Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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228
G.
V.
Orman
2.1
The
extended
Markov
property
It
is
interesting
to
see
that
the
Markov
property
can
be
extended.
The
following
theorem
will
refer
to
the
extended
Markov
property.
Theorem
3.
for
r E
K.
Proof.
Let
us
come
back
to
the
equality
(1)
before.
Now
it
will
be
proved
for
D E K
t
+.
To
this
end
the
following
equality
will
be
shown:
for
fi E
C(S),
DE
Kt+
and
0
<::
81
<
82
< ... < Sn·
But
D E K
t
+
h
for
h > 0,
so
that
by
Corollary
1
it
results
Ea(h
(XSI (et+l,w))
...
fn(Xsn
(et+hw)), D) =
Ea(Ext+,,(h
(X
S1
)
•..
fn(Xsn)),
D).
(3)
Now
one
can
observe
that
is
continuous
in
a,
if
it
is
considered
that
and
Hs : C
---+
C.
But
Xt(w)
being
right
continuous
in
t,
one
gets
as
h 1
o.
Now
,
the
equality
(2)
will
result
by
taking
the
limit
in
(3)
as
h 1
o.
In
this
way,
for
G
i
open
in
S,
the
following
equality
will
result
from
(2)
Ea(XSi(etw)
E G
1
,'"
,XSn(etw)
E G
n
,
D)
(4)
=
Ea(PXt(XSl
E G
1
,'"
,Xsn
E
Gn)
,
D),
and
now
the
Dynkin's
theorem
can
be
used.
1
2.2
The
strong
Markov
property
As
it
is
known,
the
intuitive
meaning
of
the
Markov
process
(for
example
X(t))
is
the
fact
that
such
processes
"forget"
the
past,
provided
that
t
n
-1
is
regarded
as
the
present.
Now,
the
intuitive
meaning
of
the
Markov
property
is
that
under
the
condition
that
the
path
is
known
up
to
time
t,
the
future
motion
would
be
as
if
it
started
at
the
point
Xt(w)
E S.
But
a
question
arises:
what
will
happen
if
"t"
will
be
replaced by a
random
time
a(w)?
1Theorem
(Dynkin's
formula).
Let
us
suppose
that
a is a
stopping
time
with
Ea(a)
<
00.
Then,
for
u E :D(A)
it
follows:
Ea
(~OO
AU(Xtl
dt
) =
Ea(u(Xcr))
-
u(a).
A Survey
of
the Transition Probabilities
and
Markov Property 229
Definition
3.
A
random
time
0"
:
it
-->
[0,
(Xl]
is
called a
"stopping
time"
with
respect
to
{Kt}
if
(5)
for
every
t.
But
such
a
condition
is
equivalent
to
the
condition
(6)
for
every
t.
Observation
1.
If
the
condition
(5) holds,
then
it
results
But
if
the
condition
(6) holds,
then
we
have
{O"
<:::
t}
= n
{O"
< t +
~}
E K
t+;k
n?:.m
for
every
m.
Therefore
{O"
<
t}
En
Kt+;k =
Kt
m
by
right
continuity
of
K
t.
A
trivial
example
is
the
deterministic
time
0"
==
t.
Now
the
strong
property
can
be
given
(more
details
and
proofs
can
be
seen
in
[4],
[5], [6],
[10],
[2]).
The
strong
Markov
property
is
given
in
the
theorem
below
Theorem
4.
The
following
equalities
hold
a.s.
10. P
IL
(8;11jKo-)=Px
a
(r)
a.s.
(P
IL
)
on
{O"
< (Xl},
whererEK.
2°.
EIL(F(8o-wIKo-) =
Ex"
(F)
a.s.
(P
IL
)
on
{o-
<
(Xl},
where F
is
bounded
and
K -measurable.
Note
3.
The
following
conclusion
are
true.
i.
8o-w
= 8o-(w)w
for
any
w
with
O"(w)
<
(Xl,
and
8;;lr
=
{w:
O"(w)
<
(Xl
and
8o-w
E
n.
ii.
It
is
observed
that
I"
is
arbitrary
and
for
this
reason
the
both F(8o-w)
and
Ex"
(F)
are K -measurable.
Finally,
a
variant
of
the
strong
Markov
property
is
given
below.
