Time-dependent Kondo model with a factorized initial state [Elektronische Ressource] / von Dmitry Lobaskin

Time-dependent Kondo model with a factorized initial state [Elektronische Ressource] / von Dmitry Lobaskin

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UnivLobaskinendenErlangungtvKakultätondoAugsburgMoDoktorgradesdelgenehmigtewithM.S.aRusslandFderactorizedersitätInitialzurStateeinesVderonhaftenderDissertationonDmitryaushenTimeDepFopp21mh2005ter:agPDhenDr.bS.KKehreinTZwdereitgutacündlichPrüfung:ter:DezemProf.erDr.Th....i.CONTENTS.1..Intro.KContents....3.4.........p.Hamiltonian.....Correlation.......del.4.1...e.....21.at...of..............1.1.1.NonEquilibrium.K.ondo.Eect..rium.ondo................4.3.....Non-Equilib.Mo.T.4.4...3.11Eectiv1.2..Motiv.ation3.2...............3.3............................4..ondo4the1.3.Exp.erimen.talwMotiv.ation..........4.2...............Results.........3..rium.ondo.del.the.oulouse7oint1.453Goals.of.This27WDiagonalizationorkthe.e.............31.Magnetization.............................34.Spin-spin.F12.2..Time-dep.endent.K.ondo.Mo.del.Conclusion.4.5.5436.Conclusion........................Asymptotics.the.of.Estimations44AnalyticalNonEquilib17K2.1MoEquilibriuminKKondoLimitProblem........45.Flo.Equation.h...........

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haften
TimeDep
ersit?t
enden
aus
t
eines
K
on
ondo
der
Mo
zur
del
der
with
Dissertation
a
Dmitry
F
akult?t
actorized
Univ
Initial
Augsburg
State
Erlangung
V
Doktorgrades
on

der
genehmigte

v

M.S.

Lobaskin
hen
Russland
FT

Dezem
h
?ndlic
ter:
K
PD
der
Dr.
Pr?fung:
S.
er
Kehrein
opp
Zw
ag
eitgutac
m
h
hen
ter:
21
Prof.
b
Dr.
2005
Th.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . . . .
.
.
.
.
.
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.
.
.
.
.
.
.
.
3.3
.
.
.
ondo
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ation
.
.
.
Eectiv
.
.
.
.
4
.
1.3
the
Exp
4.5
erimen
.
tal
.
Motiv
.
ation

.
.
.
.
.
.
.
.
.
.
.
.
.
4.
.
the
.
1
.
.
.
.
.
.
.
.
.
.
.
.
.
49
.
.
.
.
.
.
.
4.4
.
.
.
.
.
.
7
.
1.4
.
Goals
.
of
.
This
.
W
.
ork
Correlation
.
.
.
.
.
.
.
.
.
.
.
3.4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
rium
.
del
.
ondo
.

.
.
.
.
.
.
.
.
.
.
.
.
.
46
.
Hamiltonian
12
.
2.
.
Time-dep
.
endent
.
K
.
ondo
.
Mo

del
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Estimations
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Eect
.
ondo
.
K
.
NonEquilibrium
.
1.1
.
17
.
2.1
.
Equilibrium
.
K
34
ondo
Spin-spin
Problem
F
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
36
.
Conclusion
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17
.
2.2
.
Timedep
.
enden
.
t
.
K
.
ondo
44
Hamiltonian
NonEquilib
.
K
.
Mo
.
in
.
K
.
Limit
.
Motiv
45
1.2
Flo
.
Equation
.
h
.
4.1
.
w
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
21
.
3.
.
Non-Equilib
.
rium
.
K
4.2
ondo
e
Mo
.
del
.
at
.
the
.
T
.
oulouse
.
p
.
oint
.
1
.

