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Probing physics of Dirac cones by Landau-Zener interferometry
Jean-Noël Fuchs Lih-King Lim
Workshop on Landau-Zener Interferometry and Quantum Control in Condensed Matter, Izmir, October 2014
Gilles Montambaux Laboratoire de Physique des Solides, Orsay CNRS, Université Paris-Sud, France
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Dirac cones, from graphene to cold atoms
Jean-Noël Fuchs Lih-King Lim
Workshop on Landau-Zener Interferometry and Quantum Control in Condensed Matter, Izmir, October 2014
Gilles Montambaux Laboratoire de Physique des Solides, Orsay CNRS, Université Paris-Sud, France
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K K’
Berry phase
p
- Berry phase
-p Graphene electronic spectrum
Gilles Montambaux Laboratoire de Physique des Solides, Orsay CNRS, Université Paris-Sud, France
Dirac cones, from graphene to cold atoms
Workshop on Landau-Zener Interferometry and Quantum Control in Condensed Matter, Izmir, October 2014
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J.-N. Fuchs, M. Goerbig, F. Piéchon P. Dietl, P. Delplace, R. De Gail (PhDs) Lih-King Lim (post-doc)
p -p
« Life and death of Dirac points »
Manipulation of Dirac cones in artificial graphenes
« Artificial » graphenes
p -p 0
G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig, Eur. Phys. J. B 72, 509 (2009),
Phys. Rev. B 80, 153412 (2009)
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Outline
Motion and merging of Dirac points Modified graphene as a toy model A universal Hamiltonian, spectrum at the merging Physical realizations : 1) Microwaves in a honeycomb lattice of dielectric discs F. Mortessagne’s group, Nice (2012) 2) Graphene-like lattice of cold atoms in an optical lattice T. Esslinger’s group, ETH (2012) Landau-Zener tunneling as a probe of Dirac points More Dirac points Other artificial graphenes
p -p
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Graphene
*
0 ( )
( ) 0
f kH k
f k
1 2( ) 1ik a ik a
f k t e e
-
( ) ( )k f k
1a2
a
A B
K K’ Dirac point K K’
Q: How to move these Dirac points ?
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A: Uniaxial strain
1 2'( )ik a ik a
k te tet
't tt
' 1.5t t't t
1 2
2
3K a K a
p -
0
Dirac point
' 2t t
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Motion and merging of Dirac points
' t t
't tt
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Motion and merging of Dirac points 't t
t
« Semi-Dirac »
2' = t t
massive !
massless !
yq
xq
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't t ' 1.5t t
' 2t t ' 2.3t t
Motion and merging of Dirac points
2
2
4 42 arctan 1
3 'D
tq
t -
* 2
3m
t
“hybrid” “semi-Dirac”
cy =3
2t0
cx =
s3
µt2 ¡ t02
4
¶¡! for t0 = 2t
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Hybrid 2D electron gas : a new dispersion relation
xqyq
P. Dietl, F. Piéchon, G.M., PRL 100, 236405 (2008)
G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig, Eur. Phys. J. B 72, 509 (2009),
Phys. Rev. B 80, 153412 (2009)
Schrödinger Dirac « Semi-Dirac »
' 2t t
²n = §·(n+
1
2)eB
¸2=3²n = (n+
1
2)eB
m¤ ²n =§p2neB
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1.5't t 2't t
Berry phase
p
-p
1 | . .
2k k k k k
C
B
C
i u u dk dk p
't t
p
-p
k
( )
11
2 i ku
e
k
Topological transition
Two component wavefunction
0
p
p -
1( ) 2
(( )
2
)
2Bn
CA C eB
In a magnetic field, semiclassical quantization of trajectories
k kArg f
k
k
²n = §·(n+
1
2)eB
¸2=3²n =§
p2neB
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General description of the motion of Dirac points (with time reversal + inversion symmetry)
When changes,
*
0 ( )
( ) 0
f kH
f k
-
.
,
( ) mnik R
mn
m n
f k t e-
move
Where is the merging point?
