![FIG. 3. An example of spatial repartition of vortices for D/J =0.6 [same spin configuration as in Fig. 1{h)]. The color indicates the charge of the vortices.](/figures/fig-3-an-example-of-spatial-repartition-of-vortices-for-d-j-3pll5nq1.webp)
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XY model with weak random anisotropy in a
symmetry-breaking magnetic eld
Bernard Dieny, Bernard Barbara
To cite this version:
Bernard Dieny, Bernard Barbara. XY model with weak random anisotropy in a symmetry-breaking
magnetic eld. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American
Physical Society, 1990, 41 (16), pp.11549 - 11556. �10.1103/PhysRevB.41.11549�. �hal-01660291�
PHYSICAL
REVIEW
8
VOLUME
41,
NUMBER 16
1
JUNE
1990
XF
model
with
weak
random
anisotropy
in
a
symmetry-breaking
magnetic Seld
Bernard
Dieny
and Bernard
Barbara
Laboratoire
Louis
¹el,
Centre National
de la Recherche
Scientigque,
25 Avenue
des
Martyrs,
Boite
Postale
Xo.
166X,
38042
Grenoble
CEDEX, France
(Received 18
September
1989)
We
present a numerical
study
of
the two-dimensional
classical XY
model with
weak random
an-
isotropy
at
zero
temperature.
Zero-field
configurations obtained
by
ultrafast
cooling,
first-
magnetization
curves, and
hysteresis
loops
have
been
calculated for
different
random-anisotropy-
to-exchange
ratios. In
zero
field,
a
pinning
of vortices
by
the
random-anisotropy
field occurs.
It
prevents
the
binding
then
collapsing
of
pairs
of
opposite
charges
and
thus
leads to
a nonferromag-
netic
ground
state.
Applying
a
magnetic
field causes
a
progressive
depinning
of
vortices that
disap-
pear
in
pairs
until
saturation.
Starting
from
saturation and
decreasing the
applied
field
leads,
in
zero-field,
to
a
magnetic state of
large
remanent
magnetization.
However,
an
important aftereffect
is observed.
It should
give
smaller
remanence
after much
longer
computer
time.
Then,
the
reversal
of
magnetization
in
negative
fields
occurs
through
a
peculiar
process
that involves
the formation
and
collapse
of new kinds
of
topological
defects
(infinite
strings).
These
linear
defects are in fact
the
ultimate
stage
in the
shrinking
of
domains
oriented in the
initial direction
of
saturation. Their
col-
lapse
occurs
abruptly
through
the creation
and
propagation
in
opposite
directions,
along
the
defect,
of an
unbound vortex-antivortex
pair.
INTRODUCTION
Both the
classical
isotropic
XY
model
and the
magnetic
properties
of
random-anisotropy
systems
have
attracted
great
interest
during
the
last decade.
Very
recently
it
was
established
under
very general
conditions
that
conven-
tional
long-range order is
destroyed
in two-dimensional
(2D)
systems with continuous
symmetry
at finite
temper-
ature.
This
applies,
for
example,
to 2D
crystals,
'
mag-
nets,
superfluids,
and
superconductors. Later
Koster-
litz
and Thouless
(KT)
dealt with
these
systems
within
a
unified
theoretical
model.
They
contend that
the 2D
low-temperature
phase
is
characterized
by
a power-law
decay
in
the
pair-correlation
function
with a
temperature-dependent
exponent.
This
phase
is
charac-
terized
by
the
presence
of
pairs
of
tightly
bound defects
of
opposite
signs,
which
are
dislocations in
crystals,
spin
vortices in
magnets,
and
quantum
vortices
in
superfluids.
At
a
characteristic
temperature
Tzl,
the
system
under-
goes
a
phase
transition
in which the
pairs
of
defects
un-
bind
to create
a
new
phase
where
the correlations
de-
crease
exponentially.
The KT
prediction
for
the 2D
superfluid,
superconductor,
and XY
models
has been
confirmed
by
various
experiments
and
computer
sirnula-
tions.
Moreover, several
extensions of the
pure
classical
XY
model have
been
studied.
"
In
particular,
effects of
weak random
Dzyaloshinskii-Moriya interactions
or of
random
p-fold
symmetry-breaking
fields,
"
have been
investigated. In
these models
a
vortex
unbinding
mecha-
nism
has
been
found within the
low-temperature
phase
as
a result
of a
pinning
or
a
screening
of
their interactions
by
the random
field.
Magnetic
properties
of
random-field
systems
(random-
anisotropy
fields
of continuous
symmetry,
p-fold
symmetry-breaking
fields,
random-exchange
fields,
etc.
