A NNALES DE L ’I. H. P., SECTION A
H. S POHN R. S TÜCKL
W. W RESZINSKI
Localisation for the spin J-boson hamiltonian
Annales de l’I. H. P., section A, tome 53, n
o2 (1990), p. 225-244
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225
Localisation for the spin J-boson Hamiltonian
H. SPOHN, R. STÜCKL
W. WRESZINSKI
Sektion Physik der Universitat Munchen, theoretische Physik,
Theresienstr. 37, 8000 Munchen 2, F.R.G.
Instituto de Fisica, Universidade de Sao Paulo, Caixa de postal 20516, 01498 Sao Paulo, Brasil
Vol. 53, n° 2, 1990, Physique théorique
ABSTRACT. - We
investigate
thephase diagram
of theground
state fora
spin
Jcoupled linearly
to a Bose field. We prove, under suitable infraredconditions,
that there exists a criticalcoupling strength, (J),
above which theleft-right
symmetry of the system is broken: thespin
becomes localized.We establish lower and upper bounds on a~
(J).
Inparticular, they imply
that
rJ..c (J = oo)
agrees with the criticalcoupling strength
of the semiclassicaltheory.
RESUME. - Nous etudions le
diagramme
dephase
de 1’etat fondamental pour unspin
Jcouple
lineairement a unchamp
de Bosons. Nous montrons que sous des conditionsinfrarouges appropriees,
il existe une valeurcritique
Clc(J)
del’amplitude
ducouplage
au dessus delaquelle
lasymetrie droite-gauche
dusysteme
est brisee : Iespin
devient localise. Nous donnons des bornessuperieures
et inferieures pour Clc(J).
Ellesimpliquent
enparti-
culier que coincide avec la valeur
critique
de la theorie semiclassique.
Annales de l’Institut Henri Poincaré - Physique théorique - 0246-0211 Vol. 53/90/02/225/2U/$4.00/(0 Gauthier-Villars
1. INTRODUCTION
The
spin-boson
Hamiltonian models aspin 1/2 coupled
to a bosonicfield. It is the
prototypical example
of adissipative
quantum system. We refer to[1] ]
for a recent review. Thecoupling
between thespin
and theenvironment may be so strong that the
ground
state of the system becomes twofolddegenerate
with a brokenleft-right
symmetry. Thisphenomenon
is
necessarily
associated with thegeneration
of an infinite number ofinfrared bosons
([3], [4]).
From a quantum mechanicalpoint
of view anatural
question
to ask is whathappens
if thespin 1 /2
isreplaced by
aspin
J. Forlarge
J one can use the semiclassicaltheory ([2], [13]).
Howdoes then the quantum
regime (small J)
link up with the semiclassicalregime?
The
spin
J-boson Hamiltonian readsHere S =
(Sx, SY, SZ)
are thespin
J matrices withSY]
= i Sxplus cyclic permutations
and S. S = J(J
+1). {~ (k),
a*(k) I k
E are annihilation and creation operators in momentum space of a d-dimensional Bosefield, [a (k),
a*(k’)] = 6 (k - k’).
Since dimensionplays
noparticular role,
we setd= 1 for
simplicity.
Our results hold for any dimension. is thedispersion
relation of the Bose field and X(k) = A (k)*
are thecouplings.
