Nonequilibrium Quasiparticles 1
Nonequilibrium Quasiparticles
Anděla Kalvová Institute of Physics, v.v.i.Academy of Sciences of the Czech Republic
Nonequilibrium Quasiparticles 2
Nonequilibrium Quasiparticles
Anděla Kalvová Institute of Physics, v.v.i.Academy of Sciences of the Czech Republic
Quasi-Particle States of Electron Systems out of EquilibriumB. Velický, A. Kalvová, V. Špička; PRB 75, 195125 (2007)
Nonequilibrium Quasiparticles 3
Nonequilibrium Quasiparticles
Anděla Kalvová Institute of Physics, v.v.i.Academy of Sciences of the Czech Republic
Quasi-Particle States of Electron Systems out of EquilibriumB. Velický, A. Kalvová, V. Špička; PRB 75, 195125 (2007)
Transients in Quantum Transport A. Kalvová: Time Partitioning … Chemnitz 2005
Nonequilibrium Quasiparticles 4
- motivation
- definition
Nonequilibrium Quasiparticles
Nonequilibrium Quasiparticles 5
motivation
First slide from: Transients in Quantum Transport A. Kalvová: Time Partitioning
6
(Non-linear) quantum transport non-equilibrium problem
many-body Hamiltonian
many-body density matrix
additive operator
Many-body system
Initial state
External disturbance
H
0 0 0at ( )t t t P P
0 ( ) for U t t t ( )tU
7
0( ) for t t t
(Non-linear) quantum transport non-equilibrium problem
Many-body system
Initial state
External disturbance
Response
many-body Hamiltonian
many-body density matrix
additive operator
one-particle density matrix
H
0 0 0at ( )t t t P P
0 ( ) for U t t t ( )tU
8
(Non-linear) quantum transport non-equilibrium problem
Quantum Transport Equation a closed equation for ( )t
drift [ ( ); ]tt
generalized collision term
Many-body system
Initial state
External disturbance
Response
many-body Hamiltonian
many-body density matrix
additive operator
one-particle density matrix
H
0 0 0at ( )t t t P P
0( ) for t t t 0 ( ) for U t t t ( )tU
9
(Non-linear) quantum transport non-equilibrium problem
Quantum Transport Equation a closed equation for ( )t
drift [ ( ); ]tt
Many-body system
Initial state
External disturbance
Response
many-body Hamiltonian
many-body density matrix
additive operator
one-particle density matrix
H
0 0 0at ( )t t t P P
0( ) for t t t
QUESTIONS existence, construction of incorporation of the initial
condition
0P
interaction term
0 ( ) for U t t t ( )tU
10
a closed equation for ( )t
(Non-linear) quantum transport non-equilibrium problem
Quantum Transport Equation
drift [ ( ); ]tt
Many-body system
Initial state
External disturbance
Response
many-body Hamiltonian
many-body density matrix
additive operator
one-particle density matrix
H
0 0 0at ( )t t t P P
0( ) for t t t
QUESTIONS existence, construction of incorporation of the initial
condition
0P
interaction term
0 ( ) for U t t t ( )tU
not quasiparticle
Nonequilibrium Quasiparticles 11
quasiparticle way .... Landau theory for Fermi liquids
motivation
Nonequilibrium Quasiparticles 12
Landau theory for Fermi liquids
the original system of quantum particles with strong interactions transformed
(purely phenomenologically in physical sense) to an effective
system of quasiparticles - forming a weakly non-ideal gas
- governed by a (slightly generalized) Boltzmann equation
motivation
Nonequilibrium Quasiparticles 13
(Non-linear) quantum transport non-equilibrium problem
Many-body system
Initial state
Weak extern. disturb.
many-body Hamiltonian
m.-b. equilibrium state
additive operator
H
0 0 0at ( )t t t P P
0 ( ) for U t t t ( )tU
motivation
Nonequilibrium Quasiparticles 14
(Non-linear) quantum transport non-equilibrium problem
Many-body system
Initial state
Weak extern. disturb.
Response
many-body Hamiltonian
mb equilibrium state
additive operator
distribution function for QP
H
0 0 0at ( )t t t P P
0( ) for f t t t0 ( ) for U t t t ( )tU
motivation
Nonequilibrium Quasiparticles 15
(Non-linear) quantum transport non-equilibrium problem
Quantum Transport Eq. Quantum Boltzmann Eq.
Many-body system
Initial state
Weak extern. disturb.