)
have
been studied
theoretically
and
experimentally
in
3D
and sometimes 2D
systems.
Imry
and
Ma'
and later on
Pelcovits,
Pytte,
and
Rudnick'
showed that
an
arbitrary
weak random field
destroys ferromagnetic
order in
a
sys-
tem with
continuous
symmetry
(like
Heisenberg
or
XY
magnets)
below four
spatial
dimensions.
Then,
by
using
a
Green-function
formalism,
Chudnovsky,
'
'
Saslow,
and
Scrota'
have
investigated magnetic
properties
of
2D
amorphous
magnets
with
local
random
anisotropy
and
applied
magnetic
field.
Expressions
for the
angle-angle
correlation function
and
for
the
magnetization
law have
been derived. The
zero-field-cooled
phase
(called
corre-
lated
spin
glass)
is characterized
by
smooth
and
stochas-
tic rotations of the
magnetization
over
the
sample.
How-
ever,
topological
defects
specific
of 2D
systems
have not
been
explicitly
introduced
in
their calculation.
In
this
paper
we
present
a
numerical
study
at
zero
tern-
perature
of the classical 2D
XYmodel with weak random
anisotropy
and in the
presence
of an
applied
magnetic
field
(dipolar
effects are not taken
into account).
After
an
ultrafast
cooling
in zero field
the low-temperature
phase
exhibits both
Imry
and
Ma
domains
and
vortices which
are
pinned
by
the random
anisotropy
field
and thus
do
not
disappear
at
T=0.
Then,
applying
a
magnetic
field
leads to
a
progressive
depinning
of vortices which bind
by
pair
and
progressively
disappear
until saturation.
At
this
pairing
and
collapsing
field-induced
mechanism is super-
posed
a
progressive
polarization
of
magnetic
moments
to-
wards
the
field
direction
(ferromagnet
with
wandering
axis'
). Decreasing
the
applied
field from saturation leads
to
a
phase
of
large
remanent
magnetization
in which the
magnetization
varies
continuously
in
a
semicircle
orient-
ed
along
the saturation field.
This
phase
does
not contain
any
vortex.
However,
the
dynamics
of
relaxation in this
decreasing
field
procedure
is rather
slow
and
a
large
41
11
549
1990
The American
Physical
Society
11
550
BERNARD
DIENY AND
BERNARD BARBARA
aftereffect is
observed
in
zero field. As a
result,
a much
longer
computer
run
should lead to
a state of
lower
remanent magnetization.
%hen
a
negative
field is
ap-
plied,
a new
kind
of
topological
defect is
observed. These
defects
look
like infinite
strings
and
are characterized
by
a
linear
core
in which
the
magnetization
is
trapped
in the
initial field direction (saturation field direction) and
by
a
magnetization parallel
to the
applied
field on
their
bor-
der. The width
of the
string
decreases down to
a
few
in-
teratomic distances when
the
strength
of
the field
in-
creases
sufficiently.
Above a certain critical field the
string
abruptly
collapses through
the
creation and
propa-
gation along
the
defect
of
a
vortex-antivortex
pair.
This
paper
is divided
into four sections.
The first
one
gives
some
indications
on
the
numerical
method. The
second deals with a
comparison
of
zero-field
configurations
obtained
by
ultrafast
cooling
from
the
paramagnetic
state for different values
of the
random-
anisotropy
constant
D.
In
the
third
one,
we
present
the
first
magnetization
curves obtained
by
starting
from
the
zero-field
states discussed in
the second
part.
The last
section concerns
hysteresis
loops
with a
particular
em-
phasis
on the infinite-string
defects observed in
negative
fields.
jjj%$
,
IIL-
I
I
LL
L
—
(~)
NUMERICAL
METHOD
All the
calculations are carried out at zero
temperature
on
a
square
lattice of 100X100
spins
with
cyclic
bound-
ary
conditions. The
Hamiltonian
is
&=
—
J
g
S,
S,
Dg(n,
S,
)
——
H
gS,
,
(ij)
i I
where
(i,
j
)
indicates
a
summation
on
nearest
neighbors,
n;
is a
unit vector
of the local atomic
anisotropy
axis,
Jis
the
exchange
coupling
constant,
D is the uniaxial
anisot-
ropy
constant, and H is the
applied
magnetic
field. A cy-
cle
in
the calculation
consists
of
exploring
all of the
10000
spins
in
a random order and
setting
each
explored
spin
in its local
minimum of
energy
between
its
anisotro-
py
axis and
local-field
direction.
