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

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Electronic Multicriticality

In Bilayer Graphene

Vladimir Cvetković

National High Magnetic Field Laboratory

Florida State University

Physics Department Colloquium

Colorado School of Mines

Golden, CO, October 2, 2012



National High Magnetic Field Laboratory

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Dr. Robert E. Throckmorton

Prof. Oskar Vafek

NSF Career Grant (O. Vafek): DMR-0955561

V. Cvetkovic, R. Throckmorton, O.Vafek, Phys. Rev. B 86, 075467 (2012)


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

Graphite electronic orbitals

  • Hexagonal lattice

  • space group P63/mmc

  • Orbitals:

  • sp2 hybridization (in-plane bonds)

  • pz (layer bonding)

Massless Dirac fermions in graphene

Interesting electronic properties



Strong cohesion (useful mechanical properties)

Massless Dirac fermions in graphene

Tight binding Hamiltonian



Velocity: vF = t a ~106 m/s

Dirac cones:

Sufficient conditions:

C3v and Time reversal

Necessary conditions:

Inversion and Time reversal

(*if Spin orbit coupling is ignored)

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 pz and substrate

Exfoliation: chemical and mechanical

Scotch Tape method (Geim, Novoselov, 2004)

YouTube Graphene Making tutorial (Ozyilmaz' Group)

How to see a single atom layer?

P. Blake, et al, Appl. Phys. Lett. 91, 063124 (2007)





Ambipolar effect in Graphene

A. K. Geim & K. S. Novoselov, Nature Materials 6, 183 (2007)




  • Mobility:

  • m = 5,000 cm2/Vs (SiO2 substrate, this sample = 2007)

  • m = 30,000 cm2/Vs (SiO2 substrate, current)

  • m = 230,000 cm2/Vs (suspended)

Graphene in perpendicular magnetic field: QHE


Hall bar geometry




IQHE: Novoselov et al, Nature 2005

Room temperature IQHE: Novoselov et al, Science 2007

Graphene in perpendicular magnetic field: FQHE

FQHE: C.R. Dean et al, Nature Physics 7, 693 (2011)

Bilayer Graphene

Two layers of graphene

Bernal stacking

Tight binding Hamiltonian


Trigonal warping inBilayer Graphene

Parabolic touching is fine tuned (g3 = 0)

Tight binding Hamiltonian with g3 :


Bilayer Graphene in perpendicular magnetic field


Hall bar geometry




IQHE: Novoselov et al, Nature Physics 2, 177 (2006)

Widely tunable gap inBilayer Graphene

Y. Zhang et al, Nature 459, 820 (2009)

Trilayer Graphene

ABA and ABC stacking

Band structureABC Trilayer Graphene

Tight binding Hamiltonian

Non-interacting phases inABC Trilayer Graphene


Phase transitions, even with no interactions







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 wD = L2/2m

Only when g>0 !

Different theories at different scales (RG)

What if wD were different?

Make a small change in L:

How to keep Tc 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.

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 b-function

... and find the Tc

Electron Interactions inSingle 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




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.

Electron Interactions inBilayer Graphene

The kinetic part of the action


Short range interactions: marginal by power counting

Classified according to IR’s of D3d

Fierz identities implemented

Symmetry allowed Dirac bilinears (order parameters) in BLG

VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012)

RG in BilayerGraphene (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

Experiments on Bilayer Graphene

A.S. Mayorov, et al, Science 333, 860 (2011)

Low-energy spectrum reconstruction

RG in Bilayer Graphene (spin-1/2)

VC, R.E. Throckmorton, O. Vafek, Phys. Rev. B 86, 075467 (2012)

Finite temperature RG with trigonal warping

… used to be tanh(1/2t)

Susceptibilities (determine leading instabilities)

Forward scattering phase diagram in BLG


General phase diagram(density-density interaction)

Density-density interaction

Bare couplings in RG:

Coupling constantsfixed 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)

RG in Trilayer Graphene

Belongs to a different symmetry class

Number of independent coupling constants in Hint: 15


RG flow

Generic Phase Diagramin Trilayer Graphene

Trilayer Graphene(special interaction cases)

Hubbard model

(on-site interaction)

Forward scattering

Generic Phase Diagramin Trilayer Graphene

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