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Unconventional pairing states in a Fermi gas with
anisotropic spin-orbit coupling and Zeeman fields
Wei Yi
University of Science and Technology of China
Collaborators:Wei Zhang (RUC)Xiang-Fa Zhou (USTC)Fan Wu (USTC)
WY (USTC) Hangzhou, 2013 1 / 15
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Outline
Outline
Synthetic spin-orbit coupling
Simple picture of pairing states
Phases under the NIST SOC
Discussion and summary
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Spin-Orbit Coupling
Synthetic spin-orbit coupling
Spin-orbit coupling (SOC) in Nature
SOC within atoms
Condensed matter systems:topological phases,quantum spin Hall effects,etc...
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Spin-Orbit Coupling
Synthetic spin-orbit coupling
Spin-orbit coupling (SOC) in Nature
SOC within atoms
Condensed matter systems:topological phases,quantum spin Hall effects,etc...
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Spin-Orbit Coupling
Synthetic spin-orbit coupling in ultracold atoms
Atomic Gas Ω
𝛿
Raman lasers couple internal and external degrees of freedom
Equal Rashba (kxσx + kyσy) and Dresselhaus (kxσx − kyσy)SOC with effective Zeeman fields
Hk =~2
2m(~k + k0~xσx)2 − δ
2σx −
Ω
2σz
Y.-J. Lin, K. Jimenez-Garcıa, and I. B. Spielman, Nature 471, 83 (2011).
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Spin-Orbit Coupling
Single particle dispersion under SOC
k
−ξ+ξkε
Formation of helicity bands
BEC under SOC
Modified Fermi surface: Lifshitz-transition
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Spin-Orbit Coupling
Experiments on spin-orbit coupled degenerate Fermi gases
Lifshitz transition
Spin injection spectroscopy
P. Wang et al., Phys. Rev. Lett. 109, 095301 (2012).
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Spin-Orbit Coupling
Experiments on spin-orbit coupled degenerate Fermi gases
Lifshitz transition
Spin injection spectroscopy
L. W. Cheuk et al., Phys. Rev. Lett. 109, 095302 (2012).
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Simple Picture of Pairing States
Simple picture of pairing states
Pairing states in the absence of SOC
BCS Pairing
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Simple Picture of Pairing States
Simple picture of pairing states
Pairing states in the absence of SOC
BCS Pairing Pairing under polarization
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Simple Picture of Pairing States
Simple picture of pairing states
Pairing states in the absence of SOC
BCS Pairing FFLO Pairing
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Simple Picture of Pairing States
Simple picture of pairing states
Pairing states in the absence of SOC
BCS Pairing FFLO Pairing
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Simple Picture of Pairing States
Pairing states under SOC and Zeeman fields
Without the transverse field
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Simple Picture of Pairing States
Pairing states under SOC and Zeeman fields
Without the transverse field
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Simple Picture of Pairing States
Pairing states under SOC and Zeeman fields
Without the transverse field
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Simple Picture of Pairing States
Pairing states under SOC and Zeeman fields
Without the transverse field With a transverse field
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Simple Picture of Pairing States
Pairing states under SOC and Zeeman fields
Without the transverse field With a transverse field
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Phases under the NIST SOC
Phases under the NIST SOC
Model Hamiltonian
H =∑
k,σ=↑,↓
ξka†kσakσ +
∑k
h(a†k↑ak↓ + h.c.) + U∑
k,k′,q
a†k+q↑a†−k+q↓a−k′+q↓ak′+q↑
+∑k
[(αkx − hx)a†k↑ak↑ − (αkx − hx)a
†k↓ak↓]
BCS-type mean-field theory with
∆Q = U∑k
〈aQ−k↓ak↑〉
Multiple local minima in the thermodynamic potential Ω(∆Q,Q)Minimize the thermodynamic potential at zero temperature
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Phases under the NIST SOC
Phase diagram without the tranverse field
0 1 2 3
0
2
4
αkh/h
µ/h
VAC
FFLOy
SF
nSF2
nSF1
BCS-type mean-field calculation at zero temperature
First order boundaries imply phase separation
Various exotic pairing states: FFLOy, nSF1,nSF2
F. Wu, G.-C. Guo, W. Zhang and W. Yi, Phys. Rev. Lett. 110, 110401 (2013).
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Phases under the NIST SOC
Phase diagram with a transverse field
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
FFLOx
Instability of BCS pairing under transverse field (∂Ω/∂Qx 6= 0)
Competition between multiple FFLO states with different Q
Gapped and nodal FFLO statesF. Wu, G.-C. Guo, W. Zhang and W. Yi, Phys. Rev. Lett. 110, 110401 (2013).