For convenience we
require
is the
coupling
parameter. We normalize itby setting
The
integral
in(3)
has to be finite in order to ensure that H is bounded from below.For h =
0,
H is invariant under the discrete symmetry, T, definedby
Clearly
i2 =1. We want to understand under what conditions this left-right
symmetry isspontaneously
broken in theground
state. Weapproach
Annales de l’Iristitut Henri Poincaré - Physique théorique
the
problem by
means of an order parameter(other, equivalent, possibili-
ties are discussed in
[3], [4]),
denotedby m*,
which may be definedthrough
the
following
limitprocedure:
We confine the Bose field to a finitebox, A,
inphysical
space andimpose periodic boundary
conditions. Moreoverwe introduce a
ultraviolet-cutoff
Thek-integrals
in( 1 )
becomethen finite sums over a momentum
lattice,
denotedby
K. The Hamiltonian with these cutoffs has aunique ground
state, denotedby
~. The order parameter isgiven by
The order of limits is essential. It is part of our
proof
that these limits exist. Ifm* = 0,
then H has aunique ground
state. The T symmetry is unbroken. If m* >0,
the 03C4-symmetry isspontaneously
broken and H hasa twofold
degenerate ground
state. In thefollowing
E will bekept
fixedand we
investigate
how m*depends
on a and J.Actually,
m* isincreasing
in a. This allows us to define a critical
coupling strength,
a~(J), by
The two extreme cases,
J =1 /2
and J = oo, are well understood. For thespin 1 /2
case the centralquantity
is the effectivepotential
(note
thatW (t)
is bounded because of(2)
and 1by (3) .
Ifthen m* = 0 and hence
o~(l/2)=
oo . On the otherhand,
if the limit in(8)
is
strictly positive (or infinite),
then(x~(l/2)oo.
Forsufficiently
strongcoupling
the T-symmetry is broken. At a =(1 /2),
m* either vanishes ornot,
depending
on details([3], [4]).
On the other
hand,
forlarge
J we can use the result of Lieb[2]
whoproves that in the limit J - becomes a classical variable and the
partition
function for the Hamiltonian( 1 )
converges to thepartition
function for the semiclassical Hamiltonian
where
0~(p2Ti.
Since now x- and z-component of thespin
commute, theground
state iseasily
determined.Computing
m*through
the limit /! B
0,
one obtaines0~(00) =e, independent
of thelarge
tdecay
of the effective
potential
W(t), cf Appendix.
The
problem posed
is the behaviour of inbetween these two extreme cases. To our ownsurprise,
thespin
J-boson Hamiltonianinterpo-
lates in the
simplest possible
way: For andgeneral W (t),
we havewhere m
(J, h)
is defined in(5)
but without the limit A B 0 and msc(J, h)
isthe
corresponding quantity
obtained from the semiclassical Hamiltonian(9).
If cx>s, then has ajump discontinuity
at h= 0. For h = 0 and if adecay
conditionslightly
faster than in(8) holds,
then(J)
= oo forevery J. On the other hand if
then ~.
Presumably 03B1c(J)
isdecreasing
in J. We will prove the boundsIf the limit in
(11)
isinfinite,
thenWe expect this property to hold whenever
(11)
is satisfied. In thefollowing figure
we present a schematicphasediagram.
The
technique
to prove results as(12), (13)
is similar to thespin 1 /2
case with one extra twist however. For
spin 1/2
oneexploits
amapping
to a
ferromagnetic
one-dimensional continuumIsing
model(spin
(j(t)
= :i:1 /2)
withpair potential
a W(t).
m* becomes then the usual order parameter of spontaneousmagnetisation.
If thepair potential decays sufficiently slowly,
then theIsing
model orders and m* > O. It turns outAnnales de l’lnstitut Henri Poincaré - Physique théorique
that in the
corresponding mapping
for thespin
J model thespin magnitude
J introduces an extra dimension. The continuum model now consists of 2 J
coupled Ising-lines.
be thespin configuration
inthe j-th line,
1
~/~2J, ~~ ~t) _ ~ 1/2.
The energy of thespin configuration
in the twodimensional
volume [- P/2, f3/2] x {1~2,
...,2J }
is thenIn the t-direction the
strength
of thepotential decreases,
whereas in the J-direction the
coupling
isindependent
of the location of thepair
ofspins.
As it should
be,
the total energy isextensive, i.
e.proportional
tofiJ.
The energy
(14)
has two mechanisms forordering. If W (t) decays slowly
and if a is
sufficiently large,
then thespin
system orders in the t-direction for fixed J. On the otherhand,
for fixedP,
as J - oo the energy(14)
is ofmean field type and the system must have a mean field
phase
transition.Note that as J - 00, converges
to 6 (t - s) and,
apriori,
it isnot
quite
obvious how the two mechanisms combine.To
give
a short outline of the remainder of the paper: In Section 2 we establish themapping
between thespin
J boson Hamiltonian and thejust
mentioned system of 2 J
coupled Ising
lines. Inparticular,
we relate the order parameter m* to the spontaneousmagnetisation.
