1. Effects of Interaction and Disorder in Quantum Hall region of Dirac Fermions in 2D Graphene Donna Sheng (CSUN) In collaboration with: Hao Wang (CSUN), L. Sheng (UH), Z.Y. Weng (Tsinghua) F. D. M. Haldane (Princeton) and L. Balents (UCSB) Supported by DOE and NSF

3. Introduction: Experiment observations of “half- integer” and odd integer QHE in graphene an atomic layer of carbon atoms forming honeycomb lattice Carbon nanotubes are a beautiful material. Problem is how to make them in wafers and with reproducible properties. You can't distinguish between semiconducting and metallic nanotubes. You've got uncontrollable material. Graphene is essentially the same material, but unrolled carbon nanotubes. K. S. Novoselov et al., Nature (2005) Y. Zheng et al., Nature (2005) One of applications is to make graphene transistors of 10nm size.

4. Experiment observations of “half-integer” integer QHE in graphene K. S. Novoselov et al., Nature (2005)

5. Y. Zheng et al., Nature (2005) Columbia Univ. Theoretical work using continuous model for Dirac fermions can account/predicted such quantizations: Gusynin et al. Peres et al., Zheng and Ando Relation with band structure of honeycomb lattice model?

6. Three regions of IQHE in the energy band IQHE for Dirac fermions in the middle D.N.Sheng, Phys. Rev. B73, 233406 (2006) 

7. Effect of disorder and phase diagram: PRB 73 (2006)

8. Phase diagram Disorder splits one extended level at the center of n=0 LL to 2 critical points, leaving n=0 region an insulating phase

9. Experiment discovers  IQHE and “ =0” insulating phase Y. Zhang et. al., PRL 2006 Interaction has to be taken into account to explain the =1IQHE---Pseudospin Ferromagnet

10. delocalization of Dirac fermions at B=0 (a different issue) Transfer matrix calculation of the “finite size localization length” for quasi-1D system with width M, it indicates “delocalization” at Dirac point E_f=0 at E<1.0t E_f=0 M

11. Experiment discovers  IQHE and “ =0” insulating phase Y. Zhang et. al., PRL 2006 Interaction has to be taken into account to explain the =1 IQHE---Pseudospin Ferromagnet

12. More experiments by Z. Jiang et. al. on activation gap of IQHE For ,  E is proportional to e^2/  l

13. Interaction and pseudospin FM state: Theoretical works Nomura & MacDonald Stoner criteria for pseudospin FM Alicea & Fisher lattice effect is relevant Yang et al., Gusynin et al. Toke & Jain, Goerbig et al. continuum model, SU(2)*SU(2) symmetry Haldane’s Pseudo-Potential gives rise to incompressible state, SU(4) invariant Our motivation: detailed nature of quantum phases, quantitative behavior of systems Competitions between: Coulomb interaction, lattice, and disorder scattering effect based on exact calculations

14. Exact diagonalization using lattice model Only keep states inside the top-Landau level, large lattice size and keep a degeneracy of Ns around 20

15. Energy spectrum for pure system: PFM ordering A-sub, B-sub,  Ising PFM No pseudospin conservation for higher LL!! = 3 XY plane PFM (zero-energy states) CDW

16. the excitation energy gap (with double occupation) the excitation gap scales with 1/Ne, possibly extrapolates to zero at large Ne limit W |S_z|

17. Directly look at the transport property instead of “gaps”

18. The destruction of odd IQHE is due to the mixing of various Chern numbers

19. boundary phase Just get  at all nodes of mesh of 100-1000 points, overlap of  at nearest points  x,  y ) Chern number “IS” Hall conductance  22 0 D. J. Thouless et al 1982, J. E. Avron et al. 1883 D.N. Sheng et al., PRL 2003; Sheng, Balents, Wang Xin et al. PRB (FQHE) C is for many-body

20. The importance of mobility gap (activation gap of experiment): from direct Hall conductance and Chern number calculations States inside mobility gap the size of the Mobility gap finite size scaling confirms a finite transport gap at large size limit (more data are coming)

21. e^2  l Fluctuation of Chern numbers determine a mobility edge

22. typing Example of comparing mobility gap with experiments for 1/3 FQHE: D. N. Sheng et al (2003) PRL, Xin et al. (2005) PRB

23. Phase diagram for Odd IQHE states

24. n=3(LL=2&3) Graphene, 12/24 systerm, a/b=0.74,q * =(0,±0.595) Symmetry broken states (stripes and bubbles) in n=3 and n=4 Dirac LLs

25. ust=0.2 ust=2.4ust=3.2 Disorder-Caused Phase Transition

27. 8/24 system, LL2+3, a/b=0.65, bubble phase Structure factor:S0(q) Correlation function:G(r)

28. 8/24 system, LL2+3, a/b=0.86, stripe phase Structure factor:S0(q) Correlation function:G(r)

29. Summary: The  IQHE state is an Ising and valley polarized PFM The  IQHE state is a valley mixing, xy plane polarized PFM Critical Wc, mobility gap, and quantum phase diagram Symmetry broken states (stripes and bubbles) are predicted to be The ground state in higher (n=3 and n=4) Dirac LLs Open questions: Is there a Non-Abelian FQHE at n=1 and n=2 Dirac LLs?