Download Chern number

title
1
team
Hans Peter Büchler
Nicolai Lang
Mikhail Lukin
Norman Yao
Sebastian Huber
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motivation: topological states of matter
non-interacting,
filled band
(single particle physics)
topological insulators
integer quantum Hall effect (QHE)
(edge states, quantized σxy)
fermions
topological bands
interacting
+ flat bands
with anyonic excitations
open question: non-Abelian states?
FQHE
bosons
interacting
+ flat bands
bosonic FQHE states
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motivation: topological states of matter
non-interacting,
filled band
(single particle physics)
topological insulators
integer quantum Hall effect (QHE)
(edge states, quantized σxy)
fermions
topological bands
interacting
+ flat bands
with anyonic excitations
open question: non-Abelian states?
FQHE
bosons
interacting
+ flat bands
bosonic FQHE states
3
motivation: difficulties
topological bands
tunneling phases for neutral particles?
F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)
Y. Wang. Phys. Rev. B 86, 201101 (2012)
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motivation: experimental requirements
"natural" realization with dipoles on lattices
▷ Two-dimensional lattice
(not necessarily perfect)
square
triangular
honeycomb
...
▷ Three-level dipole
polar molecules
Rydberg atoms
NV centers
▷ Strong dipole-dipole interactions
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setup with polar molecules
one molecule
pinned at each lattice site
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setup with polar molecules
Three level dipole from rotational structure:
rotational
constant
angular
momentum
dipole
moment
external
fields
eigenstates without electric field:
see also:
N. Yao, Phys. Rev. Lett. 109, 266804 (2012)
N. Yao, Phys. Rev. Lett. 110, 185302 (2013)
S. Syzranov, arXiv:1406.0570
one molecule
pinned at each lattice site
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setup with polar molecules
Three level dipole from rotational structure:
rotational
constant
angular
momentum
dipole
moment
external
fields
eigenstates without electric field:
eigenstates with electric field:
m is still a good quantum number
see also:
N. Yao, Phys. Rev. Lett. 109, 266804 (2012)
N. Yao, Phys. Rev. Lett. 110, 185302 (2013)
S. Syzranov, arXiv:1406.0570
one molecule
pinned at each lattice site
6
setup with polar molecules
Three level dipole from rotational structure:
rotational
constant
angular
momentum
dipole
moment
external
fields
eigenstates without electric field:
eigenstates with electric field:
m is still a good quantum number
see also:
N. Yao, Phys. Rev. Lett. 109, 266804 (2012)
N. Yao, Phys. Rev. Lett. 110, 185302 (2013)
S. Syzranov, arXiv:1406.0570
one molecule
pinned at each lattice site
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dipolar exchange interactions
dipole-dipole interactions in the plane
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dipolar exchange interactions
dipole-dipole interactions in the plane
energy conserving processes:
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dipolar exchange interactions
dipole-dipole interactions in the plane
energy conserving processes:
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tunneling model
interaction term, not relevant
for single excitation dynamics
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tunneling phases
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1
3
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tunneling phases
2
1
3
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tunneling phases
2
1
3
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tunneling phases
2
1
3
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bosonic model
Describe excitations as effective particles (hardcore bosons)
Tunneling Hamiltonian
long-range tunneling
spin-orbit coupling
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bosonic model
Describe excitations as effective particles (hardcore bosons)
Tunneling Hamiltonian
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momentum space
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momentum space
spin-orbit coupling with:
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momentum space
spin-orbit coupling with:
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band structure
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band structure
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band structure
after tuning parameters
are these bands topological?
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topological invariants
Review on topological insulators:
M. Hasan and C. Kane, Rev. Mod. Phys. 82, 3045 (2010)
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topological invariants
Review on topological insulators:
M. Hasan and C. Kane, Rev. Mod. Phys. 82, 3045 (2010)
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Chern number
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Chern number
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Chern number
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Chern number as a winding number
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Chern number as a winding number
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Chern number as a winding number
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Chern number as a winding number
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Chern number as a winding number
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topological phase diagram
C as winding number
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edge states
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edge states
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edge states
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edge states
Hatsugai, Phys. Rev. Lett. 71, 3697 (1993)
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edge states
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Chern number in the disordered system
sample-averaged Chern number
density of defects
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Chern number in the disordered system
sample-averaged Chern number
density of defects
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honeycomb lattice: flat bands
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conclusion
1
2
Y. Wang, Phys. Rev. B 86, 201101 (2012)
G. Möller, Phys. Rev. Lett. 103, 105303 (2009)
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white
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Halperin state
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honeycomb lattice: topological phase diagram
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disordered system
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dispersion relation x/1 model
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Landau levels
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double-layer picture
C=1
C=1
C=1
C=2
C=1
C=1
full BZ
full BZ
1+1=2
1+1=2x1
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outline
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edge states: disorder
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