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 NNp 1 ∂ρ N =is + +  −⇒ =− ρ H(,) p q ∑∑U()() qi V qij q ∫ ∏ dVi{ (,,),pq t H} i=122mt(,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 ∂ 32σ  − +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 dEτδ()( k E−= E ) dSxτ() k E 88ππ33∫∫Evk() FEvk() x EE= F

2  assuming vk()  44ππ33k 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 = TreiH0 t/ Ateˆˆ () iH 00 t //  e iH t eiHt//ρρˆˆ(0) eiHt eiH0 t/= TrA () t () t ( )( ) II Von Neumann equation in Schrödinger picture: di ρρˆˆ()t= −  Htˆ , () dt   Von Neumann equation in Dirac (or interaction) picture:

di ˆˆd−− i iˆˆ îH00 t//iH t ˆîH00 t// iH t ρρˆÎI()t=+=− Hte00 , ()  ρ ˆ() te Ht, ρ Î () e Hte, ρ ˆ () dt dt  

iiˆˆ− ˆ iH00 t// ˆˆ iH t =H00,ρρˆÎ () t −+ e H V, () te    iiiˆˆˆi ˆ =−−Ht00,ρρˆÎ () Ht , II () V,ρρˆÎ ()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 Tr1 ()() 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 ααβ − 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  11 1 ar′′ ′ δ()x = −⇒G(, kk ,)ε − G (, kk ,) ε = 2 π i∑ k α α k δε()− εα 2πηηixi−+ xi α Kubo-Greenwood formula for DC conductivity in terms of Green functions 2 21π 32ef ∂  σ= ε −−ra′′′′′ εε ra εε − xx 2 d   kx  G(, kk ,) G (, kk ,) kx  G (, kk ,) G (, kk ,) mi2πε∫ ∂ 

Gra(, kk′′ ,)εε G (, kk ,)= Gr (, kk ′ ,) ε Ga (, kk ′ ,) εneglect quantum interference effects disorder disorder disorder

ra, 1 G(, kk′ ,)ε = isotropic scattering off short-ranged impurities disorder εε−±k i 2 τ 2 22 e  2τ 1122 2 σ= k − =eτ k δε()−= εeD ν Einstein formula ∑∑ xdisorder  xkdisorder 2πkkm iiεε−−kk22 τ εε −+ i  τ  m

PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Kubo Formula on the Computer

WW ε ∈− , m 22

conductance quantization reproduced only if all terms in the original Kubo- Greenwood formula 60 , 848 (1988) are evaluated PRL

PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula Numerical Exact Calculation of Conductivity of Graphene using Kubo Formula Numerically exact i e2 f (E ) − f (E ) 〈n | v | n'〉〈n'| v | n〉 σ = −  n n' x x evaluation of the Kubo 2 ∑ − − + η formula in exact state L n,n' En En' En En' i representation reproduces Drude conductivity together with quantum interference corrections known as weak localization

cosθ ≡ 0 22 2 2 A=++ AI A II 2 AAI II cosθ = 4 AI

Short-range scattereres Coulomb scattereres , 076602 (2007)

98 2 2 e min e σ min ≈ σ ≈

PRL hπ h

PHYS813: Quantum Statistical Mechanics Boltzmann equation and Kubo formula