The Yang-Lee edge singularity studied by a four-level quantum renormalization-group blocking method
Domain: Physics
A quantum renormalization-group technique is applied to the one-dimensional transverse field Ising model in a complex longitudinal field. The Yang-Lee edge singularity is analysed in this model in view of its analogy with the 2d classical Ising model. Previous two-level calculations are improved by retaining four levels in the blocking procedure. A more precise edge exponent σ is found in agreement with the infinite block extrapolation of the two-level calculation, but still larger in absolute value than the latest series analysis results. The occurrence of oscillations in the real part of the magnetization appears to be an artifact of the real-space method, which introduces finite size effects.
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Publié le : 28/06/2012
Langue : Français
Nombre de pages : 6
Type de la publication : Rapports et thèses
Thème :
Savoirs > Science de la nature
Source : Journal de Physique
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1075
The
Yang-Lee
edge
singularity
studied
by
a
four-level
quantum
renormalization-group
blocking
method
R.
Jullien,
K.
Uzelac
(*),
P.
Pfeuty
Laboratoire
de
Physique
des
Solides,
Université
Paris-Sud,
Centre
d’Orsay,
91405
Orsay,
France
and
P.
Moussa
Service
de
Physique
Théorique,
C.E.A.,
Orme
des
Merisiers,
91190
Gif sur
Yvette,
France
(Reçu
le 8
décembre
1980,
révisé
le
5
mars
1981,
accepté
le
23
avril 1981 )
Résumé.
2014
Une
méthode
du
groupe
de
renormalisation
pour
les
systèmes
quantiques
est
appliquée
au
modèle
d’Ising
en
champ
transverse
unidimensionnel
avec
un
champ
longitudinal
complexe.
La
singularité
de
Yang
et
Lee
est
analysée
dans
ce
modèle,
compte
tenu
de
son
analogie
avec
le
modèle
d’Ising
classique
bidimensionnel.
Des
cal-
culs
plus
anciens
à
deux
niveaux
sont
améliorés
en
retenant
quatre
niveaux
dans
le
processus
de
renormalisation.
L’exposant
de
la
singularité
est
estimé
par
03C3= - 0,220 ± 0,005,
en
accord
avec
l’extrapolation
pour
des
blocs
infinis
des
valeurs
obtenues
dans
la
méthode
à
deux
niveaux.
On
clarifie
aussi
le
problème
des
oscillations
de
la
partie
réelle
de
l’aimantation
qui
apparaissent
comme
un
artefact
de
la
méthode
dans
l’espace
réel
qui
introduit
des
effets
de
taille
finie.
Abstract.
2014
A
quantum
renormalization-group
technique
is
applied
to
the
one-dimensional
transverse
field
Ising
model
in
a
complex
longitudinal
field.
The
Yang-Lee
edge
singularity
is
analysed
in
this
model
in
view
of
its
analogy
with
the
2d
classical
Ising
model.
Previous
two-level
calculations
are
improved
by
retaining
four
levels
in
the
blocking
procedure.
A
more
precise
edge
exponent
0 3 C 3
is
found
in
agreement
with
the
infinite
block
extrapolation
of
the
two-level
calculation,
but
still
larger
in
absolute
value
than
the
latest
series
analysis
results.
The
occurrence
of
oscillations
in
the
real
part
of
the
magnetization
appears
to
be
an
artifact
of the
real-space
method,
which
intro-
duces
finite
size
effects.
J.
Physique
42
(1981)
1075-1080
AOÛT
1981,
Classification
Physics
Abstracts
05.30
-
05.50
Introduction.
-
In
1952,
Yang
and
Lee
[1]
have
shown
that,
for
simple
Ising
systems,
the
zeros
of
the
partition
function
are
distributed
on
the
unit
circle
z
=
exp(10)
in
the
complex activity
plane
where h
is
the
applied
magnetic
field.
In
the
thermo-
dynamic
limit,
a
density
of
zeros
g(6)
can
be
defined
on
the
circle
which
describes
the
behaviour
of
the
system
in
presence
of
a
magnetic
field
as
well
as
the
critical
properties
in
zero
field.
For
T
& # x 3 E ;
Tc,
g(0)
has
a
gap
of
width
2
0g(T)
and
the
edges
of
this
gap,
for
h
=
±
ihg(T),
are
branch
points
for
the
magne-
tization
M(h,
T).
