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Boltzmann Equation and Kubo Formula Branislav K Elements of Nonequilibrium Statistical Physics: Boltzmann Equation and Kubo Formula Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A. http://wiki.physics.udel.edu/phys813 PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Fundamental Quantities in Statistical Physics: Phase Space Density and Density Operator phase space density density matrix ∫ ddp qρ(,) pq = 1 Trρˆ = 1 equilibrium vs. nonequilibrium statistical physics Liouville equation von Neumann equation dρρ(,)pq∂ (,) pq dρˆ =⇒=0 {H (,),(,)pqρ pq} iH = [,]ρˆ dt ∂t dt equilibrium equilibrium ρ = ρˆ ˆ = { 0 (,),(,)pqH pq} 0 0 ,0H ensemble average ensemble average O= ∫ ddOp q(,) pqρ (,) pq OO= Tr[ρˆ ˆ ] PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Formal Derivation of Boltzmann Equation (BE) for Plasma Physics The full phase space density contains much more information than necessary → define s-particle density: N = ρ = = one-particle density f1( pqt , ,) N∫∏ dVi( p1 pq , 1 q , p 22 , q , , pNN , q ,) t i=2 N = − ρ two-particle density f2(,, pqpqt 11 2,2 ,) NN ( 1)∫∏ dVpqpqi (,,,,,,,)11 2 2 pNN q t i=3 NN!!N f= ( p , , qt ,)= dVρρ(, ,) t= (p , , qt ,)s-particle density 33 s11 s ∫ ∏ ipq ss dVi= d p ii d q (Ns−− )! is= +1 (Ns )! 2 NNp 1 ∂ρ N =is + + −⇒ =− ρ H(,) p q ∑∑U()() qi V qij q ∫ ∏ dVi{ (,,),pq t H} i=122mt(,ij )= 1 ∂ is= +1 s ∂f ∂−∂Vq() q++ f s−= ns11 sBogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy {H, fss} ∑∫ dV +1 ∂t n=1 ∂∂qpnn Assumption of molecular chaos made by Boltzmann replaces 2-particle density 3 3 Vlasov equation with a product of 1-particle densities: ndint 1 Boltzmann equation ndint 1 ∂∂Ud ∂p ∂ 32σ − +1 f =−Ω− dpd v v f(,,)(,,) pqtf p qt− f ( p′′ ,,)( qtf p ,,) qt 1∫ 2 1 2 111 121 11 1 12 1 ∂t ∂∂ q11 p mq ∂ 1 dΩ PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula BE in Condensed Matter Describes Transport of Dilute Gas of Quasiparticles For neutral or ionized gases, the BE and its range of validity, can be directly derived from the Hamiltonian for the classical gas of molecules. In condensed matter, the BE describes the distribution function for the excitation modes (quasiparticles) and not for the constituents (electrons, ion cores, ...). The definition of quasiparticles is absolutely vital for setting up BE - it effectively maps, as far as the kinetics is concerned, the quantum-mechanical many-particle system of the constituents to a semiclassical gas of excitation modes. quasielectron phonon as quasiparticle spin-wave or magnon as quasiparticle PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Heuristic Derivation of BE for Quasielectrons in Condensed Matter 1 −e 3 f( k )= frkt (, ,) = equilibrium charge density eq E=0 βε()− εF nr()= dkf(, rkt ,) e k +1 8π 3 ∫ eE −e 3 f(, r k ,) t=−+ f ( r vdt , k dt , t − dt )no scattering j() r= d kv( k ) f (, r k ,) t current density 8π 3 ∫ eE ∂f scattering f(, r k ,) t= f ( r − vdt , k + dt , t −+ dt ) dt from ∂ disorder t scattering of phonons ∂ ∂eE ∂∂ f fv++ f f = ∂∂tr ∂k ∂ tscattering ∂fV3 = dk′(1− fk ( )) w fk ( ′′ )−− (1 fk ( )) w fk ( ) 3 ∫ kk′′k k ∂t scattering (2π ) 2 assuming that phonon or impurity (or defect) perturbations are small and time-independent, = π ′ ˆ wkk′ 2 k Hk use Born approximation for the scattering rate from occupied to unoccupied Bloch state BE in condensed matter describes semi-classical transport: Effective mass approximation (which incorporates the quantum effects due to periodicity of the crystal); Born approximation for the collisions in the limit of small perturbation for the electron-phonon or electron-impurity interaction and instantaneous collisions; no memory effects (i.e., no dependence on initial conditions). PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Relaxation Time Approximation (RTA) for the Scattering Term in BE Ansatz: The rate at which a system returns to equilibrium is proportional to its deviation from equilibrium (i.e., we make the assumption that scattering merely acts to drive a non-equilibrium system back to equilibrium): momentum dependence of relaxation time is determined phenomenologically in such ∂f fk()− feq () k = − way that the dependence of the ∂t τ ()k conductivity upon the electronic density scattering agrees with experimental data If E≠0 at t<0 and at t≥0, E=0 external electric field is switched off, then for a homogeneous system we find: ∂∂ff ff− = =−eq ⇒− = =− −t/τ f feq ft( 0) feq e ∂∂ttscattering τ In the steady state transport regime induced by a time-independent homogeneous external electric field: ∂f ∂ f ef ∂∂ f fk()− feq () k =0, = 0 ⇒−E ⋅ = =− ∂∂tr ∂kk∂tscattering τ () e ∂ fk() fk()=+⋅ feq () kτ () kE ∂k PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula RTA in the Linear-Response Transport Regime PRB 60, 3963 (1999): Upon linearization BE and Poisson equation decouple For small electric field (Ohmic regime), the relaxation time approximation can be linearized: e ∂ fk() fkeq ()+⋅τ () kE fkeq () ∂k e E=⇒+ Exxˆ fk() feq kτ () kEx according to the linear Boltzmann equation, the effect of the electric field is to shift the Fermi surface by δτkxx= − eE/ elastic scattering cannot restore equilibrium, rather it would cause Fermi surface to expand → inelastic scattering (e.g., from phonons) is needed to explain relaxation PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Example: Drude Conductivity of Electrons in Disordered Metal from BE in Linearized RTA ∂ isotropic −e e ekτ () feq material j= dk v()() k f k dk v () k feq () k+ Ex ππ33∫∫ ∂ 88 kx jjyz= = 0 T →0 ∂∂fkeq () ∂∂fE feq v=−⇒ v v f() k dk = 0 −kk∫ keq = =vx −−δ () EEFx v ∂kxx ∂∂ Ek ∂ E −e ∂f j E dkv2τσ() keq = E x8π 3 xx∫ ∂E x j eevk22() vk() σ==xxdS dEτδ()( k E−= E ) dSxτ() k E 88ππ33∫∫Evk() FEvk() x EE= F 2 assuming vk() 44ππ33k spherical dSx ττ() k= kEvE ()()= kE τ ()F ∫ Evk() 33FF F FFm* Fermi EE= F surface 2 kTBF E 2 e 4π 3 kFF eEτ () Drude στ= kEFF() = nformula 3 **32= π 83π mmknF 3 PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Example: Drude Conductivity of Electrons in Doped Graphene from BE in Linearized RTA Experiment and BE with BE with delta unscreened Coulomb potential function potential Boltzmann limit