For each value
of the
parameters
D
and H
the
calculation was run
until the
greatest
variation
of
angle
of
a
spin
between two
consecu-
tive
cycles
was smaller
that
5X10
rad. As
pointed
out
by
Tobochnik and
Chester,
'
starting
from
a random
configuration and
running
the calculation
at
very
low
temperatures
generates
metastable
states
of
higher
energy
than those
obtained
using
a slow-cooling
procedure.
Thus,
for
instance,
the
configurations we
obtain
in zero
field after
ultrafast
cooling
exhibit more
vortices than
they
should if
a slow
zero-field
cooling
had been realized.
This
may
be
somewhat
disturbing
in
a
quantitative
point
of view
but is not
a
real
problem
in this
paper,
the main
purpose
of
which is to show
some
qualitative
features
of
the
magnetization
processes
in
these 20
systems
with
random
anisotropy.
ZERO-FIELD
CONFIGURATIONS
Figure
1
shows
two
configurations obtained
for the
same set of
random-anisotropy
axes
but
two different
values of
D/J
(D/J=0.
1 and
0.
6).
They
have
a
small
,
145
L/
%ii
FIG. 1. Two
spin
configurations
obtained for the same set of
random-anisotropy
axes but two
different
values
of
the
random-anisotropy-to-exchange
ratio
D/J.
(a)
D
/J
=0.
1.
(b)
D/J
=0.
6. The different thicknesses of
the arrows
simply
indi-
cate the
spins pointing
in
the
upper
(respectively
lower)
hemicir-
cle.
net
magnetization
and
contain
regions
with
local
fer-
romagnetic
order
(Imry
and
Ma
domains'
)
the
size of
which decreases
with
D/J.
These domains
are
not
specific
of two
dimensions;
they
have
already
been
stud-
ied theoretically and
experimentally
in 3D
random-
anisotropy
systems.
'
' '
The small resultant magneti-
zation observed here decreases with
D/J
and
can
be
as-
cribed to the finite size
of
the
sample
and thus
finite
num-
ber
of
Imry
and Ma
domains.
Indeed,
if the
sample
is
di-
vided into
n
ferromagnetic
Imry
and Ma
domains
ran-
domly
oriented in the XY
plane,
then a
random-walk
ar-
gurnent gives
a
resultant
magnetization
M of
the order of
(n)
'
which is
in
agreement with
our results.
Figure
1
also shows the existence of
stable
topological
defects,
namely
vortices
and
antivortices,
despite
that the
system
is
at
zero
K.
The
Kosterlitz-Thouless
theory
of
the
pure
XY
model,
as well as
numerical
simulations,
'
'
show
that a
pairing
mechanism between vortices
of
opposite
charges
occurs at
and below
the
KT
transition. At the
thermal
equilibrium
the
number
of
vortices is determined
by
a
counterbalance between
vortex-antivortex
annihila-
tions and
vortex-antivortex
pair
creations
by
thermal
ac-
41
XYMODEL
WITH
%%AK RANDOM
ANISOTROPY
IN A.
. .
11
551
tivation.
At
absolute
zero
temperature,
these
topological
defects
disappear
in the
pure
XY
model, leading
to a
com-
pletely
ferromagnetic
ground
state.
This
picture
has
to
be
modified
in the
presence
of
random
anisotropy,
where
we
observe that
stable vortices
persist
at
T=0
K.
As a
matter of
fact,
the
random-anisotropy field, although
preserving
the
continuous
symmetry
(at
least
at scale
larger
than
Imry
and Ma
domains),
creates
local
energy
barriers
preventing easy spin
rotations and
therefore
preventing
the motion of vortices
of
opposite
sign
to-
wards each
other.
Furthermore,
it introduces a
charac-
teristic cutoff
length
in
the
system
which
is of the order
of
the 2D
Imry
and Ma domain size
-(J/D).
As a
re-
sult,
the
number of
vortices
that
the
sample
can
contain
should
increase with D/J.
This
is
efFectively
observed:
the
number of vortices of
each
sign
is
equal
(periodic
boundary
conditions)
and increases
rapidly
with
D/J,
especially
around D
/J-0.
4.
For
D
/J
(
0.
2
and
D/J
)
0.
8,
it
tends to
become constant
(Fig.
2).
A
satu-
ration
of the number
of vortices for
large
D/J
can be
ex-
pected
in
such
a
system
because
(i)
in
the limit
of
infinite
anisotropy
the
size of
Imry
and Ma
domains tend to be
independent
of
D/J
(a
few interatomic distances) and
(ii)
the
pinning
of vortices
will
be
maximum
for
D)&J,
and
therefore not
very
sensitive
on
a
further
increase
of
D/J.
