2325
Energy
barriers
in
SK
spinglass
model
D.
Vertechi
and
M.
A.
Virasoro
Dipartimento
di
Fisica
dell’Università
di
Roma
«
La
Sapienza
»,
Piazzale
Aldo
Moro
2,
00185,
Roma,
Italy
INFN,
Sezione
di
Roma,
Italy
(Reçu
le
6
avril
1989,
accepté
le
17 mai
1989)
Résumé.
2014
Nous
étudions
les
hauteurs
de
barrière
séparant
des
états
métastables
pour
le
modèle
de
verre
de
spin
avec
symétrie
d’Ising
et
portée
infinie.
Une
configuration
de
barrière
correspond
à
un
col
de
{mi, i = 1, ..., N,  1 ~
mi ~ 1}
de
la
surface
d’énergie
qui
interpole
de
façon
régulière
l’énergie
dans
l’hypercube.
Pour
des
barrières
d’énergie
faibles,
qui
sont
importantes
pour
la
dynamique,
le
nombre
de
directions
descendantes
au
col
est
fini
dans
la
limite
N ~
~.
Nous
trouvons
que
ces
directions
sont
contenues
dans
un
sousespace
linéaire
de
l’hypercube
[20141,1]N
engendré
par
les
directions
pour
lesquelles
03A3j Jij mj
=
0.
Nous
avons
fait
des
simulations
numeriques
par
deux
algorithmes
différents.
Les
résultats
sont
cohérents.
Si
nous
supposons
que
les
hauteurs
de
barrière
croissent
avec
la
taille
N
du
système
comme
N03B1,
nous
trouvons
03B1
=
0,34 ±
0,08.
Abstract.
2014
The
height
of
barriers
separating
metastable
states
is
studied
for
the
infinite
range
Ising
spinglass
model.
A
barrier
configuration
corresponds
to
a
saddle
point
{mi, i
=
1,
..., N,
2014 1 ~
mi ~ 1}
of
an
energy
surface
that
smoothly
interpolates
the
energy
in
the
hypercube.
Forlow
energy
barriers,
which
are
relevant
for
the
dynamics,
the
number
of
independent
descending
directions
from
the
saddle
point
is
finite
when
N ~
~.
It
is
found
that
these
descending
directions
are
contained
in
the
linear
subspace
of the
hypercube
[20141,1]N
generated
by
directions
for
which
03A3j Jij mj = 0.
Numerical
estimates
were
performed
by
two
distinct
algorithms
which
lead
to
consistent
results.
Assuming
barrier
heights
growing
with
system
size
N
as
N03B1
we
find
03B1
=
0.34 ±
0.08.
J.
Phys.
France
50
(1989)
23252332
1cr
SEPTEMBRE
1989,
Classification
Physics
Abstracts
05.20

75.50L
1.
Introduction.
One
of
the
characteristic
features
of
spin
glasses
is
the
existence
of
many
states
of
minimum
free
energy
almost
degenerate,
separated
by
very
high
free
energy
barriers
and
unrelated
by
a
symmetry
one
to
another.
An
appropriate
infiniteranged
Ising
spinglass
model
was
proposed by
Sherrington
and
Kirkpatrick
(SK
model
[1]).
It
is
believed
that
the
SK
model
can
be
solved
by
means
of
replica
method
for
which
Parisi’s
replica
symmetry
breaking
ansatz
[2]
produces
a
stable
mean
field
solution.
Many
of
the
thermal
equilibrium
properties
of
the
model
have
been
studied
using
this
approach.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500170232500
2326
Replica
symmetry
breaking
is
physically
related
to
the
breaking
of
ergodicity
[3] ;
furthermore
one
of
the
consequences
of
Parisi’s
ansatz
is
a
particular
hierarchical
organization
of
the
states
in
phase
space
called
ultrametric
[4].
The
dynamics
of
the
SK
model
is
greatly
complicated
by
the
breaking
of
ergodicity
[5] ;
in
the
infinite
volume
limit
the
time
evolution
is
confined
to
one
of
the
available
pure
states
on
any
finite
timescale.
Many
efforts
have
been
devoted
to
the
comprehension
of
the
dynamics
on
infinite
timescales,
so
that
transitions
between
various
states
are
allowed
[6].
In
practice
the
precise
understanding
of
this
regime
requires
a
detailed
knowledge
of
properties
of
spin
glasses
at
finite
(though
large)
size,
such
as
the
height
of
the
free
energy
barriers
between
the
states,
which
is
not
available
at
the
moment.
The
complicated
structure
of the
free
energy
surface
seems
to
be
responsible
for
anomalously
slow
dynamics
[7,
8]
and
could
be
relevant
to
understand
the
dependence
of
relaxations
on
particular
waitingtimes
[9].