.
Intro
.
1.
.
CONTENTS
.
i
4.3
Contents
Results
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
53
.
Analytical
.
of
.
Asymptotics
.
.
.
.
.
.
31
.
3.2
.
Magnetization
.
.
54
.
Conclusion
.
.
.
.
.
.
.
.
.
.
27
.
3.1
.
Diagonalization
.
of
.
the
.
Eectiv
.
e
.
Hamiltonian
.
.
.
.
.
.
59
.. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
in
5.5
.
Conclusion
.
.
.
.
Mo
.
.
.
.
.
.
.
.
.
.
.
.
.
Limit
.
.
.
of
.
T
.
.
.
.
.
.
.
rem
.
.
.
.
.
85
.
.
.
66
.
.
.
.
.
endix
.

.
t
.
.
.
.
.
.
.
61
.
.
75
wledgements
6.
Fluctuation-Dissipation
Quasipa
Equilib
rticle
of
Sp
.
ectral
.
F
.

.
.
.
.
Summa
.
.
.
.
.
K
.
.
.
.
.
.
.
.
.
.
.
87
.
A.
.
r
.
.
.
.
.
P
.
64
.
Publications
.
.
.
.
.
.
79
.
6.1
FDT
Nonequilibrium
.
Sp
.
ectral
.
Densit
A
y
.
.
.
.
61
.
K
.
the
.
Fluctuation-Dissipation
74
5.
ii
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
7.
.
ry
79
.
6.2
.
Results
.
of
.
the
ondo
F
5.4
ermi
.
Liquid
.
Theory
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
App
.
93
.
Details
.
nonregula
.
sum
.
.
.
.
.
.
.
.
81
oin
6.3
oulouse
Buildup
5.3
of
.
the
95
K
.
ondo
.
Resonance
.
.
.
.
.
.
.
.
.
.
.
.
Nonequilibrium
.
Violation
.
5.2
.
.
.
.
.
.
.
.
.
.
.
.
.
105
.

.
.
.
.
82
.
6.4
Theorem
Conclusion
5.1
.
del
.
ondo
.
rium
.
Non-
.
in
.
Theo
.
the
.
Violation
.
Contents
.
.
.
107
.this
1
y
1.
other
INTRODUCTION

1.1
quan
NonEquilib
e
rium
temp
K
are
ondo
three
Eect
des
Since

early

1970s,

the
ariet


electronic
on
industry
wide
has

b
observ
een
ab
rapidly
ro
progressing
in
fol-

lo
of
wing
erturba-
the
far
Mo
a
ore's
o
La
single-
w

to
up
double
v
pro
t

main
p
Unlik
o
on
w
tunnel
er
w
ev
y
ery
thic
18
10
mon
the
ths.
insulator,
F
energy
or
w
man
tial
y
these
y
pro
ears
en
this
that
has
driv
b
sizes
een
op
satised
rise
largely
quan
b
is
y
on
scaling
and

single-w
to
nanotub
ev
a
en
of
smaller
They
dimensions.

Extrap
making
olating
o
the
able
Mo

ore's
in
la

w
are
in
in
to
phenomenon:
the
This
next
when
10-15
metallic
y
separated
ears,
insulating
it
1
b
-
ecomes
ords,
apparen
in
t
Electrons
that
ermi
the
through
scaling
en
will
terms
run
ould
in
o
to
o
ph
p
ysical
Due
limits:
sizes
limits
all
of
quan
material
start
used,
role.
but
external
also

fundamen
the
tal
ma
limits
SET
arising
equilibrium.
from
lo
the

la

ws
giv
of
wide
quan
of
tum
man



y
whic
electronics
h

b
es
ecome


These
imp
alled
ortan
on
t
es
at

v
rolled
ery
sheet
small

length
atoms.
scales.
ha
Of
e


the
transistor
ultimate
it
goal
w
then
nanometers
is
and
to
to

tain
a
t
functional
nanometers

length.

e
of
transistors,
transp

orting
based
a
an
single

electron.
tum
In
the
1985
eect.
D.
is
A
ed
v
t
erin
o
and
electro
K.
are
Likharev
b
(A
an
v
barrier
erin
out
and
nm
Likharev
k
1986)
in
prop
w
osed
just
the
atoms
idea
a
of
w.
a
at
new
F
three-terminal
energy

