Composite Fermion Groundstate of
Rashba Spin-Orbit Bosons
Alex Kamenev
Fine Theoretical Physics Institute,
School of Physics & Astronomy,
University of Minnesota
Tigran Sedrakyan
Leonid Glazman
ArXiv1208.6266
Chernogolovka QD2012
Motivation
Four-level ring coupling scheme in 87Rb
involving hyperfine states |F,mF> Ramancoupled by a total of five lasers marked σ1,
σ2, σ3, π1, and π2
Spin-orbit coupling constant
Rashba spin-orbit-coupling
Rotation + two discrete
time-reversal
parity
symmetries
Bose-Einstein condensates of Rashba bosons
Interaction Hamiltonian:
H int 
density operators

1
2
2
2
d
r
g
(
n

n
)

g
(
n

n
)
0
2




2m 
TRSB
g2  0
1 ikr  1 
k ~
e  i arg(k ) 
2
 e


Spin-density wave
g2  0
k ~
1  cos(kr) 


i
sin(
kr
)
2

Chunji Wang, Chao Gao, Chao-Ming Jian, and Hui Zhai, PRL 105,
160403 (2010)
Chemical Potential:
density
However, let us look at Rashba fermions:
Spin-orbit coupling constant
k0
2kF
k0  mv
Estimate chemical potential:
(2k0 ) (2kF ) ~ 4 2n
Near the band bottom, the
density of states diverges as
 ( E) ~ 1 / E
k F2
EF 
~ n2
2m
As in 1D!
Reminder: spinless 1D model
mean-field bosons
spinless fermions
k
0
exact bosons
Lieb - Liniger 1963
 n 0 n
k
Tonks-Girardeau
limit
 g1D
Can one Fermionize Rashba Bosons?
Yes, but…
1. Particles have spin
2. The system is 2D not 1D
Fermions with spin do interact
Eint  g (1  cos )  g
k0

2kF
Hartree
Fock
2kF
Variational Fermi surface
Ekin
 n
 k  
 k0
2
F



2
Eint  gn 2
minimizing w.r.t.
1/ 4
 n 
   2 
 gk0 
Ekin

g 3/ 2
 Eint 
n  n
k0
Berg, Rudner, and Kivelson, (2012)
2
Self-consistent Hartree-Fock for Rashba fermions
Berg, Rudner, and Kivelson, (2012)
nematic state

Elliptic Fermi surface at small density:
(k x  k0 )


2m
2m y
2
H HF
nematic fermions
k y2
k02
my  m
 m
gn
g 3/ 2

n  gn
k0
mean-field bosons
Fermionization in 2D? Chern-Simons!
bosonic
wave function
Chern-Simons
phase
fermionic
wave function
 (plus/minus) One flux quantum per particle
 Broken parity P
 Higher spin components are uniquely determined
by the projection on the lower Rashba brunch
 Fermionic wave function is Slater determinant,
minimizing kinetic and interaction energy
Chern-Simons magnetic Field
mean-field approximation
Particles with the cyclotron mass:
in a uniform magnetic field:
Integer Quantum Hall State
Particles with the cyclotron mass:
in a uniform magnetic field:
Landau levels:
One flux quanta per particle, thus n=1 filling factor: IQHE
Gapped bulk and chiral edge mode:
interacting topological insulator
Phase Diagram
chemical potential
Bose Condensate
broken R and T
Spin-Density Wave
broken R
broken R, T and P
Composite Fermions
spin anisotropic
interaction
Phase Separation
total energy per volume
Bose condensate
composite fermions
phase separation
Rashba Bosons in a Harmonic Trap
condensate
composite fermions
 high density Bose condensate in the middle and
low density composite fermions in the periphery
Conclusions
 At low density Rashba bosons exhibit
Composite Fermion groundstate
 CF state breaks R, T and P symmetries
 CF state is gaped in the bulk, but supports
gapless edge mode, realizing interacting
topological insulator
 CF equation of state:
There is an interval of densities where CF
coexists with the Bose condensate
In a trap the low-density CF fraction is pushed
to the edges of the trap
Lattice Model
In a vicinity of K and K’ points in
the Brillouin zone:
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Composite fermion groundstate of Rashba spin

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