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1 Quantum Theory of Solid State Plasma Dielectric Response Norman J. Morgenstern Horing Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA E-mail: [email protected] Abstract The quantum theory of solid state plasma dielectric response is reviewed and discussed in detail in the random phase approximation (RPA). Schwinger Action Principle (Heisenberg Picture) 2 Quantum Mechanics of both Fermions & Bosons Heisenberg Equations of Motion Equal-Time Commutation/Anticommutation Relations Hamilton Equas for Canonically Paired Quant. Operators: (upper sign for Bosons, lower for Fermions) + ; _ + ∂l, ∂r denote “left” and “right” differentiations, referring to variations δpi ; δqi commuted/anticommuted (for BE/FD) to the far left, or far right, respectively, in the variation of HH . “Second Quantized” Notation for Many-Particle Systems: ● , are the creation, annihilation operators for a particle in state “ a′ ” at time t. ; ● ● and are not hermitian, but they are canonically paired, obeying the equal-time canonical commutation/anticommutation relations ● ● (where denotes the anticommutator for Fermions, and denotes the commutator for Bosons). As they are canonically paired variables, we can associate ● in position representation, with the x spectrum continuous. 3 Variational Derivatives 4 • Mutual independence of members of a discrete set of qi , pi variables: and sums over them are denoted by ∑i. • Mutual independence of the continuum of variables at all points x (for a fixed time t): (δ symbolizes variation for members of a continuum of variables as does ∂ for a discrete set of variables), Here, plays the same role under integration over the continuum, , as does δij with respect to a discrete sum, ∑i. Hamiltonian of Many-Body System [ is the single-particle hamiltonian in x-rep.] and for particle-particle interaction, , Equation of motion for Hamilton equation: derived from the 5 6 in the first term on the right may be written as 7 For the left variation, the factor must be commuted/anti-commuted to the left of in second term, invoking a ± sign. Thus, + Dividing by & comm/anti-comm 8 Single Particle Retarded Green’s Functions Noninteracting single particle ( , but h(1) may include a local single particle potential): Retarded Green’s function: ε is always +1 for BE but it is +1 or -1 for FD for t1 > t′1 or t1 < t′1 ; (…)+ time-orders the operators placing the largest time argument on the far left. Multiplying by from the left or right to time-order for t1 ≠ t′1 and averaging in vacuum the G1ret equa is homogeneous for all times except t1 = t′1: • Verify δ-fn: integrate 0+ • → + 9 , 0+ are functional forms of time-ordered ; • Retardation is ensured by since ; • For the Dirac δ-function driving term is confirmed using the equal-time canonical comm./anticomm. relations: . Physical Interpretation of the Retarded 10 Green’s Function State of a single particle created at (drop sub H). is in a scalar product with a state describing the annihilation of the particle at , Probability amplitude for particle creation at subsequently annihilated after propagating to , : 11 Initial value problem: Obeys homog. equa. (δ ( )→ 0), with initial value by canonical comm./anticomm. relations 0+; Dynamical Content of for ∂H(1)/∂t Time Development Oper(for ∂H(1)/∂t=0):exp(- brings the times of exp( 12 =0 ), into coincidence: ) exp( Retarded one-particle Green’s Function( ), =unit step): • Expansion in single-particle energy eigenstates, : Insert unit operator I next to the time development operator ( ) • 13 The matrix element, is the single particle energy eigenfn. In x-rep., Thus, in position-time representation, • . . 14 Matrix Operator Retarded Green’s Function The operator Green’s function, , is defined by Fourier transforming T → ω + i0+, we have Using energy eigenvectors of H(1), -operator: , 15 Density of States (Dirac prescription, ) is proportional to the density D(ω) of single particle energy eigenstates (per unit energy), 16 Quantum Mechanical Statistical Ensembles I. Microcanonical Ensemble Average of Op. X for a macroscopic system of number N′ and energy E′. • Thermodynamic probability: is just the number of micro states for N′ and E′. • Entropy: [k = Boltzmann Const.] II. Grand Canonical Ensemble Avg. of Op. X 17 The normalizing denominator, • Grand Partition Fn.: EQUIVALENCE: , is the (Darwin&Fowler) (T = Kelvin temp; μ is chem. pot.). Thermodynamic Green’s Functions and 18 Spectral Structure Statistical weighting is a time displacement operator, through imaginary time provided ∂ /∂t ≡ 0 and thermodynamic equilibrium prevails. n-particle thermal Green’s fn. in x-rep. is averaged in grand canonical ensemble 19 Averaging process is done in the background of a thermal ensemble; the n creation operators creating n