It
is
referred
to
as
the
time-dependent
strong
Markov
property.
Theorem
5.
It
is
supposed
that
F(t,
w)
is
bounded
and
K[O,
(Xl]
x K -
measurable.
Then,
the
following
equality holds
(7)
Proof.
In
view
of
that
0" is K
(T-measurable
we
come
to
the
conclusion
that
F(t,
w) =
f(t)G(w).
Now,
by
considering
the
aditivity
of
both
sides
of
(7)
in
F,
the
general
case
results.
230
G.
V.
Orman
3
Application
Absorbing
and
reflecing
barri
e
rs
Let
us
consider
that
a
particle
located
on
a
straight
line
moves
along
the
line
via
random
impacts
occurring
at
times
tl,
t2, t3,
..
'.
The
particl
e
can
be
at
points
with
integral
coordinates
a, a + 1, a + 2, '
..
,b.
At
points
a
and
b
th
e
re
are
absorbing
barriers.
Each
impact
displaces
the
particle
to
the
right
with
probability
p
and
to
th
e le
ft
with
probability
q = 1 - P
so
long
as
th
e
particle
is
not
located
at
a
barrier.
If
the
particle
is
at
a
barrier
th
en,
it
remains
in
the
states
Al
and
A
n
-
I
with
probability
l.
A
similar
example
can
be
considered
for a
particle
being
in
a
random
walk,
wh
en
at
points
a
and
b
th
e
re
are
reflecting
barriers.
Th
e
conditions
remain
the
same
as
in
the
former
case
,
the
only
differenc
e
being
that
if
th
e
particle
is
at
a
barrier,
any
impact
will
transfer
it
on
e
unit
inside
th
e gap
between
th
e ba
rriers.
1.
Let
now
be
the
case
of
a
Brownian
moti
on
with
an
absorbing
barrier.
Th
e
forward
Kolmogorov's
equation
(9) for a
Brownian
motion
on
x > °
with
an
absorbing
boundary
at
x = ° is
giv
en
by
ap
1 a
2
p
in
y > °
at
2
ay2
p(O,t
,
y)
= 0,
t>O,
y > O
p(x
, t,
y)
--+
o(x
-
y)
as
t 1 0, x > 0, y > 0.
The
solution
of
such
an
initial
boundary
value
problem
is
as
follows
1
[(X
_
y
)2
(x+y)2]
p(x,t,y)
=
--
e-
~
-e-
~
.
tV21T
It
can
be
se
en
that
by
symmetry,
p(x
, t, 0) =
0.
Th
en,
it
can
be
shown
that
1 j +oo C
,,+
y)
2
!<C
e -
~
'P(x)dx
--+
'1'(
-y)
= °
tv
211'
-00
as
t 1 °
if
y > 0.
Therefore,
p(x,
t,
y)
--+
o(x
-
y)
as
t
10
for
all
x>
0,
Y >
0,
where
0 is
the
Dirac
's
o-function.
2.
Now
let
us
consider
the
Brownian
motion
on
x > °
but
with
a
reflection
barrier
at
the
origin.
The
forward
Kolmogorov
's
equation
for a
Brownian
motion
on
x > °
with
an
reflec-
tion
boundary
at
x = ° is
giv
en
by
ap
1 a
2
p
y>o
at
2
ay2'
ap(x
,t ,
y)
/
=0
ay
y=o
p(x,t,y)
--+
o(x
-
y)
with
0 -
the
Dirac's o-function.
as
In
this
cas
e
the
following
solution
is
found
t 1 0, x > 0, y > 0,
1
[(
X_
y )2 ( x
+y)
2 ]
p(x,t,y)
=
--
e-
~
+e-
~
tV21T
and
the
condition
holds
too.
ap(x,
t, y)
ax
/
-0
x=o
A Survey
of
the Transition Probabilities and Markov Property
231
Note
4.
