneling (zero temperature) or sphaleron transitions Introduction

3. Describing CME in Euclidean space

4. Charge separation in stationary states? Observables? Electric currents (allowed by torus topology) Electric currents (allowed by torus topology) Spin parts of magnetic and electric dipole moments (but is it charge separation?) Spin parts of magnetic and electric dipole moments (but is it charge separation?) Global dipole moment prohibited by torus topology Global dipole moment prohibited by torus topology Local density of electric charge Local density of electric charge

5. Charge separation in the instanton background

7. Dirac spectrum for instanton with magnetic field Motivated by recent work [ArXiv:1112.0532 Basar, Dunne, Kharzeev] What is the Dirac eigenspectrum for instanton in magnetic field? What is the Dirac eigenspectrum for instanton in magnetic field? Are there additional zero modes with different chiralities? Are there additional zero modes with different chiralities? How does the structure of the eigenmodes change? How does the structure of the eigenmodes change?

8. Dirac spectrum for instanton with magnetic field

9. IPR and localization of Dirac eigenmodes

10. IPR of low-lying Dirac eigenmodes

11. Geometric extent of the zero mode

12. Geometric extent of low-lying modes

13. Geometric structure of low-lying modes Mode Number => Mode Number => Magnetic field => Magnetic field =>

14. IPR and localization of Dirac eigenmodes There are no additional zero modes There are no additional zero modes Zero modes are extended in the direction of the magnetic field Zero modes are extended in the direction of the magnetic field Zero modes become more localized in transverse directions Zero modes become more localized in transverse directions Overall IPR only weakly depends on the magnetic field Overall IPR only weakly depends on the magnetic field Geometric parameters of higher modes weakly depend on magnetic fields Geometric parameters of higher modes weakly depend on magnetic fields

15. Charge separation at finite temperature: caloron background [work with F. Bruckmann] Caloron: A generalization of the instanton for one compact (time) direction = Finite T Caloron: A generalization of the instanton for one compact (time) direction = Finite T [Harrington, Kraan, Lee] [Harrington, Kraan, Lee] Trivial/nontrivial holonomy: stable at high/low temperatures Trivial/nontrivial holonomy: stable at high/low temperatures Strongly localized solution Strongly localized solution Action density Zero mode density

16. Charge separation at finite temperature: caloron background [work with F. Bruckmann]

17. Electric current definition for chiral fermions Current is not exactly conserved

18. Net electric current along the magnetic field 16 3 x4 lattice 16 3 x4 lattice monopole/anti-monopole distance = 8 monopole/anti-monopole distance = 8

19. Current density profile along the caloron axis B = 0.12 || caloron axis

20. Current density profile along the caloron axis B = 0.24 || caloron axis

21. Transverse profile of the current density B = 0.24 || caloron axis

22. Net electric charge 16 3 x4 lattice 16 3 x4 lattice monopole/anti-monopole distance = 8 monopole/anti-monopole distance = 8

23. Net electric charge vs. magnetic field B || caloron axis

24. Charge density profile along the caloron axis B = 0.12 || caloron axis

25. B = 0.24 || caloron axis Charge density profile along the caloron axis

26. B = 0.12 || caloron axis Transverse charge density profile@monopole

27. B = 0.12 || caloron axis Transverse charge density profile@midpoint

28. B = 0.12 || caloron axis Transverse charge density profile@anti-monopole

29. Some physical estimates Averaging over 4D orientations

30. Some physical estimates Fireball volume Collision duration

31. Conclusions Instanton Charge separation not allowed by symmetries Charge separation not allowed by symmetries Magnetic-field-induced fluctuations of electric dipole moment Magnetic-field-induced fluctuations of electric dipole moment No additional zero modes in the magnetic field No additional zero modes in the magnetic fieldCaloron Charge generation if B || caloron axis Charge generation if B || caloron axis Charge and current distributions are strongly localized Charge and current distributions are strongly localized Reasonable estimates for charge fluctuations Reasonable estimates for charge fluctuations