D D - 0
2
GD
mnt D D-and
4 possible positions in space k(1,1) (1,0) (0,1) (0,0)
0
2
( )2
xy
qf D q icq
m -Expansion near 0D
M
[ ]G
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2
2
02
( )
02
xy
xy
qicq
mH q
qicq
m
-
,
( 1) mn
mnmn
m n
t Rcy
-
12
*,
( 11
) mn
mn mn
m n
tm
R
-
,
* ( 1) mn
mn
m n
t
- *
*
2
2
02
( )
02
xy
xy
qicq
mH q
qicq
m
-
At the merging transition :
Near the transition :
*( ' 2 )t t -
* 0 * 0
xq
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This Hamiltonian describes the topological transition, the coupling between valleys
and the merging of the Dirac points
The parameter drives the topological transition
* 0 * 0
*
« universal Hamiltonian » *
*
2
2
02
( )
02
xy
xy
qicq
mH q
qicq
m
-
*
*2xq m - p
-p 0
* 0
B B2/3B
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By varying band parameters, it is possible to manipulate the Dirac points. They can move in k-space and they can even merge. The merging transition is a topological transition: 2 Dirac points evolve into a single hybrid « semi-Dirac » point and eventually a gap opens and the Fermi surface disappears. Universal description of motion and merging of Dirac points.
First summary: Manipulation of Dirac points and merging
Honeycomb Brick wall
G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig,
Phys. Rev. B 80, 153412 (2009)
it
BZ (0,1)
(1,1)(1,0) (0,0)
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Physical realizations of the merging transition
* Strained graphene
' 2t t
strain ~ 23%
Pereira, Castro Neto, Peres, PRB 2009 See also Goerbig, Fuchs, Piéchon, G.M., PRB 2008
merging is unreachable in graphene
playing with « artificial graphenes »
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Topological transition of Dirac points in a microwave experiment M. Bellec et al. PRL 111, 033902 (2013) Collaboration Fabrice Mortessagne (Nice)
Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice L.Tarruell et al. Nature, 483, 302 (2012) Tilman Esslinger (Zürich)
Microwaves
Physical realizations of the merging transition ?
Cold atoms
Merging of Dirac points in a 2D crystal G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig (PRB 2009) . “I think that it is a very long shot, given that ... the systems are yet to be realized experimentally... ….its relevance to current experiments is rather tenuous…”
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Atoms are trapped in an optical lattice potential and form an artifical crystal
Nature 483, 302 (2012)
Honeycomb Brick wall
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'i i
E te te t 1 2k.a k.a ( ) ( )'x y x yi k k a i k k a
E te tte -
Honeycomb Brick wall
't t
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t’=2 t’=1.414 t’=1
t’=2 t’=1.414 t’=1
Honeycomb
Brick wall
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Bloch oscillations = uniform motion in reciprocal space
Nature 483, 302 (2012)
dkF
dt
40K
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Bloch oscillations = uniform motion in reciprocal space
Nature 483, 302 (2012)
dkF
dt
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Bloch oscillations = uniform motion in reciprocal space
Nature 483, 302 (2012)
dkF
dt
l l 40K
l l
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How to manipulate and merge Dirac points ?
Anisotropy of the optical potential
How to detect and localize Dirac points ?
