Monte Carlo study of the two-dimensional site-diluted dipolar Ising model

Abstract

By tempered Monte Carlo simulations, we study 2D site-diluted dipolar Ising systems. Dipoles are randomly placed on a fraction x of all L^2 sites in a square lattice, and point along a common crystalline axis. For x_c< x<=1, where x_c = 0.79(5), we find an antiferromagnetic phase below a temperature which vanishes as x approaches x_c from above. At lower values of x, we study (i) distributions of the spin--glass (SG) overlap q, (ii) their relative mean square deviation Delta_q^2 and kurtosis and (iii) xi_L/L, where xi_L is a SG correlation length. From their variation with temperature and system size, we find that the paramagnetic phase covers the entire T>0 range. Our results enable us to obtain an estimate of the critical exponent associated to the correlation length at T=0, 1/nu=0.35(10). Comment: 10 LaTeX pages, 10 figures, 1 table.

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arXiv:1004.0303v1 [cond-mat.dis-nn] 2 Apr 2010
Monte Carlo study of the two–dimensional site–diluted dipolar Ising model
Juan J. Alonso1, ∗and B. All´ es2, †
1F´ ısica Aplicada I, Universidad de M´ alaga, 29071 M´ alaga, Spain
2INFN–Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
(Dated: April 5, 2010)
By tempered Monte Carlo simulations, we study 2D site–diluted dipolar Ising systems. Dipoles
are randomly placed on a fraction x of all L2sites in a square lattice, and point along a common
crystalline axis. For xc < x ≤ 1, where xc = 0.79(5), we find an antiferromagnetic phase below a
temperature which vanishes as x → xc from above. At lower values of x, we study (i) distributions
of the spin–glass (SG) overlap q, (ii) their relative mean square deviation ∆2
ξL/L, where ξL is a SG correlation length. From their variation with temperature and system size,
we find that the paramagnetic phase covers the entire T > 0 range. Our results enable us to obtain
an estimate of the critical exponent associated to the correlation length at T = 0, 1/ν = 0.35(10).
qand kurtosis and (iii)
PACS numbers: 75.10.Nr, 75.10.Hk, 75.40.Cx, 75.50.Lk
Keywords:
I. INTRODUCTION
In the last years, there has been a renewed interest in
systems of interacting dipoles (SIDs). This is in part due
to recent advances in nanoscience1which make realiza-
tions of assemblies of magnetic nanoparticles available.2,3
Empirically, these systems show a rich collective behav-
ior in which the dipole–dipole interaction plays a key role
that can be observed at low (but experimentally acces-
sible) temperatures. Dipoles forming crystalline arrays
exhibit long–range ferro or antiferromagnetic order that
depends crucially on lattice geometry4,5because of geo-
metric frustration caused by the spatial variations of the
directions of dipolar fields. Two–dimensional (2D) arrays
of cobalt–ferrite and Co nanoparticles placed on hexag-
onal arrays have been found to exhibit in–plane short–
range ferromagnetic order.6On the contrary, arrays on
a square lattice composed of MnAs ferromagnetic nan-
odisks epitaxially grown on a substrate exhibit collinear
AF patterns.7
Magnetic ordering of SIDs depends also on anisotropy.