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Phases under the NIST SOC
Phase diagram with a transverse field
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
gFFLOx
FFLOx
FFLOy
2 3 4
−5
−4
−3
−2
αkh/h
Log(−
Qx/k
h)
2 3 4
0.5
1
αkh/h
∆Q/h
2 3 40
0.1
αkh/h
Exc
itatio
n G
apInstability of BCS pairing under transverse field (∂Ω/∂Qx 6= 0)
Competition between multiple FFLO states with different Q
Gapped and nodal FFLO statesF. Wu, G.-C. Guo, W. Zhang and W. Yi, Phys. Rev. Lett. 110, 110401 (2013).
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Phases under the NIST SOC
Nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
FFLOx
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Phases under the NIST SOC
Nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
Nodal FFLOx states with topologically different gapless contours
Related to phases on the Q = 0 phase diagram
F. Wu, G.-C. Guo, W. Zhang and W. Yi (in preparation)
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Phases under the NIST SOC
Nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
0 1 2 3
0
2
4
αkh/hµ/h
N
VAC
nSF1
nSF2
SF
Nodal FFLOx states with topologically different gapless contours
Related to phases on the Q = 0 phase diagram
Ground state phases Q = 0 phases (unstable)
F. Wu, G.-C. Guo, W. Zhang and W. Yi (in preparation)
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
−2 −1 0 1 2
0
2
kx/kh
ky/kh
FFLOy
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
−2 −1 0 1 2
0
2
kx/kh
ky/kh
FFLOy
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
−2 −1 0 1 2
0
2
kx/kh
ky/kh
FFLOy
−0.5 0 0.5
−1
0
1
kx/kh
ky/kh
a
mixedWY (USTC) Hangzhou, 2013 13 / 15
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
−2 −1 0 1 2
0
2
kx/kh
ky/kh
FFLOy
−0.5 0 0.5
−1
0
1
kx/kh
ky/kh
a
−0.4 0 0.4−2
−1
0
1
2
kx/kh
ky/kh
b
ns2mixedWY (USTC) Hangzhou, 2013 13 / 15
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
−2 −1 0 1 2
0
2
kx/kh
ky/kh
FFLOy
−0.5 0 0.5
−1
0
1
kx/kh
ky/kh
a
−0.4 0 0.4−2
−1
0
1
2
kx/kh
ky/kh
b
ns2mixedWY (USTC) Hangzhou, 2013 13 / 15
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Phases under the NIST SOC
Properties of nodal FFLO states
0 1 2 3
0
2
4
αkh/h
µ/h
N
VAC
FFLOy
gFFLOx
np1
np2ns2
mixed
ns1
abc
−2 −1 0 1 2
0
2
kx/kh
ky/kh
FFLOy
−0.5 0 0.5
−1
0
1
kx/kh
ky/kh
a
−0.4 0 0.4−2
−1
0
1
2
kx/kh
ky/kh
b
−1 −0.5 0 0.5
1.25
1.35
kx/khky/kh
c
ns2mixed ns1
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Discussion and Summary
Pairing under SOC and Fermi surface asymmetry
Rashba SOC with transverse field
Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo, Phys. Rev. A 87, 031602(R) (2013)
H. Hu and X.-J. Liu, arXiv:1304.0387
Isotropic SOC (k · σ) in a 3D gas
NIST SOC in a 3D gas
L. Dong, L. Jiang, H. Hu, and H. Pu, arXiv:1211.1700
V. B. Shenoy, arXiv:1211.1831
X.-J. Liu and H. Hu, arXiv:1302.0553
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Discussion and Summary
Pairing under SOC and Fermi surface asymmetry
Rashba SOC with transverse field
Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo, Phys. Rev. A 87, 031602(R) (2013)
H. Hu and X.-J. Liu, arXiv:1304.0387
Isotropic SOC (k · σ) in a 3D gas
NIST SOC in a 3D gas
L. Dong, L. Jiang, H. Hu, and H. Pu, arXiv:1211.1700
V. B. Shenoy, arXiv:1211.1831
X.-J. Liu and H. Hu, arXiv:1302.0553
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Discussion and Summary
Pairing under SOC and Fermi surface asymmetry
Rashba SOC with transverse field
Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo, Phys. Rev. A 87, 031602(R) (2013)
