1. Geneviève Fleury and Xavier Waintal
Do metals exist in two dimensions ?
A numerical finite size scaling approach to many-body localization
Geneviève Fleury and Xavier Waintal
Nanoelectronics group, SPEC, CEA Saclay

2. Do metals exist in two dimensions ?
A numerical finite size scaling approach to many-body localization
Historical background : unexpected metallic behaviour …………………………..in low-disordered 2D samples……. .
Our approach to many-body localization : Quantum Monte Carlo ………………………………………… and finite size scaling……. .
Our results at T=0 : a huge delocalization effect, yet probably no true metal at T=0
Back to experiments : a simple mechanism to explain ……………… the metallic behaviour

3.  ANDERSON LOCALIZATION AND SCALING THEORY
Destructive interferences on impurities
 wavefunction localized on a length 
(Anderson et al, 1979)
g(L) : conductance of the system of size L
Hypothesis : one-parameter scaling  (g)=dlog(g)/dlog(L) function of g only
2D : localized states,  W …… insulator

4. ns: electronic density : sample mobility
 INTERPLAY DISORDER/INTERACTION IN THE EIGHTIES
(disorder)
EFROS-SHKLOVSKII
INSULATOR
EXPERIMENTS IN ~ 1980
« QUANTUM
COULOMB
GLASS »
INSULATOR
INSULATOR
ANDERSON LOCALIZATION
ns: electronic density : sample mobility
(interaction)
INSULATOR AA
WIGNER CRYSTAL
INSULATOR
e-/e- interactions increase the localization : still INSULATOR

5. Do e-/e- interactions induce a metallic behaviour ?
 YET EXPERIMENTALLY IN 1994 …
NEW
(Kravckenko et al, 1994)
Do e-/e- interactions induce a metallic behaviour ?
Is it a « true » metal (at T=0) or a finite temperature effect ?

6. MIT rs rs ~ 10 Conservative theories Interaction induced phenomena
 SOME THEORETICAL ATTEMPTS
Conservative theories
- temperature dependent disorder (Altshuler et al 1999, Das Sarma & Hwang 2000, …) - percolation (He & Xie 1998, Meir 1999, …) … - spin-orbit interaction (Pudalov 1997, Papadakis et al 2000, …)
Interaction induced phenomena
rs ~ 10 rs
MIT
Fermi liquid approach (in the diffusive regime)
Wigner crystal approach
- quantum melting of a Wigner crystal .. (Chakravarty et al 1999, …) phase separation, bubbles and stripes …(Spivak 2002)
- renormalization group (Finkelstein et al 1983, Castellani 1984) RG with infinite number of valleys  instability of Fermi liquid (Punnoose & Finkelstein 2005)
+ others : superconductivity (Phillips et al 1998, …), scaling theory ………….for interacting electrons (Dobrosavljevic et al 1997), …

7. Importance of materials ..(Si-MOSFETs or heterostructures)
 EXPERIMENTAL HINTS
What’s new in current 2D samples? High mobility and low density
- importance of interactions
- importance of disorder
high energy T/TF  15%: importance of excited states
Effect of a parallel magnetic field
B// destroys the metallic behaviour…  importance of spin
(Simonian et al, )
Importance of materials (Si-MOSFETs or heterostructures)
- much stronger metallic behaviour in Si-MOSFETs than in heterostructures strength and nature of the disorder (short or long range) degenerate valleys in MOSFETs

8. Do metals exist in two dimensions ?
A numerical finite size scaling approach to many-body localization
Historical background : unexpected metallic behaviour …………………………..in low-disordered 2D samples……. .
Our approach to many-body localization : Quantum Monte Carlo ………………………………………… and finite size scaling……. .
Our results at T=0 : a huge delocalization effect, yet probably no true metal at T=0
Back to experiments : a simple mechanism to explain ……………… the metallic behaviour

9. Quantum Monte Carlo at T=0 (with Fixed Node approx.)
 OUR METHOD : QUANTUM MONTE CARLO AT T=0
Ingredients : disorder, interaction, spin, valley
Quantum Monte Carlo at T=0 (with Fixed Node approx.)
Measuring localization properties : Thouless conductance g
g  diffusive constant of the center of mass (in imaginary time)
Testing the one-parameter scaling theory

10. Scaling theory remains valid with interaction  INSULATOR AT T=0
 VALIDATING THE APPROACH : SPINLESS ELECTRONS
 Polarized electrons in GaAs
Extracting the
« one parameter »
Localization by interactions
Scaling theory remains valid with interaction  INSULATOR AT T=0
 independent of L (system size)   = localization length in the ……...thermodynamic limit
 Validation of our method to ….study many-body localization

11. Do metals exist in two dimensions ?
A numerical finite size scaling approach to many-body localization
Historical background : unexpected metallic behaviour …………………………..in low-disordered 2D samples……. .
Our approach to many-body localization : Quantum Monte Carlo ………………………………………… and finite size scaling……. .
Our results at T=0 : a huge delocalization effect, yet probably no true metal at T=0
Back to experiments : a simple mechanism to explain ……………… the metallic behaviour

12.  Two-components systems are INSULATORS (at T=0)
 TWO-COMPONENTS SYSTEMS (with spins, 1 valley OR spinless, 2 valleys)
 Non polarized electrons in GaAs OR polarized electrons in Si-MOSFETs
Electronic interactions delocalise the 2D system at small rs
No visible deviation from scaling theory
 Two-components systems are INSULATORS (at T=0)

13. PRACTICALLY A METAL :    >> L
 FOUR-COMPONENTS SYSTEMS (WITH SPINS AND 2 VALLEYS)
 Non polarized electrons in Si-MOSFETs
PRACTICALLY A METAL :    >> L
IS IT A TRUE METAL (AT T=0) ? Probably NO
Dramatic delocalization by electronic interactions
  0 : METAL ?
BUT no visible deviation from scaling theory