It
is
known that
in
some
conditions
of
existence, the transition density satis-
fies the folowing two Kolmogorov's equations (they will
be
referred to as the
backward
Kolmogorov
's
equation
and respective the
forward
Kolmogorov
's
eq
u
ation).
o
-
.f~(
c:..:t,-=-:.x,--;T-,--,y~)
()
of(t,x;T,y)
1
b(
) 0
2
f(t,x;T
,y)
- =
-a
t x - - t x
ot
'ox
2'
ox
2
(8)
and
of(t
, x;
T,
y) 0 1 0
2
OT
= -
oy
[a(T,y)f(t,x;T,y)]
+ 2 oy2
[b(T,y)f(t,x;T,y)],
(9)
where
aCt
,
x),
bet,
x),
aCT,
y),
beT
, y) are
functions
satisfying
some
conditions to as-
sure the existence and the uniqueness
of
the solution
of
the equations. The forward
Kolmogorov's equation
is
also referred to as the
Fokker-Planck
equat
ion.
We emphasize that the expression "the backward Kolmogorov equation" means:
in
the "backward" variables t and
x.
Similarly, "the forward Kolmogorov equation"
must
be
understood that
it
is
considered
in
the "forward" variables T and y.
One can observe that this Application implyes stochastic differential equations
in
connection with the K olmogorov's equations
for
a Markov process; more details
and
related topics can
be
found
in
[lS},
flO},
(13).
References
[1]
BERTOIN,
J.
,
Levy
Processes
,
Cambridge
Univ.
Press
,
Cambridge,
1996.
[2]
BHARUCHA-REID
,
A.T.
,
Elements
Of
The
Theory
Of
Markov
Processes
And
Their
Applications,
Dover
Publications,
In
c.,
Mineola,
New
York, 1997.
[3]
G
NE
DENKO,
B.V.,
The
Theory
of
Probability
,
Mir
Publisher
, Moscow, 1976.
[4]
ITO
, K. ,
Selected
Papers.
Springer
, 1987.
[5]
ITO
, K. , McKEAN,
H.P.
,
Diffusion
Processes
and
their
Sample
Paths
.
Springer-
Verlag
Berlin
Heidelberg
, 1996.
[6]
IT
O,
K.,
Stochastic
Processes.
Eds
.
Ol
e E.
Barndorff-Nielsen,
Ken-iti
Sato,
Springer,
2004.
[7]
KALL
ENBE
RG
,
0.
,
Foundation
of
Modern
Probability
,
Spring
er,
New
York,
1997.
[
8]
LOEVE
,
M.,
Probability
Theory,
vol.
I,
Springer-Verlag
, New
York
, 1977.
[9]
LOEVE
, M.,
Probability
Theory,
vol.
II
,
Springer-Verlag,
New
York
,
Heidelberg
Berlin,
1978.
[10]
0KSENDAL
, B.,
Stochastic
Differential
Equations:
An
Introduction
with
Applications,
Sixth
Edition,
Springer-Verlag,
2003.
[
ll
]
0KSENDAL,
B.,
SULEM
A.,
Applied
Stochastic
Control
of
Jump
Diffusions
,
Springer
, 2007.
[12]
ORMAN
, G.
V.,
Capitole
de
matematici
aplicate,
Ed.
Albastra,
Microlnformat-
ica,
Clu
j-
Napoca,
1999.
[
13]
ORMAN
G.
V.,
Lectures
on
Stochastic
Approximation
Methods
and
Related
Topics
,
Preprint.
"
Gerhard
Mercator"
University,
Duisburg,
Germany,
2001.
[14]
ORMAN
,
G.V.,
Handbook
Of
Limit
Theorems
And
Sochastic
Approxima-
tion,
Transilvania
University
Press
,
Brasov
, 2003.
[15]
SCHUSS
Z.,
Theory
and
Application
of
Stochastic
Differential
Equations
,
J
ohn
Wiley
&
Sons
,
New
York, 1980.
[16
]
STROOCK
,
D.W.
,
Markov
Processes
from
K.
Ito
Perspective,
Princeton
Univ.
Press,
Princ
e
ton
, 2003.
232
Qualitative
Dynamics
of
Interacting
Classical
Spins
Leonidas Pantelidis
Department
of
Physics
and
Astronomy,
Hanover
College
Hanover,
IN
47243,
USA
(e-mail:
leon_pantelidis@alum.mit.
edu)
Abstract:
\Ve consider
the
classical
Heisenberg
model
(Hl'vI)
with
z-axis
anisotropy
and
external
magnetic
field.
Its
phase
space
is foliated
into
a family
of
invariant
leaves
diffeomorphic
to
(S2)A
The
flow
on
each
leaf
S is
Hamiltonian.