The finite and
nearly
constant
number of vortices
ob-
tained
for low
D/J
cannot be
explained
on the
basis
of
"normal"
metastable
equilibrium
due to
the
topological
disorder of random
anisotropy.
In
fact,
the
dynamics
for
the annihilation of
vortex-antivortex
pairs
becomes
ex-
tremely
slow when the number
of vortices tends to
zero.
The vortice number
decays
at a
rate
proportional
to
the
number of
pairs,
so
that
exponential
annihilation
re-
sults.
Consequently
a
very
large
computer
time would
be necessary
to
complete
total annihilation of
vortice
pairs
in the limit D
=0.
At finite
D,
i.e.
,
in the
presence
of
disorder,
it
is
possi-
ble to
identify
some
local
configurations
of
anisotropy
axes as
pinning
centers
for vortices. In
particular,
a
small area
over which
the
set of
anisotropy
axes has the
same cylindrical
symmetry
as
a vortex
constitutes an
en-
ergetically
favorable
location
for a
vortex
and then is a
0
~
0
0 G
~
0
P
0
~
0
0
~
~
0
0
0
~
0 ~
0
~
0
0
pinning
center.
As
an illustration of
that,
it
can
be
seen
in
Fig.
1
that
some vortices
are located
exactly
at the
same
place
for
different values
of D/J but the same
set
of
anisotropy
axes.
The
trapping
of vortice
configurations
by
disorder has
already
been discussed in several
theoreti-
cal
studies.
"
In
particular,
Rubinstein
et al. have
shown that
a
two-dimensional
XY
magnet
with random
Dzyaloshinsky-Moriya
interactions is
equivalent
to the
usual neutral
Coulomb
gas
of
vortex
charges
but
coupled
to a
quenched
distribution of
dipoles.
Below
a
charac-
teristic
freezing temperature,
the vortex
pairs
are
ripped
apart
by
the random
potential generated
by
the
quenched
dipole
array. Similarly,
Goldschmidt and
Schaub,
"
con-
sidering
the XY model with random p-fold
anisotropy,
sho~ed
that
an
unbinding
of vortices also
occurs at low
temperature
which renders the
glass
phase paramagnetic
at
large
scales.
As described
above,
such an
unbinding
mechanism
due
to
disorder
is
clearly
seen in
our
simula-
tions.
In other
respects,
it is worthwhile
to
notice that
the
spatial
repartition
of vortices is
not
homogenous
over the
sample
(see
Fig.
3}.
The vortices tend
to bind
by
pairs
but
also
to
agglomerate.
This
joins
the
point
underlined
by
Tobochnick and
Chester'
at finite
temperature,
accord-
ing
to
which,
the
more
vortices
existing
on
an area
of
the
sample,
the
easier it is
to add
other vortices in this same
area. In other
words,
considering
vortices as
particles,
the
local chemical
potential
decreases with the
density
of
vortices.
Let us now
go
further
by
considering
the two-spin
pair
correlations
(see
Fig.
4).
The
spin
configurations
show
that
(
S;
S
}
decreases
rapidly
with
distance,
this
de-
crease
being
more
pronounced
for
large
D/J
This is
a.
general
feature in 3D random-anisotropy
systems
because
the
size of
Imry
and Ma
domains'
decreases with
D/J
[1-(J/D}
in
three
dimensions).
In 2D systems
such
domains
can
also be
defined
(although
d
=2
is a
marginal
0 0
~
0
0 P
0
P
~
~
0
0
P
0
0
0
OP
0
~
0
P
~
0 0
0
~
0
0
~
P
0
~
0
0
~ 0
~
0
0
~
0
0
P
~ 0
~
0
0
0
0
P
0
P
0
p.
5
D/J
FIG. 2. Number of vortices n
of each
sign
in the
zero-field
configuration vs
D/J
for the same set
of
anisotropy
axis.
FIG.
3. An
example
of
spatial
repartition
of
vortices for
D/J
=0.
6
[same
spin
configuration as
in
Fig.
1{h)].
The
color
indicates the
charge
of the
vortices.
11
552
BERNARD
DIENY
AND
BERNARD
BARBARA
41
0
[n~
S,
Sj
po
'5V~
1
10 r
no
longer
valid.
Indeed,
for
large
r,
if all
angles
are taken
between
0
and
2',
then
((8,
—
8~))-m.
because of the
absence
of
long-range
order.
The
plot
of
Fig.
4(b)
is
then
equivalent
to m
/r
versus
lnr which
is
independent
of
D /J.
Now,
from the
plot
of
Fig.
4(a),
it is
possible
to
derive
a
ferromagnetic
correlation
length
RF
by
the
distance
over
which
(S,
-
S
)
decreases
from 1 to
exp(
—
1).
