Coulomb interaction in graphene

Outline• Graphene: One-atom thick sheet of graphite• Low-energy theory: Coulomb-interacting Dirac fermions• Theoretical research: Often neglects effect of Coulomb interactionsSee however:Gonzalez et al Nucl. Phys. B 94, ibid. PRB 99, Gorbar et al PRB 06,Khveshchenko PRB 06, Son PRB 07, Das Sarma et al PRB 2007, Vafek PRL 07,Barlas et al PRL 07, Biswas et al PRB 07, Hwang et al PRL 2007,…• Compute interaction corrections for several quantitiesRenormalization Group Hertz PRB 76, Millis PRB 1993Graphene: at a quantum critical point• Why are interaction effects negligible in optical transparency?Nair et al Science 2008

• Kinetic energyLow-energy theory of graphene• Eigenvalues ! Energy bandsEk y!!a =1.42 Angstromk x• Two inequivalent nodes: Linear dispersionEmpty!• Near nodes:E(k) " ±v | k |v = 3ta /2VelocityFilled!!!i =1,...4Spin, node• Applies at low momenta:k

!Free fermions on the honeycomb lattice• “Relativistic” low-energy HamiltonianH = % †p,i" i(p)[vp# $]" i(p)Pauli matrices.• Condensed-matter phenomena:Photoemission, Infrared spectroscopyBostwick Nat. Phys 2007; Li Nat. Phys. 2008Novel Quantum Hall effect• “Relativistic” quantum field theory phenomena!Dirac fermions in B-fieldZhang et al Nature 2005Two-componentspinorObserve v " c /300 Novoselov Nature 2005Zitterbewegung Katsnelson 2006“jittery motion” of Dirac fermions - uncertainty principleSchwinger pair production Allor et al PRD 2008• What about Coulomb interaction?Unscreenedparticle-hole production in an electric field(No Fermi surface)Next: Full Hamiltonian

H =% †p,iFull Hamiltonian: Coulomb" + 1 i(p)[vp# $]" i(p)2e 2" d 2 rd 2 r' n(r)n(r')# r - r'!Kinetic energy• Is the Coulomb interaction important?Dimensionlessinteraction strength!" = e2vhCoulomb interactionFinestructureconstant!n(r) =N# †i=1" QED= e2ch = 1137" i(r)" i(r)• Effective fine structure constant!" = 300" ! QED# 2.2• Broken relativistic invariance:Interactions: photons at velocity c!Fermions: velocity v(Not small!)Next: Dimensional analysis

H =% †p,iDimensional analysis" + 1 i(p)[vp# $]" i(p)2e 2" d 2 rd 2 r' n(r)n(r')# r - r'!Kinetic energy!• Usual 2-D electron gas: Kinetic energyKinetic energy per particle:!Coulomb interactionE KE" # Fn " n 2!" p= p22mPotential energy per particle: E Coulomb" n n " n 3 / 2!n(r) =Interparticle spacingN# †i=1" i(r)" i(r)Relative importance ofKinetic & Coulombdepends on density!• Low density:E Coulomb ! >> E KineticWigner crystal• High density:!E Coulomb

H =% †p,iDimensional analysis 2" + 1 i(p)[vp# $]" i(p)2e 2" d 2 rd 2 r' n(r)n(r')# r - r'!Kinetic energy• Dirac fermion case!n– Gas of density :Coulomb interactionE Coulomb" n 3 / 2!E Kinetic" n 3 / 2n(r) =N# †i=1" i(r)" i(r)Relative importance ofKinetic & Coulombindependent of n!• Interactions are marginal!!– No typical length scale!Perturbations break this symmetry– Applied chemical potential:" F# 1 E F– Applied Magnetic field:l B" 1 B• T = 0 Quantum critical point!!Next: RG analysis near QCP

Prescription:2) Rescale momenta & fermion fieldsIterate:!Wilsonian renormalization groupSee also Gonzalez Nucl. Phys. B 94, Khveshchenko PRB 06,Son PRB 07, Herbut et al PRL 08, …1) Trace over states in a thin high-p shell:!!!Z = Trexp "#H[$(p),$(p)]"(p) =" =1/T" > (p)" < (p)" < (p) = Z #"(bp)! !!"! ""(b) =1+ 1 • Original theory:!4 " lnbb =1Tb !!T(b) =1+ 1 "(b) #0 T(b)4 " lnb !3) New theory: same as original, with new coupling & temperature!!• Increase b: Trace over high-momentaT• , growsRenormalized theory has small coupling and high temperature†[ ]" /b < p < "p < " /b!!!Partial trace…!k yEffective theory for"k x" < (p)Next: density