"tunnel"

the
a
ev

though
tunneling

(SET)
their
transis-
w
tor.
b
T
to
w
lo
o
to
y
v
ears
the
later
oten
Theo
barrier.
dore
to
F
small
ulton
of
and

Gerald
kinds
Dolan
nonequilibrium
at

Bell

Labs
pla
in
the
the
Ev
US
small
fabricated
p

tions
h
to
a
in

bulk
and
samples
demonstrated
y
ho
e
w
transistor
it
from
op
Small
erates.
and
As
w
of
eratures
August
for
23
erating
2004,
h
Stanford
system
Univ
e
ersit
to
y
v
has
y
b
nonequilibrium
een
tum
able
yb
to
dy

It
a
not
transistor
et
from
whether
single-w
based
alled
individual

and
on
nanotubp
2
b
1.
systems
Intro
equilibrium

result
electron
since
eects
to
will
a
replace
div

o
v
timeindep
en
Another
tional
t

t
based
b
on
extension
scaled-do
t
wn
up
v
true
ersions
relaxation
of
steady

b
transistors.
resp
Only
external
one
if
thing
of
is
erturbation.

w
if

the
of

theory
of
the
miniaturization
order

the
tin
only
ues
Ev
unabated,
miss
the
a
quan
a
tum

prop
b
erties
h
of
th
electrons
statemen
will
highly
b
thermo
ecome
a


in

determining
in
the
ourier
design
quan
of

electronic
e

some
b

efore
system
the
initial
end
nonanalytically
of
F
the
enden
next
w

hnic)
F
small,
ast
up
dev
erturbation
elopmen

t
series
of
the
nano

electronics
h
is
is
already
ximate
sucien
t
t
example
motiv

ation
tioned
to
After
study
a
nonequilibrium
establishes
quan
said
tum
in
man
Although
yb
state
o
t
dy
seems
ph
equilibrium,
ysics
is
whic
is
h
state
ho
to
w
equilibrium
ev
example
er

is
enden
not
It

a
to
p
it.
e
A
in

a
,
onen
nonequilibrium
sp
man
y
yb
onse
o
a
dy

ph
external
ysics
hanged
is
t,
one
In
of
example,
the
of
most
t
fascinating
from
and
and

ev
hallenging
to

original
in

mo
standard
dern
p
ph
es
ysics,
Its
due
diagrammatic
to
applicable
b
of
oth
eakly
the
when
wide
some
v
the
ariet
is
y
get
of
F
observ
problem
ed
diagrams
phenomena
t,
and
d
dicult
if
y
to
in
diagrams

tribute
description.
if
T
ossible
ypical
often
situation
migh
of
imp
nonequilibrium
of

A
the

system
whic
sub
all

e
to
through
some
sample.
exter-

nal
time
generally
stationary
timedep
t
enden
and
t
are
p
to
erturbation.
e
If
the
the
state.
p

erturbation
a
is
is
innitesimally
enden
small
and
and,
us
therefore,
to
the
e
state
this
of
t
the
false
system
it
is
a
almost
exited
the
with
equilibrium
ect
one,
a
the

lin-
one.
ear

resp
is
onse
non
description
applied
of
timedep
an
t
externally
eld.
driv

en
e
system
mono

hromatic
is
erturbation
v
w
alid.
are
Within
terested
this
measuremen
metho
of
d
F
deviations

of
t
all
a
quan

tities
tit
from
related
their
resp
equilibrium
to
v
h
alues
p

It
alw
b
a
an
ys

b

e
at
expressed
momen
via
as
their
ell.

latter
in
for
equilibrium.
the
In
state
particular,
the
the
migh
socalled
dier

tly
theorem
the
is
one
fullled

in
e
this
en
regime.
related
It
parameters

the
generalized
problem.
susceptibilit
or
y
h
of

the
timeindep
system
t
to
erturbation
some
do
external
not
p
ork.
erturbation
nonequilibrium
with
(Keldysh
the

equilibrium
is

in
functions.

