introduction to linear response propagators and green’s functions
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Many-Body Theory Application to Electrons in Solids Charles Patterson [email protected] School of Physics, Trinity College Dublin. Introduction to Linear Response Propagators and Green’s Functions Green’s Function for Schr ö dinger Equation Functions of a Complex Variable - PowerPoint PPT PresentationTRANSCRIPT
Many-Body TheoryApplication to Electrons in Solids
Charles Patterson [email protected]
School of Physics, Trinity College Dublin
• Introduction to Linear Response• Propagators and Green’s Functions• Green’s Function for Schrödinger Equation• Functions of a Complex Variable• Contour Integrals in the Complex Plane• Schrödinger, Heisenberg, Interaction Pictures• Occupation Number Formalism• Field Operators • Wick’s Theorem• Many-Body Green’s Functions• Equation of Motion for the Green’s Function• Evaluation of the Single Loop Bubble• The Polarisation Propagator• The GW Approximation• The Bethe-Salpeter Equation• Numerical Aspects of Many-Body Theory• The GW Approximation• Hybrid Density functionals
Recommended Texts
• A Guide to Feynman Diagrams in the Many-Body Problem, 2nd Ed. R. D. Mattuck, Dover (1992).
• Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, Dover (2003).
• Many-Body Theory of Solids: An Introduction, J. C. Inkson, Plenum Press (1984).
• Green’s Functions and Condensed Matter, G. Rickaysen, Academic Press (1991).
• Mathematical Methods for Physicists, 5th (Int’l) Ed. G. B. Arfken and H. J. Weber, Academic Press (2001)
• Elements of Green’s Functions and Propagation,G. Barton, Oxford (1989).
Overview
• Propagation of single particles or holes• Zero temperature formalism (c.f. finite temperature formalism)• Applicable in first principles, model Hamiltonian, … methods• Scattering of particles and holes from each other and external potentials
– Renormalisation of particle or hole energies– Finite lifetimes for particles or hole excitations (quasiparticles)– Particle-hole bound states (excitons, plasmons, magnons, …)
• New concepts such as – N-body Green’s functions particle, hole, particle-hole, particle-particle, …
propagators – Self energy (energy renormalisation and excitation lifetime)
Overview
• Approximations to propagators derived from expansion in (Feynman) diagrams or by functional derivative technique– Wick’s Theorem for evaluating time-ordered products of operators
– Lehmann Representation to extract retarded functions
• Integral equations which arise in the new concepts– Dyson’s equation G = Go + GoG
– Bethe-Salpeter Equation = o + o v
– Effective Potential V = v + v o V
• Applications of Concepts and Methods in Many-Particle Theory – Correlation Effects and the Total Energy (No-body propagator)
– Self-Energies and the GW approximation (1-body propagator)
– Collective Excitations and the Bethe-Salpeter Equation (2-body propagator)
Classical Statistical Mechanics
• Average value of variable A• Probability distribution in phase space
Probability that system is in infinitesimal region of phase space d
Elements of position p and momenta q
Average value of variable A
Density in phase space
1),(d
),P(d
),P(),(
),P(d
),)P(,A(dA
...dddd
...dddd
ddd
)d,P(
N21
N21
qp
qp
qpqp
qp
qpqp
qqq q
ppp p
qp
qp
p1, q1
p2, q2
Classical Statistical Mechanics
• Average value of variable A in NVT Canonical Ensemble
Probability depends on total energy of state
Canonical partition function normalises
Helmholtz free energy
Average value of variable A
E)/kT-(F
F/kT
)/kT,E(-
)/kT,E(-
)/kT,E(-
)/kT,E(-
)/kT,E(-
)/kT,E(-
AedA
ez
1
lnz kTF
edz
ed
)e,A(dA
ed
e),(
e),P(
qp
qp
qp
qp
qp
qp
qp
qp
qp
Classical Statistical Mechanics
• Correspondence with Quantum Mechanics
Density operator
Variables represented as operators
Complete set of states
Expectation value of A
Trace of A
Sum of diagonal elements with probability weights
Pure state
Mixed state
0p 1p
0p 1p
a pa pA
...Tr
ˆATrAˆTrA
aA
pˆ
iji
iji
nnn
iji
ij
i
statesallnjiiin
nstatesall
n
i
ji
ii
Classical Statistical Mechanics
• Linear Response in Classical Mechanics
Hamiltonian contains Ho and perturbation B
Average value of A in absence of perturbation B = 1/kT
<A> changes in presence of perturbation B
Defines the linear response
BH-
BH-
H-
H-
H-
H-
o
o
0
o
o
o
o
o
ed
AedA
A
ed
AedAAA
ed
AedA
BHH
Classical Statistical Mechanics
• Linear Response in Classical Mechanics
Expansion of exponential with scalar exponent
Linear response function
ooo20
2
o
o
o
o
o
o
2
o
o
32
2
BAABV
UV'-VU'lim
BAABV
UV'-VU'
ed
e Bd
ed
eA d
ed
e ABd
V
UV'-VU'
e Bd V'
e ABd U'
...A3!