This
corre-
lation
length
is
plotted
versus D/J
in the inset of
Fig.
4(a).
We
note
that,
within error
bars,
RF
is
proportional
to
the cutoff
length
L,
found
in
Fig.
4(b):
RF(D/J)
=0
35L,
.
(D/J)
=14e
0.
1-
2
~
(ge)
~
1
2
0
l~ I
[n
(rj
FIG. 4. Spin-spin
correlation function
In(S,
S,
)
vs
r
for
three
values of D/J:
0.
1,
0.
3,
and
0.6. Inset:
Correlation
length
RF
vs
D/J.
(b)
Angle-angle
correlation
function
((8;
—
8,
)2lr
)
vs
1nr
for
comparison
with the
formula
given
Ref.
14
(formula
I in
the
text).
dimensionality
for
straightforward implementation
of the
qualitative
Imry
and
Ma'
argument)
and
their
size
also
decreases with D/J [1-(J/D) in two dimensions).
How-
ever,
the observed rate of the decrease
of
(S;
S
)
is
rath-
er inAuenced
by
the
presence
of vortices and also
by
the
fact that our
spin
configurations
have
a
high
degree
of
metastability.
In order to
get
some
insights
into the role
of
vortices
and
metastability,
we
have
compared
angle-
angle
correlations calculated
directly
from
angle
differences with the
expression
found,
in the absence of
vortices,
by
Chudnovsky
((8;
—
8~)
)
=
,
'(D/J)
(r/—a)
ln~L/r~,
where
r
=~r;
—
r
and
L
is
the
size of the
system.
In-
terestingly,
the
plot
of
((b,
8) )/r
versus
lnr shows
an
almost linear
behavior
for values of
r
not too close to the
interatomic
distance,
and
this
despite
the
presence
of
vor-
tices
and metastability.
In
fact,
this
agreement
with
(1)
is
only
qualitative:
the
intercept
of the line of
negative
slope
with
the abscissa is not a
constant which should be
equal
to
the size of the
sample
but takes a
much
smaller
value
and
is
a
function
of
D/J,
which
can be
put
on
the
phenomenological
form
L,
=40e
'
.
Furthermore,
the
slope,
although increasing
with
D/J,
is smaller than
—,
'(D/J),
if D/J is not too
small.
For
small
D/J,
we
ob-
serve
a constant
and
finite
slope
due
to
the
vortices per-
sisting
when
D~0
(already
discussed).
Thus,
the
pres-
ence
of vortices
and
metastability
seems
to reduce
drasti-
cally
ferromagnetic
correlations for weak random
anisot-
ropy.
For
large
values of
r,
another
regime
independent
of
D/J
is
observed
[Fig.
4(b)).
This
simply
indicates
that
the
assumption
of
small-angle rotation
in
formula
(1)
is
As
already
discussed,
RF
does
not
diverge
in the
limit
D/J~O
because
of the
slowing
down
of the
dynamics
for the low
D/J
ratio
preventing
the
complete
disappear-
ance of
vortices
in
our
calculation. It would be
worthwhile
to calculate the
correlation
length
for
D/J~O
starting
from
a
completely
disordered
state
at
different
states
of the
relaxation
process,
i.
e.
,
as the
num-
ber of
vortices decreases down to
zero. This would be
useful
in
understanding
the
quantitative
inhuence of
vor-
tices on the ferromagnetic
correlation
length.
FIRST
MAGNETIZATION CURVE
0.
5-
0.
'l
H/J
0.
2
H/J
FIG.
5. First magnetization
curves
for three
values of
D/J:
0.
1,
0.
3,
and
0.6. Inset:
Number of
vortices n
vs the
applied
field
H
/J
for the
same values
of D
/J.
Starting
from
one
of
the
zero-field
configurations
de-
scribed
above,
a
magnetic
field is
applied
in the
XY
plane.
The main characteristics of the first
magnetization
curves
(see
Fig.
5)
are the
following:
in
average,
positive
curva-
ture
in
low fields and
negative
curvature
in
large
fields,
existence of more or less
important
magnetization
jumps.
From
a qualitative
point
of
view,
two kinds of
magnetic
processes
contribute to the observed
magnetization
curve.
(1)
The
first one
is
a
progressive
depinning
of
vortices
by
the
applied
field.
The field
helps
the
magnetization
to
overcome the
energy
barriers
due to
random
anistropy
and then causes the motion of vortices of
opposite
charges
towards one another. When
two
vortices
become
suSciently
close
they
annihilate. This more
or less
abrupt
annihilating
gives
a
jump
on the
magnetization