one Bloch oscillation + Landau-Zener Tunneling
Measurement of the proportion of atoms in the upper band
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2
4
gE
c F
ZP e
p-
Landau-Zener transition
1 ZP-
F tk
gE
c
k
E
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ETH experiment
Measured transfered fraction of atoms: directions of motion
Single Dirac cone Double Dirac cone
Merging line
gapped phase merging Dirac phase Dirac phase
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Explain the experimental data using Universal Hamiltonian
1 ) Relate the parameters of the optical lattice to the parameters of the Universal Hamiltonian
VX, VXb, VY (laser intensities) Ab-initio band structure
Tight-binding model
Universal hamiltonian (Δ, cy, m* )
Lih King Lim, Jean-Noel Fuchs, G. M., PRL 108, 175303 (2012)
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Single Zener tunneling
Double Zener tunneling
yq
yq
xq
xq
2 ) Compute the inter-band tunneling probability within the Universal Hamiltonian
ZP
2 (1 )t Z ZP P P -
yq
xq
Explain the experimental data using Universal Hamiltonian
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Single Zener tunneling
Double Zener tunneling 2 (1 )t Z ZP P P -
ZP
yq
yq
xq
xq
yq
xq
2 ) Compute the inter-band tunneling probability within the Universal Hamiltonian
Explain the experimental data using Universal Hamiltonian
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functions of the laser intensities VX, VXb, VY
functions of the tight-binding couplings
(Δ, cy, m* )
2 (1 )t Z ZP P P -ZP functions of parameters of the U.H.
3 ) Back to the lasers intensities
Explain the experimental data using Universal Hamiltonian
and
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Single Dirac cone: single atom tunneling Transfer probability as a function of qx and Δ*
*
*2xq m -
E
yq
xq
yq
xq
22
**( )2
x
y
q
m
c Fy
ZP ep
-
y
ZP
xq
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Single Dirac cone: Fermi sea tunneling
Transfer probability for a cloud of finite size
Maximum slightly inside the Dirac phase xq
yq
xq
E
yq
xq
*
*2xq m -
Transfer probability as a function of qx and Δ*
y
ZP
y
ZP
22
**( )2
x
y
q
m
c Fy
ZP ep
-
xq
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Single Dirac cone: Fermi sea tunneling
Maximum slightly inside the Dirac phase
* 0
* 0
y
ZP
Transfer probability for a cloud of finite size
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Single Dirac cone Double Dirac cone
Theory
ZP
2 (1 )t Z ZP P P -
Lih King Lim, Jean-Noel Fuchs, G. M., PRL 108, 175303 (2012)
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Single Dirac cone Double Dirac cone
Experiment
Theory
ZP
2 (1 )t Z ZP P P -
Lih King Lim, Jean-Noel Fuchs, G. M., PRL 108, 175303 (2012)
1/ 2ZP
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coherent
incoherent
Probing physics of Dirac cones by Landau-Zener interferometry
Combining probability intensities gives:
xq2 paths from lower to upper band: #1 jump – stay #2 stay - jump
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Combining probability amplitudes gives
coherent
Prediction: coherent double Dirac cone -> Stückelberg interferences
Pxt = 4Px
Z(1¡PxZ) sin
2('s+'dyn+'g
2]
'dyn =2
F
Z qD
¡qD
dqxE(qx; qy)
E. Shimshoni, Y. Gefen, Ann. Phys. (1991) S. Gasparinetti et al. PRL (2011) L.-K. Lim, J.-N. Fuchs, G. M., PRL 112, 155302 (2014)
xq2 paths from lower to upper band: #1 jump – stay #2 stay - jump
Dynamical phase
Phase delay (Stokes phase)
Geometrical phase
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k
Lih King Lim, Jean-Noel Fuchs, G. M., PRL 112, 155302 (2014)
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'g = '+¡'¡ = 2'+
xq
Adiabatic impulse model : adiabatic evolution between the two LZ events
