Coulomb interaction in graphene

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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),$(p)]"(p) =" =1/T" > (p)" < (p)" < (p) = Z #"(bp)! !!"! ""(b) =1+ 1 • Original theory:!4 " lnbb =1Tb !!T(b) =1+ 1 "(b) #0 T(b)4 " lnb !3) New theory: same as original, with new coupling & temperature!!• Increase b: Trace over high-momentaT• , growsRenormalized theory has small coupling and high temperature†[ ]" /b < p < "p < " /b!!!Partial trace…!k yEffective theory for"k x" < (p)Next: density
Page 2: Outline• Graphene: One-atom thick
Page 5 and 6: !Free fermions on the honeycomb lat
Page 7: H =% †p,iDimensional analysis" +
Page 11 and 12: • Graphene: critical point at!Ren
Page 13 and 14: Recent compressibility dataJ. Marti
Page 15 and 16: Phase diagram: Crossover behaviorC
Page 17 and 18: Optical transparency of graphene•
Page 19 and 20: • Scaling: "(#,T,$) = "(#(b),T(b)
Page 21 and 22: Polarization approach"(#) = Lim q $
Page 23 and 24: Continuity:Ward Identity and conser
Page 25: Concluding remarks• Graphene: Dir

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