Response
many-body Hamiltonian
m.-b. equilibrium state
additive operator
distribution function for QP
H
0 0 0at ( )t t t P P
0( ) for f t t t0 ( ) for U t t t ( )tU
motivation
d
/ d current + influence of external fields
+ collisions
f t
Newtonchange macroscopically understandable molecular per time evolution ( ) chaos
16
d
/ d current + influence of external fields
+ collisions
f t
(Non-linear) quantum transport non-equilibrium problem
Quantum Transport Eq. Quantum Boltzmann Eq.
Many-body system
Initial state
Weak extern. disturb.
Response
many-body Hamiltonian
m.-b. equilibrium state
additive operator
distribution function for QP
H
0 0 0at ( )t t t P P
0( ) for f t t t0 ( ) for U t t t ( )tU
COLLISION INFLUENCE1. evolution to equilibrium2. time arrow = direction of the entropy growth3. forgetting of the past – after ONE collision, (during average collision time) from initial condition to “floating” initial condition
motivation
Newtonchange macroscopically understandable molecular per time evolution ( ) chaos
Nonequilibrium Quasiparticles 17
linear transport ... redistribution of the equilibrium QP (equilibrium GF)
motivation
Nonequilibrium Quasiparticles 18
motivation
non-linear transport ... redistribution of non-equilibrium QP (non-equilibrium GF) ??
linear transport ... redistribution of the equilibrium QP (equilibrium GF)
Nonequilibrium Quasiparticles 19
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
definition QP
linear transport ... redistribution of the equilibrium QP (equilibrium GF)
Nonequilibrium Quasiparticles 20
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
phenomenological constants
( )R t
linear transport ... transfer equilibrium QP (equilibrium GF)
definition QP
Nonequilibrium Quasiparticles 21
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
( )R t
formation time
linear transport ... transfer equilibrium QP (equilibrium GF)
phenomenological constants
definition QP
Nonequilibrium Quasiparticles 22
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
( )R t
renormal. constant
linear transport ... transfer equilibrium QP (equilibrium GF)
phenomenological constants
formation time
definition QP
Nonequilibrium Quasiparticles 23
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
( )R t
complex energy
linear transport ... transfer equilibrium QP (equilibrium GF)
formation time
renormal. constant
definition QP
phenomenological constants
Nonequilibrium Quasiparticles 24
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
( )R t
1i(2 )w wz E QP condition
2Q
linear transport ... transfer equilibrium QP (equilibrium GF)
complex energy
formation time
renormal. constant
definition QP
phenomenological constants
Nonequilibrium Quasiparticles 25
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
( )R t
i
0
d ( )e wz tRw MFz t t
linear transport ... transfer equilibrium QP (equilibrium GF)
definition QP
phenomenological constants
formation time
renormal. constant
complex energy
26 ... pole and residuum of the spectral representation of the equilibrium GF (one - el. propagator)
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
( , ) i exp[ i ( )]RW wG t t z t t
( )R t
i
0
d ( )e ( i 0)MF tR Rw MF MF MFz t t
... Wigner-Weisskopf (weak scatt.)
,wz Z
linear transport ... transfer equilibrium QP (equilibrium GF)
definition QP
phenomenological constants
formation time
renormal. constant
complex energy
Nonequilibrium Quasiparticles 27
non-equilibrium initial state ... arbitrary ... full description is necessary
system evolves ... the processes… very fast, fast, slow
very fast processes... smoothing, coherence loss, decay of the many-body
correlation, chaotization characteristic time
fast processes ... collisions leading to the kinetic stage of transport
characteristic time
slow processes ... compensating macroskopic inhomogeneities
characteristic time
BOGOLJUBOV POSTULATES
initial stage, kinetic, hydrodynamicIn kinetic stage …(quantum) distribution function provides complete description satisfies quantum transport equation
C H
C
hierarchy of characteristic times
H
28
from non-equilibrium to „equilibrium“
KBA (Kadanoff Baym Ansatz) ... NGF ( )... in Wigner representation ...time dependent spectral density ... NGF quantum transport equation for QP distribution
restriction ... processes… quasiclassical in time
, ,R AG G G
motivation NeQP
KB were inspired by equilibrium. Their ansatz leadsto the distribution function of QP. Their dynamical behavior is described by equation of Bolzmann type. Processes in such MB system have to be quasi-
classical in time
29
from non-equilibrium to „equilibrium“
KBA (Kadanoff Baym Ansatz) ... NGF ( )... in Wigner representation ...time dependent spectral density ... NGF quantum transport equation for QP distribution
restriction ... processes… quasiclassical in time
, ,R AG G G
GKBA (Generalized KBA)... quantum transport equation for real particles (density matrix) causal structure rapid transients ...quasiparticles
restriction ... renormalization (formation) …negligible during c
( , ) ( , ) ( )RG t t G t t t t t
motivation NeQP
In the opposite situation… in rapid transients, we construct the QTEfor real particles. The ansatz GKBA is formulated fully in time domain.