Near
this
edge,
and
on
the
circle,
the
real
part
of
the
magnetization
and
g(8)
are
pro-
portional
and
behave
like :
(*)
On
leave
from :
Institute
of
Physics
of the
University,
Zagreb,
Croatia,
Yugoslavia.
where
the
edge
exponent
Q
is
different
from
the
usual
exponent
1/ô.
The
singular
behaviour
of
g(8)
has
been
investigat-
ed
for
various
simple
systems
[2]
and
numerical
studies
have
been
done
for
d
=
2
and
d
=
3
[3,
4].
Recently,
Fisher
interpreted
the
edge
singularity
as
a
new
critical
point
(corresponding
to
a
transition
in
an
imaginary
field)
and
has
performed
field-theoretic
renormaliza-
tion-group
analysis
using
expansions
near
the
upper
critical
dimensionality d
=
6
[5].
He
also
pointed
out
the
great
universality
of a
which
does
not
depend
on
the
model
for
a
given
space
dimensionality
[5].
The
exponent
(1
is
equal
to
0.5
for
mean
field
theory,
and
u= -0.5
for d = 1
[2].
For
d = 2, J has
been
estimated
to
be
from
series
expansions.
However
series
expansion
results
require
special
analyticity
assumptions
for
the
thermodynamical
functions
and
it
is
always
useful
to
check
the
results
through
independent
method.
Here
the
problem
Article published online by
EDP Sciences
and available at
http://dx.doi.org/10.1051/jphys:019810042080107500
1076
arises
with
the
variation
with
the
temperature
[3]
of
estimate
for J
obtained
from
series
analysis.
The
best
choice
seems
to
be
the
high
temperature
case
[4],
[6],
but
one
must
be
cautious
as
it
has
been
seen
in
the
Bethe
lattice
case
[7].
We
have
recently
investigated
the
Yang-Lee
singu-
larity
of
the
2d-classical
Ising
model
indirectly
by
studying
a
one-dimensional
equivalent
quantum
mo-
del :
the
Ising
model
in
a
transverse
field
with
a
purely
imaginary
longitudinal
field
[8],
described
by
the
Hamiltonian :
where
are
Pauli
matrices.
Due
to
quantum
fluctuations,
this
model
exhibits
for h
=
0
a
phase
transition
at
T
=
0.
The
ground
state
changes
from
doublet
to
singlet
by
increasing
the
transverse
field
past
a
critical
value
(r/J)e
=
1.
For
h
=
0
this
model
is
exactly
solva-
ble
[9]
and
it
has
been
shown
[10]
that
his
critical
behaviour
is
equivalent
to
that
of
the
classical
2d
Ising
model.
The
transverse
field
r
and
the
ground
state
energy
correspond
to
the
temperature
and
the
free
energy
of
this later
respectively.
The
additional
longitudinal
held ih
has
the
same
role
in
the
two
models :
it
gives
rise
to
a
new
transition,
with
new
exponents,
corresponding
to
the
Yang-Lee
edge
singularity.
For
this
investigation
we
have
used
a
real-space
renormalization-group
blocking
procedure
[11]
retain-
ing
2
levels
(nL
=
2)
at
each
iteration.
We
have
cal-
culated
the
magnetization
and
estimated J
for
diffe-
rent
sizes
of
the
blocks ns
=
2,
3,
4,
5.
However,
the
results
were
only
slowly
improved
by
increasing
the
block
sizes,
the
convergence
of
the
exponents
being
in
(In
ns)-1.
The
present
paper
is
devoted
to
an
alter-
native,
more
efficient
improvement,
in
which
the
number
of
retained
levels
is
increased.
We
consider
here
block
of
three
sites
(n.
=
3)
and
we
retain
four
levels
(nL
=
4)
at
each
iteration.
The
value
of (1
is
considerably
improved
in
comparison
with
the
pre-
vious
calculations
[8].
Moreover
the
striking
oscilla-
tory
behaviour
of
Re
M
is
here
clearly
attributed
to
a
block
size
effect
occurring
in
the
real-space
proce-
dure.
After
summarizing
the
results
of
the
two-level
method
in
part
1,
we
present
the
four-level
method
in
part
2
and
we
discuss
the
results
in
part
3.
