1. 周黎红 中国科学院物理研究所 凝聚态理论与材料计算实验室 指导老师： 崔晓玲 arXiv:1507,01341(2015) 2015.8.2
Spin-orbit Coupled Fermi Gases in Optical lattices: Single-particle physics and SF-IN transitions
周黎红
中国科学院物理研究所 凝聚态理论与材料计算实验室
指导老师： 崔晓玲
arXiv:1507,01341(2015)

2. Outline Introduction :Spin-Orbit Coupling for neutral atom
SOC+ optical lattice: exact result vs. tight-binding model
SOC+ optical lattice: superfluidity-reentrance physics
Summary

3. Outline Introduction :Spin-Orbit Coupling for neutral atom
SOC+ optical lattice: exact result vs. tight-binding model
SOC+ optical lattice: superfluidity-reentrance physics
Summary

4. Spin-Orbit Coupling for neutral atom
Laser L Laser R
Two-photon Raman process
Single-particle Hamiltonian:
SOC Raman coupling detuning
I. B. Spielman, Nature 471,83(2011)
P.-J. Wang, J. Zhang, et al, PRL 109,095301(2012)
L.-W. Cheuk et al. PRL 109,095302(2012)
J.-Y. Zhang, et al, PRL 109,115301(2012)
Atomic gas

5. Spin-Orbit Coupling for neutral atom
Single-particle dispersion spectra (𝛿=0)
(a) Ω= (b) Ω≠0
1D topological superfluid state and Marana Fermions
Xia-Ji Liu and Hui Hu , PRA 85, (2012)
Xia-Ji Liu and Hui Hu , PRA 88, (2013)
Ehud Altman, PRL 114, (2015)

6. Outline Introduction :Spin-Orbit Coupling for neutral atom
SOC+ optical lattice: exact result vs. tight-binding model
SOC+ optical lattice: superfluidity-reentrance physics
Summary

7. Single-particle physics
Hamiltonian(1D):
Exact solution:
Field operator 𝜓 𝜎 (𝑟)= 𝑛𝑘 𝜙 𝑛𝑘 𝜎 (𝑟) 𝜓 𝑛𝑘𝜎 , 𝜙 𝑛𝑘 𝜎 𝑟 = 1 𝑉 𝐺 𝑎 𝑛𝑘 𝜎 (𝐺) 𝑒 𝑖(𝑘+𝐺)𝑟
Tight-binding model:
* Shifted Wannier basis : 𝜔 𝑛𝜎 𝑥 = 𝑒 𝑖 𝜈 𝜎 𝑞𝑥 𝜔 𝑛 (𝑥), 𝜐 ↑ =1, 𝜐 ↓ =−1
Here t= 4 𝜋 ( 𝑉 𝑂 𝐸 𝐿 ) 𝑒 −2 𝑉 0 𝐸 𝐿 , (on-site spin flip) ,
Ω ′ = Ω 𝑅 𝑑𝑥 𝜔 0↑ ∗ (𝑥) 𝜔 0↓ (x-a) (NN)
dispersions:
𝐸 ± 𝑘 𝑥 =± (2𝑡 sin ( 𝑘 𝑥 𝑎) ) 2 + (Ω+2 Ω ′ cos ( 𝑘 𝑥 𝑎) ) 2

8. Single-particle spectrum:Ω = ,Ω’=
Exact:
Model(A):
* Ω 𝑅 =0,Two different spin bands, respectively shifted right- and left-side, relate to 𝑝 𝑥 𝜎 𝑧 term, can be gauged away by
* A small Ω 𝑅 , Couples the up and down spins, energy gaps will open;
* A large Ω 𝑅 , model (A) fails to reproduce the correct spectrum;

9. exact Model (A) is not suit for large Ω 𝑅 :
exact: the spin tends to be polarized ,the spectrum becomes that a spinless particle
in lattice potential;
Model (A)：similar to small Ω 𝑅 case, with narrower band widths and larger band gap,
and higher-band contributions missed(superposition of all levels of shifted Wannier
basis {𝜔 𝑛𝜎 𝑥 },while we only take n=0 the lowest band).
Model (A)
exact

10. conventional Wannier basis : 𝜔 𝑛 (𝑥)
Here
Model (B) cannot reproduce the correct spectrum:

11. Dispersions； 𝐸 ± 𝑘 𝑥 =−2𝑡 cos ( 𝑘 𝑥 𝑎)± 4 𝑡 ′2 𝑠𝑖𝑛 2 𝑘 𝑥 𝑎 + Ω 𝑅 2
Model B is not suit
Dispersions； 𝐸 ± 𝑘 𝑥 =−2𝑡 cos ( 𝑘 𝑥 𝑎)± 4 𝑡 ′2 𝑠𝑖𝑛 2 𝑘 𝑥 𝑎 + Ω 𝑅 2
* invisible momentum shift(t’ much smaller than t);
* t’ cannot be fully gauged away , in contrast to continuous case and model (A);
higher-band contributions missed, the Wannier function not the superposition of
all levels of conventional ones for any value of Ω 𝑅 .
exact
Model B

12. A quantitative parameter regime for model (A):
𝜂 𝜐 =10%(dashed line): the validity of model (A) by requiring both 𝜂 1 and 𝜂 2 <10%
exact
Model (A)

13. Outline Introduction :Spin-Orbit Coupling for neutral atom
SOC+ optical lattice: exact result vs. tight-binding model
SOC+ optical lattice: superfluidity-reentrance
Summary

14. Band evolution in 2D 𝑉 0 =5 𝐸 𝐿 , Ω 𝑅 =0.8 𝐸 𝐿
𝐻 𝑦 = 𝑝 𝑦 2 2𝑚 + 𝑉 0 𝑐𝑜𝑠 2 ( 𝑘 𝐿 𝑦)
1st= gap (1x+1y and 2x+1y)
2nd= gap(2x+1y and 1x+2y)
1st gap-opening at certain Ω 𝑅 ，and gap between 1x and 2x increase ; 2nd gap
closing at a larger Ω 𝑅 ; the tight-binding model overestimates the first gap.
The unique high-band physics for SOC atoms in high-D lattices.

15. Superfluid-Insulator transition
The interacting Hamiltonian: Thermodynamic potential: Minimizing the 𝜅 with respect to ： ， (a) Only positive eigenvalues: insulating or normal phase ; (b) otherwise, superfluid phase Transition conditions The smallest eigenvalue ( 𝐶 𝐺𝐺′ ) = 0

16. SF-IN transition
𝑉 0 =5 𝐸 𝐿 , 𝐸 𝑏 =3 𝐸 𝐿
∗ 𝜌=2 , reentrance of SF, two competitive effects, SF to IN(spin-polarization),
and IN to SF transition(vanishing band gap);
*𝜌=1 , SF to IN transition, two cooperative effects , polarization of spin and band gap enlargement

17. occupy a large region in the trap
In a trap :
LDA with , fixed total particle number N
* Ω 𝑅 =0.1 ,SF-VAC transition
* Ω 𝑅 =0.7, flat-top density (𝜌=2,1 IN)
* large value , Ω 𝑅 =1.4,𝜌=1IN phase
occupy a large region in the trap

18. Summary SOC+lattice: single-particle physics (1D)
Limitation of tight-binding models; high-band contributions
SOC+lattice: band-evolution in 2D and SF-IN transition
SOC-induced gap closing; reentrance of SF at filling n=2; density profile in a trap
arXiv: 1507,01341(2015)

19. Thank you!