The using of GKBA is restricted too. It is possible to useit only if the formation processes during initial stage are negligible.
30
from non-equilibrium to „equilibrium“
KBA (Kadanoff Baym Ansatz) ... NGF ( )... in Wigner representation ...time dependent spectral density ... NGF quantum transport equation for QP distribution
restriction ... processes… quasiclassical in time
, ,R AG G G
GKBA (Generalized KBA)... quantum transport equation for real particles (density matrix) causal structure rapid transients ...quasiparticles
restriction ... renormalization (formation) …negligible during c
QKBA (Quasiparticle KBA) ... quantum transport equation for real particles dynamical (formation) processes …involved
( , ) ( , ) ( )RG t t G t t t t t
( , ) ( , ) ( )RwG t t G t t t t t
restriction ... if nonequilibrium QPs exist
motivation NeQP
Nonequilibrium Quasiparticles 31
definition NeQP
Any propagator … 1. should pass through a formation stage 2. then should represent a QP
In nonstationary (transient, nonequilibrium) conditions … two questions 1.? formation time ? 2.? is it possible to construct … as QP counterpart to
In transients, two processes compete permanently: formation and flight. Complex process results and we don’t know,
if it is a flight of independent quasiparticles or if this concept fails.
( , )RwG t t ( , )RG t t
??
Nonequilibrium Quasiparticles 32
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
functional relation nonequilibrium QP
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t t t t
( , ) ( , ) ( , )R R Rw QG t t G t t G t t t t t t
composition rule SGR
QP comosition rule QCR
equilibrium:
definition NeQP
Nonequilibrium Quasiparticles 33
( , ')RG t t QP formation
( , )RWZG t t
... Qt t
... Qt t
functional relation nonequilibrium QP
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t t t t
( , ) ( , ) ( , )R R Rw QG t t G t t G t t t t t t
composition rule SGR
QP comosition rule QCR
equilibrium:
definition NeQP
i /R Rt w QG G z t t logarithmic derivative ... time independent
Nonequilibrium Quasiparticles 34
nonequilibrium propagator: .... double-time function (matrix)( , ')RG t t
( , )R t tselfenergy: .... 'M mt t t t
1[ ( , )] {i ( )} ( ) ( , )R Rt MFG t t H t t t t t
definition NeQP
Nonequilibrium Quasiparticles 35
nonequilibrium propagator: .... double-time function (matrix)( , ')RG t t
( , )R t tselfenergy: .... 'M mt t t t
1[ ( , )] {i ( )} ( ) ( , )R Rt MFG t t H t t t t t
definition NeQP
free-particle Ham.+ external fields+ mean-field
Nonequilibrium Quasiparticles 36
nonequilibrium propagator: .... double-time function (matrix)( , ')RG t t
( , )R t tselfenergy: .... 'M mt t t t
1[ ( , )] {i ( )} ( ) ( , )R Rt MFG t t H t t t t t
( , ) 0 ( )RQt t t t t t
definition NeQP
NeQP (formation time condition).....
It is natural to restrict our study to the case, when a NE extension of QP formation time exists. Therefore,
we assume the existence formation time such, that the selfenergy is zero outside a double time region, which has the strip adjoing the time diagonal.
Q
Nonequilibrium Quasiparticles 37
nonequilibrium propagator: .... double-time function (matrix)( , ')RG t t
( , )R t tselfenergy: .... 'M mt t t t
1[ ( , )] {i ( )} ( ) ( , )R Rt MFG t t H t t t t t
( , ) 0 ( )RQt t t t t t
1[ ( , )] {i ( )} ( ) ( , )R Rw t MF wG t t H t t t t t
definition NeQP
NeQP (formation time condition).....
