Some recent results on computer simulations of lattice QCD

Charge-Carrier Transport in Graphene 

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 

■ Introduction: QCD and graphene 

■ Charge carriers in graphene and effective field 

theory 

■ Calculations on the hypercubic lattice 

■ Calculations on hexagonal lattice 

Workshop on QCD in strong magnetic fields 

12-16 November 2012, Trento, Italy



ArXiv:1204.0921; ArXiv:1206.0619 



theory 







ArXiv:1204.0921; ArXiv:1206.0619 



theory 







ArXiv:1204.0921; ArXiv:1206.0619 



theory 





Charge-Carrier Transport in Graphene intight 

binding model 


ArXiv:1204.0921; ArXiv:1206.0619 



theory 





QCD and Graphene

Carbon atom

Elementary structure

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

fullerenes (C 60, C 540, C 70); g) amorphous carbon; h) carbon nanotube.

Fullerene (Buckminsterfullerene) C 60 

Richard Buckminster Fuller 

1895 -1983 

The Montreal Biosphère by 

Buckminster Fuller, 1967

Fullerene C 540 

Richard Buckminster Fuller 

1895 -1983 

The Montreal Biosphère by 

Buckminster Fuller, 1967

Nanotube

Graphene

The Nobel Prize in Physics for 2010 was awarded to 

Andre Geim and Konstantin Novoselov 

"for groundbreaking experiments regarding the 

two-dimensional material graphene”

Nonrelativistic particle 

E � 

mv 

2 

2

Relativistic particle 

2 4 2 2 

E � m c � 

p c


2 4 2 2 

E � m c � p c 

Massless particle 

E � 

cp



E � m c � p c 

E � cp 

c 

Graphene E �vFp; vF 

� 

; 

300 

2 4 2 2



c 

Graphene E �vFp; vF 

� ; 

300 

� � 300� � 2.16 �1 

g 

2 4 2 2 

crit 

� � � �1.11� 0.06 Pure graphene is the insulator! 

g g 

E � m c � p c 

E � cp

Hexagonal lattice = Triangle lattice + Triangle lattice 

On such lattice nonrelativistic electrons are 

“equivalent” to massless four component Dirac 

fermions moving with 

v � 

F 

c 

300 

the effective charge is: 

g 

; 

� 300 � 2.16 �1 

� �

Graphene lattice and Brillouin zone

Wallace, P. R. (1947). "The Band Theory of 

Graphite". Physical Review 71 (9): 622. 

Semenoff, G. W. (1984). "Condensed- 

Matter Simulation of a Three-Dimensional 

Anomaly". Physical Review Letters 53 (26): 

2449.

We can vary the effective coupling in graphene! 

Graphene in the dielectric media 

Graphene on substrate 

� 

g 

� 

� 

� 

g 

2 

�g � �g 

1� 

� 

graphene 

2 

crit 

if �g ��g ( �1.11) 

graphene is the conductor 

1� 

� 

substrate

Effective theory of charge carriers in 

1. “Massless” four component Dirac fermions 

2. Fermi velocity is 

graphene 

3. The effective charge is 

4. We can vary the effective 

charge if we vary the dielectric 

permittivity of the substrate 

g 

vF�c / 300 

� 300 � 2.16 �1 

� � 

2 

�g � �g 

1� 

�

Effective field theory for graphene 

After transformation 

we can neglect A i; 

� 

� � � 300� � � g 

v 

F

On substrate 

2 

�g � �g 

1� 

� 

(2+1)D fermions 

(3+1)D Coulomb

Simulation of the effective graphene theory 

Approach 1, hypercubic lattice 

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

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

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

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

(2+1)D fermions 

(3+1)D Coulomb

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)