In Section 3 weprove a lower and in Section 4 an upper bound on the critical
coupling strength
r1c(J).
2. ORDER PARAMETER
AND FUNCTIONAL INTEGRAL REPRESENTATION
To define the order parameter we first have to introduce a cutoff
Hamiltonian, HK.
Let A c I~ be an interval oflength
thephysical
volume. We
impose periodic boundary
conditions. Let K be the set of modes in A withultraviolet-cutoff (if
necessary, zero modes arealso removed from
K).
Then the cutoff Hamiltonian isgiven by
with a suitable choice of ffik and
Àk,
cf. theproof
ofProposition
2.{ k e K}
constitute arepresentation
of the CCR.Since K
oo, thisrepresentation
isequivalent
to theSchrödinger representation.
ThereforeHK
can beregarded
as a linear operator on~f~=C~~0~,
whereis the
symmetric K -particle
Fock space. Here vN,
Ne
N,
denotes N-foldsymmetric
tensorproduct.
HK
is a finiteparticle
Hamiltoniangenerating
apositivity improving
one parameter
semigroup,
and thusHK
has aunique ground
stateWe define the order parameter
by
We will prove below that the sequence in
( 16)
is monotoneincreasing
andthat
m (h)
decreasesmonotonically
to m * .We want to express
m (h)
as anexpectation
value with respect toa stochastic process on the time interval
[ - ~i j2, P/2] taking
values in{ - J, ...,J}.
For this purpose we construct first the measuregenerated by
exp(t
E Here and in what follows we will work in the Sz-basis. In this basis theground
state of Sx isgiven by
Annales de l’lnstitut Henri Poincaré - Physique théorique
LOCALISATION FOR THE SPIN
Let rl1 be the set of
piecewise
constantpaths
on[
-03B2/2, fl/2] taking
valuesin
{- J, ..., J}.
LetS(.)
be apath
in rl1 withjumps
at- 03B2 t1 ... tn 03B2
and with the valueS (t) = m; e ( - J, ... , J )
for2 2
to = - fl/2, = 03B2/2.
Weassign
toS ( . )
theweight
where
are the matrix elements of S~ in the SZ-basis. The so defined
(unnormalized)
measure on r~ is denoted
by (S).
Let us define an action functional
by
where
This is a Riemann sum with limit
compare with
(7). Expectation
values with respect to the normalizedmeasure - 1 exp [
-J
are denotedby . )J (Ø, K).
PROPOSITION 1. - Let
B}I K,
h be theground
stateo, f ’ HK.
ThenProof. -
LetH~
be the Hamiltonian(15)
with a=/!=0. This is the Hamiltonian of aspin
J andI K I independent
harmonic oscillators. Itsground
state,0,~
is theproduct
of°0 and K
harmonic oscillatorground
states. Since s- lim exp
[ (HK - EK, 0)]
= h, theorthogonal projec-
o - 00
’
tion on 03A8K, h, and since > 0
by positivity,
we havee - SZ e - can be rewritten as a functional
integral.
TheJ
free process is a
product
of(S)
andindependent
Ornstein-Uhlen- beck processes. The action isgiven by
where the
qk ( . )
are Ornstein-Uhlenbeckpaths
on the time interval[ - [i/2, P/2].
The bosonicdegrees
of freedom can beintegrated
out, com- pare with[3, 5].
The net result isIt turns out that the limit J - oo can be better controlled in a system of 2 J
coupled Ising lines,
which we introduce next. As an additional bonus this system makes it easy to prove correlationinequalities.
The 2 Jcoupled Ising
lines can be viewed as a quantum version of Griffiths’ method ofanalogue
systems,[6].
For 1
_ j _ 2 J
leto~(.)
be apiecewise
constantpath
on[ - [3/2, P/2]
withvalues ±1/2. By
d03BD03B2(03C3j) we denoted 03B2 (S)
for J=1 /2.
