1. Pure Fulde-Ferrel-Larkin-Ovchinnikov state in optical lattices of off-diagonal confinement 高先龙 2011.8.5 兰州 Collaborators: Reza Asgari, 汪泾泾，陈阿海

2. 金华 八月五月

3. 框架 Intro: 1D system of FFLO phase Confinement: Diagonal confinement versus Off-diagonal confinement Results: Pure FFLO state Conclusions

4. 1D system of FFLO phase Introduction

5. 实验物理学家

6. 理论物理工作者

7. Why 1D: Non-Fermi liquid 1D

8. BCS: Δ(r) ∝ const FF: Δ(r) ∝ exp(iq ⋅ r) LO: Δ(r) ∝ cos(q ⋅ r) Hunt for the Elusive FFLO State Attractive Fermi systems, spin polarization and superfluidity are enemies Conventional: a partially polarized Fermi gas undergoes macroscopic phase separation into a polarized normal region and an unpolarized superfluid region FFLO state: Unconventional superfluid state when, in which fermion pairs with nonzero momentum form a spatially modulated inhomogeneous superfluid phase, [Fulde & Ferrell (1964); Larkin & Ovchinnikov (1964)]

9. 1D Exotic phase:FFLO Bosonization: Yang Phys. Rev. B 63, 140511 (2001); Zhao & Liu PRA (2008) Bethe-ansatz: Orso, PRL98 (2007); Hu, Liu, Drummond, PRL98 (2007); Guan, Batchelor, Lee, Bortz, PRB (2007) DMRG: E. Feiguin and F. Heidrich-Meisner, PRB (2007); M. Tezuka and M. Ueda, PRL (2008); M. Rizzi, et al, PRB (2008) QMC: M. Casula, D. M. Ceperley, and E. J. Mueller, PRA (2008) DFT: Gao Xianlong & Reza Asgari, PRA (2008) Related: mass imbalanced Fermi Hubbard model, B Wang, et al., PRA (2009) SJ Gu, PRB (200?); Cazalilla and Giamarchi, PRL (2005)

10. Why FFLO in cold atom? Condensed matter systems: FFLO physics is obscured by impurities, orbital effects, and spin-orbit coupling. Ultracold atomic systems: the interaction, lattice, and polarization can be chosen at will.

11. Characterization of the FFLO phase Pairing at finite k; nonzero pairing momentum, q 0 =  k F ≠ 0 oscillating pairing function, F~cos(  k F x). oscillations in order parameter Δ(r) Fulde-Ferrell vs Larkin-Ovchinnikov Translational & rotational invariance broken

12. Suggestions for the experimental observation of the FFLO state Image density profiles of : search for oscillations, absorption imaging; phase-contrast imaging technique RF-spectroscopy: Kinunnen et al. PRL 96, 110403 (2006) Rapid-sweep-method, time-of-flight: peaks at finite velocities. Noise correlations: density of states: RF spectroscopy Greiner et al. PRL 94, 110401 (2005) Altman et al. Phys. Rev. A 70, 013603 (2004) Luescher et al. PRA (2009) Yang, PRB (2001)

13. Inhomogeneous FFLO state in 1D

15. The 1D attractive Hubbard model: Phase diagram Bethe ansatz, Phases: I. Empty lattice II. (n < 1, p = 1): Fully polarized III. (n = 1, p = 1): Fully polarized IV. (n < 1, p < 1): Less than half-filled, partially polarized: FFLO V. (n < 1, p = 0): no polarization, fully paired Essler’s Book, The One-Dimensional Hubbard Model, 2005

16. Diagonal confinement versus Off-diagonal confinement Confinement:

17. DC: 1D-Pairing at finite Q & Spatial decay

18. DC: Power-law decay of correlations, spatial oscillations

19. The asymmetric Hubbard model “BCS” “FFLO” ( cf. B. Wang et al., PRA79, 2009 ) 1 component gas Spin-independent hopping

20. The asymmetric Hubbard model superconducting correlations ‘incommensurate’ densities unequal hoppings: the model is no longer integrable, hence use DMRG ‘commensurate’ densities

21. The attractive Gaudin model Yang Phys. Rev. B 63, 140511 (2001); Orso, Phys. Rev. Lett. 98, 070402 (2007).

22. The attractive Gaudin model Yang Phys. Rev. B 63, 140511 (2001); Orso, Phys. Rev. Lett. 98, 070402 (2007) ； XW Guan, PRA

23. Predictions from field theory and LDA The attractive Gaudin model: in a trap Partially-polarized phase associated with FFLO state Yang Phys. Rev. B 63, 140511 (2001); Bethe-Ansatz + local density approximation: Two-phase structures: centre partially polarized; edge either fully paired or fully polarized. Orso, PRL 98, 070402 (2007) Hu, Liu & Drummond, PRL 98, 070403 (2007) Guan, Batchelor, Lee & Bortz, PRB 76, 085120 (2007).

24. Predictions from BA and LDA The attractive Gaudin model: in a trap Mean field theory vs. exact solution

25. The attractive Gaudin model: in a trap

27. FFLO---Experimental Results 6 Li Liao et al., Nature 467, 567 (2010)

28. FFLO---Experimental Results 6 Li No unambiguous demonstration for FFLO state is obtained in cold atomic systems until now!

29. Liao et al., Nature 467, 567(2010) 一维系统

30. Phases induced by external potential M. Rigol et al., PRL (2003) ; G. Xianlong et al., PRL (2007) U< ０ U ＞０

31. Pure state possible? through different designing harmonic trapping

32. Results Pure FFLO state

33. Predictions from Bethe-ansatz based DFT: N=36

36. Critical FFLO state in a 1D attractive Fermi gas Pure FFLO state occurs only at the critical polarization!

37. The effect of disorder on the 1D attractive Fermi gas Wang Jingjing, Gao Xianlong, JPB (2011)

38. speckle intensity the spatial (auto)correlation FFLO-BCS phase could change to FFLO-N phase while increasing disorder The effect of disorder on the 1D attractive Fermi gas

39. Off-diagonal confinement harmonic trapping t=0

40. Phase diagram in DC system M.P.A. Fisher et al., PRB 40,546 (1989) Phase Diagram

41. The model

42. Phase diagram

43. Particle-hole symmetry

44. Pairing correlations 均匀体系 非均匀体系

45. N=80

46. N=70

47. Spin-spin correlations detectable in a non-destructive way via spatially resolved quantum polarization spectroscopy.

48. Spin-spin correlations.

49. Conclusions We show that the off-diagonal confinement supports a region of pure FFLO state, thus provides an ideal system to detect the FFLO state in 1D systems. deviates from linear relations Magnetic structure factor shows a kink related to finite FFLO momentum Note for helpful ALPS (Algorithms and Libraries for Physics Simulations) http://alps.comp-phys.org/

50. Team 感谢： NSFC 的支持

51. Thanks for your attention