1. Electron-hole duality and vortex formation in quantum dots Matti Manninen Jyväskylä Matti Koskinen Jyväskylä Stephanie Reimann Lund Yongle Yu Lund Maria Tureblad Lund Susanne Viefers Oslo Ben Mottelson Nordita

2. DOUBLE BARRIER SOURCE DRAIN QUANTUM DOT Island with ELECTRONS Tarucha et al., PRL, 1996 ”ARTIFICIAL ATOM” InGaAs (5% In) well AlGaAs (22%) barriers Tarucha et al., PRL, 1996, NTT Research Labs Trapped fermions in quasi – 2D

3. Electron Number N Magnetic Field B Shel l Stru ctur e 2 6 12 Conductance Gate voltage

4. Reimann, Koskinen, Manninen, Mottelson, Phys. Rev. Lett. (1999)

5. PHASE DIAGRAM S.M. Reimann, M. Manninen, M. Koskinen and B. Mottelson, PRL 83, 3270 (1999)

6. Tarucha et al., PRL, 1996 Oosterkamp et al., PRL, 2001 Coulomb blockade spectra in a magnetic field T.H. Oosterkamp et al, PRL 82, 2931 (1999)

7. H. Saarikoski (2003) PhD Thesis, Helsinki Univ. Of Technology, 2003 6-electron quantum dot at high magnetic fields in CSDFT

9. We can do exact many-particle calculations For small systems. Can we understand vortex formation in Rotating systems?

10. We solve exactly the problem of 20 interacting polarized (spinless) electrons or spinless bosons in a two-dimensional harmonic potential and analyze the rotational spectrum (yrast spectrum)

12. Result for 20 electrons E(M) - f(M) M

13. Harmonic oscillator in two dimensions single particle level structure angular momentum -4 -3 -2 -1 0 1 2 3 4 energy

14. Harmonic oscillator in two dimensions single particle level structure angular momentum -4 -3 -2 -1 0 1 2 3 4 energy Ground state of six polarized electrons

15. Harmonic oscillator in two dimensions single particle level structure angular momentum -4 -3 -2 -1 0 1 2 3 4 energy Rotational state of six polarized electrons

16. -4 -3 -2 -1 0 1 2 3 4

17. Radial density

18. -4 -3 -2 -1 0 1 2 3 4 Radial density

19. -4 -3 -2 -1 0 1 2 3 4 11111100000 Total density of 20 electrons: Maximum Density Droplet (MDD) Radial density

20. -4 -3 -2 -1 0 1 2 3 4 11111100000 Total density of 20 electrons: Maximum Density Droplet (MDD) Radial density NEW NOTATION

21. Radial density: Total density of 20 electrons “Chamon-Wen edge” 11111110111111111111100000 Surface reconstruction:

22. Radial density: Total density of 20 electrons “3 vortices” 1111110001111111111111100000 Vortex formation: Generator of 3 vortices

24. 111111111111111111110000000 011111111111111111111000000

25. 111111000111111111111110000

26. Particle-hole dualism for spinless electrons

27. Particle-hole dualism and vortices 1111110001111111111111100000 3-vortex state: 1111101010111111111111100000 1111110010111111111111010000 A B C + + +... Same state for holes: 0000001110000000000000011111 A ++ 0000010101000000000000011111 B ++...

28. ~ For holes: 0000001110000000000000011111 A ++ 0000010101000000000000011111 B ++... 0000001110000000000000 A ++ 0000010101000000000000 B ++... ~ q = 7 Holes localized in a ring ~ ~

29. Radial density: Total density of 20 electrons “3 vortices” 1111110001111111111111100000 Vortex formation: Generator of 3 vortices

30. Radial density: Total density of 3 holes as “3 vortices” 1111110001111111111111100000 Vortex formation: Generator of 3 vortices 00000011100000000000000 particles holes

31. 20 electrons energy angular momentum

32. 20 electrons energy angular momentum Fit a smooth function f(m)=a+bm+cm² to the yrast line f(m)=a+bm+cm²

34. “cusps”

35. E(m)-f(m) Difference spectrum N = 20

36. Compare to the spectrum of holes

37. Yrast spectrum of 20 spinless electrons energy Angular momentum

38. Take part of the spectrum

39. E = E(m) + 0.09 m

40. Change to spectrum of holes: m(holes) = 253 - m Spectrum for 20 electrons

41. Spectrum for 20 particles (3 holes) Spectrum for 3 particles Spectrum for 20 electrons 3 holes

42. Spectrum for 20 particles (3 holes) Spectrum for 3 particles Center of mass excitations Spectrum for 20 electrons 3 holes

43. Spectrum for 20 particles (3 holes) Spectrum for 3 particles Center of mass excitations period of 3 oscillations

44. Pair-correlation for 3 particles

45. N = 3

47. Compare pair-correlation of 3 particles to that of 3 holes in 20 electron system

48. N = 3 N = 20 hole-hole correlation m=16m=17m=18 m=231m=230 m=229

49. Overlaps ?

50. Spectrum for 20 electrons 3 holes Overlap between the particle and hole states

51. Pair correlation of three holes corresponding to the three vortices particles N=20, L=232 (filling factor=0.82) holes N=3, L=18 (filling factor=1/7, q=7) Radial density:

52. Hole-hole correlation for N = 30, M = 555 (Coulomb interaction) Ground state First excited state Six vortices have two stable coonfigurations

53. How about bosons ?

57. Is there a single-particle explanation ?

59. 229 11111110001111111111111000000000 M 3-vortex region in the mean field approach: 228 229 energy

60. 229 11111110001111111111111000000000 230 11111101001111111111111000000000 M 3-vortex region in the mean field approach: 228 229 energy 230

61. 229 11111110001111111111111000000000 230 11111101001111111111111000000000 231 11111100101111111111111000000000 M 3-vortex region in the mean field approach: 228 229 energy 230231

62. 229 11111110001111111111111000000000 230 11111101001111111111111000000000 231 11111100101111111111111000000000 232 11111100011111111111111000000000 M 3-vortex region in the mean field approach: 228 229 energy 230231232 Cusps appear at 'compact' states

63. N = 20 N = 4 (holes) Difference spectrum fitted with 1 parameter 1v 2v 3v 4v

64. The oscillations in the spectrum can be explained with the mean field model but the pair-correlation not

65. Exact result Mean field result Pair correlation for 3 holes

66. Conclusion: Universal vortex formation Particle-hole duality Pair-correlation functions (revealed by energy spectrum)

67. Conclusions: Yrast spectrum shows vortix-rings as oscillations Spectrum is dominated by holes in Fermi see Bosons have similar vortex formation (what are holes for bosons?)