1. Qiang Gu Ferromagnetism in Bose Systems Department of Physics
Workshop on Cold Atom Physics and Quantum Information for Young Researchers
Ferromagnetism in Bose Systems
Qiang Gu
Department of Physics
University of Science and Technology Beijing (USTB)
Taiyuan, July 4, 2007

2. Outline Historical introduction to ferromagnetism
Curie point of ferromagnetic Bose gas
Thermodynamics of ferromagnetic Bose gas
Spin dynamics in ferromagnetic condensates
Brief summary

3. Historical introduction to ferromagnetism

4. Historical introduction
b.c. Earliest known reference to magnetism
Thales (台利斯)--Greek philosopher & mathematician
In China: magnetite (cishi) mentioned in Guanzi by
Guan Zhong (管仲) before 200b.c.
First known application
Shen Kuo(沈括) Meng Chhi Pi Than
lodestone used as compass by Chinese since 7th-8th)
In Europe: compass first mentioned by Neckam in 1187
P. Mohn, Magnetism in the Solid State: An Introduction (Springer-Verlag, Berlin, 2003)

5. Historical introduction
Late 19th ~ early 20th Properties of magnets
P. Curie Linear T-dependence of inverse susceptibility
P. Langevin Theoretical explanation of Curie law
& Langevin diamagnetism
P. Weiss Molecular field theory of ferromagnetism
1928 Heisenberg Model
Neel Generalied Weiss theory for antiferromagnets
Curie law
Curie-Weiss law (1907)

6. Magnetic order in insulators
Antiferromagnetic order
Ferromagnetic order
Heisenberg model
I > 0 Antiferromagnetic
I < 0 Ferromagnetic

7. Magnetic order in insulators
Weiss Molecular-field (Mean-field) theory (1907)

8. Magnetic order in insulators
The ferromagnetic phase transition

9. Itinerant ferromagnetism in Fermi gas
Ideal fermi gas: Pauli paramagnetism 1927
where is the density of state at Fermi surface
is the Bohr magneton
due to the intrinsic magnetic moment of electrons

10. Altogether, free electron gas is paramagnetic
Itinerant ferromagnetism in Fermi gas
Ideal Fermi gas: Landau diamagnetism 1930
due to the quantization of orbital motions of charged particles
Altogether, free electron gas is paramagnetic

11. Itinerant ferromagnetism in Fermi gas
Itinerant ferromagnetism (Stoner Mean-field theory, 1936)
Ferromagnetic molecular field energy
where is the exchange interaction
is the magnetization

12. Itinerant ferromagnetism in Fermi gas
Increase in Band energy
when
The Stoner criterion:

13. Itinerant ferromagnetism in Fermi gas
The transition temperature
where
Note:
is very difficult to obtained theoretically.

14. Ferromagnetism in spinor bosons
Prototypical Bose system: 4He
scalar particles
does not display magnetism at all
Alkali atoms: 23Na, 87Rb,...
Atomic bosons have (hyperfine) spin degree of
freedom
Atomic bosons now can be confined in purely
optical traps
Q. Gu, Ferromagnetic phase transition in spinor Bose gases, Chapter 6,
Progress in Ferromagnetism Research (Nova Science Publishers, New York, 2006)

15. Spinor-1 Bose condensate
Ferromagnetism in spinor bosons
Optical trapping:
Focused
laser
BEC or cold fermions
All spin states are trapped,
releasing the hyperfine spin degrees of freedom
Spinor-1 Bose condensate
Stamper-Kurn et al., Phys. Rev. Lett. 80, 2027 (1998);
Stenger et al., Nature 396, 345 (1998).

16. Ferromagnetism in spinor bosons
Ground state of spinor Bose gases
Effective interactions between F=1 atoms
C2>0 Polar state Na
C2<0 Ferromagnetic state Rb
Ho, Phys. Rev. Lett. 81, 742 (1998)
Ohmi and Machida, J. Phys. Soc. Jpn 67, 1822 (1998)

17. Ferromagnetism in spinor bosons
Mechanism for generating ferromagnetic couplings
Spin-flip scattering
Super-exchange process
Magnetic dipolar interaction
Burke and Boh., Phys. Rev. A 59, 1303 (1999)
Yang and Li, Inter. J. Mod. Phys, B 17, 1027 (2003)
Gu, Phys. Rev. A 68, (2003)

18. Ferromagnetism in spinor bosons
FM phase transition induced by FM couplings TF
BEC: intrinsic phase transition in bosons TC
Competing of Two energy scale TF & TC
TF > TC for large I
TF < TC for small I
Is that true?