1.
Outline
of
the
two-level
method.
-
Let
us
sum-
marize
briefly
the
main
technical
points
of
our
pre-
vious
study
[8].
An
iterative
method
has
been
applied
to
hamiltonian
[2]
as
follows :
(i)
the
chain
is
cut
into
independent
blocks
of
ns
sites
and
the
hamiltonian
for
an
isolated
block
is
solved
exactly.
A
technical
difficulty
comes
from
the
fact
that
the
hamiltonian
is
no
longer
hermitian
in
presence
of
an
imaginary
field,
but
still
symmetric.
Therefore,
we
just
use
another
definition
of
the
scalar
product.
(ii)
A
new
block
spin
is
defined
by
retaining
only
two
eigenvalues
in
each
block.
In
order
to
recover
the
right h
=
0
limit
[11],
we
retain
those
with
lowest
real
parts.
(iii)
From
the
expression
for
the
matrix
elements
of
the
old
spin
operators
between
these
two
states
the
following
spin
recursion
relation
holds :
which
relates
the
pth
spin
in
the
block
with
the
new
block
spin
operators
Sj".
A
similar
expression
can
be
derived
for
S },p.
This
relation
is
used
for
p
=
1
and
p
= ns
to
rewrite
the
original
interblock
interaction
-
rcx
ex
-
JStns
Sj+ 1,1.
(iv)
Then,
after
a
suitable
rotation
of
Sj
in
the
x-z
plane
the
hamiltonian
recovers
a
similar
form
but
with
new
parameters
J,
f,
h.
The
procedure
is
repeated
until
the
parameters
J(n),
r (n),
h(n)
reach
one
of the
stable
fixed
points.
In
the
parameter
space
(T /J,
h/T )
the
stable
fixed
points
are
separated
by
a
critical
curve
starting
from
((r/Y)c
0)
and
ending
at
a
new
fixed
point
h
=
r
=
oo.
The
magnetization
components
in
the
ground
state
are
then
evaluated
by
expressing
the
original
averag-
ed
quantities
Sfl(o) >
and
St(O) )
as
a
function
of
S j(n»,
S j(n» in
the
stable
fixed
points,
which
are
known
quantities.
The
results.
depend
on
the
position
of
the
spin
inside
the
block
during
the
itera-
tive
process
as
a
consequence
of
the
blocking
proce-
dure.
As
it
was
shown
in
reference
[8],
the
best
result
is
obtained
by
considering
a
spin
always
at
the
centre
(for ns
odd)
or
near
the
centre
(for
ns
even)
of the
block,
to
avoid
edge
effects.
We
obtained
Q
= -
0.28,
-
0.30, -
0.26, -
0.29
respectively
for ns
=
2,
3,
4,
5.
An
alternative
procedure
consists
in
renormalizing
an
effective
operator
averaged
over
the
block
Although
the
values
obtained
for (1
are
less
accurate,
they
will
display
a
monotonic
convergence
with
ns,
allowing
an
infinite
size
extrapolation.
This
calcula-
tion
has
only
been
done
recently
[12]
and
we
present
in
figure
1
the
results
for
7,
together
with
the
results
for
f3
(exponent
of
the
magnetization
for
h
=
0)
as
a
func-
tion
of
(In
n. ,)-’,
which
is
the
convergence
law
gene-
rally
observed
in
this
method
[13].
The
values
obtain-
ed
for
n.
odd
and
even
are
represented
by
squares
and
triangles
respectively
and
they
correspond
to
that
calculated
in
references
[8]
and
[11].
The
values
obtain-
ed
by
averaging
at
each
step
are
represented
by
the
1077
Fig.
1 .
-
Plot
of
the
exponents P
and
a,
calculated
by
the
two-
level
method,
as
a
function
of
1/ln
n.
when
using
two
different
pro-
cedures :
by
renormalizing
a
spin
at
the
centre
(squares)
or
near
the
centre
(triangles)
of
the
block,
or
by
renormalizing
a
block-averag-
ed
operator
(dots).
dots.
One
can
see
for fi
that
the
values
align
quite
well
with
the
exact
result fl
=
0.125
for ns
-
oo.
For
a
the
corresponding
extrapolation
would
give :
This
final
range
for J
is
the
best
estimation
that
we
can
deduce
from
our
two-level
calculations.