additional particles at with tracing their joint dynamical propagation characteristics to , where they are annihilated by the n annihilation operators; yielding the amplitude for this process with account of their correlated motions due to interparticle interactions 20 Single Particle Thermal G-fn. • and (± means upper sign for BE; lower sign for FD) • Using cyclic invariance of Trace & using translation oper. through imaginary time Time Rep: Freq. Rep: ≠0 as time , ; 21 Spectral Weight Fn. Define: ≡ and where f(ω) is the BE or FD equilib. distrib. These results can be understood in terms of a periodicity/antiperiodicity condition on the Green’s function in imaginary time. Defining a slightly modified set of Green’s functions as Periodic/Antiperiodic Thermal Green’s Functions ; Matsubara Fourier Series, (BE) or odd(FD) integers. ( Matsubara F.S. Coeff. ; = even ) 22 23 Spectral Weight and Matsubara Fourier Series • ; • • • • = Multivalued. Unique solution with (i)These discrete values at ; (ii)Analytic everywhere off real z-axis;(iii)Goes to 0 as Baym & z→∞ along any ray in upper or lower half planesMermin Thermal G1-Equa. With 2-Body Interaction • 24 Noninteracting Spectral Weight • • 25 26 Ordinary Hartree & Fock Approx. (Equilib.) H • ′; 2′) = • GHF (1, 2; 1 2 • • where ; n(x) = 27 Nonequilibrium Green’s Functions: ∂H/∂t ≠ 0 I. Physical: NO Periodicity • Time Dev. Op.: • Iterate: • Time-Ordered Exp: (Time Development Op.) Periodic/Antiperiodic Nonequilib. G-Fn. • Periodicity: (depends t, t′ separately) • Lim →-∞ G1(1, 1′; to) = • Var. Diff: of G1 • Var. Diff: of 28 Nonequilib. G-Fn. Eq. of Motion 29 where • Eff. Pot: . (Drop δ/δU) Time-Dep. Hartree Approx-Nonequilibrium Linearize: 30 RPA Dynamic, Nonlocal Screening Function K(1,2) ′ ′ ′ = U(1) 31 RPA Polarizability α(1,2) • K= ε-1: 32 , • , • where • Matsubara FS Coeff: . Ring Diagram I • • 33 Ring Diagram II • • 34 Ring Diagram III – 2D – Momentum Rep. for Graphene 35 R(q, ω+iδ) = where - μ is the energy of stateφλ(q) measured from μ; n ≡ f is the Fermi distrib.; g is degeneracy; and A = area (2D), with (λ = ± 1 for ± energies) = (1 + λλ′ cosθ ), for Graphene This is analogous to the Lindhard-3D and Stern-2D ring diagrams for normal systems, and their generalization to include B. Density – Density Correlation Fn. Def: Exact: : : = ± iRε-1 Def: Do = ± iR ( →0, no interact; Bare Density Autocorr. Fn.) D = Doε-1 (Screened Density Autocorr. Fn.) 36 Particle-Hole Excitation Spectrum I 37 _ _ _ Notation: πo ≡ + iDo ≡ R ; πRPA ≡ + iD • NORMAL 2DEG; T=0; B=0 Bare Spectrum (a) Screened Spectrum (b) Density plot of Im π(q,ω). (a) corresponds to non-interacting polarization of a 2DEG, whereas (b) accounts also for electron interactions in the RPA (R. Roldan, M.O. Goerbig & J.N. Fuchs, arXiv: 0909.2825[cond-mat.-mes-hall] 9 Nov 2009) 38 Particle-Hole Excitation Spectrum II • Normal 2DEG; T = 0; B ≠ 0 (lB =[eB]-1/2= magnetic length) ; ∑′ = ∑n=max(0,N NF - 1 F – m) R. Roldan, et al, arXiv:0909.2825 Bare Spectrum Screened Spectrum (a) (b) (a) and (b) show the imaginary part of the non-interacting and RPA polarization functions, respectively, of a 2DEG in a magnetic field. In (a) and (b), NF = 3 and δ = 0.2ωc Particle-Hole Excitation Spectrum IIIa 39 • DOPED GRAPHENE ; 2D ; T = 0 ; B = 0 Bare Spectrum Zero-field particle-hole excitation spectrum for doped graphene. (a) Possible intraband (I) and interband (II) singlepair excitations in doped graphene. The excitations close to the Fermi energy may have a wave-vector transfer comprised between q = 0 (Ia) and q = 2qF (Ib), (b) Spectral function Im π(q0,ω) in the wave-vector/energy plane. The regions corresponding to intra- and interband excitations are denoted by (I) and (II), respectively. Particle-Hole Excitation Spectrum IIIb 40 • DOPED GRAPHENE ; 2D ; T = 0 ; B = 0 (c) Screened Spectrum M.O. Goerbig, arXiv: 1004.3396v1 [cond-mat.-meshall] 20 Apr 2010 (c) Spectral function Im πRPA(q,ω) for doped graphene in the wavevector/energy plane. The electronelectron interactions are taken into account within the RPA. Particle-Hole Excitation Spectrum IVa 41 • DOPED GRAPHENE ; 2D ; T = 0 ; B ≠ 0 (ω′ = 21/2 vF/lB) (Fλn, λ′n′ are Graphene form factors playing the role of the chirality factor for B = 0) Bare particle-hole excitation spectrum for graphene in a perpendicular magnetic field. We have chosen NF = 3 in the CB and a LL broadening _ of δ = 0.05vF h / lB. Bare Spectrum Particle-Hole Excitation Spectrum IVb 42 •DOPED GRAPHENE ; 2D ; T = 0 ; B ≠ 0 ; Screened Spectrum Screened Spectrum M.O. Goerbig, arXiv: 1004.3396v1 [cond-mat.-meshall] 20 Apr 2010 Screened particle-hole excitation spectrum for graphene in a perpendicular magnetic field. The Coulomb interaction is taken into account within the RPA. We have chosen NF = 3 in the CB and a LL broadening of δ = 0.05vF h/lB.