For
the
isotropic
Hl'vI
with
zero
field,
the
manifold
:F
of
longitudinal
fixed
points
(LFPs)
intersects
each
leaf
S orthogonally.
In
addition,
we show
that
the
ferromagnetic
(FR)
state
and
the
antiferromagnetic
(AF)
state
with
non-zero
total
spin
are
both
stable
LFPs.
This
is a
direct
implication
of
a
Lemma
which
extends
Lyapunov
stability
from
an
invariant
subspace
to
the
whole
leaf
by
exploiting
the
rotational
symmetry.
The
lemma
does
not
apply
in
the
case
of
zero
total
spin,
and
indeed,
the
AF
state
on
an
equal-spins
leaf
is
shown
to
be
unstable.
PACS
numbers:
05.45.-a
L
Introduction
The
Heisenberg model (HM)
is
a simplified model of magnetic ordering in materials
[1,
2,
3,
4]
which
has
drawn
considerable
attention
for a long time. Theoretical
and
experimental studies on
the
quantum
version
of
the
HM are
abundanq.
The
classical
HM
has also been of interest,
both
as a model of ferromagnetic
and
other
phenomena
and
as a
testground
for
methods
of nonlinear dynamics. However, cla.'isical Heisenberg
spin
systems, especially chains, have been mostly
studied
in
the
continuum
limit§.
From
the
point
of view of
quantitative
analytical dynamics,
the
discrete HM
is
barely
tractable
with
even a simple Heisenberg ring (closed chain
of
spins
with
equal nearest-neighbor only interactions) being non-integrable.
The
only known exact
solutions for
the
HM on a general periodic lattice arc
the
(propagating) ferromagnetic
(FR)
and
antiferromagnetic (AF) nonlinear spin waves (NLSWs),
and
some special
planar
solutions
[32,
33, 34].
These
arc finite-amplitude generalizations of
the
small-
amplitude
linear spin waves,
the
latter
being
approximate
periodic solutions of
the
nonlinear
equations
of
motion in
the
vicinity
of
the
FR
and
AF
fixed
points
(FPs)
II.
:j:
Some important work in the field can be found in
[5,
6,
7,
8,
9,
10,
11,
12, 13, 14,
15,
16, 17, 18, 19,
20, 21, 22, 23, 24,
251
and references therein, although this list
is
by
no
means comprehensive.
§ For
instance, see [26,27, 28,
29,
30,
311
and references therein.
II
Also termed as critical points, equilibrium points, or steady states.
Qualitative Dynamics a/Interacting Classical SpillS 233
Certain
linear combinations
of
NLSWs havc also been found
to
satisfy
the
equations
of Illotion
[35],
although
these
solutions do not, in general, reside on
the
leaf (shell)
of
equal spins ("off-shell" NLSW s). In all
the
aforementioned solutions
the
spins have
constant
and
equal (in each sublattice, in
the
AF
case) z-components,
that
is, all known
solutions are
of
a
rather
special type.
This
paper
focuses
on
the
qualitative dynamics
of
the
classical HM. In section
2,
we
introduce
the
general classical HM
with
z-axis anisotropy
and
magnetic field. In
section
3, we discuss
its
phase space
structure
and
in
particular
the
foliation
of
the
phase space into invariant leaves
with
Hamiltonian
flow. In section 4,
we
discuss
the
usual
FPs
of
the
flow
and
point
out
that
they
form manifolds which intersect
the
leaves
orthogonally. Section 5
addresses
the
question
of
stability
of
FPs.
This
question
is
answered in
the
special
but
interesting cases
of
the
FR
and
AF
states
on
the
basis of
a general
lemma
which
extends
Lyapunov stability from
an
invariant subspace
to
the
whole leaf
by
exploiting
the
rotational
symmetry
of
the
model.
2.
The
classical
HM
The
standard
quantum
HM
Hamiltonian
reads
A A
H = -
LgjlSJ'
S[-
LBJ'
Sj,
(2.1 )
j<l
where
gjl
are
the
exchange constants
and
the
second
term
,
the
Zeeman
term,
arises in
the
presence of
an
external
magnetic field
B.