Another
interesting
feature
of
the
two-level
calcula-
tions
is
the
oscillating
character
of
Re
M
as
a
func-
tion
of
the
field.
In
reference
[8]
we
were
unable
to
distinguish
clearly
between
an
intrinsic
property
of
g(8)
or
an
artifact
of the
renormalization-group
proce-
dure.
2.
The
four-level
method.
-
For
a
system
which
has
complex
energy
levels,
it
is
difficult
to
estimate
a
priori
the
approximation
made
by
selecting
the
two
levels
of
lowest
real
parts.
For
this
reason,
as
well
as
for
a
simple
reason
of
improvement
of
the
precision,
it
is
useful
to
extend
the
previous
calculations
to
the
case
where
a
greater
number
of levels
is
retained.
In
this
new
approach
we
consider
blocks
of
three
sites
(ns
=
3)
and
four
levels
(nL
=
4).
At
each
itera-
tion,
the
hamiltonian
takes
the
general
form
where
D,
A,
B
are
4
x
4
symmetric
matrices
of complex
elements.
D
is
diagonal
and
contains
the
four
energy
levels
E1,
E2,
E3,
E4
of
site
i.
A
and
B
contain
the
elements
of
the
couplings.
Let
us
first
describe
the
initial
step
which
consists
in
grouping
the
spins
two
by
two,
as
schematically
represented
in
figure
2,
so
that
the
initial
hamilto-
nian
(2)
can
be
written
in
the
form
(6).
At
each
new
site
i
stay
two
spins
S i , ;
and
SZ,i.
The
site-hamiltonian
is
diagonalized
exactly
in
order
to
find
the
four
energy
levels
attached
to
site
i.
We
observe
that
three
states
(labelled
1,
2,
3)
are
symmetric,
i.e.
they
belong
to
the
symmetric
space
constructed
with
, while
one
state
with
the
energy
E4
=
+
J,
is
antisymmetric.
The
initial
diagonal
matrix
Di0)
contains
the
four
exact
energies
E1,
E2, E3,
E4.
The
initial
expressions
for
Ai
and
Bi
are
the
4 x 4
matricial
representations
of
flSî,i
and
JÎSg,;
respectively
in
the
four-level
base
of
the
site
i.
The
elements
of
Ai
and
Bi
are
equal
up
to
a
minus
sign
occurring
in
the
elements
involving
the
site-antisymmetric
state.
To
give
an
example,
in
the
case
h
=
0,
D!O),
Ai0),
B!O)
take
the
form
1078
Fig.
2.
-
Sketch
of the
initial
step :
hamiltonian
(2)
is
written
under
form
(6)
by
grouping
the
initial
spins
two
by
two.
In
the
general
case
(h #
0),
a
computer
diagonaliza-
tion
is
performed
for
the
3
x
3
symmetrical
subspace.
The
matrix
is
then
complex
symmetric
and
the
eigen-
vectors
are
normalized
by
dividing
by
the
square-
root
of the
sum
of the
simple
squares
of the
coordinates
(this
norm
is
generally
complex).
The
hamiltonian
(2)
being
written
under
form
(6),
the
iterative
process
can
start.
Let
us
explain
the
main
points
of
the
procedure
which
follows
closely
that
described
in
part
1
for
the
two
level
case.
We
first
solve
exactly
the
hamiltonian
for
a
block j
of
three
sites j,
1 ;
j,
2 ; j,
3 :
z
The
diagonalization
is
simplified
by
observing
that
the
complete
space
of
dimensionality
4
x
4
x
4
=
64
can
be
divided
into
two
subspaces :
a
« symmetric »
subspace
of
dimensionality
38
and
an
« antisymme-
tric »
subspace
of
dimensionality
28
constructed
with
block-states
having
an
even
or
an
odd
number
of
antisymmetric
site
states
respectively.
Then
the
new
four-level
base
for
the
block
is
defined
as
follows :
from
the
complete
set
of
64
states
we
extract
only
the
three
lowest
levels
of
the
symmetric
subspace
and
the
ground
state
of
the
antisymmetric
subspace.
Those
four
levels
are
normalized
as
before.
Their
energies
give
the
elements
of
the
new
matrix
Dj
for
the
block j.