NeQP propagator
Nonequilibrium Quasiparticles 38
NeQP propagator ( , ')RwG t t
SGR ...satisfied time local optical potential ( )w t
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t
1( , ) independent
( , )R
t w Rw
G t tG t t
t
time diagonal; ( ) ( i 0)R Rw w w t t t
( , ) i exp[ i d { ( ) ( )}t
Rw MF
t
G t t T u H u u
definition NeQP
Nonequilibrium Quasiparticles 39
NeQP propagator ( , ')RwG t t
SGR ...satisfied time local optical potential ( )w t
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t
time diagonal; ( ) ( i 0)R Rw w w t t t
( , ) i exp[ i d { ( ) ( )}t
Rw MF
t
G t t T u H u u
definition NeQP
1( , ) independent
( , )R
t w Rw
G t tG t t
t
Nonequilibrium Quasiparticles 40
NeQP propagator ( , ')RwG t t
SGR ...satisfied time local optical potential ( )w t
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t
time diagonal; ( ) ( i 0)R Rw w w t t t
( , ) i exp[ i d { ( ) ( )}t
Rw MF
t
G t t T u H u u
definition NeQP
mean-field propagator
unitary evolution independ. „particles“in external and internal fields
1( , ) independent
( , )R
t w Rw
G t tG t t
t
Nonequilibrium Quasiparticles 41
NeQP propagator ( , ')RwG t t
SGR ...satisfied time local optical potential ( )w t
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t
1( , ) independent
( , )R
t w Rw
G t t tG t t
time diagonal; ( ) ( i 0)R Rw w w t t t
( , ) i exp[ i d { ( ) ( )}t
Rw MF
t
G t t T u H u u
definition NeQP
Nonequilibrium Quasiparticles 42
NeQP propagator ( , ')RwG t t
SGR ...satisfied time local optical potential ( )w t
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t
1( , ) independent
( , )R
t w Rw
G t t tG t t
time diagonal; ( ) ( i 0)R Rw w w t t t
( , ) i exp[ i d { ( ) ( )}t
Rw MF
t
G t t T u H u u
definition NeQP
NeQP propagator
Nonequilibrium Quasiparticles 43
NeQP propagator ( , ')RwG t t
SGR ...satisfied time local optical potential ( )w t
( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t
1( , ) independent
( , )R
t w Rw
G t t tG t t
time diagonal; ( ) ( i 0)R Rw w w t t t
( , ) i exp[ i d { ( ) ( )}t
Rw MF
t
G t t T u H u u
definition NeQP
NeQP propagator
not a unitary evolution ( eff. single particle. nonhermit. H)flight of independent NeQP
Nonequilibrium Quasiparticles 44
starting point Dyson equations for
to find RwG closed equation for ( )w t ... Phase Equation
,R RwG G
( , ) i ( , ) ( , )
d d ( , )( )( , ) ( , )
R R Rw
t tR R R Rw w
t t
t t t
G t t G t t G t t
t t G t t t t G t t
(subtraction)
renormalized quasiparticle composition rule (RQCR)
definition NeQP
Nonequilibrium Quasiparticles 45
starting point Dyson equations for
to find RwG closed equation for ( )w t ... Phase Equation
,R RwG G
( , ) i ( , ) ( , )
d d ( , )( )( , ) ( , )
R R Rw
t tR R R Rw w
t t
t t t
G t t G t t G t t
t t G t t t t G t t
(subtraction)
renormalized quasiparticle composition rule (RQCR)
definition NeQP
Nonequilibrium Quasiparticles 46
starting point Dyson equations for
to find RwG closed equation for ( )w t ... Phase Equation
,R RwG G
( , ) i ( , ) ( , )
d d ( , )( )( , ) ( , )
R R Rw
t tR R R Rw w
t t
t t t
G t t G t t G t t
t t G t t t t G t t
(subtraction)
renormalized quasiparticle composition rule (RQCR)
definition NeQP
QP composition rule (QCR)
Nonequilibrium Quasiparticles 47
starting point Dyson equations for
to find RwG closed equation for ( )w t ... Phase Equation
,R RwG G
( , ) i ( , ) ( , )
d d ( , )( )( , ) ( , )
R R Rw
t tR R R Rw w
t t
t t t
G t t G t t G t t
t t G t t t t G t t
(subtraction)
renormalized quasiparticle composition rule (RQCR)
definition NeQP
renorm. vertex correction
Nonequilibrium Quasiparticles 48
starting point Dyson equations for
to find RwG closed equation for ( )w t ... Phase Equation
,R RwG G
( , ) i ( , ) ( , )
d d ( , )( )( , ) ( , )
R R Rw
t tR R R Rw w
t t
t t t
G t t G t t G t t
t t G t t t t G t t
(subtraction)
renormalized quasiparticle composition rule (RQCR)
definition NeQP
slide Transients in Quantum Transport B. Velický: Semigroup Property of Propagators ...