Relaxation time + Born approximation , 109 (2009) 81 eigenstates of clean graphene 0 RMP σ xx ∝ n scattering potential PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Kubo Linear-Response Theory for Time-Dependent Density Matrix ρˆ = ε αα equilibrium density matrix ˆ 0 ∑ f ()α for noninteracting H0 α= εαα α fermions in GCE Hˆˆ= H + Ht ˆ() ⇒=ρρˆˆ + ρ ˆ()t 0 1 01 in linear response system is ∂ρˆ driven slightly away from i 1 ≈+ Htˆˆ(),ρρˆˆ () t Ht (), equilibrium by a small ∂t 01 10 perturbation pAˆ − etˆ () e ne2 ˆ=i ⇒=ˆ −ˆˆ = + ˆ current density operator is the sum of j()te∑∑j()t pˆ i Aj()tpd () t j () t paramagnetic and diamagnetic term iim mm ˆˆ1 ˆ 2 H=∑(pAˆ i −+ et()) Vext 2m i ˆˆˆˆ ˆ ˆ assume external field is small H≈− H0jAp () t ⋅ () t =+ H01 Ht () and keep only terms linear in vector potential t i −−ˆˆ− ρˆ= − iH00( t t ')/ ˆ ρρ ˆˆiH( t t ')/ −∞ = solution 11()t∫ e H('),(') t t e dt ' 1 ()0 −∞ ∂ρˆ ()t i ii 1 =−ˆˆρˆ −+ ρρ ˆˆ ˆ H1(), t 0 H 01 () t () t 1 () tH 0 check by direct ∂t differentiation PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Von Neumann Equation in Interaction Picture Split Hamiltonian of quantum system into free (unperturbed) term and external perturbation assumed to be small: ˆˆˆ HHV=0 + Expectation value is the same in Schrödinger and Dirac (or interaction) picture: ˆˆ= ρρ= ˆ−iHtˆˆ//iHt At() Tr At ()ˆˆ () t Tr Ate () (0) e ˆ−− ˆˆ− ˆˆˆ = TreiH0 t/ Ateˆˆ () iH 00 t // e iH t eiHt//ρρˆˆ(0) eiHt eiH0 t/= TrA () t () t ( )( ) II Von Neumann equation in Schrödinger picture: di ρρˆˆ()t= − Htˆ , () dt Von Neumann equation in Dirac (or interaction) picture: di ˆˆd−− i iˆˆ ˆiH00 t//iH t ˆˆiH00 t// iH t ρρˆˆII()t=+=− Hte00 , () ρ ˆ() te Ht, ρ ˆI () e Hte, ρ ˆ () dt dt iiˆˆ− ˆ iH00 t// ˆˆ iH t =H00,ρρˆˆI () t −+ e H V, () te iiiˆˆˆi ˆ =−−Ht00,ρρˆˆI () Ht , II () V,ρρˆˆI ()t= − VtII , () PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Kubo Formula for AC Conductivity as Current-Current Correlation Function t 2 i −−ˆˆ′′− ne ˆ=ρˆ ˆ=−−′′iH00( t t )/ˆ ρρ ˆˆiH( t t )/ ˆˆ j()t Tr1 ()() t j t∫ dt e H10( t ), e jAp Tr ()t 0 −∞ m retarded current-current correlation function ∞ 2 ˆ ˆne ˆ i ˆˆ j()t=− Tr dttt′ Π− ( ′′ )AA ( t ) + () tρθˆ0 Π− ( tt′′′ ) =−( tt −) jjpp(), t ( t ) ∫ 0 −∞ m ∂AEˆ ()t ˆ ()ω Ohm law connecting Eˆˆ()t =−⇒=⇒ Aˆˆ()ω jE ()ω = σω ()() ω Fourier transformed ∂tiω current and electric field 2 ∞ i ne i ˆˆ itω σω()=Π+ () ω Π=− () ω jjpp (),(0)teAC conductivity ∫ 0 ω m 0 2 e it(εε− )/ −− it ( εε )/ ˆˆ = ε αˆˆ ββ ααβ − αβ jjpp(t ), (0) ∑ f()α px pe x e 0 m αβ, 2 e ff()εεαβ− () Π=()ωα 2∑ pˆ x ββpˆ x α m αβ, εαβ−+ ε ωη +i 2 11 e 2 = −πδ ⇒Π ω =− π α β ε− ε δ ε− ε + ω Pix() ( ) 2i∑ pˆ x f()()()α f β αβ xi+ η x m αβ, PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Kubo Formula and Kubo-Greewood Formula for DC Conductivity 2 e 1 2 σ(0)= 2 π lim αpˆ x β ff()()( εα− ε β δε αβ −+ ε ω) ω→0 ∑ m ω αβ, 2 ef2 ∂ = ε αˆ β− δε −− ε δε ε Kubo formula for DC conductivity in hd∫ ∑ px ()()ββ m αβ, ∂ε exact eigenstate representation 1 kkαα ′ Gˆˆra,=⇒==Gra ,,(, kk′′ ,)ε kGra k ∑ ε−±Hi η α εε−±α i η
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