The
new
matrices
Aj
and
Bj+ 1
are
obtained
by
rewritting
the
old
interblock
coupling
A j,ns
Bj + 1, 1
in
the
new
block-bases.
Aj
and
Bj
are
thus
given
by
When
the
symbo1 l3
designs
the
change
of
basis.
The
structure
of
the
four-level
basis
insures
that
A(n)
and
B (n)
keep
their
symmetry
properties
under
renormalization.
Then
the
procedure
is
repeated
until
the
elements
of
D(n),
A(n),
B(n)
reach
stable
values.
In
principle,
the
x-component
of
the
magnetization
can
be
evaluated
by
computing
at
each
step
the
4 x 4
representation
of
an
initial
operator
Si
of
(2)
in
the
successive
basis.
As
already
observed
in
the
previous
work
[8],
the
result
depends
on
the
position
in
the
block.
This
problem
is
avoided
by
computing
instead
a
« block-averaged »
operator
C (n).
In
the
initial
step
C (0)
is
defined
as
the
4
x
4
matrix
representation
of 1(Stl
+
Six2).
Then
the
renormalization
of
C (n)
is
given
by
z
We
have
observed
that
this
is
the
best
choice.
Taking
only
the
renormalization
of
S x
for
the
centre
of
the
block,
as
in
reference
[8],
would
give
meaning-
less
results
due
to
the
presence
of
an
antisymmetric
state
in
the
four-level
basis.
We
observe
that
the
diagonal
element
C(l, 1)
is
always
purely
imaginary.
When
the
fixed
point
is
reached,
this
diagonal
element
C(l, 1)
represents
the
imaginary
part
of
the
magnetization.
(This
corres-
ponds
to
the
linear
response
in
a
purely
imaginary
longitudinal
field.)
Moreover,
we
observe
that
the
off-diagonal
element
C(l,
2)
is
real.
C(l,
2)
represents
the
real
part
of
the
magnetization
only
when
the
two
lowest
states
1
and
2
are
asymptotically
degenerated
in
the
fixed
point,
i.e.
only
in
the
low-r
phase
as
it
will
be
seen
below.
Before
discussing
the
results,
let
us
emphasize
that
in
this
four-level
method,
the
parameter
space
is
enlarged
compared
with
the
two-level
method.
How-
ever
the
results
are
discussed
in
the
two-dimensional
space
r/J,
h/J
which
corresponds
to
the
parameters
of
the
initial
Hamiltonian.
z
3.
Results.
-
Let
us
first
present
briefly
the
results
for
h = 0.
The
location
of
the
transition
is
found
to
be
(r /J)e
=
0.969
997...
i.e.
-
3
° ô
lower
than
the
exact
value
(F/J)e
=
1
[9].
The
exponent
for
the
x-component
of
the
magnetization
below
the
transi-
tion
is
found
to
be
extremely
good : 03B2=0.124 ±
0.001,
the
exact
value
being
=
0.125.
We
have
calculated
the
other
exponents
v,
s,
z, 1
characteristic
of
the
transition,
as
in
reference
[11].
v
gives
the
divergence
index
of
the
coherence
length
when
approaching
the
transition. s
gives
the
opening
of
the
gap
above
the
transition.
z
is
the
dynamical
exponent
describing
how
the
gap
renormalizes
under
length
scale
at
the
transition. il
gives
the
power
law
decay
of
the
x-x
correlation
function
at
the
transition.
We
have
found
to
be
compared
with
the
exact
values
[9]
We
have
checked
that
the
scaling
laws
s
=
vz,
2 fi
= v 11
are
verified
within
the
error
bars.
These
results,
espe-
1079
cially
for
{3,
show
clearly
that
the
convergence
in
nL
is
much
faster
than
the
convergence
in
ns.
In
the
two
level
scheme,
we
never
reached
this
accuracy,
even
with
the
largest
block
used
(ns
=
7),
which
was
involving
larger
computation.
The
results
for
non-zero
imaginary
field ih
can
be
discussed
in
the
plane
of
the
initial
parameters
r/J,
h/J.
We
can
determine
a
critical
line
(full
line
in
Fig.
3.
-
Phase
diagram
in
the
plane
r/J,
h/J.
