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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: [email protected]
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