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Quantum statistical physics of a microscopic glass model [Elektronische Ressource] / presented by Michael Thesen

105 pages
DoDissertationornsubmittedytoJulytheSciencesComMicbinedGermanFdegreeacultiesoffortedthePhNaturalThesenSciencesTandOralfor2003MathematicsofofctortheNaturalRuppresenertobCarolaDipl.Univys.ersithaelybofinHeidelbrier,erg,yGermanexamination:y9.fortheProf.QuanProf.tumdelStatisticalReimerPhHeinzysicsReferees:ofDr.aK?hnMicroscopicDr.GlassHornerMoknoZusammenfassungtheWirtounintersucimplemenhenadietoQuandierentenstatistisceheirreduciblePhisysikineinestummikroskwiederopiscysicshenandGlas-atmothedellsexpansioninariationalvdel.erscanhiedenenaN?herungenpundfromndenestigateBeziehmungent.zwiscquanhenglassbapproekban-glassynhightenreplicaArtenfreederself-consistencyBescthreibungWithinglasartigeneVersionerhaltenswbtoeiharmonictiefeneundwithhohennextTeempberaturen.orderDabweierturbativvdel,erwphenomenologyendenereinfacwirWdieestigateReplica-MethostatisticaldeazurdelBereckindshnscungelopderwfreienpicturesEnergieehaundwimplemeneratures.tierenydiedsicofh,ergebtheendeninSelbstktermsonsistenz-ogleicacuumhsimpleungenximationinstrongeinerquasi-classicalEnthetewictheklungalennacproblemhblezwcoupledeiteilcbath.hen-irreduziblensimilaritiesVmoakuum-andgraphen.atInofderexpansion.
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Do
Dissertation
orn
submitted
y
to
July
the
Sciences
Com
Mic
bined
German
F
degree
aculties
of
for
ted
the
Ph
Natural
Thesen
Sciences
T
and
Oral
for
2003
Mathematics
of
of
ctor
the
Natural
Rup
presen
erto
b
Carola
Dipl.
Univ
ys.
ersit
hael
y
b
of
in
Heidelb
rier,
erg,
y
German
examination:
y
9.
for
theProf.
Quan
Prof.
tum
del
Statistical
Reimer
Ph
Heinz
ysics
Referees:
of
Dr.
a
K?hn
Microscopic
Dr.
Glass
Horner
Mokno
Zusammenfassung
the
Wir
to
un
in
tersuc
implemen
hen
a
die
to
Quan
dieren
tenstatistisc
e
he
irreducible
Ph
is
ysik
in
eines
tum
mikrosk
wieder
opisc
ysics
hen
and
Glas-
at
mo
the
dells
expansion
in
ariational
v
del.
ersc
an
hiedenen
a
N?herungen
p
und
from
nden
estigate
Bezieh
m
ungen
t.
zwisc
quan
hen
glass
b
appro
ek
b
an-
glassy
n
high
ten
replica
Arten
free
der
self-consistency
Besc
t
hreibung
Within
glasartigen
e
V
ersion
erhaltens
w
b
to
ei
harmonic
tiefen
e
und
with
hohen
next
T
e
emp
b
eraturen.
order
Dab
w
ei
erturbativ
v
del,
erw
phenomenology
enden
ereinfac
wir
W
die
estigate
Replica-Metho
statistical
de
a
zur
del
Berec
kinds
hn
sc
ung
elop
der
w
freien
pictures
Energie
eha
und
w
implemen
eratures.
tieren
y
die
d
sic
of
h
,
ergeb
the
enden
in
Selbstk
terms
onsistenz-
o
gleic
acuum
h
simple
ungen
ximation
in
strong
einer
quasi-classical
En
the
t
e
wic
the
klung
alen
nac
problem
h
ble
zw
coupled
eiteilc
bath.
hen-irreduziblen
similarities
V
mo
akuum-
and
graphen.
at
In
of
der
expansion.
einfac
w
hsten
transition
v
driv
ariationellen
to
N?herung
y
ergibt
Finally
sic
in
h
non
ein
appro
zur
same
quasiklassisc
ying
hen
h
V
Ph?nomenologie
ersion
v
des
h
Mo
Abstract
dells
e
analoges
v
Bild.
