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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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
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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
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
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
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)
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
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 ?
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
Motion and merging of Dirac points
' t t
't tt
Motion and merging of Dirac points 't t
t
« Semi-Dirac »
2' = t t
massive !
massless !
yq
xq
'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
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
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
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
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
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
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)
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 »
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…”
Atoms are trapped in an optical lattice potential and form an artifical crystal
Nature 483, 302 (2012)
Honeycomb Brick wall
'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
t’=2 t’=1.414 t’=1
t’=2 t’=1.414 t’=1
Honeycomb
Brick wall
Bloch oscillations = uniform motion in reciprocal space
Nature 483, 302 (2012)
dkF
dt
40K
Bloch oscillations = uniform motion in reciprocal space
Nature 483, 302 (2012)
dkF
dt
Bloch oscillations = uniform motion in reciprocal space
Nature 483, 302 (2012)
dkF
dt
l l 40K
l l
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
2
4
gE
c F
ZP e
p-
Landau-Zener transition
1 ZP-
F tk
gE
c
k
E
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
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)
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
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
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
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
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
Single Dirac cone: Fermi sea tunneling
Maximum slightly inside the Dirac phase
* 0
* 0
y
ZP
Transfer probability for a cloud of finite size
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)
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
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
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
k
Lih King Lim, Jean-Noel Fuchs, G. M., PRL 112, 155302 (2014)
'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)
'+
'¡
'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)
'+
'¡
'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
The geometric phase depends on
* The chiralities of the cones
The geometric phase depends on
* The chiralities of the cones
M
M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
M
-- M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
cf. Haldane model PRL 1988
M
M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
* The trajectory
M
M
The geometric phase depends on
* The chiralities of the cones
* The sign of the gap M
* The trajectory
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±
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±
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
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
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