1 / 76

I. EP L 8 4 , 4 0001 ( 2008 )

I. EP L 8 4 , 4 0001 ( 2008 ). 1. Dilute Hard Sphere Bose Gas. Bogoliubov 1947 J. Phys. JSSR II, 23. * I am indebted to L. D. Landau for this important remark.

thi
Download Presentation

I. EP L 8 4 , 4 0001 ( 2008 )

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. I. • EPL 84, 40001 (2008)

  2. 1. Dilute Hard Sphere Bose Gas

  3. Bogoliubov 1947 • J. Phys. JSSR II, 23

  4. * I am indebted to L. D. Landau for • this important remark.

  5. Bogoliubov expanded in powers of the potential V(r), which lead to divergence for hard sphere potentials. Landau told him to use thescattering lengtha.

  6. Lee, Huang, Yang 1957 • Phys. Rev.106, 1135

  7. 2. Pseudopotential Method

  8. Key idea:To replace theboundary condition (thatφ= 0 when two particles touch) with a “pseudopotential”, and thendo usual perturbation.

  9. One particle scattering from one fixed hard sphere.

  10. Thus we haveas potential to replace the boundary condition.

  11. This potential gives the correct wave function for r > a, and satisfies the correct boundary condition.

  12. Notice that this potential gives the divergent term inside r < a for the wave function.

  13. Two particles:

  14. Many particles: • Huang & Yang, PR 105, 767 (1957); • Huang, Yang & Luttinger, PR 105, • 776 (1957).

  15. For particle j = 1, it sees the other N−1 spheres. Each contributes potential

  16.    

  17. The wave length for particle 1 is very long, hence the total V for particle 1 is

  18. 3. Field Theory

  19. Momentum representation of particles: = occupation # of particles with

  20. Diagonal elements of V' is

  21. Off diagonal elements from :Only those where 2 of the four α,β,μ,ν are zero are important.

  22. Perturbation theory leads to: • Justification of “Bogoliubov Transformation” • Wave Function of Ground State • Correction term = constant • Excitation Spectrum

  23. Wave Function for Ground State • • (k, −k) pairs of correlated • particles floating in a sea of • particles with k = 0. • • average # of pairs = • • Correlation length

  24. Second order perturbation energy = ∞

  25. Divergence is due to the fact that the summand at large k becomes

  26. which is of course equal to (constant)in coordinate space.

  27. This is precisely the term in the 1st order perturbation wave function.

  28. To suppress this term, we use pseudopotentialFermi (1936), Breit (1947)

  29. • If operate on , gives zero

  30. Excitation Spectrum

  31. Pair Distribution Function D(r12)

  32. Generalization to other dimensions

  33. Dimension 1

  34. Shrink each link:• N sticks with a=0 in box L − Na• Fermion ground state

  35. Girardeau, J. Math. Phys. 1, 516 (1960).Lieb and Liniger, PR 130, 1605 (1963).Yang and Yang, J. Math. Phys. 10, 1115 (1969).van Amerongen et al. PRL 100 090402 (2008)

  36. Dimension 4:

  37. (PP4)

  38. Depletion of k = 0 state (14)

  39. Dimension 5: (16)

  40. Linear divergence at large k.

  41. Dimension 2:Many papers reviewed inA. Posazhennikova, RMP. 78, 1111 (2006).M. Schick, PR A3, 1067 (1971).

  42. Dim 3Dim 2

More Related