Inparticular,
ifpendent
of the initial and final values of~(.).
(1) Note that due to our boundary conditions expectation values are taken in the harmonic
oscillator ground states rather than over thermal states as in [5} or [3], compare with equation (5.47) in [5].
Annales de l’Institut Henri Poincaré - Physique théorique
Proof -
LetS (t)
take values mi in the intervals2J
-
to = -
03B2/2,
tn + 1 =03B2/2.
Itsweight under 03A0
d03BD03B2(03C3j) is of the formj= 1
where
u (mo)
is the number of ways mocan be realized and p
(m, m’)
is the number of ways m’ can be obtainedgiven
m,weighted by E/2
J(the
factor1 /2
is the propernormalisation).
-III ~ ....,... ,
Comparing
with(19)
the claim follows fromAs a
Consequence
of Lemma 1 we havefor any
(bounded) function f
onI’~.
2J
The 2J
coupled Ising
lineshave n dv~ «(J j)
as free measure and in termsj= 1
of the the action
(20)
readswhere we use o as a short hand for
(03C31,...,03C32 J). Expectations
withrespect to the normalized
measure- 1 03A0 d03BD03B2(03C3j)
are denotedby . > K).
The functional
(27)
isexplicitly ferromagnetic.
Also each dv~(a)
can beapproximated by
discreteIsing spin
chains withferromagnetic interactions,
see
[3].
Therefore the 2 Jcoupled Ising
lines is aferromagnetic spin
model.PROPOSITION 2. - The limits
(16)
and(17)
exist and m* agrees with the spontaneousmagnetisation of
the 2 Jcoupled Ising
lines. Furthermore thelimits 03B2
- oo and K ~ R commute,.. 2 J
Proof - By Proposition
1 and Lemma1,
Let us first prove
that 1 03A3 03C3j(0) > (03B2, K)
increasesmonotonically
as~7=1
We choose the discretisation of o
(k)
and À(k)
such thatVVK (t)
approx- imatesmonotonically
from below for all Let kEK andkl, k2
be in the closed interval of
length
g2 ~
with center at k such that!~!
o
(k1)~03C9 (k’) and |03BB (k2) |~| 03BB (k’) for
all k’ in thecorresponding
interval.Let and kEK. Then
~
I ~~ ~ (k~) 2
(&’)) 11 for all t. Since(21 )
is a Riemann sum approx-imating
theintegral (22),
this choice amounts inapproximating
theintegral monotonically
from below as for allBy
Griffiths’ secondinequality,
the samemono tonicity
property holds then for1 2J 1 2J
- E CF,(0))(P,K)
for allP>0. Therefore J 03A3 o,(0))(P,K)
is mon-otone
increasing
in K also in thelimit P
-+ oo and m(h)
is well defined.1 2J The limits
and 03B2 ~ ~
commuteJj=i
increases
monotonocally with P
for all K because eachIsing
line has freeboundary
conditions at t = ~p/2.
Again by
Griffiths’ secondinequality, m (h)
decreases with h.Therefore, m* = lim m (h)
is well defined. It is known that this m* agrees with theh 0
spontaneous
magnetisation
definedby taking
the infinite volume limit with " + "boundary
conditions([3], [4]).
0Annales de Henri Poincaré - Physique theorique
We have shown that
ground
stateexpectations
of thespin
J-bosonHamiltonian agree with the
expectations
for the 2 Jcoupled Ising
lines.Because
they
areferromagnetic,
the "infinite volume"limit P
-~ oo andthe limit K - R exist. From now on we
study
the 2 Jcoupled Ising
linesand
adapt
our notationaccordingly. (.) (a)
denotes infinite volumeexpectations,
where the brackets indicate thecoupling
parameter.Finally
we note thatA (0-) originates
in the Euclidean action of aHamiltonian.
Integrating
out the bosonicdegrees
of freedom in a system of 2 Jindependent spins coupled linearly
to a harmonic latticeyields
theeffective action A
(a).