• Chemical potential scaling:µ(b) =Electron density• Fermion density: adjustable via doping or external gate V g– Impose chemical potential!Intrinsic:Electron doped:!Hole doped:!!µ " V gµ = 0,n = 0 (No Fermi surface)µ > 0,n > 0µ < 0,n < 0(e Fermi surface)(h Fermi surface)Geim & Novoselov Nat. Mat. 2007µb1+ 1 4 " lnb Grows under RG!• Density scaling relation:!n(µ,T,") = b #2 n(µ(b),T(b),"(b))What we wantIn renormalized system• Need: Choice for brenormalization condition!!"(b)smallT(b) largeNext: RG condition

• Graphene: critical point at!Renormalization conditionT = µ = B = n = 0Magnetic field• Relevant perturbations: Grow under RG, leave vicinity of QCP– Terminate RG flow when perturbation reaches UV scale (bandwidth)Electroniccompressibility" #1 = $µ$nAssumeµ

Compressibility at low T• Finite density, low T: RG conditionn(b*) = n 0" a #2Renormalized densitylattice scale" #1 $ v% '!4 n 1+ & 4 ln n 0)( n*,+See Also: Hwang et al PRL 07n0 1015 ! 2# 4"cmFree Dirac FermionsLog correction depends on density!• Note: Result is alwaysLeading-order…$v " v&1+ # % 4 ln(...) '(• Interactions: Alter velocity in n or T dependent way!–Log prefactors: NOT small) At low energies,velocity growsweakly parameter dependent!–Different experiments should measure distinct velocitiesNext: Recent exp’ts

Recent compressibility dataJ. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing,A. Yacoby, Nat. Phys. (2007)• Scanning Single electron transistor: Locally measure" #1 = $µ$nSubstrate: Altersdielectric const." #1 $ v!" = e2#vh% '4 n 1+ & 4 ln n *0)( n ,+!Graphene in vacuum" = 0 Doesn’t fit….!!• Interactions necessary to understand data" = 5.5–Best fit• Difficult to observe ln(n) dependence!–Uncertainty in velocityNext: Heat capacity

• Two regimesF(T,",µ) = b #2!– Fermi liquid ( µ >> T )Metallic ! C " #TFree energy & specific heatC = "T # 2 F#T 2TT(b) F(T(b),"(b),µ(b))n" #%v 1+ $ 4 ln n (0' *& n )electron orhole F.S.• Scaling:Vafek PRL 07!– Dirac ! liquid ( µ

Phase diagram: Crossover behaviorC " T 2log 2 T!C " !T• ashed line: Crossover between two regimesNext: Transparency

Aperture partiallycovered by grapheneOptical transparency of graphene• Recent Experiments: Graphene nearly transparentNair et al, Science 2008: “Fine structure constant defines visual transparencyof graphene”See also Kuzmenko et alPRL 2008 -- graphiteTransmittance: t = 97.7%1Measures conductivity in• Theory*: t(") =(1+ 2#$(") /c) 2 optical regime!!*Stauber et al PRB 2008Next: Are interactions important?!

Optical transparency of graphene• Nair et al results: Consistent with noninteracting Dirac Fermions!!!• Transmission!!0.980.97t0.96400t(") =!5006001(1+ #$ QED/2) 2! !!"(nm)• Question: Why can we neglect Coulomb interaction?!– No log prefactors in"(#)700!" 0= e24h“Universal”Ludwig et al PRB 94Fine structure constant!– Small perturbative correctionNext: Are interactions important?!

