Unconventional pairing states in a Fermi gas with...

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

Outline

Outline

Synthetic spin-orbit coupling

Simple picture of pairing states

Phases under the NIST SOC

Discussion and summary


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...


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).


Spin-Orbit Coupling

Single particle dispersion under SOC

k

−ξ+ξkε

Formation of helicity bands

BEC under SOC

Modified Fermi surface: Lifshitz-transition


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).


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).


Simple Picture of Pairing States


Pairing states in the absence of SOC

BCS Pairing





BCS Pairing Pairing under polarization





BCS Pairing FFLO Pairing



Pairing states under SOC and Zeeman fields

Without the transverse field



Pairing states under SOC and Zeeman fields

Without the transverse field With a transverse field




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



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).



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).



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).



Nodal FFLO states

0 1 2 3

0

2

4

αkh/h

µ/h

N

VAC

FFLOy

gFFLOx

FFLOx



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)



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)



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




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 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



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



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


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



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!


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