Spinglass
models
have
many
aspects
in
common
with combinatorial
optimization
problems
[10] ;
the
investigation
of
free
energy
barriers
in
SK
model
may
favour
a
detailed
comprehension
of
the
structure
of
some
of
these
problems,
possibly
explaining
the
efficiency
of
«
simulated
annealing » algorithms.
Energy
barriers
introduce
in
a
natural
way
a
hierarchical
organization
of
the
energy
minima
[7]
which
could
be
related
to
the
ultrametric
organization
of
states
in
phase
space.
A
numerical
evaluation
of
energy
barriers
between
metastable
states
in
SK
model
at
zero
temperature
has
been
reported
in
a
recent
paper
by
Nemoto
[11] ;
the
analysis
is
carried
out
considering
all
the
metastable
states
in
a
given
sample
and
averaging
over
samples.
However
the
data
in
reference
[11]
do
not
show
a
significant
scaling
form
for
barrier
heights
since
only
rather small
systems
are
considered
(up
to
24
spins).
In
the
present
work
we
extend
the
analysis
of
energy
barriers
to
systems
of
greater
size
(up
to
96
spins),
considering only
low
energy
metastable
states.
We
use
two
distinct
algorithms
to
evaluate
barrier
heights
which
lead
to
consistent
results.
We
find
that
in
a
barrier
configuration
the
descending
directions
of the
energy
surface
are
contained
in
the
linear
subspace
of
the
hypercube
[1,1 ]N
generated
by
directions
with
a
small
(0(1/
IN»
magnetic
field.
From
the simulations
we
derive
that
the
average
energy
barrier
between
metastable
states
is
a
nondecreasing
function
of
the
Hamming
distance ;
the
connection
of
this
result
with
the ultrametric
property
of
states
is
examined.
Assuming
barrier
heights
increasing
with
system
size
N
as
N ",
we
estimate
a
=
0.34 ± 0.08
for
our
numerical
data.
2.
Energy
barriers
for
oi =
±
1.
We
consider
zero
temperature
solutions
of
TAP
equations
[12]
to
identify
metastable
states.
Setting
the
external
field
equal
to
zero
TAP
equations
and
energy
E
are
written
as :
where
oj
= ±
1
and
{Jij}
are
independent
random
Gaussian
variables
with
zero
mean
and
variance
1 /N,
N
being
the
number
of
spins.
We
limit
our
investigation
to
the
low
energy
solutions
in
each
system,
choosing
respectively
the
5, 8,
20
lowest
lying
metastable
states
for
systems
with
N
=
48,
64,
96.
The
algorithm
used
to
find
these
solutions
consists
in
performing
deterministic
descents
starting
from
many
randomly
generated
configurations.
2327
Given
these
states
we
try
to
determine
the
energy
barrier
Eab
separating
the
states
a, b,
defined
by
where
a
path
is
a
sequence
of
configurations
connected
by
onespinflip
passages.
The
minimization
is
taken
over
all
possible
paths
connecting
the
states.
Our
main
approach
to
this
complicated
combinatorial
optimization
problem
consists
in
using
a
deterministic
descent
algorithm
to
carry
out
the
minimization.
As
in
Nemoto
[11]
one
introduces
the
onespinflip
operator
Pi,
defined
by
Pi ai = 
ai.
A
path
connecting
the
states
a, b
can
then
be
described
as
a
product
of
Pi
satisfying
thé
relation :
where n
is
the
length
of
the
path.
Starting
from
a
randomly
generated
minimum
length path,
our
iterative
improvement
algorithm
is
based
upon
two
elementary
moves ;
the
first
one
consists in
commuting
the
operators
which
lead
to
the
highest configuration
in
the
path
if
this
operation
reduces
the
barrier.
The
second
one
consists
in
adding
to
the
path
two
new
identical
operators
in
the
region
of
maximum
height,
choosing
the
pair
which
gives
the
maximum
reduction
of
the
barrier.
We
have
considered
more
complicated
« deformations »
of
the
path
but,
as
a
preliminary
analysis
of the
results
did
not
show
significant
improvements
while
the
increase
in
computertime
was
considerable,
we
decided
to
drop
them.
It
is
obvious
that
barriers
defined
by
(2.3)
satisfy
the
inequality
so
that
barriers
are
described
by
a
hierarchical
tree
indexed
by
{Eab} .
So,
after
the
evaluationof
Eab
for
each
pair
of metastable
states
considered,
we
constructed
the
minimal
spanning
tree
[13]
to
obtain
{Eab} .
3.
Barriers
in
a
smoothed
energy
surface.
A
different
approach
to
the
problem
of
energy
barriers
consists in
looking
for
the saddle
points
of
a
smoothed
energy
surface
given
by :
which
corresponds
to
TAP
free
energy
[12]
without
the reaction
term.
We
point
out
that
the
mi’s
are
not
the
actual
finite
temperature
magnetization,
neither the
F
is
the
correct
free
energy
but
just
an
auxiliary
function.