Unfortunately
a
,
w
the

linear
erturbation,
resp
summation
onse
to
formalism
nite
is
of
only
p
v
theory
alid
sucien
while
to
the
an
p
result.
erturbation
or
do
strongcoupling
es
the
not
of
driv
are
e
ergen
the
and
system
metho
far
is
from
applicable
equilibrium.
one
F
manage
or
sum
example,
all
during
whic
measuring

of
most.
transp
en
ort
it
prop
p
erties
the

is
y
appro
,
and
thermoresistivit
t
y
some
,
ortan

features
an
a
applied
solution.
bias

v
of
oltage
h

problem
a
h
nonequilibrium
orates

ab
t
v
to
men
o
wTK
is
rium
is
K
turn,
ondo
resp
Eect
system
3
during
diculties
the
of
v
the
equilibration

other
description
the
together
to
with
resp
the
go

e
imp
ou
ortance
are
in
the
nano
in
electronics
orthogonal
is
o
the
impurit
nonequilibrium
erties
K
v
ondo
equilibrium
eect.
t
Study
reac
of
one
this
in
one
enough
of
ujisa
the
arises
paradigm
and
mo

del
,
of
general
the
ards

measurable
matter
initial
ph
system.
ysics

will
tioned
inevitably
prop
help
In
to
of
ac
bias
hiev
ondo
e
enough
m
ed
uc
together
h
This
b
measuremen
etter
a
understanding
electric
of
That
the
ys

On
of
h
electronic
ev
prop
(F
erties
Elzerman,
of
aruc
nanoscaled
en

enden

sample
man

y-b
t
o
t
dy
(or,
,
a

asp
e
initial
and
t
spin
external
eects.
erimen
The
initially
equilibrium

K
b
ondo
state
eect,
for
as
vides
w
to
e
e
ha
erties
v
eigh
e
to
men

tioned,
erimen
is
measuremen
the
ort
paradigm
applied
mo
oltage
del
the
of

the
is

ph
matter
longer
ph
y
ysics.
ondo
It
the
describ
theory
es
unimp
the
a
in
Due

size
b
one
et
high
w
inside
een
from
magnetic
wh
impurit
alw
y
the
em
regime
b
other
edded

in

to
reac
a
for
metal
v
and
hi,
the
der

ema,
band
a,
F
and
ermi
enho
sea.
Similar
F
the
or

the
to
an
ogan,
tiferro-
2004).
magnetic
p

relev
at
a
temp
imp
eratures
is
lo
imp
w
alen
er
ving)
than
t
the
One
socalled
is
K
of
ondo
of
temp
its
erature
to
NonEquilib
with
1.1.
oir.
is


y
electrons
that
tend
able
to
from
screen
of
the
a
impurit
migh
y
ev
spin.
an
Since
the
it

is
h
not
pro
p
additional
ossible
tribution
to
ab
form
v
a
men
b
prop
ound
with
state
w
from
t
the
ortional
impurit
the
y
y
spin
tration.
and
exp
single
t

the
electron,
t
in
transp

prop
leads
an
to
nite
a
v

tries
man

yb
K
o

dy
the
scattering
oltage
state
high

the
screening
ysics

no
made
describ
of
b

the
electrons.
K
F
eect
or
with
the
linear
impurit
onse
y
.
spin
is
equal
ortan
to
for
one
bulk
half
ts.
ground
to
state

of
of
the
sample
system

b
h
ecomes
enough
a
elds
singlet.
to
Its
far
energy
equilibrium.
is
is
prop
y
ortional
is
to
a
the
in
K
linear
ondo
onse
temp
there.
erature
the
and
hand,
dep
miniature
ends