19. Curie point of ferromagnetic Bose gas
Gu and Klemm, Phys. Rev. A 68, (R) (2003)
Gu, Bongs and Sengstock, Phys. Rev. A 70, (2004)

20. Phase transitions Hamiltonian
The first term describes a free Bose gas, where
The second term describes FM couplings
Mean-field approximation

21. Phase transitions Mean-field Hamiltonian
We consider a homogeneous spinor Bose gas by using the grand canonical ensembles.

22. Phase transitions
Mean-field equations

23. Phase transitions Mean-field equations
Here we suppose only spin-1 bosons can condense.

24. Phase transitions The polylogarithm function
Asymptotic behaviors (a<<1):
The value of a

25. Phase transitions
Phase diagram
Acrobat Document

26. Phase transitions
Phase diagram
Acrobat Document

27. Phase transitions
Kis-Szabo et al., Phys. Rev. A 72, (2005)

28. Phase transitions
Wolters, Gelderen, Stoof, 2006, Itinerant ferromagnetism in an ultracold Bose gas

29. Other theories
P. Soltan-Panahi , A. Pelster, and H. Kleinert, 2006, unpublished
Isoshima, Ohmi, and machida, J. Phys. Soc. Jpn. 69, 3864 (2000)

30. Other theories Spin conservation! W. zhang, S. Yi, and L. You,
Phys. Rev. A 70, (2004)
Spin conservation!

31. Experiment
Sadler et al., Nature 443, 312 (2006)

32. Thermodynamics in FM spinor Bose gas
Tao, Wang, Qin and Gu, unpublished

33. Background
P. Soltan-Panahi , A. Pelster, and H. Kleinert, 2006, unpublished

34. Basic formula
The free energy
The internal energy

35. Basic formula
The specific heat
The magnetic susceptibility

36. Free energy
The free energy

37. Free energy
t=0.5

38. Specific heat
The specific heat

39. Specific heat

40. Susceptibility
The magnetic susceptibility

41. Susceptibility

42. Spin dynamics of ferromagnetic condensates
Gu and Qiu, PRL 98, (2007)

43. Spin-1 condensates The Hamiltonian Spin dynamics : Internal JT
H. Schmaljohann et al., Appl. Phys. B 79, 1001 (2004);
Romano and de Passos, PR A 70, (2004)

44. Spin-1 condensates Single mode approximation The reduced Hamiltonian
with

45. Spin-1 condensates Contour plot of the energy
H. Schmaljohann et al., Appl. Phys. B 79, 1001 (2004)

46. Coherent spin mixing in Spin-1 Bose condensate
Spin-1 condensates
Coherent spin mixing in Spin-1 Bose condensate
M.-S.Chang et al.. Nature Physics,1,111(2005)

47. Spin-1 condensates
Romano and de Passos, PRA 70, (2004)

48. Spin-1 condensates with domains
The schematic view of domain inside a ferromagnet

49. Spin-1 condensates with domains
Phase separation without dissipation 1 -1
- the overlap constant

50. Spin-1 condensates with domains
The field annihilation operator replaced by its expectation value
the normalized distribution

51. Spin-1 condensates with domains
Our Hamiltonian:
where

52. The spin dynamics part of Hamiltonian
Spin-1 condensates with domains
The spin dynamics part of Hamiltonian
Integrate out the spatial degree of freedom

53. The reduced Hamiltonian
Spin-1 condensates with domains
the overlap factor
The reduced Hamiltonian

54. Spin-1 condensates with domains
For the unmagnetized state

55. Spin-1 condensates with domains
Equations of motion
Bloch relaxation

56. Phase Diagram under different overlap factor
Spin-1 condensates with domains
Phase Diagram under different overlap factor

57. Spin dynamics under different relaxation time
Spin-1 condensates with domains
Spin dynamics under different relaxation time
Gu and Qiu, PRL 98, (2007)

58. Spin-1 condensates with domains
For initially magnetized (m≠0) case:
a=0.5
a=0.1
Wu, Qiu, Gu, unpublished

59. Brief summary

60. Summary Phase diagram: 3 different statistics T T Bose Gase TF TFf I
Weiss Theory 1907
Stoner Theory 1936
Gu and Klemm, PRA68, (R) (2003) T T
Bose Gase TF
TFf TF I I0 I
Fermi Gas
Ferromagnetic Insulator
Fermi Gas
Insulator

61. Summary Thermodynamics Specific heat: two transition points
different behaviors
Susceptibility:
Tao, Wang, Qin and Gu, unpublished

62. Summary Coherence dynamics of domain formation
Gu and Qiu, PRL 98, (2007)

63. 谢谢！
Many thanks!