Persistent currents in two dimension: New regimes induced by the interplay between electronic

correlations and disorderZoltán Ádám Németh

Jean-Louis Pichard

CEA - Saclay, Service de Physique de l'Etat Condensé

Outline:

• Overview of the strongly correlated 2D electron-gas problem;

• Introduction of the numerical model: fewinteracting particles on a lattice

• Persistent current maps with disorder

References:

Z.Á. Németh and J.-L. Pichard, Eur. Phys. J. B 45, 111 (2005)

J.-L. Pichard and Z.Á. Németh, J. Phys. IV France 131, 155 (2005)

• Fermi: weakly-interacting quantum particles• Wigner: strongly interacting particles

Quantum solid state physics

TWO LENGTH SCALES:• average interparticle spacing a• Bohr-radius aB

Dimensionless scaling parameter: rs= a/aB

Weak interaction limit

• Electrons localized in k-space (Fermi liquid behavior).

( ) [ ]NRyrOra

raE s

ss

Fermi ×++= ln221

0

a1 = 2.0a2 = –1.6972

M. Gell-Mann and K. A. BruecknerPhys. Rev. 106, 364 (1957)

0→srhigh density limit,kinetic energy

exchange energy

Strong interaction limit

• Electrons localized in real space (Wigner crystal)

( ) [ ]NRyrOrc

rcE s

ss

Wigner ×++= − 22/32/31

0

W. J. Carr Jr., Phys. Rev. 122, 1437 (1961)

c1 = –2.21c3/2 = 1.63

low density limit, ∞→srclassical elctrostatic energy

quantum zero-point motion

• Fixed node Monte-Carlo method:

B. Tanatar and D. M. CeperleyPRB 39, 5005 (1989)

Quantum Monte-Carlo

solid

polarized liquid

rs

GS energy

rs = 37±5

unpolarized liquid

Semiconductor heterostructures

Since the ’70s it is possible to fabricate 2D electron gas in semiconductor devices.

Electron density and rs are varied through voltage gates.Example:

quantum point-contact

Unexpected metallic behavior in 2D

In ultra-clean heterostructures:rs can reach ≈ 40

Observed metallic behavior at intermediate rs

• S.V. Kravchenko et al., PRB 50, 8039 (1994)

• J. Yoon et al., PRL 82, 1744 (1999)

resistivity

temperature

low density

intermediate density

Hybrid phase in QMCDensity-density correlation function

Hybrid phase: nodal structure of

Slater determinants in the crystal potential:

mixed liquid-solid behaviorH. Falakshahi, X. Waintal: Phys. Rev. Lett. 94, 046801 (2004)

rs

Theory for intermediate rsStill mainly speculations...

• Andreev-Lifshitz « supersolid » state (relation with He-physics)

• Inhomogeneous phases, stripes and bubbles (B. Spivak)

Relation with the physics of high-Tc cuprates and with Hubbardmodel (high lattice filling, contact interaction).

Lattice modelN spinless fermions on L × L square lattice with periodic

boundary conditions (lattice spacing s):

seU

2

=2

2

2mst h

=N

raar l

Bs π2

→=t

ULrl =

the t and U parameters of this Hamiltonian:

THE rs AND rl PARAMETERS:

discrete Laplacian

jj

jji

ij

ji

i,jji nW

dnnUccN-t ˆˆˆ

2ˆˆ4H ∑+∑⋅+⎟

⎠⎞

⎜⎝⎛ ∑⋅=

≠ ><

+ ε

disorder

Persistent current

02

)0,1(0Im2)( ΨΨ=Φ Φ+

+ Lijj

longj eccj π

)()0()(~1000 Φ−=Φ=Φ∆= ∑ EEEj

LI

j

longjlong

Longitudinal and transverse currents

•local:

•total:

Transverse current is analogous.

Persistent currents with disorder

• Effects of an infinitesimal disorder: new lattice perturbative regime

– Ballistic motion– Coulomb Guided Stripes– Localization if the Wigner crystal

Can be relevant in realmaterials

e.g. Cobalt-oxides(NaxCoO2)

• effective mass: m*/m = 175• relative dielectric constant: εr = 20• lattice spacing s = 2.85 Å• carrier density depends on Na+

concentration

Lemmens et al., cond-mat/0309186

Strong interaction, lattice regimes

Lattice perturbation theory when t → 0

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ +−=⎟

⎠⎞⎜

⎝⎛ Φ=Φ yxeff K

LNKtE coscos22

0 π

( ) ( )yxeff KKtE coscos20 +−==Φ

( ) 2

20 9

20~L

tEEI efflatticeπ

≈⎟⎠⎞⎜

⎝⎛ Φ=Φ−=Φ

where teffeff describes the rigid hopping of the three particle« molecule »