On the one hand, dipolar–dipolar interactions create ef-
fective anisotropies that in square lattices, for example,
push spins to lie on the plane of the lattice.8On the
other hand, magnetocrystalline site–anisotropy energies
of the crystallites that form the nanoparticles are often
greater than dipolar–dipolar interparticle energies. This
is the case of the arrays of MnAs ferromagnetic nan-
odisks we mention above, that behave as a system of Ising
dipoles with their magnetic moment rigidly aligned along
the in–plane crystalline easy axes of the nanodisks.7In
such a case, the resulting magnetic order depends on the
competition of dipolar and anisotropic energies. Crys-
talline Ising dipolar systems (IDSs) are reasonable mod-
els for these planar systems.9Some ferroelectrics,10insu-
lating magnetic salts as LiHoF4, as well as some three–
dimensional (3D) crystals of organometallic molecules11
are known to be well described by arrays of IDSs.9,12
SIDs in disordered spatial arrangements are particu-
larly interesting. The presence of spatial disorder, to-
gether with the geometric frustration generated by dipo-
lar interactions, gives rise to random frustration that may
result in SG behavior. In fact, some non–equilibrium SG
behavior (like time dependent susceptibilities and mem-
ory effects) has been observed in experiments with sys-
tems of randomly placed nanoparticles or very diluted
magnetic crystals.10,13Furthermore, Monte Carlo (MC)
simulations have given clear evidence of the existence of
a transition at finite temperature TSGfrom a paramag-
netic to an equilibrium SG phase in systems of randomly
oriented axis dipoles (RADs) placed either on fully oc-
cupied or on diluted simple cubic (SC) 3D lattices, and
TSG= 0 instead for 2D square lattices.14Recent numer-
ical work has reported a SG transition in a model of par-
allel axis dipoles (PADs) placed on a lattice that approx-
imates that of the diluted15LiHoxY1−xF4, a material
for which such a transition has been reported,16(albeit
not without some controversy17). By MC simulation the
whole phase diagram of site–diluted PADs placed on a
3D SC lattice has been obtained18as a function of the
concentration x. It includes a SG phase for 0 < x ? 0.65
which, strikingly, has been found to behave marginally,
that is, it has quasi–long range order, as in the 2D XY
model.19This is contrary to theoretical expectations,20,21
that SG systems with long–range interactions may be-
have as short–range Edwards–Anderson (EA) models,22
which in 3D are believed to have a SG phase with a non–
vanishing order parameter (according to the RSB23or
droplet24pictures of SGs).
Then, in order to get a deeper understanding of SG
systems beyond the already extensively studied random–
bond models with short–range interactions, it makes
sense to analyze the behavior of the 2D PAD model and
compare it with the short–range 2D EA model. The lat-
ter had been found to have an algebraic divergence at
TSG = 0 with critical exponent251/ν = 0.50(5), al-
though more recent simulations for larger systems and
lower temperatures give a value of 1/ν = 0.29(4) for
Gaussian interactions.26
Our purpose is the study by MC simulations the phase

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diagram of a site–diluted system of magnetic dipoles.
They are placed at random on the sites of a square lat-
tice and point up or down along a given principal axis.
Since in the limit of low concentrations every detail of
the lattice is expected to become irrelevant,18our results
have direct connection with some of the work we describe
above. Our intention is to search for the temperature TSG
of a possible SG transition and study the related diver-
gence of the correlation length. Further, we aim to study
whether the diluted PAD model belongs to the same uni-
versality class recently conjectured,27though not reliably
shown by MC simulations,28for the set of 2D EA Ising
models with varying quenched disorder.
The plan of the paper is as follows. In Section II we de-
fine the model and give details on the parallel tempered
Monte Carlo (TMC) algorithm29used for updating. We
also define the quantities we calculate. They include the
spin overlap30q and a correlation length22,31ξL. In Sec-
tion IIIA we give results for the dipolar AF phase for
x > xc, where xc = 0.79(5), as well as for its nature
and boundary. In Section IIIB, numerical results are
shown for distributions of q and ξL/L at x = 0.2 and 0.5.
We examine the evidence against the existence of a finite
temperature SG phase transition when x < xc: (i) the
mean values ?| q |? and ?q2? decrease faster than alge-
braically with L as L increases for T/x ? 0.3, (ii) double
peaked, but wide, distributions of q/?| q |? change with L
for temperatures as low as T/x = 0.4, and (iii) kurtosis
and ξL/L decrease with L at all T and do not cross, as
it would be expected for a finite temperature transition.
Scaling plots for g and ξL/L are given in Section IIIC.
Our results are consistent with a ratio ξL/L that diverges
with exponent 1/ν = 0.35(10). Results are summarized
in Section IV.
II. MODEL, METHOD, AND MEASURED
QUANTITIES
A. Model
We treat site–diluted systems of Ising magnetic dipoles
(also named spins in this paper) on a 2D square lattice.
At each lattice site a dipole is placed with probability x.
Then, the number N of spins on the lattice is less than
L2(L is the lateral size of the lattice) approximately
by a factor x. Site i is said occupied if it contains one
spin. All dipoles are parallel and point along the Y axis
of the lattice. This axis shall be called spin axis. The
Hamiltonian is given by,
H =1
2
?
i?=j
Tijσiσj, (1)
where the sum runs over all occupied sites i and j except
i = j, σi= ±1 on any occupied site i and
Tij= εa(a/rij)3(1 − 3y2
ij/r2
ij). (2)
If rijis the vector joining sites i and j, then rij= ?rij? is
its modulus and yijits Y component. εais an energy and
a is the lattice spacing. In the following all temperatures
and energies shall be given respectively in units of εa/kB
(kBis the Boltzmann constant) and εa.