Approach 2, 2D hexagonal lattice, Hamiltonian 

^ ^ ^ 

H �H�H tb I


Lattice 

geometry 

^ ^ ^ 

H �H�H tb I 

Coulomb 

interaction


Lattice 

geometry 

^ ^ ^ 


Coulomb 



^ ^ ^ 


v � 

F 

{ ˆ ˆ � 

� 

a � , X , a� 

', 

Y} 

� �� ' X , Y 

c 

300


^ ^ ^ 


Coulomb 



^ ^ ^ 

H �H�H tb I

Fermion condensate as a function 

of substrate dielectric permittivity 

Approach 1 Approach 2 

Hypercubic lattice Hexagonal lattice

Conductivity as a function of 

substrate dielectric permittivity 

Approach 1 Approach 2 

Hypercubic lattice Hexagonal lattice

Perpendicular magnetic field 

substrate 

H 

H 

graphene 

Graphene changes its properties when an external magnetic field 

is applied, we can numerically simulate all that

Fermion condensate as the function 

of substrate dielectric permittivity at 

finite magnetic field

Substrate dielectric permittivity 

- Magnetic field phase diagram 

Approach 1 (preliminary) 

??? 

???

What can be done in 

the field theory approach 

Magnetic field 

Finite temperature 

Impurities 

2-3-4 layers 

Conductivity 

n 

E�vF Viscosity – Entropy 

Optical properties 

Critical indices 

Conductivity of nanotube

Parallel magnetic field (Aleiner, Kharzeev, Tsvelik 2007) 

substrate 

H 

graphene 

Graphene changes its properties when an external magnetic field 

is applied, we can numerically simulate all that 

H

Trajectory of the magnetic head 

Ferromagnetic substrate 

graphene 

magnetic head 

Along the trajectory of the magnetic head graphene becomes 

the conductor! 

We can draw (construct) chips! All that we can simulate on 

computers

Mobius carbon is a topological insulator? 

ArXiv: 0906.1634

Dependence of conductivity on the 

2R 

radius of nanotube and 

on magnetic field 

B

PRL 108, 086804(2012) 

PHYSICAL REVIEW LETTERS 

24 FEBRUARY 2012 

Graphyne 

Competition for Graphene: Graphynes with 

Direction-Dependent Dirac Cones 

Daniel Malko, Christian Neiss, FrancescVines, 

and Andreas Gorling

Tuesday 13 November 16:30 - … 

Discussion Sessions 

Gerald Dunne and Yoshimasa Hidaka 

Landau-level structure in QCD. Questions: To what extent is the Landau-level 

picture applicable in QCD as an interacting theory? Is the LLL approximation valid 

for strong magnetic fields, and does physics reduce to a 1+1 dimensional theory 

here? If so, does the Mermin-Wagner theorem become effective? Is this visible in 

the Dirac eigenmodes? 

Wednesday 14 November 16:30 - … 

Andreas Schmitt and Ingo Kirsch 

Introduction to the ads/qft approach and comparison to lattice. Questions: Is there 

any input from the lattice side, that could be used to fix certain free parameters of 

the holographic approach, and what are the observables that we can compare?

Thursday 15 November 16:30 - … 

Discussion Sessions 

Dmitri Kharzeev and Vladimir Skokov 

Chiral magnetic effect. Questions: Is there consensus about CME signatures in 

experimental <strong>results</strong> from heavy ion colliders? What are the possible 

interpretations of these data? What are the motivated theoretical suggestions for 

ALICE to measure from the CME-interested community (becoming especially 

actual after ALICE upgrade)? Is it resonable to look at higher order sin-harmonics 

and their correlators? Is it reasonable to study their averages not over set of all 

event but over some subsets?

09:00 - 09:40 Edward Shuryak 

QCD topology near and above T_c 

Friday 16 November 

Talk of Eduardo Fraga => Wednesday 

09:40 - 10:20 Discussion session 

Maxim Chernodub, Yoshimasa Hidaka, Arata Yamamoto 

Superconducting vacuum in strong magnetic field, does it exist or not? 

10:20 - 10:40 Coffee break 

10:40 - 11:20 Final discussion 

All participants of the workshop

Some recent results on computer simulations of lattice QCD

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