• Kubo formula:Conductivity scaling"(#,T) = 1 # j x(q,#) j x($q,$#)current-current correlatorq %0• Scaling of!!!j µ(q,") : Constrained by a Ward identity Gross, Les Houches, 1975rj (q,") = b rj R(bq,"(b))"(b) = " T(b)exactT• Quantum field theory: No anomalous dimension of conserved currents• Conductivity scaling:–Valid to all orders in perturbation theory!"(#,T,$) = "(#(b),T(b),$(b))No factor…!Next: Higher-order correction

• Scaling: "(#,T,$) = "(#(b),T(b),$(b))Conductivity of clean graphenePreviously:"(b) # 0in renormalized theory• Perturbation theory in renormalized:!"(#,T,$) % " 0(#,T) + $(b)" 1(#,T) + $(b) 2 " 2(#,T) + ...!• Low-T regime: "(b*) = v# "(b*) #!1+ " 4&)(" = e24h 1+ c 1# +!(( 1+ 1 +# ln v$ /%!( ) +' 4 *tiny correction forkeep…" # 0"ln( v$ /%)Herbut, Juricic, Vafek, PRL 08Mishchenko Europhys. Lett. 08; NOT small in optical range!• What ! is the coefficient ?!c 1Calculate some diagrams…!Next: Two different results!

Transparency:t(") =!!!Herbut, Juricic, Vafek:c 1=25 " 6#$ 0.51212!!!Conductivity of clean graphene0.980.97t0.96400!1(1+ 2#$(") /c) 2500!600• What is the source of this discrepancy?"(nm)!Kubo formula: "(#) = 1 !# Lim j (q,#) j (%q,%#)q $0 x x#"(#) = Lim q $0!!&)("(#) = e24h 1+ c 1$ +(1+ 1 +( $ ln( v% /#)+' 4 *700Mishchenko:Two ways to compute!19 " 6#c 1= $ 0.01212Almost indistinguishable fromnoninteracting resultVia polarization:!charge densityq 2%(q,#)%(&q,&#)Next: Polarization…

Polarization approach"(#) = Lim q $0#q 2%(q,#)%(&q,&#)Diagrams inperturbationtheory….!Density vertexSelf energyVertex correctionResult:– Diagrams separately finite forv" # $19 " 6#– Sum: c 1= $ 0.012 Mishchenko’s result12!(Almost indistinguishable fromnoninteracting result)!

Kubo formula approach"(#) = 1 # Lim q $0j x(q,#) j x(%q,%#)SimilarDiagrams…!Current vertexSelf energyVertex correctionProblem:– Diagrams separately divergent forv" # $– Sum is finite… depends on regulation!

Continuity:Ward Identity and conservation laws" # j + $%$t = 0DEFINE" = j 0andQ = (",q x,q y)Q µj µ(Q) = 0!Diagrams must satisfy!!Q µj µ(Q) j "(#Q) = 0at each order in!"interactionHow to regulate divergent diagrams?!!Restrict momenta of Green functions:!!V ( p)Q µj µ(Q) j "(#Q) $ 0 Ward identity violated(simple to show!)!propagator!G( p,") # G( p,")$(% & p)Restrict ! momenta of Coulomb interaction: V ( p) " V (p)#($ % p)Q µj µ(Q) j "(#Q) = 0Ward identity preserved!G( p,")step functionConductivity has25 " 6#c 1= $ 0.51212c 1=19 " 6#12$ 0.012!!!Agrees with polarization approach….

Optical transparency: Best hope for observing Coulomb…• High frequency: Coupling&)(" = e24h 1+ c 1# +(1+ 1 +( # ln( v$ /%)+' 4 *!"(#)close to bare value!" # 2.2• Unfortunately:c 1" 0.0125Probably not observable!!Mishchenko : Europhys. Lett. 08!• Kubo formula: Sum of divergent diagrams depends on regularization• Restrict momenta in Green functionsViolates charge conservation• Restrict transferred momenta Coulomb• Modify Coulombe 2r " e2r 1#$withObeys conservation" # 0 Obeys conservationPhys. Rev. B 80, 193411 (2009)!!Next: Concluding Remarks

Concluding remarks• Graphene: Dirac fermions with Coulomb interaction (Marginal)– Interaction effects: Log corrections to free case (Dirac fermions)• Renormalization group: Scaling equations for various quantities– Specific heat, compressibility, diamagnetic susceptibility, dielectricfunction, …– Interactions only renormalize velocity:• Optical transparency probes conductivitylarge!• We resolved discrepancy with earlier Kubo formula calculation!(Leading order)&)(" = e24h 1+ c 1# +(!1+ 1 +( # ln( v$ /%)+' 4 *– Regularization method must preserve conservation laws!"%v " v'1+ # ln v$ /T& 4!But[ ]c 1=(*)19 " 6#$ 0.01212Small correction!Mishchenko Europhys. Lett. 2008