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

WY (USTC) Hangzhou, 2013 2 / 15

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...

WY (USTC) Hangzhou, 2013 3 / 15

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...

WY (USTC) Hangzhou, 2013 3 / 15

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).

WY (USTC) Hangzhou, 2013 4 / 15

Spin-Orbit Coupling

Single particle dispersion under SOC

k

−ξ+ξkε

Formation of helicity bands

BEC under SOC

Modified Fermi surface: Lifshitz-transition

WY (USTC) Hangzhou, 2013 5 / 15

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).

WY (USTC) Hangzhou, 2013 6 / 15

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).

WY (USTC) Hangzhou, 2013 6 / 15

Simple Picture of Pairing States

Simple picture of pairing states

Pairing states in the absence of SOC

BCS Pairing

WY (USTC) Hangzhou, 2013 7 / 15

Simple Picture of Pairing States

Simple picture of pairing states

Pairing states in the absence of SOC

BCS Pairing Pairing under polarization

WY (USTC) Hangzhou, 2013 7 / 15

Simple Picture of Pairing States

Simple picture of pairing states

Pairing states in the absence of SOC

BCS Pairing FFLO Pairing

WY (USTC) Hangzhou, 2013 7 / 15

Simple Picture of Pairing States

Simple picture of pairing states

Pairing states in the absence of SOC

BCS Pairing FFLO Pairing

WY (USTC) Hangzhou, 2013 7 / 15

Simple Picture of Pairing States

Pairing states under SOC and Zeeman fields

Without the transverse field

WY (USTC) Hangzhou, 2013 8 / 15

Simple Picture of Pairing States

Pairing states under SOC and Zeeman fields

Without the transverse field

WY (USTC) Hangzhou, 2013 8 / 15

Simple Picture of Pairing States

Pairing states under SOC and Zeeman fields

Without the transverse field

WY (USTC) Hangzhou, 2013 8 / 15

Simple Picture of Pairing States

Pairing states under SOC and Zeeman fields

Without the transverse field With a transverse field

WY (USTC) Hangzhou, 2013 8 / 15

Simple Picture of Pairing States

Pairing states under SOC and Zeeman fields

Without the transverse field With a transverse field

WY (USTC) Hangzhou, 2013 8 / 15

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

WY (USTC) Hangzhou, 2013 9 / 15

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).

WY (USTC) Hangzhou, 2013 10 / 15

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).

WY (USTC) Hangzhou, 2013 11 / 15

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).

WY (USTC) Hangzhou, 2013 11 / 15

Phases under the NIST SOC

Nodal FFLO states

0 1 2 3

0

2

4

αkh/h

µ/h

N

VAC

FFLOy

gFFLOx

FFLOx

WY (USTC) Hangzhou, 2013 12 / 15

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)

WY (USTC) Hangzhou, 2013 12 / 15

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)

WY (USTC) Hangzhou, 2013 12 / 15

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

WY (USTC) Hangzhou, 2013 13 / 15

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

WY (USTC) Hangzhou, 2013 13 / 15

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

WY (USTC) Hangzhou, 2013 13 / 15

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

WY (USTC) Hangzhou, 2013 13 / 15

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

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

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

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

WY (USTC) Hangzhou, 2013 13 / 15

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

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

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

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

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

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

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