H. Hu and X.-J. Liu, arXiv:1304.0387
Isotropic SOC (k · σ) in a 3D gas
NIST SOC in a 3D gas
L. Dong, L. Jiang, H. Hu, and H. Pu, arXiv:1211.1700
V. B. Shenoy, arXiv:1211.1831
X.-J. Liu and H. Hu, arXiv:1302.0553
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Discussion and Summary
Pairing under SOC and Fermi surface asymmetry
Rashba SOC with transverse field
Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo, Phys. Rev. A 87, 031602(R) (2013)
H. Hu and X.-J. Liu, arXiv:1304.0387
Isotropic SOC (k · σ) in a 3D gas
NIST SOC in a 3D gas
L. Dong, L. Jiang, H. Hu, and H. Pu, arXiv:1211.1700
V. B. Shenoy, arXiv:1211.1831
X.-J. Liu and H. Hu, arXiv:1302.0553
nFFLO1
g-FFLON
nFFLO2
nFFLO1
0.0 0.5 1.0 1.5 2.0-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Α khh
Μh -0.4 -0.2 0.0 0.2 0.4
-1.0
-0.5
0.0
0.5
1.0
kxkh
k zk h
muh=0.1600, Αkhh=1.500, deltah=0.3646,kkh=-0.2122
-0.4 -0.2 0.0 0.2 0.4
-1.0
-0.5
0.0
0.5
1.0
kxkh
k zk h
muh=0.2000, Αkhh=1.5000, deltah=0.3838,kkh=-0.2064,
L. Dong, L. Jiang, and H. Pu, arXiv:1302.1189
X.-F. Zhou, G.-C. Guo, W. Zhang and W. Yi, arXiv:1302.1303
WY (USTC) Hangzhou, 2013 14 / 15
![Page 37: Unconventional pairing states in a Fermi gas with ...zimp.zju.edu.cn/~iccqm/workshop2013/PPT/YiWei.pdf · x states with topologically di erent gapless contours Related to phases on](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f8951086fc4a20098786796/html5/thumbnails/37.jpg)
Discussion and Summary
Pairing under SOC and Fermi surface asymmetry
Rashba SOC with transverse field
Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo, Phys. Rev. A 87, 031602(R) (2013)
H. Hu and X.-J. Liu, arXiv:1304.0387
Isotropic SOC (k · σ) in a 3D gas
NIST SOC in a 3D gas
L. Dong, L. Jiang, H. Hu, and H. Pu, arXiv:1211.1700
V. B. Shenoy, arXiv:1211.1831
X.-J. Liu and H. Hu, arXiv:1302.0553
WY (USTC) Hangzhou, 2013 14 / 15
![Page 38: Unconventional pairing states in a Fermi gas with ...zimp.zju.edu.cn/~iccqm/workshop2013/PPT/YiWei.pdf · x states with topologically di erent gapless contours Related to phases on](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f8951086fc4a20098786796/html5/thumbnails/38.jpg)
Discussion and Summary
Summary
Rich phase structure in a Fermi gas under the NIST SOC
Pairing under SOC and Fermi surface asymmetry leads tocompeting FFLO phases
Experimental detection:in situ density profiles, momentum-resolved rf spectroscopy, etc.
Posters:
’Unconventional superfluid in a two-dimensional Fermi gas with anisotropic spin-orbit coupling and Zeeman
fields’, Fan Wu
’Exotic pairing states in a Fermi gas with three-dimensional spin-orbit coupling’, Xiang-Fa Zhou
THANK YOU!
WY (USTC) Hangzhou, 2013 15 / 15
![Page 39: Unconventional pairing states in a Fermi gas with ...zimp.zju.edu.cn/~iccqm/workshop2013/PPT/YiWei.pdf · x states with topologically di erent gapless contours Related to phases on](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f8951086fc4a20098786796/html5/thumbnails/39.jpg)
Discussion and Summary
Summary
Rich phase structure in a Fermi gas under the NIST SOC
Pairing under SOC and Fermi surface asymmetry leads tocompeting FFLO phases
Experimental detection:in situ density profiles, momentum-resolved rf spectroscopy, etc.
Posters:
’Unconventional superfluid in a two-dimensional Fermi gas with anisotropic spin-orbit coupling and Zeeman
fields’, Fan Wu
’Exotic pairing states in a Fermi gas with three-dimensional spin-orbit coupling’, Xiang-Fa Zhou
THANK YOU!
WY (USTC) Hangzhou, 2013 15 / 15