We
can
also
introduce
a longitudinal
anisotropy
L'l.
by
setting
Si'
S[
==
S;-.
st +
L'l.SjSF
==
(SjSF +
sysn
+
L'l.SjS{
.
(2.2)
In a mean-
field approach, in
particular
in a
time-dependent
Hartree-Fock approximation,
the
average spins
Uj,
j = 1
...
A
interact
according
to
the
first order system of A coupled
quadratic
equations
[34]
~~j
=
Uj
x
(tgjl(ut
+
L'l.l1tZ)
+
Bi)
, j =
1,
...
,A,
1=1
(2.3)
which
constitute
the
classical HM.
Taking
the
dot
product
of
both
sides
of
(2.3)
with
Uj,
we
find
that
the
lengths
of
the
individual spin vectors arc first integrals of
the
motion,
that
is
IlJ
=
57.
j =
1,
...
,A
,
(2.4)
where
the
5j
are
arbitrary
non-negative real constants. Additional
constants
of
motion
can
be
found by inspecting
the
sum
of
equations (2.3),
d A A A
dt
L
Uj
=
(L'l.
-
1)
L
gj[(Uj
x
I1tZ)
+ L
Uj
x B j .
j=1 j.l=1 j=1
(2.5)
234 L. Pantelidis
One
sees
that,
in
the
isotropic case
L).
= 1
and
for a homogeneous magnetic field,
the
total
spin
v =
L.~'=l
Uj
has
constant
length
and
precesses
about
the
applied field. In
the
absence
of
an
external
field,
both
the
direction
and
the
magnitude
of
the
total
spin
arc conserved.
If
L).
#
1,
only
the
z-component of
the
total
spin
is
conserved, provided
the
magnetic field
is
7:ero
or
parallel
to
the
z-direction.
In passing,
note
that
(2.3) is invariant
under
the
rescaling
transformation
(lit,
g,
B)
-+
J(llt,
g,
B)
(2.6)
where J
is
a non-zero real
constant.
Hence,
an
overall renormalization of
the
exchange
and
ficld
strengths
does
not
affect
the
dynamics. Jdoreover,
the
dynamics essentially
depcnd
only
on
the
rclative spin lengths since (2.3)
is
invariant
under
the
rescaling
(lit,
Uj,
B)
-+
s(l/t,
Uj,
B)
(2.7)
for any non-zero real
constant
s.
3.
Phase
space
structure
of
the
HM
Spins of zero
length
are zero vectors and, according
to
(2.3),
they
neither
change
themselves
nor
affect
the
non-zero spins. Therefore. A
can
be
considered
to
be
the
number
of non-7:ero spins
and
so
the
total
phase
space of
the
HM
is
the
open
submanifold
P =
{(Uj)
E
IR:JA
:uJ
>
0,
\lj
E
{I,
...
,
A}}
(3.1 )
of
1R
3A
If
we
consider
the
Coo
surjection
its
Jacobian
matrix
has
non-zero
orthogonal
rows so
rankn
=
A.
Hence n is a submersion
and
therefore it induces
the
codimension-A, class
Coo
foliation
{Sh)
hSj
)EJP:~'
where
and
P =
(~)S(sJ)'
By definition of a foliation,
the
leaves
S(8j)
are disjoint
and
connected
(see Figure 1).
In
addition, as fibers of a submersion,
they
are 2A-dimensional closed
regular submanifolds
of
P. Since
they
are also
bounded,
Sis})
are
compact
regular
submanifolds
of
P.
For A = 1 we recover
the
well-known result
that
the
sphere 52
of
radius s
is
a 2-dimensional
compact
regular submanifold
of
1R3~.
This
immediately implies
that
(5
2
)A
is
a 2A-dimensional
compact
regular submanifold of
1R
3A
Because
there
is
a
unique 2A-dimensional regular submanifold
structure
on
the
set
S,
we
conclude
that
S
is
diffeomorphic
to
(5
2
)A
'If
It
is also well-known
that
all spheres
of
non-zero radii
are
diffeomorphic
and
that,
since
dim
52
=
2 < 4,
there
is a
unique
differentiable
structure
on 52.
