1. Vladimir Cvetković Physics Department Colloquium Colorado School of Mines Golden, CO, October 2, 2012 Electronic Multicriticality In Bilayer Graphene National High Magnetic Field Laboratory Florida State University

2. National High Magnetic Field Laboratory Superconductivity http://www.magnet.fsu.edu/mediacenter/seminars/winterschool2013/ TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

3. Collaborators Dr. Robert E. Throckmorton Prof. Oskar Vafek V. Cvetkovic, R. Throckmorton, O.Vafek, Phys. Rev. B 86, 075467 (2012) NSF Career Grant (O. Vafek): DMR-0955561

4. Graphite Carbon allotrope Greek (γράφω) to write Graphite: a soft, crystalline form of carbon. It is gray to black, opaque, and has a metallic luster. Graphite occurs naturally in metamorphic rocks such as marble, schist, and gneiss. U.S. Geological Survey Mohs scale 1-2

5. Graphite electronic orbitals Orbitals: sp 2 hybridization (in-plane bonds) p z (layer bonding) Hexagonal lattice space group P6 3 /mmc

6. Massless Dirac fermions in graphene  bond Strong cohesion (useful mechanical properties)  bond Interesting electronic properties

7. Massless Dirac fermions in graphene Sufficient conditions: C 3v and Time reversal Necessary conditions: Inversion and Time reversal (*if Spin orbit coupling is ignored) Dirac cones: Tight binding Hamiltonian where Spectrum Velocity: v F = t a ~10 6 m/s

8. Graphene fabrication Obstacle: Mermin-Wagner theorem Fluctuations disrupt long range crystalline order in 2D at any finite temperature Epitaxially grown graphene on metal substrates (1970): Hybridization between p z and substrate Exfoliation: chemical and mechanical Scotch Tape method (Geim, Novoselov, 2004)

9. YouTube Graphene Making tutorial (Ozyilmaz' Group)

10. How to see a single atom layer? Si SiO 2 300nm graphene P. Blake, et al, Appl. Phys. Lett. 91, 063124 (2007)

11. Ambipolar effect in Graphene A. K. Geim & K. S. Novoselov, Nature Materials 6, 183 (2007) I sd VgVg Graphene Mobility:  = 5,000 cm 2 /Vs (SiO2 substrate, this sample = 2007)  = 30,000 cm 2 /Vs (SiO2 substrate, current)  = 230,000 cm 2 /Vs (suspended)

12. Graphene in perpendicular magnetic field: QHE I sd VgVg Graphene H Hall bar geometry IQHE: Novoselov et al, Nature 2005 Room temperature IQHE: Novoselov et al, Science 2007

13. Graphene in perpendicular magnetic field: FQHE FQHE: C.R. Dean et al, Nature Physics 7, 693 (2011)

14. Bilayer Graphene Two layers of graphene Bernal stacking Tight binding Hamiltonian Spectrum

15. Trigonal warping in Bilayer Graphene Parabolic touching is fine tuned (  3 = 0) Tight binding Hamiltonian with  3 : Vorticity:

16. Bilayer Graphene in perpendicular magnetic field I sd VgVg BLG H Hall bar geometry IQHE: Novoselov et al, Nature Physics 2, 177 (2006)

17. Widely tunable gap in Bilayer Graphene Y. Zhang et al, Nature 459, 820 (2009)

18. Trilayer Graphene ABA and ABC stacking

19. Band structure ABC Trilayer Graphene Tight binding Hamiltonian

20. Non-interacting phases in ABC Trilayer Graphene Phase transitions, even with no interactions Spectrum:  3+3+ 9-9- 3-3-  c2  c1

21. Electron interactions (Mean Field) An example: Bardeen-Cooper-Schrieffer Hamiltonian (one band, short range) Superconducting order parameter Decouple the interaction into quadratic part and neglect fluctuations The transition temperature Debye frequency  D =  2 /2m Only when g>0 ! 0

22. Different theories at different scales (RG) What if  D were different?Make a small change in  : How to keep T c the same? This example shows that the interaction is different at different scales. The main idea of the renormalization group (RG): select certain degrees of freedom (e.g., high energy modes, high momenta modes, internal degrees of freedom in a block of spins... ) treat them as a perturbation the remaining degrees of freedom are described by the same theory, but the parameters (couplings, masses, etc) are changed Our example (BCS): treat high momentum modes perturbatively (one- loop RG)... but RG is much more powerful and versatile than what is shown here.

23. Finite temperature RG Revisit our example (BCS) Treat fast modes perturbatively The change in the coupling constant The effective temperature also changes In this simple example we can solve the  -function... and find the T c

24. Electron Interactions in Single Layer Graphene Rich and open problem, nevertheless in zero magnetic field: Short-range interactions: irrelevant (in the RG sense) when weak. As a consequence, the perturbation theory about the non- interacting state becomes increasingly more accurate at energies near the Dirac point Coulomb interactions: marginally irrelevant (in the RG sense) when weak semimetal*insulatorQCP O. Vafek, M.J. Case, Phys. Rev. B 77, 033410 (2008) In either case, a critical strength of e-e interaction must be exceeded for a phase transition into a different phase to occur. Hence, this is strong coupling problem.

25. Electron Interactions in Bilayer Graphene Short range interactions: marginal by power counting Classified according to IR’s of D 3d The kinetic part of the action where Fierz identities implemented

26. Symmetry allowed Dirac bilinears (order parameters) in BLG VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012)

27. RG in Bilayer Graphene (no spin) Fierz identities reduce no of independent couplings to 4 O. Vafek, K. Yang, Phys. Rev. B 81, 041401(R) (2010) O. Vafek, Phys. Rev. B 82, 205106 (2010) Susceptibilities (leading instabilities, all orders tracked simultaneously) Possible leading instabilities: nematic, quantum anomalous Hall, layer- polarized, Kekule current, superconducting

28. Experiments on Bilayer Graphene A.S. Mayorov, et al, Science 333, 860 (2011) Low-energy spectrum reconstruction

29. RG in Bilayer Graphene (spin-1/2) Finite temperature RG with trigonal warping VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012) Susceptibilities (determine leading instabilities) … used to be tanh(1/2t)

30. Forward scattering phase diagram in BLG Only

31. General phase diagram (density-density interaction) Density-density interaction Bare couplings in RG:

32. Coupling constants fixed ratios In the limit the ratios of g’s are fixed The leading instability depends on the ratios (stable ray) Stable flows: Target plane Ferromagnet Quantum anomalous Hall Loop current state Electronic density instability (phase segregation)

33. RG in Trilayer Graphene Belongs to a different symmetry class Number of independent coupling constants in H int : 15 Spectrum RG flow

34. Generic Phase Diagram in Trilayer Graphene

35. Trilayer Graphene (special interaction cases) Forward scattering Hubbard model (on-site interaction)

36. Generic Phase Diagram in Trilayer Graphene