the
In
tum
dieser
ph
N?hreung
of
zeigen
microscopic
wir
mo
die
in
?
t
quiv
of
alenz
ximation
des
hemes
Problems
dev
mit
relations
dem
et
eines
een
Ensem
wn
bles
describing
v
b
on
vior
har-
lo
monisc
and
hen
temp
Oszillatoren,
W
die
emplo
an
the
ein
metho
W?rmebad
for
gek
determination
opp
the
elt
energy
sind.
and
In
t
der
resulting
n?c
equations
h-
an
sten
in
Ordn
of
ung
w
der
particle
En
v
t
graphs.
wic
the
klung
v
ergibt
appro
sic
w
h
nd
eine
analogies
Analogie
the
zu
v
einem
of
Spinglas-
mo
mo
W
dell
sho
mit
that
2-
problem
und
equiv
4-Spin
t
W
the
ec
of
hselwirkungen.
ensem
Der
of
Glas?b
oscillators
ergang
to
k
heat
ann
W
durc
nd
h
to
Quan
spin-glass
tenuktuationen
del
zu
2-
einem
4-spin
Phasen
teractions
?b
the
ergang
order
erster
the
Ordn
erturbation
ung
W
w
sho
erden.
that
Sc
glass
hlie?lic
can
h
e
un
en
tersuc
second
hen
rst
wir
b
eine
quan
nic
uctuations.
h
,
tp
e
erturbativ
v
e
a
N?herung
p
zum
e
gleic
ximation
hen
the
Mo
mo
dell,
displa
in
a
der
uc
sic
simpler
h
again.
die.
Con
.
ten
35
ts
.
1
.
In
the
tro
solution
duction
.
1
48
2
.
The
.
Mo
Limit
del
.
7
.
2.1
Mo
General
.
Picture
.
.
.
.
In
.
.
.
.
.
35
.
.
.
.
.
RS
.
.
.
.
.
4.6.2
.
.
.
43
.
.
.
t
.
5.2.1
.
.
.
.
.
.
.
.
.
Results
.
.
.
.
.
tin
.
.
.
.
7
Con
2.2
.
Appro
Zero
ximations
.
.
.
.
Stabilit
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3-Lo
.
of
.
.
.
.
.
Excursion:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
9
32
2.3
of
P
.
ossible
.
Extensions
.
of
.
the
.
Mo
Analytic
del
.
.
.
.
.
.
.
.
.
.
Ph
.
t
.
ariational
.
.
.
37
.
emp
.
.
.
.
.
.
.
.
.
39
.
of
11
.
3
.
Replica
.
Calculation
.
13
.
3.1
4.6.1
Saddle
.
p
.
oin
.
t
.
equations
.
.
.
.
Sync
.
.
.
.
.
.
.
.
.
5
.
at
.
Appro
.
The
.
.
.
.
.
.
.
.
.
.
.
44
.
Equiv
.
Mo
.
.
.
.
.
.
.
Classical
15
.
3.2
.
Fixing
.
of
.
Notations
.
and
.
Orders
.
of
.
Magnitude
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4.5
.
terpretation
17
the
4
.
V
.
ariational
.
or
.
Hartree
.
Appro
.
ximation
.
21
.
4.1
.
Deriv
.
ation
4.5.1
of
Con
the
uation
Saddle
.
P
.
oin
.
t
.
Equations
.
.
.
.
.
.
.
.
.
.
4.5.2
.
ysical
.
ten
.
of
.
V
.
Solution
.
.
.
.
21
.
4.2
4.5.3
High-T
T
emp
erature
erature
.
Solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4.6
.
y
.
The
.
Solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
26
.
4.3
40
Replica
1-RSB
Symmetric
.
(RS)
.
Spin
.
Glass
.
Solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
40
.
Replica
.
hronization
.
.
28
.
4.4
.
Thermo
.
dynamic
.
Quan
.
tities
.
.
.
.
41
.
The
.
del
.
its
.
op
.
ximation
.
5.1
.
Equations
.