3. LOWER BOUNDS ON THE CRITICAL COUPLING Let us differentiate the
pair
correlation for h = 0 with respect to a.Using
the Lebowitz
inequality
we obtain for the infinite volumeexpectations
for 1
~7~~
2J. (
crj(0)
ak(t) ~ (ex)
is boundedby
the solution of the differ- entialequation corresponding
to(29)
with initial conditionThus we have
where
W (m)
andG (m)
are the Fourier transformsof W (t)
4
respectively. (30)
is valid aslong
as 1 > aW (c~) G (~)
for all m. SinceW (w)
andG (o)
take their maximumat o=0,
this meansat the mean field bound
PROPOSITION 3. - then ~*=0.
If the interaction
decays
faster than t - 2 for t - ~, we can use the energy-entropy argument of[3]
and[7]
to provePROPOSITION 4. - and all J.
4. UPPER BOUNDS ON THE CRITICAL COUPLING We state the main result of our
investigation.
THEOREM 1. - Let lim
t2 W (t)>0. Then for
anyJ~1/2
there exists a a+(J)
such thatThe bound E _
(J)
is an obvious consequence ofProposition
3.Our
proof
of oc+(J)
00 and(33)
is divided into two steps. We firstpartition
the system into blocks oflength 8
anddecouple
the free measure(this yields
a lower bound onm*).
Themagnetisations
per block formthen a standard
spin
model over the one dimensional lattice.Applying
Wells’
inequality,
itsmagnetisation
is bounded belowby
themagnetisation
of a ::l: 1
Ising spin
system- a well understood model[12].
To obtain useful bounds we have to control the apriori
distribution of themagnetisation
in a
single block,
inparticular
its behavior forlarge
J. This is carriedthrough
in step two. The crucialpoint
there is that forsufficiently large coupling
thesingle
block has a mean fieldphase
transition as J - aJ.Therefore the
single
site measure cannot concentrate at zero as J -~ oo . Annales de l’Institut Henri Poincaré - Physique théoriqueLOCALISATION FOR THE SPIN
STEP 1. - We
change
the time-scale in the action(27) by setting
t’ =t/J.
2 J
Let
(3‘
=P/J,
then the free process,fl dvw (o~),
refers topaths
on the timej=i
interval
[ - [3’/2, P’/2]
and the action isgiven by
With the new scale the action is
explicitly extensive,
i. e.proportional
to(34)
has a mean field interaction in the"spatial" direction,
{ - J, ...,J}.
In the timedirection, [ - [i’ j2, P’/2],
the interactionstrength,
JW
(J t I),
becomes strong andshortranged
as J increases with total(inte- grated) strength independent
of J.We
partition
the interval[ - PV2, P’/2]
into intervals oflength Õ, indepen-
dent of J. For notational convenience we set
P’=N8
with NEN. For2J
Then
W~-~W(~).
As in Section 2 letS(~)= ~
and define thej=i
magnetisation
per volume in the block Iby
By d03C6J (Ml)
we denote the distribution ofMl
underHere
dJlo (S)
is the measure on rsgenerated by
exp(Eð S")
with freeboundary
conditions as defined in Section 2 and Z is the normalisation constant. If obvious from the context we will supress the Jdependence
ofdcpJ. Let ~ . > (J) (a)
denoteexpectations
with respect to the normalizedmeasure
Since
compared to ~ . )(ex) ferromagnetic
interactions have beendecreased, m* >_- lim (ex).
0
To control the width of the
single
site measure in the limit J -~ 00 weuse the
following
property.PROPOSITION 5. - For each ex> E there exists a v >
0, independent of J,
anda 03B41
> 0 suchthat for
all Õ >b 1
This
proposition
will beproved
in step two.Let
( . > 1 (ex’)
denoteexpectations
with respect to the normalizedIsing
measure
We
apply
Wells’inequality [3, 9]
to(38). By Proposition
5 there existsthen a
independent of J,
such thatThe
phase diagram
of theIsing
model(40)
forN -+- 00, equivalent 03B2’
- ~, withcoupling
oc’ = eL J2 b2u2
is discussed in[12].
