Transcript Dark Solitons with Majorana Fermions in spin imbalance

Dark Solitons with Majorana Fermions in
Spin-Orbit-Coupled Fermi Gases
毛力
Wuhan University
Aug. 7th, 2014 Hangzhou
Collaborators: Yong Xu, Biao Wu and Chuangwei Zhang
2014-8-7
Outline
Introduction to Majorana fermions(MFs).
Requirements for MFs in condensed matter
physics.
Exotic properties.
Dark solitons and MFs.
Discussion.
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Majorana Fermion: possible physical systems
Electron
Spin ½
Antiparticle: positron
?
Majorana Fermion
Spin ½
Antiparticle: itself
ˆ  ˆ†
Neutrino
Dark matter
Solid State
Non-Abelian exchange statistics
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Topological quantum computation
Majorana fermions: increasing simplicity
 Fractional Quantum Hall (NPB, 360, 362. NPB, 479,529. PRL94,166802…)




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Chiral p-wave superconductors (PRL 100, 027001. PRL 99 220502…)
Topological insulator/s-wave superconductor(PRL 100, 096407…)
Spin-orbit coupled semiconductor/s -wave superconductor(PRL 104, 040502…)
Spin-orbit coupled Fermi gases (PRL 101, 160401. PRL 91, 090402…)
Arguments and Comments about solid state platform
Near-zero-energy end states in topologically
trivial spin-orbit coupled superconducting nanowires
with a smooth confinement,
G. Kells, D. Meidan, and P. W. Brouwer, arXiv:1207.3607
Phys. Rev. B 86, 100503(R) (2012)
Zero-bias peaks in spin-orbit coupled superconducting
wires with and without Majorana end-states
J. Liu, et al, Phys. Rev. Lett. 109, 267002 (2012).
Zero-voltage conductance peak from weak antilocalization in a Majorana nanowire,
D. I. Pikulin, J. P. Dahlhaus, M. Wimmer, and C. W. J. Beenakker, arXiv:1206.6687,
New J. Phys. 14 125011(2012)
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Majorana fermions: in cold atoms
It is possible, but need:
 Spin orbit coupling(SOC).
SOC
Hsp   (kx y  k y x )
 Topological defect such as vortex (2D), wire ends(1D)
and dark solitons.
Wire ends
 Zeeman field.
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Majorana fermions: topological defects
With any particle
cˆ†:
cˆ  cˆ†
ˆ1 
,
2
cˆ  cˆ†
ˆ2 
2i
{ˆm , ˆn†}   mn , n, m  1, 2
(a)
1,2
Local
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1
(b)
None local
2
Majorana fermions: topological defects
Order parameter of
p-wave superfluid:
( P)  0 ( Px  iPy )
Vortices
Order parameter amplitude
suppressed at the core
Low energy normal bound
states in the core
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Nature 2005
Majorana fermions: zero energy states
 Bogoliubov-de Gennes equation
   2  2 / 2m  




Η 
*
2 2




/
2
m




1

En   n  0
2

s-wave superfluid


 p  ip superfluid
x
y
Particle hole symmetry:
En  n0
E0
aˆ  aˆ E  aˆ  aˆ0
†
E
†
0
Zero energy states are the Majorana fermionic states
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Majorana fermions: Zeeman field and effective p wave
Effective pairing in the lowest band.
eff  aˆk aˆk  is eik
Effective p-wave
Zeeman field requirement:
Vz   2   2
J. D. Sau et. al, PRL 104, 040502
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i 3k
eff  is e
Effective f-wave
L. Mao et. al, PRL 108, 177001,
PRL 106, 157003
Majorana fermions: an example
S-wave + spin orbit coupling + Zeeman field
+vortex
Vortex core state
Li Mao and C. Zhang, PRB, 82, 174506 (2010)
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Vortex edge state
Majorana fermions: properties
Local disorders or impurities:
Increasing the mini-gap!
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Majorana fermions: the interests
Chiral p-wave superfluid
fermion
2
1
 A pair of vortices supports one zero-energy fermionic mode
c   1  i 2  / 2
c, c  0, c, c  1
c’s can be occupied by fermions
0
Degenerate states
Non-local Occupation
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1  c 0
Majorana fermions: the interests
4
3
Vortex
1
2
Exchange
Chiral p-wave superfluid
 1,  2
 
 
0  exp i  0 , 1  exp  i  1
 4
 4
Exchange  2 ,  3
z
1
y
 00  i 11 
00 
2
Non-Abelian Statistics of Majorana Vortices
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Generation of Spin-Orbit Coupling: Experiment
Spin-orbit coupling
for BEC
Rb87
0
1

F  1, mF  1
BEC
BEC
1
2
Heff  px z   x

2

F  1, mF  0
Lin et al., Nature 2011
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Generation of Spin-Orbit Coupling: Experiment
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Jing Zhang group, PRL 2012
40K
atoms
Martin Zwierlein group, PRL 2012
6Li
atoms
Another defect: dark solitons
BEC
atoms
S. Burger et al, PRL, 83, 5198 (1999)
J. Denschlag et al, Sicence, 287, 97 (2000)
B. P. Anderson et al, PRL, 86, 2926 (2001)
Strongly interacting Fermi superfluids:
atoms
T. Yefsah et al, Nature, 499, 426 (2013)
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Another defect: dark solitons

phase difference
MFs?
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MFs with dark solitons?
Our model:
With

phase difference in the initial guess of pairing:
( x)   0tanh( x)
e x  e x
  0  x  x
e e
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Yong Xu*, Li Mao*, Biao Wu and Chuanwei Zhang, arXiv:1401.3777.
*contribution equally
Solitons in spin imbalanced Fermi gas
xs 
/ m
  mg / n(0)
2
  xs / N a1D
g  2
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2
/ ma1D
Zero energy state
10
0
-10
10
0
-10
Zero energy state and real space distribution
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Phase diagram
TS
NS TS
LDA
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m
TS
Possible detecting signatures
Density contrast: P  n  n
Vz  0.3, 0.7, 0.768E f
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Interactions
Interaction between two MFs:
So,
t
i cos( / 2) 1 2
2
 (t) Means none zero interaction!
R. G. Scott, et. al, PRL 106, 185301(2011)
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Discussion
 Dark solitons can exist in the spin
imbalanced Fermi gas.
 They can support Majorana fermions.
 Oscillation of dark solitons induce interaction
to MFs.
 Dynamics
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Thanks!
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