nonanalytically
elds
on
b
the
easily
strength
hed
of
en
the
small

bias
whic
oltages
h

demonstrates
Hanson,
breakdo
an
wn
Wiel,
of
Wijpk
the
F

w
v
T
en
ha
tional
K
p
w
erturbation
v
theory
2002).
.
problem
The
while
K
timedep
ondo
t
eect
eld
manifests
applied
itself
the
in
(K
man
Amasha
y
Kastner
thermo
All

h
and
erturbations
transp
extremely
ort
an
prop
for
erties
smallsized

Another
h
ortan
as
example
sp
the

after
heat,
osing
magnetic
equiv
susceptibilit
tly
y
remo
,



y
to
,
system.

particular

ect
of
the
formation
problem
of
an
a
preparation
quasib
a
ounded
and
state
subsequen
of
relaxation

w
electrons
equilibrium
in
an
the
reserv

It
y
a
of

the
man
impurit
exp
y
ts
site,
a
the
observ
densit
is
y

of
the
states
degrees
gets
freedom.
enhancemen
h
t


state
K
t
ondo
e
resonance)
en
at
to
the
equilibrium
F
of
ermi

lev
Natural
el.

This,

in
itsof
4
Leggett,
1.
.
Intro
els.

e
systems
matter
will
of
b
at
e
of
a
1.2
sp
spin
eed
w
of
oson
the
and
equilibration
results
of
t
observ
prop
ables
In
of

in
nonequilibrium
terest.
problem
In
of
the
e
nonequilibrium
the
K
Fisher,
ondo
onds
eect
it
language


et
h
in
a

problem
in
tak

es
impurit
place
their
when
w
the
y
impurit
tally
y

spin
has
is
ecial
initially
mapp

del
from
del
the
h
F

ermi
In
sea
already
and
y
then
erger
at
prepared
some
originally
time
K
the
y

time
is
et


hed
v
on.
el
Whereas
mo
initial
v
state
T
of


are
band
arious
electrons
transp
is
of
the
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unp
and
erturb
la
ed
next
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ermi
discuss
sea,
problem
the
exp

an
hing

on
of
the
ondo

long
leads
applying
to
K
the
b
formation
on
of
oson
the
also
K
paradigm
ondo
the

ysics

es

tum
man
the
yb
el
o
form
dy
w

b
state.
v
The
Dorsey
sp
and
eed
In
of
the
its
state
buildup
the
denes
at

els
onding
mo
relaxation
denite
rate
pro
of
at
thermo
starts

hopping
and
een
transp
socalled
ort
ximation
prop
(1987)
erties.
deriv
Next
the
nonequilibrium

problem
K
is
language)
the
sp
equilibration
of
of
t
the
p
initially
oulouse
frozen
of
impurit
terests
y

spin
v
(e.g.
thermo
b
and
y
ort
a
erties
strong
the
magnetic
y
eld)
degree
after
freedom,
the
particularly

relaxation
t
ws.
is
the
released.
t
In
o
this
w

will
the
wh
initial
this
state
is
of
and
the
erimen

relev
band
t.
is
Theo
a
Motivation
p
study
olarized
the
p
K
oten
eect

a
state
history
b
After
ecause
sp
of
transformation
absence
ondo
of

spin
e
ip
ed
pro
the

b
In
mo
the

follo
one
wing
the
thesis
mo
w
of
e

are
ph
fo
whic

describ
on
dissipativ
ab
quan
o

v
of
e
t
men
olev
tioned
system.
nonequilibrium
this
t
ulation
yp
problem
es
as
of

the
y
initial
Chakra
preparation,
art
namely:
,
I)
,
the
Garg
impurit
Zw
y
(1987).
spin
spinb
is
language
initially
nonequilibrium

initial
from

the
to
F
particle
ermi
situated
sea
one
and
lev
then
(for
at
ondo
some
del
time
means
the
impurit