1A
2!
1A1e
V V'
U U'
V
UV'-VU'
V
U
B-H-
B-H-
B-H-
B-H-
B-H-
B-H-
B-H-
B-H-
A
Classical Statistical Mechanics
• Example of application of Static Linear Response
Molecule in gas phase with permanent dipole moment
-p.E term in Hamiltonian
<px>o vanishes in absence of field
p is magnitude of permanent moment
P is polarisation
is molecular susceptibility of gas phase
kT3
p
EkT3
pE
V3
NpP
3
pp
Epp
EpppEppEppp
EpB-
pA
2
x
2
x
2
x
2
o
2x
xo
2xx
xoxoxo
2xoxxoxoxxxx
xx
x
Classical Statistical Mechanics
• Time-dependent linear response
Relaxation after static perturbation in time domain
B(0)A(t)
B(0)A(t)A(t)
ed
e B(0)1A(t)d
ed
e A(t)dA(t)
Bt)(HH
0
o
o
o
H-
H-
H-
H-
• Consider a system where a steady perturbation –B is applied from t→-• The perturbation is switched off abruptly at t = 0• To obtain <A(t)> we must use the perturbed system density at t = 0 (full H)
(-t)
t
1
0
Classical Statistical Mechanics
• Time-dependent linear response• Onsager regression hypothesis
The way in which spontaneous fluctuations in a system relax back to
equilibrium is the same as the way in which a perturbed system relaxes
back to equilibrium, so long as the perturbation is small
Relaxation after static perturbation in time domain(t)(0)pp(t)
E(t)(0)pp(t)p
(t)(0)p(0)Ep(t)p
B(0)A(t)
xx
xxxx
xxxx
kT
B(0)A(t))(χ d)(χ dA(t)
0 t 0)f(t'
0 t)f(t'
t τ0 t' τ- t'dt'dτ
t'tτ
))f(t't'-(tχdt'A(t)
)t'-(tχ)t'(t,χ
t' t 0)t'(t,χ
))f(t't'(t,χdt'A(t)
t
AB
t
AB
t
-
AB
ABAB
AB
-
AB
Classical Statistical Mechanics
• Time-dependent linear response
Define the response function AB
Causality requires this for t < t’
Response depends only on time difference
Introduce change of variable
f(t’) switches off at t’ = 0
Linear response in terms of response function
Classical Statistical Mechanics
• Linear response function
Linear response function is correlation function
0 0)(χ
0 )(AB(0)-)(χ
(t)AB(0)B(0)A(t)dt
d
rule Leibniz' (t)χ)(χ ddt
d
B(0)A(t))(χ d
AB
.
AB
.
AB
t
AB
t
AB
Classical Statistical Mechanics
• Example: Mobility of particle in a fluid
Phenomenological relation
Force acting on charged particle in uniform field
Linear response approach
Need to know velocity auto-correlation function
Mobility
0
xx
..
0
xxx
0
x
.
x
0
x
.
xx
0
xvx
t
-
xvxx
xx
xx
)((0)vv d
τ)(tBA(t)τ)(t)B(tA )((0)vv dF
0τ)A(t)B(tdt
d )((0)vx dF
)(vx(0) dF (t)v
)(χ dF)t'-(tχdt'F(t)v
kxρ(x)φ(x)HqEF
F (t)v
xx
Classical Statistical Mechanics
• Example: Electric conductivity
Hamiltonian for charged particle in vector potential, A
Omit nonlinear A2 term
Define current density
Hamiltonian as Ho + perturbing part
t),(.t,dc
1HH
)'-((t)et),(c
1 U
2m
pH
AO.mc
eHH
UA2mc
e.
mc
e
2m
pH
Uc
e
2m
1H
o
.
pot.
2
o
2o
pot.2
2
22
pot.
2
rj)r A(r
rrvrj
AE
pA
pA
Ap
-
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
Equation of motion
Set driving force to zero
Equation of motion in absence of driving force
Solution in absence of driving force
A, depend on initial conditions
Required for EoM to be satisfied – defines 1
t/2
t/2
eδ)tsin(2
-δ)tcos(A(t)x
4-
δ)tsin(Aex(t)
0xxx
0 F(t)
F(t)x mxmxm
111
.
22o
21
1
2o
...
2o
...
)t'-(t )t'-G(tdt
d
dt
d 2o2
2
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
We can view the continuous force applied to a mass on a spring as
a sequence of delta function impulses. If we know the response of
the system to a single impulse, provided the system is linear, we
can immediately write down the solution in terms of the impulse
response function.
