Probing physics of Dirac cones by Landau-Zener interferometry · one Bloch oscillation +...

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



K K’

Berry phase

p

- Berry phase

-p Graphene electronic spectrum


Dirac cones, from graphene to cold atoms


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)


'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 -


'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


Nature 483, 302 (2012)

dkF

dt


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


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


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


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


Theory

ZP

2 (1 )t Z ZP P P -



Experiment

Theory

ZP

2 (1 )t Z ZP P P -


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

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

k


'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

M

M



* The sign of the gap M

M

-- M




cf. Haldane model PRL 1988

M

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


« 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


« 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


« 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

-p p 0

p p 2p

Many new realisations in « artificial graphenes »

(p,p) scenario of merging

Probing physics of Dirac cones by Landau-Zener interferometry · one Bloch oscillation +...

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