Iflim
t2 W (t) > 0,
then theIsing
model ordersprovided a’, equivalently
(x,t ~ 00
is
large enough.
This proves(32).
Let us chose anarbitrary
a > E and let limt2 W (t) = oo .
Then the nearestneighbor coupling, J2 W (J t), diverges
i - m
as J - oo .
Furthermore,
for Jsufficiently large,
Therefore,
E cx+(J)
cxprovided
J islarge enough.
DSTEP 2
(Proof
ofProposition 5). -
We have toinvestigate
thesingle
block measures
d03C6J
in the limit J - ~.Substituting
JW(J t) by
õ(t) (which
Annales de l’lnstitut Henri Poincaré - Physique théorique
gives
anegligible error)
we obtain the mean fieldproblem
In more familiar cases the
single
site space consistsonly
of twopoints,
Here we must deal with the a
priori
measureFortunately
such
general
mean field systems have been studied before. In[10]
thesingle
site space is a bounded volume in IRdequipped
with theLebesgue
measure. The
proof
in[10]
has to be modifiedonly slightly
in order toapply
to(43).
Beforedoing
so let usexplain
the main result of[10].
Let p be a bounded
density
relative to0 ~ p _
a, with normalisationthe entropy
and the free energy
F (p)
is bounded from below. Let-4Y ~
be the set ofp’s minimizing
F.For each p we can build the
product
measureNow let us choose a
subsequence
J -~ oo such that PJ convergesweakly
to
(p.
Sincep
must bepermutation invariant,
the theorem of Hewitt andSavage
ensures that(p
can bedecomposed
intoproduct
measures asThe main result of
[10]
is that thedecomposition
measure,~r (dp; cp),
isconcentrated on In
particular, along
the chosensubsequence,
Thus the
proof
ofProposition
5 isaccomplished by studying
the minimaof the free energy functional
(46).
Let us now introduce some notation. We write
rij2
for rs ifJ = 1 /2.
Let ~ be the set of all
probability
measures on which areinvariant under
permutations.
This means,... for any measurable sets
Al’...’ An C r~/2’
alln e N and all
permutations 1t of {1,
...,~}.
Let!7 a
c y be the set of allpermutation
invariant measuresd~
on such that there exist densities..., bounded above
by ak
for somea > 0,
andsatisfying
LEMMA 2. - The sequence
of
measures, has weak limitpoints
in~a
as J ~ ~. Each limit
point, (p,
can bedecomposed
into extremal measuressuch that the
decomposition
measure is concentrated onIf ~ (dp; (p)
denotes the
decomposition
measure, thenProof -
We first prove that the sequence of measures PJ has weak limitpoints
in Y. We cannotadopt
the argument of[10]
since is not compact. Instead weapply
results of[ 11 ], chapter 4,
inparticular Proposi-
tion 4. 7 and
Example
1. We have to check that for all 1__ j __ 2 J
is bounded
uniformly
in J. This is obvious since(51)
is boundedby 8/2 (in
theterminology
of[11] ]
this means that the interaction isabsolutly summable).
The
Lipschitz continuity
used in[10]
isreplaced by
Here we have used that
where Z is the normalisation constant. Let
Zo =
Then we haveAnnales de l’Institut Henri Poincaré - Physique théorique
LOCALISATION FOR THE SPIN
This
replaces
thecorresponding
estimate(2.6)
in[10].
Furthermore there exist constantsC, a > 0, independent
of Jand k,
such that for allk _ 2 J
This
replaces
Lemma 2 in[10].
Along
thegiven subsequence,
convergesweakly
to alimit he
whichis the
marginal
of(p
on thesites {1, ... , k ~ .
The main technical tool in[10]
is to make sure that also the entropyof ff
converges to the entropyof
fk.
For this weak convergence is notenough.
In[10]
the uniformLipschitz continuity
of the densitiesf2 Jk
was used. This is substituted hereby (55). By
the theorem of Arzela Ascoli itimplies
the existence ofpointwise
convergentsubsequences of
as J - 00 on compact sets. Sinceby
weak convergence the limit isunique, --~ fk
almostsurely.
