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Quantum Theory of Solid State Plasma Dielectric Response

Norman J. Morgenstern HoringDepartment of Physics and Engineering Physics,

Stevens Institute of Technology, Hoboken, New Jersey 07030, USAE-mail: nhoring@stevens.edu

Abstract

The quantum theory of solid state plasma dielectric response is reviewed and discussed in detail in the random phase approximation (RPA).

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+

Schwinger Action Principle (Heisenberg Picture)

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 .

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● , 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.

“Second Quantized” Notation for Many-Particle Systems:

4Variational Derivatives• 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 derived from the Hamilton equation:

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in the first term on the right may be written as

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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

+

8Single Particle Retarded Green’s FunctionsNoninteracting 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 G1

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

9• Verify δ-fn: integrate → + ,

• are functional forms of time-ordered ;

• Retardation is ensured by since ;

• Forthe Dirac δ-function driving term is confirmed using the equal-time canonical comm./anticomm. relations:

.

0+ 0+

10Physical 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 :

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Initial value problem:

Obeys homog. equa. (δ ( )→ 0), with initial value by canonical comm./anticomm. relations

0+;

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Dynamical Content of for ∂H(1)/∂t =0Time 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 ( )

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The matrix element,

is the single particle energy eigenfn. In x-rep., .Thus, in position-time representation,

•.

14Matrix 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), ,

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(Dirac prescription, )

is proportional to the density D(ω) of single particle energy eigenstates (per unit energy),

Density of States

16Quantum Mechanical Statistical EnsemblesI. 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.]

17II. 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.).

18Thermodynamic 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

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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

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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

≡

22Periodic/Antiperiodic Thermal Green’s Functions

; Matsubara Fourier Series, ; = even (BE)

or odd(FD) integers. ( )

Matsubara F.S. Coeff.

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Baym &Mermin

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

24Thermal G1-Equa. With 2-Body Interaction

•

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Noninteracting Spectral Weight

•

•

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; n(x) =

•

• GHF(1, 2; 1′; 2′) =

•

• where

Ordinary Hartree & Fock Approx. (Equilib.)H

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27Nonequilibrium Green’s Functions: ∂H/∂t ≠ 0

I. Physical:NO Periodicity

• Time Dev. Op.:

• Iterate:

• Time-Ordered Exp: (Time Development Op.)

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Periodic/Antiperiodic Nonequilib. G-Fn.

• Periodicity:(depends t, t′ separately)

• Lim →-∞ G1(1, 1′; to) =

• Var. Diff:

• Var. Diff:

of G1

of

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where .

• Eff. Pot: (Drop δ/δU)

Nonequilib. G-Fn. Eq. of Motion

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Linearize:

Time-Dep. Hartree Approx-Nonequilibrium

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= U(1)

RPA Dynamic, Nonlocal Screening Function K(1,2)

′ ′ ′

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• K= ε-1: ,

•

,

• where .

• Matsubara FS Coeff:

RPA Polarizability α(1,2)

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•

•

Ring Diagram I

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•

•

Ring Diagram II

35Ring Diagram III – 2D – Momentum Rep. for Graphene

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)

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

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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.)

37Particle-Hole Excitation Spectrum I Notation: πo ≡ + iDo ≡ R ; πRPA ≡ + iD• NORMAL 2DEG; T=0; B=0 Bare

(a) (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)

_ _

_

Screened Spectrum Spectrum

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• 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

Particle-Hole Excitation Spectrum II

Screened

Spectrum Spectrum

; ∑′ = ∑NF - 1

n=max(0,NF – m)

R. Roldan, et al, arXiv:0909.2825

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• 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) 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 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 IIIa

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• DOPED GRAPHENE ; 2D ; T = 0 ; B = 0

Screened

(c) Spectral function Im πRPA(q,ω) for doped graphene in the wavevector/energy plane. The electron-electron interactions are taken into account within the RPA.

Particle-Hole Excitation Spectrum IIIb

M.O. Goerbig, arXiv: 1004.3396v1 [cond-mat.-mes-hall] 20 Apr 2010

(c)

Spectrum

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• 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 Spectrum

Particle-Hole Excitation Spectrum IVa

Bare particle-hole excitation spectrum for graphene in a perpendicular magneticfield. We have chosen NF = 3 in the CB and a LL broadening of δ = 0.05vF h / lB.

_

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Screened

Particle-Hole Excitation Spectrum IVb

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 inthe CB and a LL broadening of δ = 0.05vF h/lB.

•DOPED GRAPHENE ; 2D ; T = 0 ; B ≠ 0 ; Screened Spectrum

Spectrum

M.O. Goerbig, arXiv: 1004.3396v1 [cond-mat.-mes-hall] 20 Apr 2010