Spontaneous Time Reversal
Breaking of Majorana Fermions in
Dipolar Systems
Yi Li and Congjun Wu
University of California, San Diego
APS March Meeting
Dallas, TX; Mar. 22, 2011
Ultracold Dipolar Fermion Molecules
Shoulder
by Shoulder
d
Head to Tail
r
Anisotropy
Repulsive
Attractive
•
Experiments
40K87Rb polar molecules
[Ni et. al, 2008, etc …]
•
Unconventional Single component p-wave Cooper pairing
Anisotropy
channel dominant pairing [Baranov et. al. 2002]
 ci 

H   (ci
ci )H BdG   ,
c 
i
 i 
H BdG

1
 
2  (  0 / p F )e i i z
i
(  0 / p F ) e i z 
,



  p 2 /(2m)   ( z ).
Majorana Fermions in the Tube Lattices
of Dipolar Fermion Molecules
z-direction: 1D tube of pz pairing fermions
Andreev bound states localized at ends z0 with energy zero.
 ( z )  sin( k z ( z  z 0 )) e
 ( z  z0 ) / 
 ( z )  sin( k z ( z  z 0 )) e
( z  z0 ) / 
z
y
x
1 

 , z  z 0 ;
 i
1
  , z  z 0 ;
i
Kitaev, 2000;
Tewari, et al, 2007;
Alicea, et al, 2010;
etc ...
Dispersionless in kx and ky
Each x-y plane: Andreev Bound States
→ 1D or 2D Majorana fermions lattices.
Josephson Coupling between
Parallel Dipolar Molecule Tubes
2 '  2  
z
L>>ξ
J majorana
J majorana
J
L>>ξ

J
2
leading
order
H t   J cos(   2 )
 J majoranai 1 2 sin(

L>>ξ
H t   J cos(     2 ' )
   2
2
)
 J majorana i  1 2 cos(
   2 '
)
2
[Kitaev, 2000; Yakovenko et al, 2004;
Fu and Kane, 2009; Xu and Fu, 2010]
1D Su-Schrieffer-Heeger Model
CH
CH
CH
CH
CH
H SSH  t electron
CH
CH
CH
 (c j ci  ci c j )   e ph


i , j 

K ph
2
 (u j  ui ) 
1
2
i , j 
CH
2M

CH
CH
CH
CH
 (c j ci  ci c j )(u j  ui )


i , j 
2
pi
i
Electron-phonon interaction → Dimerization (modulation of the lattice distortion)
H SSH    (telection  (1)  t )(c j ci  ci c j )  2 NK phu ,  t  2 e  phu

i
CH
CH
i , j 
CH
CH

CH
CH
2
CH
CH
CH
CH
CH
CH
CH
1D Chain of Majorana States

  
Ht   J
  2

 cos(i   j  Aij )  J majorana  sin(
i , j 
2 e J majorana
(

2
i   j  Aij
2
i , j 
  A
) i i j , Aij  const .
) sin( ka )  JN cos(    A )
2
  A
cos(
) sin( ka )  JN sin(    A ))
2
E ( k )   J majorana sin(
j(k ) 
  3

j
Majorana - Superfluid phase coupling → (modulation of the phases)

     2   3
 
 
 

 
  ( 1)   Aij
i
Ht  J
 cos(   ( 1)   A )  J
 sin(
i
ij
i , j 
E ( k )   JN cos(    A )  J majorana
2
sin (
  A
2
majorana
i , j 
2
) cos (

2
2
) sin ( ka )  cos (
2
2
  A
2
2
) i i j .
) sin (

2
2
) cos ( ka )
2D Majorana Honeycomb Lattice
Ht   J
 cos(
i , j 
i
  j )  iJ majorana  sin(
i   j
i , j 
2
A  J majorana | sin(
) i j
=
uij  sign(sin(
 iA  uij  i j
i , j 
Assuming constant superfluid phase difference |Δθ|,
the system prefers a vortex free state.


) |,
2
)),
2
   i   j .
[Lieb, 1994; Kitaev, 2006]
0
 0 0
 0 0
 0
 0 0
 0 0
0
 0 0
0
 0
 0
 0 0
 0 0
0
E ( q )   J 2 N cos(   )
 0
0
 0 0
0
 0
 0 0
 0 
 4 J majorana sin
|  |
|e
ik  n1
e
ik  n 2
1|
2
  J 2 N cos(   )
 0
 4 J majorana sin
 0
 0 0
n2
n1
 0
|  |
2
2
1
1
3


4  cos( k x )   4 cos( k x ) cos(
ky) 1
2
2
2


2D Majorana Square Lattice
θ=π/2
θ=0
θ=π
θ=3π/2
Fermion vortex state (vorticity =1 )
Majorana π-flux state
Conclusion
 pz-wave Cooper pairing in dipolar Fermion molecule systems
Majorana fermions at z-direction ends of the dipolar tube lattices
 Interaction between Majorana fermions and fermion superfluid phases
Spontaneous TR breaking phases on Majorana lattices
 Theoretical Challenges
Further numerical simulation of the TR breaking Phases of
Majorana lattices
 Experimental Challenges
Stability and manipulation of dipolar tube lattices
Thank you!
1D Chain of Majorana States
1D Majorana Chain with current j   
     2   3




 

   
Ht   J
  3
 2  
 cos(
i , j 
i
  j )  J majorana  sin(
i   j
2
i , j 
) i i j .
majorana - superfluid phase coupling → (modulation of the phases)
c.f. 1D Su Schrieffer Heeger Model
CH
CH
H SSH
CH CH
CH CH
CH CH
CH CH
CH CH
CH CH
CH
CH
CH
CH CH
CH
CH
CH CH
CH CH
CH
K ph
1




2
 t election  (c j ci  ci c j )   e  ph  (c j ci  ci c j )(u j  ui ) 
(u j  ui ) 

2 i , j 
2M
i , j 
i , j 

i
2
pi
electron-phonon interaction → dimerization (modulation of the lattice distortion)
H SSH    (telection  (1) 2 e  phu )(c j ci  ci c j )  2 NK phu
i
i , j 


2