Composite Fermion Groundstate of Rashba Spin-Orbit Bosons Alex Kamenev Fine Theoretical Physics...
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![Page 1: Composite Fermion Groundstate of Rashba Spin-Orbit Bosons Alex Kamenev Fine Theoretical Physics Institute, School of Physics & Astronomy, University of.](https://reader030.fdocuments.us/reader030/viewer/2022032709/56649eb05503460f94bb581a/html5/thumbnails/1.jpg)
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
Chernogolovka QD2012
ArXiv1208.6266
![Page 2: Composite Fermion Groundstate of Rashba Spin-Orbit Bosons Alex Kamenev Fine Theoretical Physics Institute, School of Physics & Astronomy, University of.](https://reader030.fdocuments.us/reader030/viewer/2022032709/56649eb05503460f94bb581a/html5/thumbnails/2.jpg)
Four-level ring coupling scheme in 87Rbinvolving hyperfine states |F,mF> Raman-coupled by a total of five lasers marked σ1, σ2, σ3, π1, and π2
Spin-orbit coupling constant
Motivation
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Rashba spin-orbit-coupling
Rotation + two discrete symmetries
time-reversal
parity
![Page 4: Composite Fermion Groundstate of Rashba Spin-Orbit Bosons Alex Kamenev Fine Theoretical Physics Institute, School of Physics & Astronomy, University of.](https://reader030.fdocuments.us/reader030/viewer/2022032709/56649eb05503460f94bb581a/html5/thumbnails/4.jpg)
Bose-Einstein condensates of Rashba bosons
Interaction Hamiltonian:
22
20
2int )()(
2
1nngnngrd
mH
02 g
)arg(
1
2
1~ ki
ikrk e
e
TRSB
02 g
)sin(
)cos(
2
1~
kri
krk
Spin-density wave
Chunji Wang, Chao Gao, Chao-Ming Jian, and Hui Zhai, PRL 105, 160403 (2010)
density operators
![Page 5: Composite Fermion Groundstate of Rashba Spin-Orbit Bosons Alex Kamenev Fine Theoretical Physics Institute, School of Physics & Astronomy, University of.](https://reader030.fdocuments.us/reader030/viewer/2022032709/56649eb05503460f94bb581a/html5/thumbnails/5.jpg)
However, let us look at Rashba fermions:
nkk F2
0 4~)2()2( 2
2
~2
nm
kE FF Estimate chemical potential:
As in 1D!Near the band bottom, the density of states diverges as EE /1~)(
Chemical Potential:
density
mvk 0
Spin-orbit coupling constant
Fk20k
![Page 6: Composite Fermion Groundstate of Rashba Spin-Orbit Bosons Alex Kamenev Fine Theoretical Physics Institute, School of Physics & Astronomy, University of.](https://reader030.fdocuments.us/reader030/viewer/2022032709/56649eb05503460f94bb581a/html5/thumbnails/6.jpg)
Reminder: spinless 1D model
spinless fermions
mean-field bosons
exact bosons Lieb - Liniger 1963
Tonks-Girardeau limit
Dg1
k0
k0 nn
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Can one Fermionize Rashba Bosons?
Yes, but…
1. Particles have spin
2. The system is 2D not 1D
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Fermions with spin do interact
Fk2
2int )cos1( ggE
Hartree Fock
0k
2
0
2kin
kn
kE F2
int gnE
Variational Fermi surface
Berg, Rudner, and Kivelson, (2012)
4/1
20
gk
n 2/3
0intkin n
k
gEE
minimizing w.r.t.
Fk2
n
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Self-consistent Hartree-Fock for Rashba fermions
Elliptic Fermi surface at small density:
Berg, Rudner, and Kivelson, (2012) nematic state
y
yxHF m
k
m
kkH
22
)(22
0
mgn
kmmy
20
gnnk
g 2/3
0
mean-field bosonsnematic fermions
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Fermionization in 2D? Chern-Simons!
fermionic wave function
bosonic wave function
Chern-Simons phase
(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
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Chern-Simons magnetic Field
mean-field approximation
Particles with the cyclotron mass:
in a uniform magnetic field:
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Integer Quantum Hall State
Landau levels:
One flux quanta per particle, thus =1 filling factor: IQHE
Particles with the cyclotron mass:
in a uniform magnetic field:
Gapped bulk and chiral edge mode: interacting topological insulator
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Phase Diagram
spin anisotropic interaction
chemical potential
Spin-Density Wave
Bose Condensate
Composite Fermions
broken R
broken R and T
broken R, T and P
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Phase Separation
total energy per volume
composite fermions
Bose condensate
phase separation
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Rashba Bosons in a Harmonic Trap
condensate
composite fermions
high density Bose condensate in the middle and low density composite fermions in the periphery
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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
Conclusions
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Lattice Model
In a vicinity of K and K’ points in the Brillouin zone: