Download Quantum Theory of Solid State Plasma Dielectric Response

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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)
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 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.
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Variational Derivatives
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• 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
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6

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
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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+
•
→
+
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,
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
,
:
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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 (
)
•
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The matrix element,
is the single particle energy eigenfn. In x-rep.,
Thus, in position-time representation,
•
.
.
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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:
,
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Density of States
(Dirac prescription,
)

is proportional to the density D(ω) of single particle
energy eigenstates (per unit energy),

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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
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
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
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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)
•
Using cyclic invariance of Trace & using
translation oper. through imaginary time
Time Rep:
Freq. Rep:
≠0
as time
,
;
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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
)
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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
•


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Noninteracting Spectral Weight
•
•


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Ordinary Hartree & Fock Approx. (Equilib.)

H
•
′; 2′) =
• GHF
(1,
2;
1
2
•
• where
; n(x) =
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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
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Nonequilib. G-Fn. Eq. of Motion
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

where
• Eff. Pot:
.
(Drop δ/δU)
Time-Dep. Hartree Approx-Nonequilibrium


 Linearize:
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RPA Dynamic, Nonlocal Screening Function K(1,2)




′
′
′ = U(1)
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RPA Polarizability α(1,2)
• K= ε-1:
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,
•

,
• where
• Matsubara
FS Coeff:
.
Ring Diagram I
•
•
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Ring Diagram II
•
•
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Ring Diagram III – 2D – Momentum Rep.
for Graphene
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 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.)
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Particle-Hole Excitation Spectrum I
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_

_
_

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)
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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
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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) 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
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• 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
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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 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
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•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.