■ Charge carriers in graphene and effective field

theory

■ Calculations on hypercubic lattice

■ Calculations on hexagonal lattice

BNL 25 June 2012

Numerical study of the monolayer graphene phase diagram

P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov

ArXiv:1204.0921; ArXiv:1206.0619

■ Charge carriers in graphene and effective field

theory

■ Calculations on hypercubic lattice

■ Calculations on hexagonal lattice

BNL 25 June 2012

Numerical study of the monolayer graphene phase diagram

P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov

ArXiv:1204.0921; ArXiv:1206.0619

■ Charge carriers in graphene and effective field

theory

■ Calculations on hypercubic lattice

■ Calculations on hexagonal lattice

BNL 25 June 2012

Numerical study of the monolayer graphene phase diagram

P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov

ArXiv:1204.0921; ArXiv:1206.0619

■ Charge carriers in graphene and effective field

theory

■ Calculations on hypercubic lattice

■ Calculations on hexagonal lattice

BNL 25 June 2012

Numerical study of the monolayer graphene phase diagram

P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A. Zubkov, V.V. Braguta, M.I. Polikarpov

ArXiv:1204.0921; ArXiv:1206.0619

QCD and Graphene

Carbon atom

Some allotropes of carbon: a) diamond; b) graphite; c)lonsdaleite; d–f)

fullerenes (C60, C540, C70); g) amorphous carbon; h) carbon nanotube.

Graphene

Hexagonal lattice = triangular lattice + triangular lattice

Tight binding Hamiltonian

0.142a nm

2.7eV

hopping parameter

lattice spacing

2 4 2 2

Relativistic particle

Massless particle

Charge carrier in Graphene

;300

F F

E m c p c

E cp

cE v p v

we can neglect Ai;

Effective field theory for graphene Four component Dirac fermions + Coulomb field

After transformation

we can neglect Ai;

Effective field theory for graphene Four component Dirac fermions + Coulomb field

After transformation

we can neglect Ai;

Effective field theory for graphene Four component Dirac fermions + Coulomb field

After transformation

2

1g g

Graphene on substrate

g

g

Graphene in the dielectric media

substrate

graphene

2if ( 1.11) graphene is insulator (?)

1

crit

g g

We can vary the effective coupling in graphene!

There exists the additional renormalization T.O. Wehling et al. arXiv: 1101.4007

(2+1)D fermions

(3+1)D Coulomb

2

1g g

On substrate

Effective theory of charge carriers in graphene

/ 300Fv c

300 2.16 1g

2

1g g

1. “Massless” four component Dirac fermions

2. Fermi velocity is

3. The effective charge is

4. We can vary the effective charge if we vary

the dielectric permittivity of the substrate

Vacuum ε=1

SiO2 ε ~ 3.9

SiC ε ~ 10.0 There exists the additional renormalization

T.O. Wehling et al. arXiv: 1101.4007

Simulation of the effective graphene theory Approach 1, hypercubic lattice

(2+1)D fermions

(3+1)D Coulomb

J. E. Drut, T. A. Lahde, and E. Tolo (2009-2011)

P.V. Buividovich, O.V. Pavlovsky, M.V. Ulybyshev, E.V. Luschevskaya, M.A.

Zubkov, V.V. Braguta, M.I. P. (2012)

W. Armour, S. Hands, and C. Strouthos (2008-2011)

Simulation of the effective graphene theory Approach 2, 2D hexagonal lattice and

rectangular lattice in z and time dimensions

R. Brower, C. Rebbi, and D. Schaich (2011-2012)

P.V. Buividovich, M.I.P. (2012)

Hybrid Monte-Carlo algorithm

for fermions and heat bath for

gauge field

Fermion condensate as the function

of substrate dielectric permittivity

Approach 1 Approach 2

Hypercubic lattice Hexagonal lattice

Fermion condensate as the function

of substrate dielectric permittivity

Approach 1 Approach 2

Hypercubic lattice Hexagonal lattice

Second order phase transition?

Fermion condensate as the function

of substrate dielectric permittivity

Approach 1 Approach 2

Hypercubic lattice Hexagonal lattice

Order of the phase transition?

Fermion condensate as the function

of substrate dielectric permittivity

Hexagonal lattice (Approach 2)

Order of the phase transition?

Crossover? Connected part of the susceptibility of

the fermion condensate (no volume

dependence!) Crossover?

Phase diagram Temperature

- dielectric permittivity

Hexagonal lattice (Approach 2)

e 4

T

0

Conductivity as a function of

substrate dielectric permittivity

Approach 1 Approach 2

Hypercubic lattice Hexagonal lattice

substrate

graphene

HH

H

Graphene changes its properties when an external magnetic field

is applied, we can numerically simulate all that

Perpendicular magnetic field

Substrate dielectric permittivity

- Magnetic field phase diagram Approach 1 hypercubic lattice (preliminary)

Quark condensate vs permittivity for various values of magnetic field

Substrate dielectric permittivity

- Magnetic field phase diagram Approach 1 hypercubic lattice (preliminary)

Quark condensate suscepsibility vs permittivity for various values of magnetic field

Substrate dielectric permittivity

- Magnetic field phase diagram Approach 1 hypercubic lattice (preliminary)

Conductivity at finite magnetic field Approach 1 hypercubic lattice (very preliminary)

Main Results (hypercubic and hexagonal lattices)

4 1 4 1

Main Results (hypercubic and hexagonal lattices)

e 4

T

0

Magnetic field

Finite temperature

Impurities

2-3-4 layers

Conductivity

Viscosity – Entropy

Optical properties

Critical indices

Our plans (hypercubic and hexagonal lattices)

2

FE v

Hexagonal lattice = triangular lattice + triangular lattice

Tight binding Hamiltonian

0.142a nm

2.7eV

hopping parameter

lattice spacing

YXYX aa ,',', }ˆ,ˆ{

Vacuum

Charge operator

Redefinition of creation and annihilation operators

Free Hamiltonian with regularization

(staggered potential m)

3

1

( ) aiq e

a

q e

( )

3

2

F

F

E q v q

v a

1 0m

Eigenvalues of the regularized TB Hamiltonian

2.7eV 0.142a nm

( )

3/ 300

2

F

F

E q v q

v a c

Fermi velocity (velocity at Fermi point)

Hamiltonian with Coulomb interaction

^ ^ ^ ^

tb I emH H H H

4 4 4

4 4 3

4 4 4

1

L =18 ; 0.1; T = 0.56 = 1.51 eV = 1.8 10 K

L =18 ; 0.2; T = 0.28 = 0.76 eV = 8.8 10 K

L =24 ; 0.1; T = 0.42 = 1.13 eV = 1.3 10 K

t

TL

Green functions M. V. Ulybyshev, M. A. Zubkov; arXiv:1205.0888

Deep in the Semi-metal phase

arXiv:1205.0888

M. V. Ulybyshev, M. A. Zubkov

Deep in the insulator phase no dependence on energy different time slices do

not correlate energy of the fermion excitation is

infinite