14. Do metals exist in two dimensions ?
A numerical finite size scaling approach to many-body localization
Historical background : unexpected metallic behaviour …………………………..in low-disordered 2D samples……. .
Our approach to many-body localization : Quantum Monte Carlo ………………………………………… and finite size scaling……. .
Our results at T=0 : a huge delocalization effect, yet probably no true metal at T=0
Back to experiments : a simple mechanism to explain ……………… the metallic behaviour

15. Tpolarization .(Monte Carlo) Bpolarization (Monte Carlo)
(Altshuler et al, 2000)
Characteristic temperature : Tc~0.15TF
Temperature effect
Tpolarization .(Monte Carlo)
(Altshuler et al, 2000)
Characteristic temperature : Tc~0.15TF
Temperature effect
 ENERGY SCALE OF THE OBSERVED METALLIC BEHAVIOR
Bpolarization (Monte Carlo)
In-plane magnetic field effect
(Mertes et al, 1999)
 Characteristic energy = polarization energy without any adjustable parameters !!!

16.  ENERGY SCALE OF THE OBSERVED METALLIC BEHAVIOR
 Energy scale = polarization energy, checked in various samples
 An « high » energy physics is probed in experiments

17. The polarized system conducts much less than the non-polarized one
 A SIMPLE MECHANISM TO EXPLAIN EXPERIMENTS AT FINITE T
With interaction, it is the opposite !
Without interaction, the polarized system is less localized than the non polarized one
Without interaction, the polarized system is less localized than the non polarized one
The polarized system conducts much less than the non-polarized one

18. 1/NP - 1/P  QUANTITATIVE AGREEMENT WITH EXPERIMENTAL FEATURES
- characteristic T and in-plane B : the polarization energy
- density range for the observed metallic behavior
Metallic phase
1/NP - 1/P
- importance of materials  strength of the disorder  valley degeneracy in MOSFETs: delocalisation much more dramatic

19. Finally, do metals exist in 2D (at T=0) ?
 CONCLUSION
Finally, do metals exist in 2D (at T=0) ?
 1-C (spinless, one valley) : NO (and correlations localize the 2D gas)
2-C (with spin OR 2 valleys) : NO (but correlations delocalize the 2D gas)
4-C (with spin AND 2 valleys ) : PROBABLY NO …………………………………… (but correlations delocalize drastically the 2D gas)
What happens in experiments ?
With interactions, the polarized system conducts much less than the non-polarized one
A genuine non perturbative effect of interactions. Yet, probably not a « true » metal at T=0

20. How to calculate the Thouless conductance in QMC ?
 MEASURING LOCALIZATION: THOULESS CONDUCTANCE
Persistent current :
Thouless conductance :
 Thouless conductance = good measure of the localization properties of the system
How to calculate the Thouless conductance in QMC ?
gx  diffusive constant of the center of mass (in imaginary time) along the x direction

21. log g0 independent of kFl and rs
 TESTING THE ONE-PARAMETER SCALING THEORY (spinless e-)
Diffusive regime (Ohm’s law)  g independent of L
log g0 independent of kFl and rs
Localized regime g=g0exp(-L/)   = localization length
Data with or without interactions
Scaling function
Diffusive regime
Localized regime

22. SIMPLE MODEL TO ADD TEMPERATURE
 TOWARD A COMPREHENSIVE QUANTITATIVE PICTURE ;;;;;(work in progress)
QMC RESULTS AT T= Ep(rs, kFl) polarization energies (rs, kFl, p) localization lengths
SIMPLE MODEL TO ADD TEMPERATURE
Resistivity (ns, T, B//)
Good agreement with experimental data for B<Bp (one adjustable parameter per curve: L phase coherence length)
First check : magnetoconductance at fixed density, for various T ……………(compared to data of Simonian et al 1998)

23. The 2C ground state is not delocalized (contrary to 4C)
 METAL / INSULATOR TRANSITION IN HETEROSTRUCTURES
The 2C ground state is not delocalized (contrary to 4C)
The temperature has two conflicting effects
it activates transport mechanims (VRH)
it populates the more localized excited polarized states

24.  FINITE SIZE EFFECTS
from the long range part of the Coulomb interaction  Ewald summation
from the definition of the Fermi surface
No disorder High rs=40  N40
With disorder Low rs  Nmin=16 (1C)
For , the situation is even more favourable: C : Nmin= C : Nmin= C : Nmin=36 4C 1C 2C

25.  POLARIZATION ENERGY

26. 4C  SUSCEPTIBILITY AT ZERO TEMPERATURE
Without disorder: comparison to numerical results of Senatore et al 2008 4C
With disorder : no agreement with experimental data
With temperature (in experiments) :
and Hc increases with T

27.  LOCALIZATION LENGTH AS A FUNCTION OF POLARIZATION P
Apparent breakdown of one parameter scaling theory at finite temperature

28.  IS THE 4C SYSTEM A « TRUE » METAL AT T=0 ?
PROBABLY NO
- no visible deviation to one-parameter scaling
- localization lengths as a function of disorder
Without interaction:
rs=0
With interaction:

29.  PRINCIPE DE NOTRE METHODE MONTE CARLO QUANTIQUE
(Methode Green Function Monte Carlo, Trivedi et Ceperley, PRB, 1990)
Approximation Fixed Node : ………..H  H FN de signe constant  pas de « problème de signe »
donc 0  0FN de meme structure nodale que G
- Notre choix de fonctions guides G
G(R) = Slater x Jastrow , proche du fondamental 0
G « liquide » : solution exacte sans interaction (rs = 0)
G « Hartree » : solution avec interaction (rs  0), en champ moyen