Qualitative Dynamics
of
Interacting Classical Spins 235
It
can
be shown
[36]
that
S
is
a symplectic manifold
on
which
the
spins satisfy
the
angular
momcntum
Lie algebra
and
their
motion
is
a
Hamiltonian
flow
with
Hamiltonian
h : S
-t
lR
dcrived
by
replacing in (2.1)
thc
operators
S with c:lassical spins
u:
A A
h =
~
""""'
gzu
. uz
~
""""'
B . u
~.1.1
~
J J
(3.2)
j<z
j=]
Moreover, since S is a connected manifold,
any
other
Hamiltonian
9
generating
the
same flow, i.e., such
that
Vg
=
Vi"
differs from h
only
by a
constant
[37].
Note
that
the
classical HM
is
not
a mechanical system, in
the
sense
that
the
Lagrangian
is
not
quadratic
in velocities.
In
what
follows,
we
will consider only
the
isotropic case
with
zero magnetic
field,
although
our
analysis could be
extended
to
deal
with
more general cases.
The
Hamiltonian
h
is
of
course conserved as
it
does
not
depend
on
time
explicitly. As
we
saw in section (2),
the
total
spin
v
is
also conserved.
By
comparison
with
the
quantum
case, we expect
that
the
total
spin v generates
the
rotations
of
the
spins in
lR
3
,
which
makes it
the
c:lassical
total
angular
momentum
of
the
system. Indeed, as one
can
verify,
where
Ret
is
the
infinitesimal
rotation
about
the
a-axis
parametrized
by
f.
The
time-
translation
and
rotation
arc continuous symmetries of
the
Hamiltonian
and
thus
of
the
equations
of
motion. We
can
also see
that
directly from (2.3).
These
arc Niiether
symmetries as
they
are
generated
by
Hamiltonian
vector fields,
that
is, vector fields
which leave
the
2-form invariant+:
LvJ2
= Lv"" n =
O.
Notice
that
space inversion
is a discrete
symmetry
of
the
Hamiltonian
but
does
not
leave
the
equations
of motion
invariant
as
the
cross
product
in (2.3) is a pseudovector. A maximal
set
of known first
integrals in
invol'Ution (pairwise commuting) has only 3 elements, e.g.,
{h,
u
2
, U
Z
},
and
so
the
system
is
far from integrable in
the
general case. (For early numerical evidence
of non-ingrability see [32].)
The
trivial case
of
2 spins
and
the
cases
of
the
3-
and
4-
spin Heisenberg rings are exceptions. Note
that
the
constants
(2.4)
cannot
be
related
to
Hamiltonian
symmetries since
they
define
the
very phase space S
on
which
the
equations
of
motion (2.3) have
Hamiltonian
form.
Each
leaf S
is
covered by a
4-parameter
family
of
invariant subspaces
~E,v'
The
generic
~
is
a (2A
~
4)-dimensional submanifold of
S.
However. since
the
parameters
E
and
v are nonlinear functions of
the
intrinsic coordinates of S
they
are
not
always
independent,
and
on
the
other
hand
additional
conserved
quantities
may
be present.
Therefore, one
may
find
~s
of
lower
or
higher
(than
2A
~
4)
dimensionality
and
even
~s
that
are
not
submanifolds. A simple example is
the
FR
state
(all A spins parallel
to
a given direction) which forms a O-dimensional submanifold
by
itself.
+ Non-Niiethcr
symmetries
arc
automorphisms
which preserve
the
equations
of
motion
but
not
the
2-fonn.
Both
Nc)ether and nOll-Noether sYlnnretries are associated
with
constants
of
Illation.
236
L.
Pantelidis
s
I
I
Figure
1.
The phase space structure for the discrete classical HM. The 3A-dimensional
phase space
is foliated into
2A-dimens
ion
al,
connected, compact,
,·eg
·ular, invariant
submanif
olds
S.
The 2
A
-
1
(11.
+ 2)-dimension
al
submanifolds
of
LFPs, denoted
by
F,
are connected, unbound
ed,
and intersect every leaf S orthogonally along an
52
4.
The
manifold
of
LFPs
From
the
equations of IlIotion (2.3),
we
can
immediately see
th
at
a configuration of
spins
is
stationary
iff (if and only if) the spin and
the
vector in parenthesis, which is
the mean field in
the
Hartree-Fock approximation, are linearly dependent
at
every site.
