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■ 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
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■ 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
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■ 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
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■ 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
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QCD and Graphene
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Carbon atom
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Some allotropes of carbon: a) diamond; b) graphite; c)lonsdaleite; d–f)
fullerenes (C60, C540, C70); g) amorphous carbon; h) carbon nanotube.
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Graphene
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Hexagonal lattice = triangular lattice + triangular lattice
Tight binding Hamiltonian
0.142a nm
2.7eV
hopping parameter
lattice spacing
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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
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we can neglect Ai;
Effective field theory for graphene Four component Dirac fermions + Coulomb field
After transformation
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we can neglect Ai;
Effective field theory for graphene Four component Dirac fermions + Coulomb field
After transformation
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we can neglect Ai;
Effective field theory for graphene Four component Dirac fermions + Coulomb field
After transformation
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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
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(2+1)D fermions
(3+1)D Coulomb
2
1g g
On substrate
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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
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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)
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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
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Fermion condensate as the function
of substrate dielectric permittivity
Approach 1 Approach 2
Hypercubic lattice Hexagonal lattice
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Fermion condensate as the function
of substrate dielectric permittivity
Approach 1 Approach 2
Hypercubic lattice Hexagonal lattice
Second order phase transition?
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Fermion condensate as the function
of substrate dielectric permittivity
Approach 1 Approach 2
Hypercubic lattice Hexagonal lattice
Order of the phase transition?
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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?
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Phase diagram Temperature
- dielectric permittivity
Hexagonal lattice (Approach 2)
e 4
T
0
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Conductivity as a function of
substrate dielectric permittivity
Approach 1 Approach 2
Hypercubic lattice Hexagonal lattice
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substrate
graphene
HH
H
Graphene changes its properties when an external magnetic field
is applied, we can numerically simulate all that
Perpendicular magnetic field
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Substrate dielectric permittivity
- Magnetic field phase diagram Approach 1 hypercubic lattice (preliminary)
Quark condensate vs permittivity for various values of magnetic field
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Substrate dielectric permittivity
- Magnetic field phase diagram Approach 1 hypercubic lattice (preliminary)
Quark condensate suscepsibility vs permittivity for various values of magnetic field
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Substrate dielectric permittivity
- Magnetic field phase diagram Approach 1 hypercubic lattice (preliminary)
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Conductivity at finite magnetic field Approach 1 hypercubic lattice (very preliminary)
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Main Results (hypercubic and hexagonal lattices)
4 1 4 1
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Main Results (hypercubic and hexagonal lattices)
e 4
T
0
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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
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Hexagonal lattice = triangular lattice + triangular lattice
Tight binding Hamiltonian
0.142a nm
2.7eV
hopping parameter
lattice spacing
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YXYX aa ,',', }ˆ,ˆ{
Vacuum
Charge operator
Redefinition of creation and annihilation operators
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Free Hamiltonian with regularization
(staggered potential m)
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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
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2.7eV 0.142a nm
( )
3/ 300
2
F
F
E q v q
v a c
Fermi velocity (velocity at Fermi point)
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Hamiltonian with Coulomb interaction
^ ^ ^ ^
tb I emH H H H
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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
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Green functions M. V. Ulybyshev, M. A. Zubkov; arXiv:1205.0888
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Deep in the Semi-metal phase
arXiv:1205.0888
M. V. Ulybyshev, M. A. Zubkov
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Deep in the insulator phase no dependence on energy different time slices do
not correlate energy of the fermion excitation is
infinite