1. Spin-Orbit Coupling Effects in Bilayer and Optical Lattice Systems
纪安春
首都师范大学物理系

2. Collaborator Qing Sun (首都师范大学) Lin Wen (重庆师范大学) W.-M. Liu (物理所)
Liang He & Walter Hofstetter
(Goethe–Universität, Germany)
G. Juzeliunas
(Vilnius University, Lithuania)

3. Outline I. Bilayer Spin-Orbit Coupling
II. Optical Lattices subjected to Spin-Orbit Coupling
1) SO coupled –U Hubbard model
2) SO coupled +U Hubbard model

4. I. Bilayer Spin-Orbit Coupling
1D SOC实验实现
Effective single particle Hamiltonian：
Y.-J. Lin et al., Nature 471, 83 (2011)
J.-Y. Zhang et al., PRL 109, (2012)
P. J. Wang et al., PRL (2012)
L. W. Cheuk et al., PRL 109, (2012)

5. 2D SOC：a Challenge! Non-Abelian Gauge Potentials for Ultracold Atoms
with Degenerate Dark States
J. Ruseckas et al., PRL 95, (2005)
\omega_1,2,3 表示的是一束raman光，角度是theta和phi,他分别在x,y,z三个方向进行投影。
Tripod-coupling scheme

6. Drawback：
The two dark states are
not the ground states!

7. Realistic Rashba and Dresselhaus
spin-orbit coupling for neutral atoms
D. L. Campbell, G. Juzeliunas, and I. B. Spielman, PRA 84, (2011)
Drawback：
i) The total phase should
be fixed to be \pi!
ii) The four internal states
are not easy to be
Implemented.

8. Experimental realization of a two-dimensional synthetic spin-orbit coupling in ultracold Fermi gases
Jing Zhang group, arXiv:
Three Energy level scheme:
Transition loop:

9. Challenge：
i) Gap of the Dirac cone can not be opened.
ii) many-body physics is a challenge.

10. Bilayer Spin-Orbit Coupling
Q. Sun, L. Wen, W.-M. Liu, G. Juzeliunas, and A.-C. Ji,
PRA 91, (2015)

11. Energy level loop：
In plane mometum:
Effective single particle Hamiltonian：

12. Single-particle spectrum
Lowest branch
Many-body Ground States
Variation Ansatz

13. Ground state phase diagram
PW: plane-wave
SP-I: normal stripe
SP-II: time-breaking stripe
FSL: fractionalized skyrmion lattice

14. SP-I: Conventional
stripe
SP-II:
Time-symmetry breaking stripe

15. FSL: Fractional Skyrmion lattices

16. Drawback：
i) 2D SOC, but not the Rashba-type SOC!
ii) The experimental scheme is not easy!

17. Further Improvements：
i) Gap of the Dirac cone can be easily opened.
ii) The scheme includes only two internal states, which is easy for many-body physics.
Relevant results are in preparation!

18. II. Optical Lattices subjected to Spin-Orbit Coupling
K. Osterloh et al. PRL 95, (2005).

19. Non-Abelian gauge potentials acting on cold atoms in optical lattices
K. Osterloh et. al., PRL 95, (2005)

20. 1) SO coupled –U Fermi-Hubbard model
Coventional -U Fermi-Hamiltonian
R. Micnas et al., Rev. Mod. Phys. 62, 113 (1990).

21. SO coupled -U Fermi-Hamiltonian
Q. Sun, G.-B. Zhu, W.-M. Liu, and A.-C. Ji,
PRA 88, (2013)
Two-body bound state solution

22. Results of SO coupled -U Fermi-Hamiltonian

23. Comments i). 在格点当中，SOC既能压制配对，也能增强配对。这是SOC与Lattice联合作用的结果，跟连续体系不同。
ii). 随着掺杂的增加，体系从Dirac cone主导的物理演变为SOC诱导的BCS-BEC crossover物理。

24. 2) SO coupled +U Hubbard model
For large U limit, , implement second-order perturbation:
Dzyaloshinskii-Moriya (DM) interactions
W. S. Cole et al., PRL 109, (2012)
J. Radic et al., PRL 109, (2012)

25. W. S. Cole et al., PRL 109, (2012)
J. Radic et al., PRL 109, (2012)

26. Superfluid-Mott insulator Transitions of Bosons in
SO Coupled Optical Lattices
Liang He, An-Chun Ji, and Walter Hofstetter, to appear in PRA (2015)
Spin configuration：

27. Phase diagram of the Superfluid-Mott insulator Transitions
HSF: homogenous superfluid

28. The effective Hamiltonian of chargon
Slave-boson Theory
chargon
Introduce
spinon
constraint
The effective Hamiltonian of chargon

29. Metal-Mott insulator Transitions of Fermions in
SO Coupled Optical Lattices
X. Zhang, W. Wu, G. Li, L. Wen, Q. Sun, and A.-C. Ji,
New journal of Physics 17, (2015)
SO coupled Fermi-Hubbard model:

30. Cluster Dynamical mean-Field theory (CDMFT)
Weiss function of the cluster
(1)
Dyson equation:
(2)
Calculation procedure:
impurity solver
(1)
(2)

31. Phase diagram in the whole parameter regime
X. Zhang, W. Wu, G. Li, L. Wen, Q. Sun, and A.-C. Ji,
New journal of Physics 17, (2015)

32. \alpha=0
\alpha=1.0
\alpha=1.0
\alpha=1.5

33. Drawback：need further size scaling!
Non-magnetic insulator:
candidate of algebraic spin liqiud
Drawback：need further size scaling!

34. Quantum spin liquid emerging in 2D fermion system?
Z. Y. Meng et al., Nature 464, 847 (2010)

35. Summary i). 双层体系是实现2D Rashba SOC 一个很好的
candidate，可以研究2D 自旋轨道耦合的玻色子和费米子多体效应。
ii). 对于光晶格体系，自旋轨道耦合的Hubbard模型具有丰富的强关联物理。

36. Thank you !