49
Discussion of the RSG rule
1. universal vertex, derived with almost no effort and no specific properties of the GF
2. off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future
3. Similar to the Dyson Equation, but NO free GF
4. In fact, looks pretty much like a linear response
we would like to understand all these features
'
( , ') i ( , ) ( , ')
d d ( , ) ( , ) , ')
'
(
R R R
t tR R R
t t
t t t
G t t G t t G t t
t t G t t t t G t t
Semi-group property...FZÚ 14.2.2006
50
Discussion of the RSG rule
1. universal vertex, derived with almost no effort and no specific properties of the GF
2. off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future
3. Similar to the Dyson Equation, but NO free GF
4. In fact, looks pretty much like a linear response
we would like to understand all these features
'
( , ') i ( , ) ( , ')
d d ( , ) ( , ) , ')
'
(
R R R
t tR R R
t t
t t t
G t t G t t G t t
t t G t t t t G t t
Semi-group property...FZÚ 14.2.2006
Nonequilibrium Quasiparticles 51
starting point Dyson equations for
to find RwG closed equation for ( )w t ... Phase Equation
,R RwG G
( , ) i ( , ) ( , )
d d ( , )( )( , ) ( , )
R R Rw
t tR R R Rw w
t t
t t t
G t t G t t G t t
t t G t t t t G t t
(subtraction)
renormalized quasiparticle composition rule (RQCR)
definition NeQP
self-consistent condition for cancellation of the vertex
explicit equation for ( )w t
Nonequilibrium Quasiparticles 52
ot
summary of notation of times :
,m Mt t ... minimum, maximum
,t t ...evolution from tto t ( , ,m Mt t t t )
time- inhomogeneity
t
... time between ,t t
NEW... time, where QCR is valid
... min t
( , ) ( , ) ( , )R R RwG t t G t t G t t
1i ( , ) ( ) ( )
( , )R
t MF wRG t t H t t
G t t
independent of ,t t
definition NeQP
Nonequilibrium Quasiparticles 53
ot
,m Mt t ... minimum, maximum
,t t ...evolution from tto t ( , ,m Mt t t t )
time inhomogeneity
t
... time between ,t t
NEW... time, where QCR is valid
... min t
( , ) ( , ) ( , )R R RwG t t G t t G t t
1i ( , ) ( ) ( )
( , )R
t MF wRG t t H t t
G t t
definition NeQP
implication: for fixed many exist... typically ... identical NeQP... ... independent
ot tmax ( )o o
mt t t t t
( , )R owG t t
t
phase equation
independent of ,t t
summary of notation of times :
Nonequilibrium Quasiparticles 54
cesta k fázové rovnici
1[ ( , )] {i ( )} ( ) ( , )R Rt MFG t t H t t t t t
how to get the phase equation for an optical potential ( )w t
definition NeQP phase equation
Nonequilibrium Quasiparticles 55
i ( , ) ( ) ( , ) d ( , ) ( , )t
R R R Rt MF
t
G t t H t G t t t t t G t t
how to get the phase equation for an optical potential ( )w t
definition NeQP
( )w t
definition NeQP phase equation
Nonequilibrium Quasiparticles 56
i ( , ) ( ) ( , ) d ( , ) ( , )t
R R R Rt MF
t
G t t H t G t t t t t G t t
i ( , ) 1( ) d ( , ) ( , )
( , ) ( , )
tRR Rt
MFR Rt
G t tH t t t t G t t
G t t G t t
( )t t definice...
how to get the phase equation for an optical potential ( )w t
definition NeQP phase equation
Nonequilibrium Quasiparticles 57
i ( , ) ( ) ( , ) d ( , ) ( , )t
R R R Rt MF
t
G t t H t G t t t t t G t t
i ( , ) 1( ) d ( , ) ( , )
( , ) ( , )
tRR Rt
MFR Rt
G t tH t t t t G t t
G t t G t t
( )t t definice...
i ( , )( ) ( )
( , )
Rt
MF tR
G t tH t t
G t t
how to get the phase equation for an optical potential ( )w t
definition NeQP phase equation
Nonequilibrium Quasiparticles 58
1( ) d ( , ) ( , )
( , )
tR R
t Rt
t t t t G t tG t t
how to get the phase equation for an optical potential ( )w t
definition NeQP phase equation
Nonequilibrium Quasiparticles 59
? ( )w t ... ot t ?