Becauseof
(54)
thisimplies
the convergence ofentropies.
The energy of the "state"p is
given by (44)
sinceJW (J t) - 6 (t)
as J - 00. The remainder of theproof
is identical to[10].
DLet us
write (.)p
forexpectations
withrespect
to the measurep
( cr) dvs ( cr)
and let m(t) _ ~
cr(t) ~P.
p is astationary point
of the free energy functional F(p)
iffClearly,
the weakcoupling
solution is] -
1 withm
(t)
= 0 for all t. To proveProposition
5 we have to show that there areabsolute minima
of F (p)
withLEMMA 3. - For a E there exists a
§o>0
such that for all 03B4 >Õo
po is theunique
minimum of F(p).
For a>8 there exists
a 8i>0
such that for all8>8i
po is an unstablestationary point
of F(p).
Proof. - By inserting (56)
intoF (p)
we obtain the functionalSince the
stationary points
ofF (p)
andF (m ( . ))
with~(~)=(cr(~))p
agree, we
only
have toinvestigate
the absolute minima ofF (m ( . )).
The
quadratic
variation ofF
with respect to m( . )
at m(. )
= 0 isgiven by
For J =
1 /2
we have°0= 1 2(11) )2
1 and thusThe Fourier coefficients of
8 (~) - -~ "’ ~ - 8/2 ~ ~ 8/2,
aregiven by
with 03C9n
= 03C0 n/03B4,
n E Z.(59)
ispositive
for all 03C9n if a E and 03B4 islarge enough.
Thisimplies stability.
Uniqueness
followsby
a contraction argumentanalogous
to the onegiven
in[10]
in theproof
of Theorem 3. We remark thatnonuniqueness
also contradicts
Proposition
3 because the argument in step 1 wouldyield
m*>0 for r:t E.
If a > E, then
(59)
isnegative for I
smallenough
and 8sufficiently large.
From this we conclude that thequadratic
variation ofF
atm ( . ) = 0
is not
positive
definite. 0Proof of Proposition
5. - Let a > E and letcp
be any weak limitpoint
of PJ as J -~ oo.
Then, along
thegiven subsequence,
lim cpJ isgiven
by (49). Suppose
that Then and hencefor almost all which contradicts Lemma 3. Hence cpJ has to be bounded away from zero
uniformly
in J. DThe
proof
ofProposition
5 has theCOROLLARY. - For a > E we have
independent of
the choiceof
W(t).
Annales clo I’Iti.witut Henri Poincaré - Physique théorique
LOCALISATION FOR THE SPIN
APPENDIX
As an
example
weexplain
how to calculateground
stateexpectations
of 1 J
SZ in the semiclassical limit J ~ ~. We introduce the cutoff Hamil- Jtonian
corresponding
to the semiclassical Hamiltonian(9),
where we suppress the K
dependence
ofH~
in our notation. This Hamil- tonian is defined onYf K,
sc -[/2 Q ~ K~
where[/2
is the twosphere.
Diagonalizing H~
one finds that itsground
state energy for fixed e and p isgiven by
By H~
we denote the Hamiltonian(61)
with all termsexcept L wk ak
akk e K
multiplied by (J + 1 )/J.
Itsground
state energy for fixed e and p isgiven by
Thus the
ground
stateenergies
ofH ~
are determinedby
Taking
thelimit P
- oo inequation (5.4)
of[2] yields
then the boundsfor all 11 ~ O. In
(65)
one can take the limit K -~ IR. We have limg+ (e, cp) -
g ~(e, cp)
* g(e, cp)-
ThusTaking
the limit J - oo andthen 11 ?
0 for the lowerand 11
B 0 for theupper bound in
(65) yields
then
g (~, ~p)
has aunique
minimum(90 (h),
Po(h))
and(67) yields
Let h=0. If then
~(9,(p)
has theunique
minimumand
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( Manuscript received January 10th 1990.)
Annales de l’Institut Henri Poincaré - Physique théorique