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Graphene, topological
insulators and Weyl
semimetals
Pavel Buividovich
(Regensburg)
Why these condmat systems?
They are very similar to relativistic strongly coupled QFT
• Dirac/Weyl points
• Quantum anomalies
• Strong coupling
• Spontaneous symmetry breaking
•
•
•
•
Much simpler than QCD (the most interesting SC QFT)
Relatively easy to realize in practice (table-top vs LHC)
We (LQCD) can contribute to these fields of CondMat
We can learn something new
 new lattice actions
 new algorithms
 new observables/analysis tools
BUT BEWARE: ENTROPY
QCD
Small (Log 1)
CondMat
Large (all materials)
VS
COMPLEXITY
Large (Millenium problem)
Small (mean-field often enough)
Instantaneous approximation
Typical values of vF ~ c/300
(Graphene)
Typical sample size ~ 100 nm
(1000 lattice units)
Propagation time ~ 10-16 s
(Typical energy ~ 100 eV)
Magnetic interactions ~ vF2
Coulomb interactions are
more important by factor
~1/vF2
Graphene ABC
• Graphene: 2D carbon crystal with
hexagonal lattice
• a = 0.142 nm – Lattice spacing
• π orbitals are valence orbitals (1 electron
per atom)
• Binding energy κ ~ 2.7 eV
• σ orbitals create chemical bonds
Geometry of hexagonal lattice
Two simple
rhombic
sublattices
А and В
Periodic boundary
conditions on the
Euclidean torus:
Tight-binding model of Graphene
Or The Standard Model of Graphene
“Staggered” potential m distinguishes
even/odd lattice sites
Physical implementation
of staggered potential
Graphene
Boron Nitride
Spectrum of quasiparticles in graphene
Consider the non-Interacting tight-binding model !!!
One-particle
Hamiltonian
Eigenmodes are just the plain waves:
3
 (k ) 
e
a 1
Eigenvalues:
 
i k e a
Spectrum of quasiparticles in graphene
Close to the «Dirac points»:
“Staggered potential” m = Dirac mass
Spectrum of quasiparticles in graphene
Dirac points are only covered by discrete lattice
momenta if the lattice size is a multiple of three
Dirac fermions
Dirac fermions
Near the Dirac points
Dirac fermions
«Valley» magnetic field`
Mechanical strain: hopping amplitudes change
«Valley» magnetic field
[N. Levy et. al., Science 329 (2010), 544]
Symmetries of the free Hamiltonian
2 Fermi-points
Х
2 sublattices
=
4 components of the
Dirac spinor
Chiral U(4) symmetry
(massless fermions):
right
left
( L, R, L , R )
(  A , B , A , B )
Discrete Z2 symmetry
between sublattices
А
В
U(1) x U(1) symmetry: conservation
of currents with different spins
Particles and holes
• Each lattice site can be occupied by two
electrons (with opposite spin)
• The ground states is electrically neutral
• One electron (for instance )
at each lattice site
• «Dirac Sea»:
hole =
absence of electron
in the state
Lattice QFT of Graphene
Redefined creation/
annihilation operators
Charge
operator
Standard QFT vacuum
Electromagnetic interactions
Link variables
(Peierls Substitution)
Conjugate momenta
= Electric field
Lattice
Hamiltonian
(Electric part)
Electrostatic interactions
Effective Coulomb coupling constant
α ~ 1/137 1/vF ~ 2 (vF ~ 1/300)
Strongly coupled theory!!!
Magnetic+retardation effects suppressed
Dielectric permittivity:
• Suspended graphene
ε = 1.0
• Silicon Dioxide SiO2
ε ~ 3.9
V (r ) 
2e
2
(   1) r
 
2
 1
• Silicon Carbide SiC
ε ~ 10.0
Lattice simulations
of the tight-binding model
Lattice Hamiltonian from the beginning
Fermion doubling is physical
Perturbation theory in 1D (Euclidean time)
• No UV diverging diagrams
• Renormalization is not important
• Not so important to have exact chirality
• No sign problem at neutrality
• HMC simulations are possible
Chiral symmetry breaking in graphene
Symmetry group of the low-energy theory is U(4). Various
channels of the symmetry breaking are possible. Two of them
are studied at the moment. They correspond to 2 different
nonzero condensates:
- antifferromagnetic condensate
- excitonic condensate
From microscopic point of view, these situations correspond to
different spatial ordering of the electrons in graphene.
Antiferromagnetic condensate:
opposite spin of electrons on
different sublattices
Excitonic condensate:
opposite charges on sublattices
Chiral symmetry breaking in graphene:
analytical study
1) E. V. Gorbar et. al., Phys. Rev. B
66 (2002), 045108.
αс = 1,47
2) O. V. Gamayun et. al., Phys. Rev. B 81 (2010), 075429.
αс = 0,92
3), 4)..... reported results in the region αс = 0,7...3,0
D. T. Son, Phys. Rev. B 75 (2007) 235423: large-N analysis:
Excitonic condensate
P. V. Buividovich et. al., Phys. Rev. B 86 (2012), 045107.
Joaquín E. Drut, Timo A. Lähde, Phys. Rev. B 79, 165425 (2009)
All calculations were performed on the lattice with 204 sites
Graphene conductivity: theory
and experiment
Lattice calculations: phase
transition at ε=4
Experiment: D. C. Elias et. al.,
Nature Phys, 7, (2011), 701;
No evidence of the phase
transition
Path integral representation
Partition function:
Introduction of fermionic coherent states:
Using the following relations:
and Hubbard-Stratonovich transformation:
Fermionic action and (no) sign problem
No sign problem!
At half-filling
Antiferromagnetic phase transition
P. V. Buividovich, M. I. Polikarpov, Phys. Rev. B 86 (2012) 245117
Comparison of the potentials
«Screening» of Coulomb interaction at small distances
Condensate with modified potentials
Screened potential
Coulomb potential
Ulybyshev, Buividovich, Katsnelson, Polikarpov, Phys. Rev. Lett. 111, 056801 (2013)
Phase diagram
Short-range interaction
Influence of the short-range interactions on the excitonic phase transition:
O.V. Gamayun et. al. Phys. Rev. B 81, 075429 (2010).
Short-range repulsion suppresses formation of the excitonic condensate.
Antiferromagnetic
phase
Excitonic phase
Long-range
interaction
Graphene with vacancies
• Hoppings are equal to zero for all links connecting vacant
site with its neighbors.
• Charge of the site is also zero.
• Approximately corresponds to Hydrogen adatoms.
• Midgap states, power-law decay of wavefunctions
Nonzero density of states near Fermi-points:
• Cooper instability
• AFM/Excitonic condensates
What about other defects?
???
Electron spin near vacancies
Graphene in strong magnetic fields
What is the relevant ground state for B ~ 15 T?
Spin is not polarized…
Kekule distortion:
superlattice structure
Skyrmions

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