( ) d ( , ) exp[ i d { ( ) ( )}]
1d ( , ) ( , )
( , )
exp[ i d { ( ) ( )}]
o
o
t tR
t MF t
tt
tR R
R ot
t
MF t
t
t t t t T u H u u
t t t G t tG t t
T u H u u
definition NeQP phase equation
to find RwG closed equation for ( )w t ... Phase Equation
Nonequilibrium Quasiparticles 60
? ( )w t ... ot t ?
( ) d ( , ) exp[ i d { ( ) ( )}]
1d ( , ) ( , )
( , )
exp[ i d { ( ) ( )}]
o
o
t tR
t MF t
tt
tR R
R ot
t
MF t
t
t t t t T u H u u
t t t G t tG t t
T u H u u
( )oQt t t
phase equation
definition NeQP phase equation
far enough from ot
Nonequilibrium Quasiparticles 61
( ) d ( , ) exp[ i d { ( ) ( )}]
1d ( , ) ( , )
( , )
exp[ i d { ( ) ( )}]
o
o
t tR
t MF t
tt
tR R
R ot
t
MF t
t
t t t t T u H u u
t t t G t tG t t
T u H u u
definition NeQP phase equation
Initialcondition
? ( )w t ... ot t ?
Nonequilibrium Quasiparticles 62
( ) d ( , ) exp[ i d { ( ) ( )}]
1d ( , ) ( , )
( , )
exp[ i d { ( ) ( )}]
o
o
t tR
t MF t
tt
tR R
R ot
t
MF t
t
t t t t T u H u u
t t t G t tG t t
T u H u u
definition NeQP phase equation
It is natural to restrict our study to the case, when a NE extension of QP formation time exists. Therefore,
we assume the existence formation time such, that the selfenergy is zero outside a double time region, which has the strip adjoing the time diagonal.
Initialcondition
Q
Nonequilibrium Quasiparticles 63
? ( )w t ... ot t ?
( )
( ) d ( , ) exp[ i d { ( ) ( )}]
d ( , )
e
1( , )
xp[ i d
(
{ ( ) ( )}]
, )o o
o
o
Q
t tR
t MF t
tt
tR
R ott
R
F
t
t
M t
t
t
G t tG t t
t t t T u H u u
t t t
T u H u u
initial condition
definition NeQP phase equation
Nonequilibrium Quasiparticles 64
( )w t ... ot t
( )
( ) d ( , ) exp[ i d { ( ) ( )}]
1d ( , ) ( , )
( , )
exp[ i d { ( ) ( )}]
o
o
o oQ
t tR
w MF w
tt
tR R
R ott t
t
MF w
t
t t t t T u H u u
t t t G t tG t t
T u H u u
closed phase equation
definition NeQP phase equation
Nonequilibrium Quasiparticles 65
( ) d ( , ) exp[ i d { ( ) ( )}]o
t tR
w MF w
tt
t t t t T u H u u
why the term... phase equation ?
why ... and not ? ( )w t
Qcondition for derivation of the phase equation... existence
( , ) 0 ( )RQt t t t t t ... „dominant strip“
... analogue to the dominant strip for ... existence ( , )t t C
,Q C … „forgotten times“, ... decay of init. corr. (formation of the kinetic stage) ... phase eq. doesn´t depend on (formation of QP)
CQ t
definition NeQP phase equation
( , )RwG t t
( )oQt t t
Nonequilibrium Quasiparticles 66
( ) d ( , ) exp[ i d { ( ) ( )}]o
t tR
w MF w
tt
t t t t T u H u u
... semilocal in time ( ...effective time locality)( ) ( )MF wH t t
- anticausal time ordering (near future influence)
- past influence ...closed phase equation
definition NeQP phase equation
( )oQt t t
Nonequilibrium Quasiparticles 67
Conclusion:
consistent definition of the NeQP
NeQP ... two properties (equivalent), used for definition-corresponding propagator satisfies QP composition rule-corresponding propagator contains optical potencial ( )w t
existence of the NeQP ... dependence on retard. selfenergy ( , ')R t t short formation time narrow „dominant strip“ in ( , ')R t t
optical potential ( )w t and NE retarded selfenergy ( , ')R t t ...
... connected with the phase eq. For its solution ...necessity to know initial conditions... reflect time period of formation Q
QP composition rule ... „free term“ in (transf.) Dyson equationvanishing of the vertex correction validity of the phase equation
definition NeQP phase equation