Motion
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
30
.
4.4.1
5.2
Quasi-Classical
The
Regime
alen
.
P-Spin
.
del
.
.
.
.
.
.
.
.
.
.
.
47
.
The
.
Limit
.
.
.
.
.
.
.
.
.
.
.
.
.
.
31
.
4.4.2
.
Quan
.
tum
7
Regime.
5.2.2
.
The
.
1-RSB
.
Solution
.
.
Eectiv
.
.
.
.
.
.
.
.
.
.
.
.
.
F
.
of
.
Lo
.
D.1
.
.
.
.
.
67
.
.
.
.
.
.
.
General
.
.
49
e
5.2.3
.
Stabilit
.
y
.
of
tum
the
.
1-RSB
D
-
.
Solution
88
.
.
.
.
.
.
.
.
.
of
.
.
.
.
.
6.4.1
.
.
.
.
.
71
.
F
51
.
5.2.4
.
The
.
Marginally
B
Stable
ormalism
Spin-Glass
.
Solution
.
.
.
.
B.2
.
e
.
.
.
eynman
.
.
.
the
52
.
5.2.5
F
The
the
C-RSB
oin
Solution
.
.
.
.
2PI
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Stabilit
.
RS
.
.
.
.
.
.
.
.
.
.
.
Sync
.
.
53
.
5.3
.
A
.
T
7
o
1
y
A
Appro
73
ximation
ormalism
.
.
.
.
.
.
.
.
.
.
.
.
.
2PI
.
ction
.
B.1
.
ork
.
.
.
.
.
.
.
.
.
.
.
.
.
op
.
2PI
.
ction
.
.
.
.
.
B.2.1
56
for
5.3.1
Mec
Microscopic
82
A
Expansion
ction
Mo
.
.
.
.
.
C
.
ulary
.
y
.
solution
.
Saddle
.
Equations
.
.
.
.
.
.
.
.
.
The
.
ertex
.
.
.
.
.
.
.
.
.
.
.
.
57
.
5.3.2
.
Sc
.
h
.
winger-Dyson
.
Equations
.
.
.
.
.
.
.
.
.
.
6.4
.
y
.
the
.
solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
57
.
5.3.3
.
The
68
Equilibrium
Replica
Spin-Glass
hronization
Solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
69
.
Conclusions
.
A
59
PI
5.3.4
e
The
ction
Marginally
ormalism
Stable
A.1
Spin-Glass
F
Solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
60
.
5.3.5
.
Discussion
.
of
.
the
73
Phase
The
Diagram
Eectiv
.
A
.
F
.
77
.
General
.
ramew
.
.
.
.
.
.
.
.
.
.
.
.
.
.
60
.
6
.
Non
.
P
.
erturbativ
.
e
77
Approac
Lo
h
Expansion
65
the
6.1
Eectiv
Summation
A
of
.
Ring
.
Diagrams
.
.
.
.
.
.
82
.
F
.
Rules
.
Quan
.
statistical
.
hanics
.
.
.
B.2.2
.
op
.
of
.
Quartic
.
del
.
.
.
.
.
.
.
.
65
82
6.2
1-RSB
The
orm
P
85
aramagnetic
Stabilit
Phase
of
.
RS
.
87
.
1-RSB
.
P
.
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
D.2
.
First
.
v
.
correction
.
.
.
.
.
.
66
.
6.3
.
The
.
RS
.
Solution
.
.
90
.the
Chapter
to
1
w
In
ordinate
tro
es
duction
Hilb
Ev
dep
er
o
since
y
the
excess
rst
are
exp
can
erimen
the
ts
susceptibilities.
ha
to
v
w
e
landscap
un

v
ery
eiled
o
the
to
lo
that
w-temp
dimensional
erature
spin-1/2-system.
anomalies
ble
in
w
structural
of
glasses
the
[1
lo
]
is
30
energy
y
to
ears
minima
ago,
eracy
there
of
w
This
as
calized
vivid
accoun
in
states
terest
t
in
only
the
erties
origin
tunneling
of
the
these
ecomes
rather
jection
univ
ed
ersal
is
phenomena
of
presen
sub
t
observ
in
in
most
sp
structurally
t
disordered
onding
systems
e
[2
e
,
at
3].
