Quantum Theory of Solid State Plasma Dielectric Response

1. Quantum Theory of Solid State Plasma Dielectric Response Abstract The quantum theory of solid state plasma dielectric response is reviewed and discussed in detail in the random phase approximation (RPA). Norman J. Morgenstern Horing Department of Physics and Engineering Physics, StevensInstitute of Technology, Hoboken, New Jersey 07030, USA E-mail: nhoring@stevens.edu

2. Schwinger Action Principle (Heisenberg Picture) • Quantum Mechanics of both Fermions & Bosons • Heisenberg Equations of Motion • Equal-Time Commutation/Anticommutation Relations • HamiltonEquas for Canonically Paired Quant. Operators: (upper sign for Bosons, lower for Fermions) + + _ ; ∂l, ∂rdenote “left” and “right” differentiations, referring to variations δpi; δqicommuted/anticommuted(for BE/FD) to the far left, or far right, respectively, in the variation of HH .

3. “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.

4. Variational Derivatives • Mutual independence of members of a discrete set of qi, pivariables: 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 δijwith respect to a discrete sum, ∑i.

5. Hamiltonian of Many-Body System [ is the single-particle hamiltonian in x-rep.] and for particle-particle interaction, , • Equation of motion for derived from the Hamilton equation:

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 G1retequa is homogeneous for all times except t1 = t′1:

9. 0+ 0+ • Verify δ-fn: integrate → + , • 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: .

10. Physical Interpretation of the Retarded 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+;

12. Dynamical Content of for ∂H(1)/∂t =0 • Time Development Oper(for ∂H(1)/∂t=0):exp(- ), brings the times of into coincidence: exp( ) 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 -operator: Using energy eigenvectors of H(1), ,

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 • 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.]

17. Grand Canonical Ensemble Avg. of Op. X The normalizing denominator, , is the • Grand Partition Fn.: • EQUIVALENCE: (Darwin&Fowler) (T = Kelvin temp; μ is chem. pot.).

18. Thermodynamic Green’s Functions and 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) • ≠ 0 Using cyclic invariance of Trace & using as time translation oper. through imaginary time , Time Rep: ; Freq. Rep:

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

22. Periodic/Antiperiodic Thermal Green’s Functions • ; • Matsubara Fourier Series, ; = even (BE) or odd(FD) integers. ( ) • Matsubara F.S. Coeff.

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 z→∞ along any ray in upper or lower half planes Baym & Mermin

24. Thermal G1-Equa. With 2-Body Interaction

25. Noninteracting Spectral Weight

26. Ordinary Hartree & Fock Approx. (Equilib.) H • ; n(x) = • GHF(1, 2; 1′; 2′) = • where H 2

27. Nonequilibrium Green’s Functions: ∂H/∂t ≠ 0 • I. Physical: NO Periodicity • Time Dev. Op.: • Iterate: • Time-Ordered Exp: (Time Development Op.)

28. Periodic/AntiperiodicNonequilib. G-Fn. • Periodicity: (depends t, t′separately) • Lim →-∞G1(1, 1′; to) = • Var. Diff: • Var. Diff: of G1 of

29. Nonequilib. G-Fn. Eq. of Motion where . • Eff. Pot: (Drop δ/δU)

30. Time-Dep. Hartree Approx-Nonequilibrium • Linearize:

31. RPA Dynamic, Nonlocal Screening Function K(1,2) ′ ′ ′ • = U(1)

32. RPA Polarizabilityα(1,2) • K= ε-1: , , • where . • Matsubara FS Coeff:

33. Ring Diagram I

34. Ring Diagram II

35. Ring Diagram III – 2D – Momentum Rep. for Graphene • R(q, ω+iδ) = where - μis 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) This is analogous to the Lindhard-3D and Stern-2D ring diagrams for normal systems, and their generalization to include B. = (1 + λλ′cosθ ), for Graphene

36. 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.)

37. Particle-Hole Excitation Spectrum I _ _ _ • Notation: πo ≡ + iDo ≡ R ; πRPA ≡ + iD • NORMAL 2DEG; T=0; B=0 • Bare (a) (b) Density plot of Imπ(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) • Screened Spectrum Spectrum

38. Particle-Hole Excitation Spectrum II ; ∑′ = ∑ NF - 1 • Normal 2DEG; T = 0; B ≠ 0 (lB =[eB]-1/2= magnetic length) • Bare (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 n=max(0,NF – m) R. Roldan, et al, arXiv:0909.2825 • Screened Spectrum Spectrum

39. Particle-Hole Excitation Spectrum IIIa • DOPED GRAPHENE ; 2D ; T = 0 ; B = 0 • Bare Spectrum Zero-field particle-hole excitation spectrum for doped graphene. (a) Possibleintraband (I) and interband (II) single-pair 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 Imπ(q0,ω) in the wave-vector/energy plane. The regions corresponding to intra- and interband excitations are denoted by (I) and (II), respectively.

40. Particle-Hole Excitation Spectrum IIIb (c) M.O. Goerbig, arXiv: 1004.3396v1 [cond-mat.-mes-hall] 20 Apr 2010 • DOPED GRAPHENE ; 2D ; T = 0 ; B = 0 • Screened (c) Spectral function ImπRPA(q,ω) for doped graphene in the wavevector/energy plane. The electron-electron interactions are taken into account within the RPA. Spectrum

41. Particle-Hole Excitation Spectrum IVa • DOPED GRAPHENE ; 2D ; T = 0 ; B ≠ 0 (ω′ = 21/2vF/lB) (Fλn, λ′n′ are Graphene form factors playing the role of the chirality factor for B = 0) • Bare Spectrum 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 = 0.05vFh / lB. _

42. Particle-Hole Excitation Spectrum IVb • DOPED GRAPHENE ; 2D ; T = 0 ; B ≠ 0 ; Screened Spectrum • Screened M.O. Goerbig, arXiv: 1004.3396v1 [cond-mat.-mes-hall] 20 Apr 2010 Spectrum 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.