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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 VS COMPLEXITYQCD Small (Log 1) Large (Millenium problem) CondMat Large (all materials) Small (mean-field often enough)
Instantaneous approximationTypical 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 ~ vF
2
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
Two simplerhombic
sublattices А and В
Geometry of hexagonal lattice
Periodic boundary conditions on the Euclidean torus:
Tight-binding model of GrapheneOr The Standard Model of Graphene
“Staggered” potential m distinguishes even/odd lattice sites
Spectrum of quasiparticles in grapheneConsider the non-Interacting tight-binding model !!!
Eigenmodes are just the plain waves:
Eigenvalues:
3
1
)(a
eki aek
One-particleHamiltonian
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
2 Fermi-points Х 2 sublattices= 4 components of the Dirac spinor
),,,( RLRL
),,,( BABA
Chiral U(4) symmetry (massless fermions): right left
Discrete Z2 symmetry between sublattices
А В
Symmetries of the free Hamiltonian
U(1) x U(1) symmetry: conservation of currents with different spins
• 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
Particles and holes
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
r
erV
)1(
2)(
2
Dielectric permittivity:
• Suspended graphene
ε = 1.0• Silicon Dioxide SiO2
ε ~ 3.9• Silicon Carbide SiC
ε ~ 10.01
2
Effective Coulomb coupling constantα ~ 1/137 1/vF ~ 2 (vF ~ 1/300)
Strongly coupled theory!!!Magnetic+retardation effects suppressed
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 grapheneSymmetry 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 sublatticesExcitonic condensate:opposite charges on sublattices
Chiral symmetry breaking in graphene: analytical study
1) E. V. Gorbar et. al., Phys. Rev. B 66 (2002), 045108. α
с = 1,47
2) O. V. Gamayun et. al., Phys. Rev. B 81 (2010), 075429.α
с = 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
Experiment: D. C. Elias et. al., Nature Phys, 7, (2011), 701;
No evidence of the phase transition
Lattice calculations: phase transition at ε=4
Path integral representationPartition function:
Introduction of fermionic coherent states:
Using the following relations:
and Hubbard-Stratonovich transformation:
Antiferromagnetic phase transition
P. V. Buividovich, M. I. Polikarpov, Phys. Rev. B 86 (2012) 245117
Condensate with modified potentials
Ulybyshev, Buividovich, Katsnelson, Polikarpov, Phys. Rev. Lett. 111, 056801 (2013)
Screened potential
Coulomb potential
Long-range interaction
Sh
ort
-ran
ge
inte
ract
ion
Excitonic phase
Antiferromagnetic phase
Phase diagramInfluence 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.
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?
???