Due to the long–range nature of the dipolar interac-
tions, we are able to simulate on rather small lattice sizes
(L ≤ 32).
Strength Tij is the usual long–range dipole–dipole in-
teraction. Note that Tijsigns are not distributed at ran-
dom, but depend on the orientation of rijvectors on the
lattice. Randomness in our model arises only through the
introduction of the probability x for placing dipoles. This
is to be contrasted with random–bond EA Ising models
with bond strengths Jij = εij/rµ
at random from a bimodal or Gaussian distribution with
zero mean21and µ is a real exponent. This is why PADs
exhibit AF order at high concentration in contrast to
these models that do not. Similar statements apply when
our PAD model is compared with a random–axes dipolar
model (RAD), in which Ising dipoles lie along directions
chosen at random for each site.14
ijwhere εij are chosen
B.Method
Periodic boundary conditions (PBC) are imposed.
Spins on occupied sites i have been allowed to interact
only with spins j within an L × L squared box centered
on site i. This method unambiguously defines the vector
rij to be used in (2) and also excludes interactions with
spins belonging to the repeated copies of the lattice that
appear beyond the boundary. Because of the long–range
nature of dipolar interactions, contributions from beyond
this box would have been taken into account (for exam-
ple by means of Ewald’s summations32) if spins were to
form ferromagnetic domains. They do not do so in our
PAD model. In all simulations presented in this work we
have found TχF? 1, where χF is the ferromagnetic sus-
ceptibility. Therefore those contributions do not affect
the thermodynamic limit regardless of the kind of phase
the system lies (paramagnetic, AF or SG). Some details
on this point are found in.18
In order to circumvent large energy barriers that could
slow down the evolution of the system, in particular
from certain states representing minima of the energy
(mainly at low temperatures), we have used the TMC
algorithm.29It consists in running in parallel a set of
n identical systems at equally spaced temperatures Ti,
given by Ti= T1−(i−1)∆T (i = 1,··· ,n and ∆T > 0)
where each system i is cyclically allowed to exchange its
state with system i + 1. Each system evolves indepen-
dently by use of the standard single–spin–flip Metropolis
algorithm33and whenever a single flip is accepted, all
dipolar fields throughout the entire lattice are updated.
In detail the procedure is as follows:14,18(1) a cycle on
i is run from i = 1 to i = n; (2) when the cycle arrives at
system i, 8 Metropolis steps are applied on it; (3) next,

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TABLE I: Simulation parameters. x is the probability for
sites to be occupied with a magnetic dipole; L is the lateral
lattice size; ∆T is the temperature step in the TMC runs; T1
and Tn are the highest and lowest temperatures, respectively;
Nr is the number of pairs of quenched disordered samples; t0
is the number of MC sweeps. The measuring time interval is
[t0,2t0] in all cases.
x = 0.2, ∆T = 0.02, T1= 0.6, t0= 4 × 107
8 12 16
L
20 2432
Tn
Nr
0.040.040.040.040.04 0.08
2400 5501500650 700 200
x = 0.5, ∆T = 0.05, T1= 2, Tn = 0.1, t0= 8 × 106
8 16 20
L
24
Nr
25002500 350250
x = 0.6, Tn= 0.2, t0= 4 × 106
16 20
L
8 24
T1
∆T
3322
0.20.2 0.10.1
Nr
1200 300300 300
x = 0.7, ∆T = 0.1, T1= 2, Tn= 0.2
1620
8 × 1058 × 1054 × 1064 × 106
L
8 24
t0
Nr
4200 2200 400100
x = 0.8, ∆T = 0.1, T1= 3, Tn= 0.2
1620
8 × 1058 × 1054 × 1064 × 106
L
824
t0
Nr
45001200500 350
x = 0.86, ∆T = 0.1, T1= 3, Tn = 0.2, t0= 8 × 105
81620
L
24
Nr
3000 40030070
x = 0.9, ∆T = 0.1, T1= 3, Tn= 0.2, t0= 8 × 105
8 1620
L
24
Nr
2000 250 250800
a chance is given to systems i and i+1 to exchange their
configurations (note that at this moment system i + 1
has undergone 8 Metropolis steps less than system i).
