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Vladimir Cvetković

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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

http://www.magnet.fsu.edu/mediacenter/seminars/winterschool2013/

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

- Hexagonal lattice
- space group P63/mmc

- Orbitals:
- sp2 hybridization (in-plane bonds)
- pz (layer bonding)

Interesting electronic properties

p-bond

s-bond

Strong cohesion (useful mechanical properties)

Tight binding Hamiltonian

where

Spectrum

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)

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)

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

graphene

300nm

SiO2

Si

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

Isd

Graphene

Vg

- 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)

Isd

Hall bar geometry

H

Graphene

Vg

IQHE: Novoselov et al, Nature 2005

Room temperature IQHE: Novoselov et al, Science 2007

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

Two layers of graphene

Bernal stacking

Tight binding Hamiltonian

Spectrum

Parabolic touching is fine tuned (g3 = 0)

Tight binding Hamiltonian with g3 :

Vorticity:

Isd

Hall bar geometry

H

BLG

Vg

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

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

ABA and ABC stacking

Tight binding Hamiltonian

Spectrum:

Phase transitions, even with no interactions

D

3-

9-

3+

Dc2

Dc1

An example: Bardeen-Cooper-Schrieffer Hamiltonian (one band, short range)

Superconducting order parameter

Decouple the interaction into quadratic part and neglect fluctuations

0

The transition temperature

Debye frequency wD = L2/2m

Only when g>0 !

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.

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

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*

QCP

insulator

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.

The kinetic part of the action

where

Short range interactions: marginal by power counting

Classified according to IR’s of D3d

Fierz identities implemented

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

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

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

Low-energy spectrum reconstruction

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)

Only

Density-density interaction

Bare couplings in RG:

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)

Belongs to a different symmetry class

Number of independent coupling constants in Hint: 15

Spectrum

RG flow

Hubbard model

(on-site interaction)

Forward scattering