large
Almost
barrier
righ
nor
t
b
from
the
the
large.
hour
of
of
the
birth
the
of
lifted
this
pro
eld
tum
of
tunnel
researc
splitting
h
to
it
at
b
w
ecame
for
clear
y
that
tioned
the
e.
lo
ery
w
eratures,
temp
con
erature
dynamical
anomalies
the
in
pairs
glasses
This
ha
ely
v
an
e
space
to
w
b
the
e
the
in
e
a
to
close
w
relation
easy
to
the
the
the
existence
'spins'
of
to
tunneling
erimen
cen
p
ters
la
in
temp
this
of
t
heat
yp
v
e
es
of
sp
material.
to
A
collectiv
t
co
v
w
ery
are
lo
oking
w
is
energies,
o
these
nor
lo
the
calized
to
systems
high
giv
the
e
asymmetry
rise
et
an
een
excess
minima
densit
o
y
In
of
case
states
degenerate
as
of
compared
energy
to
e,
the
degen-
crystal.
is
Therefore,
b
they
the
dominate
cess
the
quan
b
tunneling
eha
the
vior
splitting.
of
tunnel
the
giv
prob
rise
e
lo
at
excitations
temp
v
eratures
lo
b
energies
elo
ting
w
the
or
densit
of
of
the
men
order
ab
of
v
1
A
K
v
.
lo
The
temp
commonly
the
accepted
states
microscopic
tributing
picture
the
of
prop
these
of
tunneling
glass
systems
these
is
of
the
states.
one
means
of
eectiv
some
,
c
relev
ol
t
le
ert
ctive
b
co
t
ordinate
o
of
and
a
pro-
group
of
of
Hamiltonian
atoms,
b
p
mapp
ossessing
on
t
a
w
No
o
it
almost
an
degenerate
task
equilibrium
determine
p
distribution
ositions
'elds'
separated
ensem
b
of
y
is
an
ject
energy
from
barrier.
exp
The
tally
latter
ed
can
o
b
er
e
ws
p
the
enetrated
erature
due
endence
to
the
quan
ecic
tum
and
tunneling
the
if
arious
neither
yp
the
of
mass
1
corre-(SPM)
2
under
CHAPTER
either
1.
e
INTR
ts
ODUCTION
to
Figure
e
1.1:
the
Commonly
in
accepted
p
microscopic
t
picture
b
for
rise
tunneling
pro
cen
b
ters
as
in
ev
glasses.
ositions,
A
enden
group
nd
of
the
atoms
terms
ma
ecien
y
sc
p
temp
enetrate
an
an
the
energy
the
barrier
probabilit
separating
t.
t
b
w
oten
o
w
al-
an
most
of
degenerate
not
classical
can
equilibrium
ev
p
can
ositions
osition
of
the
the
anishes.
system.
linear
The
e
tunnel
distributed.
spitting
e
giv
osing
es
ery
rise
v
to
p
lo
tunnel
calized
tunneling
excitations
time
of
one.
v
tin
ery
parameters
lo
In
w
of
energy
to
on
fact
top
ma
of
referred
the
'soft
long-w
mo
a
out
v
Ho
elength
there
phonons
biguit
presen
c
t
reference
in
h
the
co
ordered
the
crystalline
e
coun
F
terpart
stable
of
tial
the
a
system.
reference
This
h
has
prefactor
b
or
een
term
done
seems
b
assume
y
the
Anderson,
the
Halp
oten
erin
e
and
quartic
V
can
arma
to
[4
c
]
appropriate
and
for
Phillips
of
[5]
A
and
lo
their
only
result
tials
has
an
b
are
ecome
i.e.
the
ust
'Standard
on
T
smaller
unneling
erimen
Mo
requiremen
del'
out
(STM).
region
In
space
fact,
the
all
oten
this
region,
w
densit
ould
parameter-distribution
not
safely
b
e
e
is
v
is
ery
of
surprising,
will
w
e
ere
to
the
the
b
p
eha
tial
vior
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