The
full
line
repre-
sents
the
critical
curve
(transition
in
the
infinite
chain).
The
dashed
line
represents
the
transition
in
a
finite
block
of six
spins.
figure
3)
starting
from
(r/J)c ~
0.97
for h
=
0
and
going
to
infinity
in
the
asymptotic
direction h
=
r.
If
the
initial
values
of
r/J,
h/J
are
chosen
in
region
1
on
the
right
of
this
line,
the
energy
levels
remain
real
through
the
iteration
and
the
fixed
point
corresponds
to
a
singlet
ground
state
well
separated
from
the
other
states
by
a
gap.
This
corresponds
to
a
trivial
stable
fixed
point
Fiv --->
oo,
h
=
0,
in
the
plane.
In
this
singlet
the
real
part
of
the
x
magnetization
is
zero.
Thus
the
whole
region
1
has
the
same
charac-
teristics
as
the
disordered
line
r / J
& # x 3 E ;
(r / J)e
for
h=o.
If
the
initial
values
of
r/J,
h/J
are
instead
chosen
in
the
other
regions
(II
or
III),
the
real
parts
of
the
two
lowest
levels
merge
in
a
doublet
at
the
fixed
point.
This
situation
corresponds
to
a
new
trivial
fixed
point
r/J
=
0,
h/J
=
oc
in
the
plane.
Then
in
these
regions,
the
real
part
of
the
magnetization
is
non-zero
and
is
given
by
the
(1,
2)
elements
of
the
matrix
C.
When
looking
more
carefully
to
the
behaviour
of
the
levels
through
the
iterations,
it
appears
that
we
can
define
another
line
(dashed
line
in
figure
3)
separating
region
II
and
region
III.
In
region
II
the
levels
are
real
at
the
beginning
and
become
complex
at
a
given
step
of
the
iteration,
while
in
region
III
the
levels
are
complex
already
after
the
first
step,
their
real
part
forming
a
doublet.
This
dashed
line
corresponds
to
a
transition
in
the
finite
system
of
three
sites
consi-
dered
at
the
beginning :
i.e.
a
finite
chain
of
six
spins.
We
have
to
remember
that
a
finite
quantum
block
corresponds
to
a
one-dimensional
strip
in
the
classi-
cal
analog
and
that
an
infinite
classical
Ising
chain
in
an
imaginary
field
has
a
transition
at
a
non-zero
temperature.
This
transition
is
characterized
here
by
the
appearance
of
imaginary
parts
in
the
energy
levels
i.e.
by
a
vanishing
norm.
We
have
verified
that,
after
only
one
step,
the
real
part
of
the
magne-
tization
diverges
when
approaching
by
the
left
the
dashed
line
with
an
exponent
equal
to -
0.5,
exactly
the
edge
exponent
in
ld.
Let
us
focus
now
our
attention
on
the
full
line
which
corresponds
to
the
transition
in
the
infinite
quantum
chain.
We
have
calculated
the
real
part
and
the
imaginary
part
of
the
magnetization
when
crossing
this
line.
A
numerical
result
is
given
in
figure
4
where
Real
M,,
and
Im
M,,
has
been
plotted
as
a
function
of
h/J
for
r /J
=
1.5.
As
in
the
previous
work
we
observe
some
oscillations
in
both
Re
Mx
and
Im
Mx
above
the
transition.
The
exponent a
has
been
evaluated
precisely
from
the
imaginary
part
below
the
transition
which
does
not
show
any
oscillations.
We
have
obtained
Fig.
4.
-
Plot
of Re
Mx
and
Im
Mx
as
a
function
of hl J
for
1-IJ
=
1.5.
This
result
is
much
more
precise
than
that
of
the
two-
level
study
and
it
is
consistent
with
the
extrapolation
for n.
~
oo
given
by
(5).
This
result
for J
should
be
compared
with
the
series
expansion
estimations
for
the
classical
Ising
model
in
two
dimensions :
Q
= -
0.12
±
0.05
[3]
and
Q
= -
0.163
±
0.003
[4].
We
find
a
definitively
larger
absolute
value.
A
possible
explanation
of
the
discrepancy
could
be
the
influence
of
the
transition
for
the
finite
block :
note
that
the
dashed
line
approa-
ches
the
full
line
asymptotically.