is the phase accumulated between the two LZ events
Pyt = 4P
yZ(1¡P
yZ) sin
2('s+'dyn+'g
2]geometric phase
S.N. Shevchenko et al. , Phys. Rep. (2010)
S. Gasparinetti et al. PRL (2011)
'+
'¡
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'g = '+¡'¡ = 2'+
Adiabatic impulse model : adiabatic evolution between the two LZ events
is the phase accumulated between the two LZ events
'+
'¡
Pyt = 4P
yZ(1¡P
yZ) sin
2('s+'dyn+'g
2]geometric phase
S.N. Shevchenko et al. , Phys. Rep. (2010)
S. Gasparinetti et al. PRL (2011)
'+
'¡
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'g = '+¡'¡ H(t)jÃa(t)i = ²(t)jÃa(t)i
H(t)jÃ(t)i = i~@tjÃ(t)i
'+ = 'g=2 =
Z tf
ti
hÃji@tÃi]dt+ arghÃa(ti)jÃa(tf)i
jÃ(ti)i=jÃa(ti)
jÃ(tf)i = ei~
R tfti[¡²t+ihÃj@tÃi]dt jÃa(tf)i
hÃa(ti)jÃ(tf)i = ei~
R tfti[¡²t+hÃji@tÃi]dt hÃa(ti)jÃa(tf)i
J. Samuel and R. Bhandari, PRL (1988)
G.G. de Polavieja and E. Sjöqvist, Am. J. Phys. (1998)
'g = ¡
2
Adiabatic evolution
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The geometric phase depends on
* The chiralities of the cones
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The geometric phase depends on
* The chiralities of the cones
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M
M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
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M
-- M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
cf. Haldane model PRL 1988
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M
M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
* The trajectory
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M
M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
* The trajectory
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M M
M M
0
--M M
--M M p
M M
M M
M M
--M M
¡2 atanD
M
2 atanM
D
p
¡2 atanD
M
2 atanM
D
0
The eightfold way …
« graphene » « bilayer »
« Haldane »
D shortest distance to the Dirac point
¹Â±
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M M
M M
0
--M M
--M M p
M M
M M
M M
--M M
p
0
The eightfold way …
« graphene »
BN
« bilayer »
« Haldane »
¡2 atanD
M
2 atanM
D
¡2 atanD
M
2 atanM
D
(t2 ¡ 1)¾x +D¾y +M¾z
(t2 ¡ 1)¾x +Dt¾y +M¾z
(t2 ¡ 1)¾x +D¾y +Mt¾z
(t2 ¡ 1)¾x +Dt¾y +Mt¾z
¹Â±
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M M
M M
0
--M M
--M M p
M M
M M
--M M
M M
p
0
The eightfold way …
« bilayer »
« Haldane »
d shortest distance to the Dirac point 'g = ¡2¹ arctan
·D
M(1 + ¹Â±)
¸¹
+¡+
+¡¡
+++
++¡
¡¡+
¡¡¡¡++
¡+¡
¡2 atanD
M
2 atanM
D
¡2 atanD
M
2 atanM
D
¹Â±
« graphene »
BN
![Page 52: Probing physics of Dirac cones by Landau-Zener interferometry · one Bloch oscillation + Landau-Zener Tunneling Measurement of the proportion of atoms in the upper band . 2 4 E g](https://reader034.vdocument.in/reader034/viewer/2022052005/6018fa18bbe49a6a581c0b8a/html5/thumbnails/52.jpg)
M M
M M
0
--M M
--M M p
M M
M M
--M M
p
0
The eightfold way …
« Haldane »
(t2 ¡ 1)¾x +D¾y +M¾z
(t2 ¡ 1)¾x +Dt¾y +M¾z
(t2 ¡ 1)¾x +D¾y +Mt¾z
(t2 ¡ 1)¾x +Dt¾y +Mt¾z
¡2 atanD
M
2 atanM
D
2 atanM
D
'g = ¡
2
« graphene »
BN
![Page 53: Probing physics of Dirac cones by Landau-Zener interferometry · one Bloch oscillation + Landau-Zener Tunneling Measurement of the proportion of atoms in the upper band . 2 4 E g](https://reader034.vdocument.in/reader034/viewer/2022052005/6018fa18bbe49a6a581c0b8a/html5/thumbnails/53.jpg)
Conclusions and perspectives Universal description of motion and merging of Dirac points in 2D crystals
(-p,p) merging : hybrid semi-Dirac spectrum Cold atoms : Landau-Zener probe of the Dirac points Interference effects Condensed matter : New thermodynamic and transport properties Interaction effects : from Dirac to Schrödinger
-p p 0
p p 2p
Many new realisations in « artificial graphenes »
(p,p) scenario of merging