spin
is


then
hed
some
on;
it
I
its
I)
with
the
b
impurit
w
y
lev
spin
Using
is
nonin
frozen
appro
for
Leggett
some
al.
time
ha
(e.g.
e
b
ed
y
for
a
lev
strong
o
magnetic
(magnetization
eld)
the
and
ondo
at
del
a
exactly

the
time
ecial

alue
t
the
is

remo

v
oulouse
ed.
oin
A
(T
sp
1969)),
ecial
tdef
P(t)) = hS (t)i∼ exp − .z
character. time scale
′C(t,t)
def′ ′C(t,t) = hS (t)S (t)i.z z
′t t
′τ = t−t
′ ′ ′ 2C(t,t) =C(t−t)∝ 1/(t−t) .
1
C(t,0) = P(t)
2

magnetization
oth
in
One
the
system
whole
en
parameter
true
range
almost
at
In
all
or
time
y
scales.
Wingreen,
Another
r
op

en

question
ortan
is
in
b
this
eha
Ho
vior
hand,
of
e
the
the
impurit
the
y
one
spinspin


tly
function
functions
(1.1)
w
time
ends
with
one
fast
the
tially
K
onen
F
exp
longtime
dened
giving
as
there
zero
the
to
(1.3)
es
(1998),
go
observ
(magnetization)
their
el
w
lev
y
the
Using
of
the

the
o
that
that
algebr
is
spinspin
result
are
main
from
Their
time
space.
b
parameter
ones.
whole
this
the
ens
in
the
ximately
and
appro
most
and
questions
5
ph
(1.2)
of
This
mo
general
with
t
theory
w
w
otime
y
denition
eha
is
umerical
suitable
no
in
ev
and

out
asymptotically
of
pro
equilibrium
the
situations.
ha
Times
and
Motivation
the
and
ev

w
Theo
v
are
ossible
the
suggested
times
ork
of
Pustilnik,
the
Langreth
rst
an
and
t
the
t

applicable
measuremen
observ
ts
follo
of
will
the
alize
impurit
longtime
y
of
spin.
b
While
times
the
sucien
system
far
is
the
in
hing
equilibrium

the
should
timetranslation
ecome
in
equilibrium
v
Ho
ariance
fast
is
pro
presen
happ
t
dep
and
on
all
explicit

setup
functions
is
dep
the
end
imp
only
t
on
of
the
nonequilibrium
time
ysics.
dierence

b
the
et
ondo
w
del
een
equilibrium,
t
the
w
ermiliquid
o
one
measuremen
sho
ts



as
of
w
function
result
b
This
results
vior
n
the
exact
and
are
tially
er,
of
w

h.
(1.1).
formfactor

using
in
it
is
v
w
e
h
On
exp
other
tial
v
y
who
in
Saleur

Lesage
of
b
b
later
vior
it
(1.4)

exp
should
1.2.
y
ondo
onen
at
out
F
equilibrium
energy
to
they
P
v
ular
in
terest

ho


the
an

onen
er

e

that
to
relaxation
longtime
system
eha
ables
when
zero
system
era-
ables
out
olv
equilibrium
to
ens
ard
if
equilibrium
w
alues.
e
p
timeindep
scenario
b
as
ior
in
at
w
nite
b

Nordlander,
er
Meir,
e
and
h,
(1999).
its
as
is
example
enden
heigh
If
of
o
K
e
resonance
.
the
In
ermi
our
,

ha
there
e
is
tro
one
the
sp
of

ee
p
temp
oin
atur
t

on
the
the
of
time
observ
axis
at

temp
the
ture
time
of
of
happ

as
hing
it
on
ere
the
quilibrium

endent
Th
ehav-
us,
but
timetranslation
some
in
ee
v
temp
ariance
atur
is
whic
violated
in
immediately
turn,
and
timedep
timedep
t.
enden
ab
t
v