The exchange is accepted with probability PTMC= 1 if
δE = Ei−Ei+1< 0 or PTMC= exp(−∆βδE) otherwise.
Here Ei is the numerical value of Hamiltonian (1) for
system i and ∆β = 1/Ti+1−1/Ti; (4) 8 Metropolis steps
are applied on system i + 1 (regardless of the fact that
the previous exchange have or have not been performed);
(5) the above exchange is tried between systems i+1 and
i+2; (6) the cycle ends after the 8 Monte Carlo steps for
i = n, after which no configuration exchange is tried.
Since18in 3D TSG∼ x and the purpose of TMC is to
overcome energy barriers that could be as high as TSG,
then we found necessary to choose the highest temper-
ature T1 ? 2x. It is also important to take ∆T small
enough to allow frequent state exchanges between sys-
tems. This is fulfilled by taking ∆T ? T/√csN where
cs is the specific heat per spin. We choose appropriate
values for ∆T from inspection of plots (not shown) of
the specific heat vs T in preliminary simulations of the
smaller systems.18
Initially the n configurations were completely disor-
dered. For details on how we chose equilibration times t0
see Section IIC. Time t0is particularly large outside the
AF region, varying from at least 4 × 106MC sweeps for
x = 0.7 and a number of dipoles N ≥ 300 up to 4 × 107
sweeps for x = 0.2 and N = 200. Instead, t0in the AF
zone is as low as 8 × 105for x ≥ 0.86. Thermal averages
were calculated over the time range [t0,2t0]. We further
averaged over Nr samples with different realizations of
disorder. Each realization was run twice to permit the
calculation of overlapping parameters (see Section IIC).
Values of the parameters for all TMC runs are given in
Table I.
C. Measured quantities
Measurements were performed after two averagings:
first over thermalized configurations and secondly over
different realizations of the quenched disorder.
We begin by presenting the specific heat. It was ex-
tracted from the slope of the energy as a function of the
temperature.
As for the staggered magnetization, also for a PAD
model on the square lattice we find appropriate to define
it as5
m = N−1?
i
(−1)xiσi, (3)
where xi is the X coordinate of site i. We calculated
the probability distribution P(m), as well as the mo-
ments m1= ?|m|?, m2= ?m2?, and m4= ?m4?, where
?...? stands for the above–defined double averages. From
these moments we calculated the kurtosis (known also as
Binder’s cumulant) of P(m) as gm= (3−m4/m2
these quantities have proven to be good signatures for
possible AF–paramagnetic phase transitions.
In order to look for SG behavior, we also calculated the
Edwards–Anderson overlap parameter between two inde-
pendent equilibrium configurations obtained from a pair
of identical replicas evolving independently in time,30
q = N−1?
2)/2. All
j
φj, (4)
where
φj= σ(1)
jσ(2)
j
, (5)
σ(1)
j
as (1) and (2). Clearly, q is a measure of the spin config-
uration overlap between the two replicas. As done for m,
we also calculated the probability distribution P(q) as
well as the moments q1= ?|q|?, q2= ?q2?, and q4= ?q4?.
The SG susceptibility χSGis given by Nq2. Finally, we
also make use of the relative mean square deviation of q,
∆2
and σ(2)
j
being the spins on site j of replicas labelled
q= q2/q2
1− 1, and kurtosis g = (3 − q4/q2
2)/2.

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4
102
103
104
105
106
107
t
0.0
0.2
0.4
0.6
0.8
q2, q2
T=0.01
T=0.03
T=0.01
T=0.03
~
q2
q2
~
FIG. 1: (Color online) Semilog plots of ? q2(t0,t) and q2vs time
t (in MC sweeps) for systems of 24×24 sites and concentration
x = 0.5 at the values of T shown in the legend. Here, q2comes
from averages over time, starting from an initial random spin
configuration at t = 0. Here t0 = 8 × 106MC sweeps. Data
points at time t from an average over the time interval [t,1.2t]
and over 500 system samples.