G(t-t’) is the Green’s functionDirac delta function is a unit impulse function
Velocity of oscillator when given an unit impulse at t = 0
t/2et)sin(2
-t)cos(m
1(t)x 111
1
.
)F(t'
m
e )t'(tsindt'x(t)
e )t(tsin Am
1x(t)
e )t'(tsin2
)t'(tcosm
1(t)
e )t'(tsinm
1x(t)
m
1
m
)F(t'dt'
m
)F(t'lim(t)dtxlim)t'(x)t'(x
t
0 1
1
ii1
ii
1
1111
11
t'
t'0
t'
t'
..
0
..
)t'-(t2Γ-
)t-(t2Γ-
)t'-(t2Γ-.
x
)t'-(t
2Γ-
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
Unit impulse F(t’) = 1
t t+ t t+ t'
Position following unit impulseVelocity following unit impulsePosition following impulse seriesPosition following continuous force
)F(t')t't(m
e)t'(tsin
2-)t'(tcos dt'
)F(t')t'G(tdt'dt
d(t)x
)F(t')t'G(tdt'x(t)
)t'-(t )t'-G(tdt
d
dt
d
)t't(m
e )t'(tsin )t'-G(t
1111
t
t.
t
2o2
2
1
1
)t'-(t2Γ-
)t'-(t2Γ-
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
Green’s function
Defining relation for G
Position, from G for a given F
Velocity, from G for a given F
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
Change of variable
Example: impulse at t’=0
Recover original solution from impulse response function
)t(m
et)sin(
2-t)cos((t)x
)-(t)(m
e)sin(
2-)cos( d(t)x
)-(t)(t')F(t'
)-F(t)(m
e)sin(
2-)cos(ω d(t)x
0t t' - t'dt'd t't
)F(t')t't(mω
e)t'(tωsin
2-)t'(tωcosω dt'(t)x
1111
.
1111
0
.
1111
0
.
1111
t.
2Γt-
2Γ-
2Γ-
)t'-(t2Γ-
ii
i ii
i
ii
22o
22o
.
1111
0
.
tt
-t2Γ-
-tt'
e
dt
dRe
m
1e Re
m
1x
e )(m
e)sin(
2-)cos( dRe(t)x
eRe e Re)F(t'
Classical Statistical Mechanics
Example: Periodic forcing
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Impulse Response Function
If we set the lower limit in the integral dt’ to 0 instead of – we obtain additional transient terms which depend on initial conditions
Classical Statistical Mechanics
2Γt-
2Γt-
2Γ-
2)Γ(t-
x
x2Γt-
et)sin(2
-t)cos(m
kT(t)x(0)x
et)sin(2
-t)cos(1
m
kT(t)x
m
kT(0)x
2
(0)xm
2
kTE
e)sin(2
-)cos(2
e)(tsin2
-)(tcos
et)sin(2
-t)cos(dt)(t(t)vvdt
1111
..
1111
...
1111
111
111
00
xx
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Evaluate velocity auto-correlation function
T
t')t'(T
'
e f *f dT
ee )(t'f )t'-(Tf dt'dt)()g(g
dT dt t'-T tt'tT
e )(t'f dt'e (t)fdt )()g(g
e (t)fdt )(g
f*f)'(tf)t'-(tf dt'f*f
21
2121
2121
11
122121
i
ii
titi
ti
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
• Parseval’s Theorem
Convolution of f1 and f2
Fourier transform of f1
Product of Fourier transforms
Fourier transform of convolution
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
)().()( )(mV
e
e)(e
)sin(2
-)cos(dmV
e)G(
)-(tE)(e)sin(2
-)cos(m
kT
V
e)-(t).()((0)
V
e
)()((0).c
)-(td
V
e(t)
)((0))G( )'((t)et),(
))F(G()X( )-F(t)G( d)-F(t)G( d)F(t')t'G(tdt'x(t)
22o
2
1111
2
1111
22
0
.2
.
0
t
2Γ-
2Γ-
EJ
Evv
vvA
j
ABrrvrj
i
i
i
Classical Statistical Mechanics
• Example: Electric conductivity of harmonic oscillator e.g. phonon
Same result as Drude when o tends to zero
22
2
22
2
2
2
22
Drude
22
2
oSHO
1
1
m
e1
1
m
e
1
1
m
e
1
1
m
e)(
m
e)0;(
ii
i
i
i
Classical Statistical Mechanics
• Conclusions
• Linear response functions, e.g. transport coefficients, are derived
from correlation functions
• The correlation function is independent of the external stimulus
(Onsager)
• The reponse function contains the step function () to satisfy
causality