statemen
functions
is
dep
and
end
to
on
other
b
ables,
oth

times
ws
explicitly
one
.
never
One
e
ma
the
y

only
de
exp
ay
ect
the
that
whent ∝ ~/k TK B K
TK

since
v
at
group
an
results
y
ysics.
nite
initial
temp
bias
erature
in
the
as

librium


ys
n
exp
these
onen
generalized
tially
a
.
hiller
This
v
is
vior
one
e
of
umerical
the
tum
most
for
imp
hnically
ortan
b
t
for
issue
as
to
since
b
equilibrium
e
the
solv
umerical
ed.
orks
It

is
nonequilibrium
also

tigh
v
tly
ortan

whic
with
applied
the
del

1997),
theorem
ec
since
2004).
imaginary
e
and
of
real
w
parts
ery
of
of
the
whereas
F
resolution
ourier
ics.
transform
other
of
with
(1.2)
m
are
need


b
There
y
the
this

relation.
for
If
Sc
it
b
is

violated
1995,
then
jumdar,
the
exact

del
of
some
the
phenomena
eectiv
results
e
to
temp
of
erature
the
serv

es
ha
exactly
een
as
the
a
ondo
measure
the
of
group
its
ymatrix
violation.
hollw
Th
2005),
us,
teCarlo
one

ma

y

ask

the
highly
follo
relaxation.
wing
er,
questions:
is
whether
since
the
b

is
theorem

is
needs
fullled
high
or

not;
longtime
ho
tation
w
ds
to

dene
steady
the
v
eectiv
ev
e
h
temp
metho
erature
b
if
are
it
in
is
er-
violated
few
in
addition
sp
result
ecied
whic
nonequilibrium
e


to
of
what
o
exten
et
t
n
this
of
denition
hiller
is
2000,
applicable.

Finally
hiller
,
1998,
one
hiller
is
1998)
also
for
in
ondo
terested
the
in
bias

is
of
of
transp
w
ort
y
prop
the
erties,
from

b
h
this
as
most

problems
y
state
,
Suitable
whic
hes
h
h
are
v
most
b
easily
already
measured
to
in
equi-
exp
K
erimen
mo
t.
are
The
n
K
renormalization
ondo
(Costi
problem
densit
is
renormalization
the

strongcoupling
o
problem
k
and
quan
it
Mon
is
(Egger
already
In
dicult
these
b
hnics
y
b
itself,
applied
without
hanged
an
the
y
lation
timedep
the
endence.

With
state
nonequilibrium
Ho
initial
ev


it
it
b
v
ecomes
dicult
a
the
time-dep
um
endent
er
str
eigenstates

limited
oupling
y
mo
time
del
one
for
a
whic
ery
h
energy
no
while
general
hing
metho
the
ds
dynam-
exist.
Implemen
Results
of
obtained
metho
b
to
y
problems
v
h
arious
nonequilibrium
appro
state
ximate
applied
analytic
oltage

is
hnics,
en

uc
h
harder
as
all
time
ds
dep
to
enden
e
t
and

not
appro
applicable
ximation
their
(Nordlander
v
et
sions.
al.
are
1999,
exact
Nordlander,
in
Wingreen,
to
Meir
Leggett's
and
for
Langreth
magnetization
2000,
h
Plihal,
serv
Langreth
as
and
b
Nordlander
hmark
2000)
adjusting
ha
n
v
meth-
e
ds.

hiller
existence
al
of
a
time
um
scale
er
eriment
w
exp

in
and
function
held
elation
Sc
orr
and

held

Sc
Intro
and
(1.5)
held
related
Ma
to
Sc
the
and
K
held
ondo
got
temp
solution
erature
the
1.
K
6
mo
sp
with
p
nite
t
oltage
the
at
space.
et,

relev
oin
an
of
t
parameter
for
Y
the
there
buildup
no
of

the
these
K
as
ondo
ell
resonance.
an
Another
exact
line
regarding
of


er
k
nonequilibrium
of
equilibrium
this
eha
problem
in
is
one
to
the
use
imp
dieren
t
t
of
n
solid
umerical
ph
metho
ds.