22

63

491296

πULtteff =

Example: the persistent current

-t

-t

-t∑⋅+⎟⎠⎞

⎜⎝⎛ ∑⋅=

≠ ><

+

jiij

ji

i,jji d

nnUccN-t2

4H

perturbation

rigid hopping

Ballsitic motion (BWM)

N=3 L=9 W=0.01 t=1 U=300

∑ +−−=',

'4Hjj

jjeffCouleff CCtENt

331~ −

−N

N

N

eff LU

tt

Effective Hamiltonian:

'jj CC+

Density mapPersistent current map

Coulomb Guided Stripes (CGS)

N=3 L=9 W=1 t=1 U=1000

13/2

3/2

0

)2cos(~)(~ −

ΦΦ∆ L

Leff

long Wt

EIπ

331~ −

−N

N

N

eff LU

tt

Disorder correction in the effective Hamiltonian:

∑∑ +−−= +

jjj

jjjjeffCouleff NWCCtENt ˆ'4H

',' ε

longtrans II =transverse current:

current of a collective motion

Localized Wigner Molceule (LWM)

N=3 L=6 W=20 t=1 U=1000

1

33

0)2cos(~)(~ −

− ΦΦ∆ L

LL

long ULtEI π

Standard perturbation therory:

current of independent particles

0=transItransverse current:

Numerical check of the different regimes

ballistic

localized

stripe

L=6, t=1 L=6, W=1

localized

stripe

longitudinallongitudinal & transverse

Phase diagram for weak disorder

62

2

LUt

tWstripe ∝

2/32/1

2/1

LUt

tWloc ∝

localized

stripe

ballistic

solidliq. hyb.

Critical lines:

continuum lattice

In case of long-range interaction, lattice models without disorder

exhibit a latice-continuum transition.

Continuum perturbation theory when rs → ∞

Zero point motion in the vibrating mode of the molecule

( ) ⇒++∇+∇+∇−≈ XMXrrh ˆ

2H 2

322

21

2

elEm

( ) ( )26

25

24

23

6

12

22

4102

H χχχχχα α

+++++∂∂

−≈ ∑=

BBEm elh

3

2

246

DeB π

=

( )TLelEE ωω ++== h0K mB

L20

=ωmB

T8

=ω

Longitudinal modes Transverse modes

2nd order expansion around equilibrium

where

D

Limit for the zero point motion

E0: ground state energyFN: scaling function

Lattice behavior:

Harmonic vibration of the solid in the continuum:

),0,()0,,(),,(),,(

0

00

tULEtULEtULEtULFN =

=−=

N = 3L = 6

L = 9

electrostatic energy

lrF 2327.03 =

NtEE el 40 =−

L = 12L = 15L = 18

rl = UL/t

Example:Three spinless fermions on L × L lattice

Persistent currents with disorder

Do we see similar thing with disorder?

• Effect of an intermediate disorder in the continuum limit:– Coulomb guided stripes– Level crossing and supersolid behavior

N=3 L=9 W=1 t=1 U=50

Parallel density and current

Continuum version

of the Coulomb

guided stripe

N=3 L=9 W=1 t=1 U=1000

Coulomb guided

stripe on a lattice

N=3 L=9 W=1 t=1 U=7

Legett’s rule:

1D motion diamagnetic

means even number of

particles

Sign of supersolidSign of supersolid

N=3 L=9 W=1 t=1 U=15

Crossover regime

Disconnected current

and density

N=3 L=9 W=1 t=1 U=7

Legett’s rule:

1D motion diamagnetic

means even number of

particles

Sign of supersolidSign of supersolid

N=3 L=9 W=1 t=1 U=15

Crossover regime

Disconnected current

and density

Level crossing & strong disorder

paramagnetic

diamagentic

Strong disorder

UWglass ~Crossover:

The presence of a

level crossing is

specific to N.

Weak disorder

Phase diagram (N=3)

continuum lattice

localized

stripe

ballistic

glass

solidliq. hyb.

Conclusion

• In the presence of a weak disorder, we haveidentified for large U/t three lattice regimes,characterized by different power-law decaysas a function of U, t and W, L, N.

• The physics of the continuum is also affected by disorder (signatures of the supersolid).