A
stationary
point
for
function
F
satisfies
the
equations :
with
In the
limit
13 +
oo,
with hi
finite
equation
(3.2)
becomes :
2328
as
in
(2.1).
So
the
minima
of
the
F
coincide
with
zero
temperature
solutions
of
TAP
equations
when
/3 >
oo
while
for
small
/3
the
surface
becomes
smooth
enough
to
allow
a
numerical
search
for
saddle
point
configurations.
At
a
fixed
value
for
/3
(typically
f3
= 14,8,,)
we
numerically
minimized
the function
whose
absolute
minima
with
G
=
0
corresponds
to
stationary
points
of the functionF.
Given
a
stationary
point
we
established
through
the
diagonalization
of
the
Hessian
whether
we
had
a
minimum
or
a
saddle
point,
obtaining
both,
up
to
saddle
points
with
three
negative eigenvalues.
4.
Characterization of barrier
configurations.
To
recover
the
connection with
the
srcinal
spin
system
we
looked
at f3 +
00
limit
of
saddle
point configurations
of function
F
(Eq.
(3.1)).
As
a
first
step
we
notice
that
while
equation
(3.2)
is
related
to
stationary
points,
equation
(3.4)
only
describes
the
minima
of the
F,
suggesting
that
to
have
saddle
points
one
needs hi
+
0
in
some
site
when
f3 +
oo
(numerical
simulations
were
found
in
agreement
with
this
scheme).
In
fact
the
stationarity
condition
is
satisfied,
in
f3 +
oo
limit,
if
The
generic
element
of the
Hessian
matrix of function
F
is :
where
8ij
is
the
Kronecker
delta.
The
structure
of
the
Hessian
is
rather
simple
when 6 >
oo ;
in fact
variables
satisfying
equation
(3.4)
lead
to
diagonal
terms
positive
and
divergent
while
variables
satisfying
equation
(4.1)
lead
to
vanishing
diagonal
terms.
Since
the
contribution of
nondiagonal
terms
to
the
eigenvalues
is
finite,
the
search
for
negative
eigenvalues
can
be
restricted
to
the
linear
subspace
generated
by
variables
satisfying
(4.1).
The
submatrix
of
the
Hessian
related
to
these
variables
has
zero
trace
so
that
there
must
be
eigenvalues
of
both
positive
and
negative
sign
(having
all
the
eigenvalues
equal
to
zero
is
an
event
of
zero
probability).
So the
union
of
the
eigenspaces
related
to
the
negative
eigenvalues
o f
the
Hessian
is
a
linear
subspace
strictly
contained
in
the
space
of
variables with
hi
=
0.
It
is
an
interesting
remark
[14]
that
similar
results
can
be
obtained
directly
in
the
discrete
system ;
in
fact,
using
definition
(2.3),
a
barrier
configuration
must
be
higher
than
the
configuration
which
preceeds
or
follows
it
in
the
path
and
differ
from
them
just
in
one
site ;
calling
these
sites
1
and
2
this
property
many
be
written
as :
with
2329
The
stability
of
the
path
with
respect
to
commutation
of inversion
operators
gives :
Equations
(4.3)
and
(4.5)
lead
to :
and
N
is
even
one
has :
For
Gaussian
distributed
couplings
equation
(4.6)
gives :
while,
in
general,
one
has hi
=
0 (1 )
for
low
energy
configurations
[15].
5.
Exhaustive
search
for
stationary
points.
It
is
worth
noting
that
the considerations
of
the
previous
section
allow,
in
principle,
a
complete
enumeration
of
the
stationary
points
of function
F
in 8 >
00
limit
for
any
finite
system.
In
fact
assuming
condition
(4.1)
satisfied
in
P
sites
(sites
1,..., P
for
simplicity)
and
equation
(3.4)
valid
in
the other
sites
one
has :
with
so
that
variables
satisfying
(4.1)
must
solve
the
linear
system
(5.1).
Then,
given
a
spin
configuration
in
N  P
sites,
one
must
solve
(5.1) ;
if
the
solution
satisfies
1 mi
1
1
for
i
= 1,
...,
P
and
each
of the
ri
in
the other
sites
has
the
same
sign
of
the
corresponding
spin
one
has
found
a
stationary
point.
Performing
these
steps
for
all
possible spin
configuration
of
a
given
choice of
sites
and
for
all
possible
choices
of
sites
gives
all
the
stationary
points
at
a
fixed
P.
Repeating
this
procedure
for
P = 0,..., N
one
completely
determines
all
the
stationary
points
of the function
for
a
given
set
of
Jij’s.
Following
the
same
line
of
reasoning
as
before
one
may
obtain
an
analytic
computation
of
the
number
of
stationary
points
with
a
given
a
=
P /N
and
f
=
F/N
when
N +
ao
(1) ; one
must
consider :
(1)
We
gratefully
acknowledge
B.
Derrida
for
crucial
observations
about
this
point.