In
the
isotropic zero-field case
that
we
examine,
this
condition is
met
when a
ll
spins
are collinear. Therefore, on each
S,
there
are
2
A
-
1
disjoint spheres 52 of fixed points
of
the
flow.
Eac
h
sp
here is generated by ro
tations
of a configur
at
ion with one
sp
in in
a given direction
an
d
the
rest parallel or anti-p
ara
llel
to
that
direction. We will refer
to these poi
nts
as longitudinal fixed points (LFPs). These are the only equilibria for
Qualitative Dynamics
of
Interacting Classical Spins 237
generic exchange constants. However, additional
st
ationary
states
of non-collinear spins
can exist, depending on
the
exchange constants (e.g., planar fixed points for closed
Heisenberg chains etc.) By varying
the
lengths of
the
spins,
we
see
that
each
LFP
belongs
to
an
(A
+ 2)-dimensional submanifold F of
LFPs
within
the
3A-dimensional
phase space.
Clearly, there are 2
A
-
1
such submanifolds
and
they
are disjoint. Moreover,
every
F is connected, unbouncled, and intersects every leaf S along
an
52
Since
the
lengths of
the
spins are
constant
on a shell S
we
expect
that
F
and
S are orthogonal.
This
orthogonality can be formally
demonstrated
[36].
5.
Stability
of
LFPs
To investigate the stability of LFPs,
we
can use
the
following
lemma
which
is
implied
by
Lyapunov's
direct method:
Lemma
5.1.
Consider
the
Hamiltonian
system
(M,
0,
h).
If
x is a
strict
local
extremum
point
of
the
Hamiltonian
h,
then x is a stable
FP
of
the flow.
Incl
eed,
there
are a
few
in
te
r
est.
ing cases where
LFPs
do ext.remize
the
Hamiltonian.
The
problem
though
is
that,
due
to
the
50(3)
symmet.ry of
the
model, these
extrema
are
not
strict
on S
and
therefore
Lemma
5.1 does
not
apply. Luckily,
it.
is
possible
to
overcome t.his difficulty by finding an invariant subspace r of S in which
the
above
extrema
of
the
Hamiltonian
are
strict.
and
thus
stable
FPs,
and
th
en
extend
t.his
stability
to
S by using the
symmetry
of
the
model.
This
idea is formalized in a general theorem
[3
6].
In this
pap
er,
we
will tailor t.his
strategy
directly
to
t.he
model
at
hand:
Lemma
5.2.
If
a
LFP
x is a strict local
extremum
of
the
Ham
iltonian h
in
where n is
some
unit
'Vector,
then
it
is Lya.punov sta.ble
in
S(s}J'
Proof. Let x be
the
LFP
, r.r
the
r-manifold
through
x,
B(x,
f)
the
ball in S cent.ered
at
.T
with radius f >
0,
and
F
t
the
flow
on
S.
According
to
Lemma 5.1, since :r is a
strict
local extremum in r
x
,
x
is
st.able in r
x
'
Thus
, 3E],E2,E3 : E
/2
>
E]
>
2E2
>
4E3
> 0,
and
(5.1)
Since
the
total
spin v is a continuous function on
Sand
Vx
# 0,
315
1
E (0,
(3)
:
Vy
#
0,
Vy E
B(x,
bd. Then,
there
exists a continuous function R :
B(x,
15])
--750(3),
where
Ry
is a
rotation
on
the
plane of
the
unit
vectors
Tt
x
,
Tty
such
that
Tty
= RTtx ·
This
implies
that
the
function
B(x,
15
1
)
3 y
--)
Z =
(EBJ=I
Ry)x
E
ry
is
also continuous.
Therefore,
315
E (O,bd :
z(B(x,
b))
c
B(z(x),
(3) = B(.T, (3). Now, if y E B(:r,
b),
then
d(y, z)
:::;
d(y
,
x)
+ d(
z,
x)
<
15
+
E3
<
15
1
+
fa
<
2E3
<
E2
,
thus
y E
B(z,
(2) n r
z
·
Since
Ry
==
EB.7=1
Ry
is
a
symmetry
transformat.ion,
[Ry,
F
t
]
=
0,
and
so, acting
wit.h
Ry
on (5.1) yiclds: B(Z,EI)
nr
z
:)
F
t
(B(z,E2)
nrz)
3 Ft(y) =
y(t).
As a result,