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 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

Physical implementation of staggered potential

Boron Nitride

Graphene

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

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]

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:

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

«Screening» of Coulomb interaction at small distances

Comparison of the potentials

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?

???

Electron spin near vacancies

Graphene in strong magnetic fieldsWhat is the relevant ground state for B ~ 15 T? Spin is not polarized…

Kekule distortion: superlattice structure

Skyrmions