Thus
the
present
result
could
be
understood
as
intermediate
between
the
finite
block
value
6
= -
0.5
and
an
infinite
sys-
tem
value
which
could
be
J - -
0.12, -
0.16.
This
interpretation
might
be
supported
by
new
results
of a
completely
independent
and
different
renormaliza-
tion-group
method
applied
to
the
same
model
[14].
Another
point
of
discussion
is
the
oscillatory
character
of
the
magnetization
observed
in
figure
4.
As
in
reference
[8]
these
oscillations
are
periodic
in
In
and
can
be
analysed
through
the
1080
formula :
with
x
=
2.4.
The
interpretation
suggested
in
refe-
rence
[8]
that
the
oscillations
could
be
due
to
an
arti-
fact
of
the
renormalization-group
procedure
is
here
justified
since
this
period
is
now
close
to
the
expect-
ed
value
In
the
preceding
calculation
[8]
the
agreement
was
considerably
worse.
Thus
the
present
four-level
calculation
improves
the
previous
one,
and
gives
a
more
precise
value
of
( 1 .
The
oscillation
phenomenon
is
clarified
and
appears
to
be
a
finite
block
effect.
The
discrepancy
between
our
results
and
series
analysis
remains
an
open
ques-
tion.
This
point
will
be
discussed
in
[14]
in
which
a
different
renormalization-group
approach
displays
different
result
according
to
whether
the
real
part
on
the
imaginary
part
is
renormalized.
References
[1]
YANG,
C.
N.
and
LEE,
T.
D.,
Phys.
Rev.
87
(1952)
404;
LEE,
T.
D.
and
YANG,
C.
N.,
Phys.
Rev.
87
(1952)
410.
[2]
SUZUKI,
M.,
Prog.
Theor.
Phys.
38
(1967)
1225 ;
BESSIS,
J.
D.,
DROUFFE,
J.
M.
and
MOUSSA,
P.,
J.
Phys.
A
9
(1976)
2105 ;
KURTZE,
D.
A.
and
FISHER,
M.
E.,
J.
Stat.
Phys.
19
(1978)
205 ;
BAKER,
G.,
FISHER,
M.
E.
and
MOUSSA,
P.,
Phys.
Rev.
Lett.
42
(1979)
615.
[3]
KORTMAN,
P.
J.
and
GRIFFITHS,
R.
B.,
Phys.
Rev.
Lett.
27
(1971)
1439.
[4]
KURTZE,
D.
A.
and
FISHER,
M.
E.,
Phys.
Rev. B
20
(1979)
2785.
[5]
FISHER,
M.
E.,
Phys.
Rev.
Lett.
40
(1978)
1610.
[6]
GAUNT,
D.,
Phys.
Rev.
179
(1969)
174.
[7]
MOUSSA,
P.,
Springer
series
in
Synergetics,
in
press.
[8]
UZELAC,
K.,
PFEUTY,
P.
and
JULLIEN,
R.,
Phys.
Rev.
Lett.
43
(1979)
805 ;
UZELAC,
K.,
JULLIEN,
R.
and
PFEUTY,
P.,
Phys.
Rev. B
22
(1980) 436.
[9]
PFEUTY,
P.,
Ann.
Phys.
(N.
Y.)
57
(1970)
79.
[10]
SUZUKI,
M.,
Prog.
Theor.
Phys.
56
(1976)
1454.
[11]
DRELL,
S.
D.,
WEINSTEIN,
M.,
YANKIELOVICZ,
S.,
Phys.
Rev.
D
14
(1976)
487 ;
JULLIEN,
R.,
PFEUTY,
P.,
FIELDS,
J.
N.
and
DONIACH,
S.,
Phys.
Rev. B
18
(1978)
3568.
[12]
UZELAC,
K.,
JULLIEN,
R.,
PFEUTY,
P.
and
MOUSSA,
P.,
Springer
series
in
Synergetics,
in
press.
[13]
PFEUTY,
P.,
JULLIEN,
R.
and
PENSON,
K.
A.,
unpublished;
UZELAC,
K.,
Thèse,
Orsay
(1980).
JULLIEN,
R.,
Can.
J.
Phys.
59
(1981)
605.
[14]
UZELAC,
K.
and
JULLIEN,
R.,
J.
Phys.
A
14
(1981)
L-151.
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