Let us explain now how t0 was extracted. To make
sure that equilibrium was reached, plots of q2and energy
(average of H) were made over time intervals [t,1.2t],
not starting at t = t0, as we do everywhere else, but
starting at t = 0, from an initial random spin configu-
ration. Semilog plots of q2 versus t displayed in Fig. 1
for x = 0.5, L = 24 and low temperatures show that
a stationary state is reached only after some millions of
MC sweeps. In order to check whether this state is truly
in equilibrium, we define a time dependent spin overlap
? q, not among pairs of identical systems, but among spin
t0and t1= t0+ t of the same TMC run,
? q(t0,t) = N−1?
configurations of the same system at two different times
j
σj(t0)σj(t0+ t). (6)
Let ? q2(t0,t) = ?(? q(t0,t))2?. Suppose thermal equilib-
t → t′provided that t0? t′. Plots of ? q2(t0,t) vs t, for
in Fig. 1. Note that both quantities, q2 and ? q2 become
der to obtain equilibrium results, we have always chosen
sufficiently large values of t0 to make sure that, within
errors, ? q2(t0,t) = q2for t ? t0. All values of t0are given
In addition, we calculated a so–called correlation
length for finite systems,
rium is reached at a time t′. Then, ? q2(t0,t) → q2 as
10−5t0< t < t0and t0= 8 × 106MC sweeps, are shown
approximately equal when t ? 106MC sweeps. In or-
in Table I.
ξx,L=
1
2sin(k/2)
?
q2
?| q(k) |2?− 1
?1/2
, (7)
0 0.2 0.40.6 0.81
x
0.0
0.5
1.0
1.5
2.0
T
0.20.4 0.6
x
0.81
0.1
1.0
m1
AF
paramagnet
T=0.2
FIG. 2: (Color online) Phase diagram of the 2D PAD model.
◦ stand for the N´ eel temperature TN and × for temperatures
below which we cannot completely rule out a SG phase (see
Section III). The full line for the phase boundary between the
paramagnetic and AF phases is a fit to the data points given
by TAF ≃ 4.5(x − xc)1/2, where xc = 0.79. In the inset, m1
versus x for T = 0.2. ◦, ?, ⋄, and △, stand for L = 24,20,16,
and 8 respectively.
where
k = ?k?,q(k) = N−1?
j
φjeik·rj, (8)
rj is the position of site j, and k = (2π/L,0). Recall
that this system is anisotropic, as interactions along the
spin axis are twice as large as in the perpendicular direc-
tion. Then, one could define a correlation length along
the Y axis, ξy,L, by choosing k = (0,2π/L). We have
found that ξx,Lis more convenient because it is less af-
fected by finite–size effects than ξy,L. In order to com-
pare with similar quantities defined for isotropic systems
like the short–range 2D EA Ising model, we define also
ξL= (ξx,L+ξy,L)/2. In contrast to P(q) and its first mo-
ments, ξLtakes into account spatial variations of the EA
overlap q and shows a good signature of SG transition.
Its use has become customary in recent SG work.22,31
Analogous expressions define the AF correlation length
ξ(m)
L
by substituting φjfor ψj= (−1)xiσiin Eqs.(7–8).
It is worth mentioning that in the ξL/L → 0 limit, ξL
is, up to a multiplicative constant, the spatial correlation
length of ?φ0φr?. Therefore in the paramagnetic phase
we can think of ξ∞, the L → ∞ limit of ξL, as the true
correlation length of a macroscopic system. On the con-
trary, if there is strong long–range order with short–range
order fluctuations (as predicted for the droplet model24
for 3D SGs), q2?= 0 (that is, ?φ0φr? does not vanish as
r → ∞) and ?φ0φr? − ?φ0??φr? would be short–range. It
then follows from its definition Eq.(7) that ξL ∼ L2in
2D. Following current usage, we shall nevertheless refer
to ξLas the “correlation length” .

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5
0
0.5
1
T
1.5
2
0.3
0.9
q2
0
0.5
1
T
0.0
0.3
0.6
0.9
q2
0.2
0.4
0.6
m2
0.4
0.8
m2
(a) x=0.7(b) x=0.86
(d) x=0.86 (c) x=0.7
FIG. 3: (Color online) (a) Squared staggered magnetization
m2 vs T for x = 0.7.Icons ◦, ?, ⋄, and △ stand for
L = 24,20,16 and 8 respectively. Lines are only guides to
the eye. Note that m2 decreases with L at all temperatures
consistently with absence of AF order. (b) Same as in (a)
but for x = 0.86. Note that m2 grows with L at low tem-
perature, indicating an AF phase. (c) Same as in (a) but
for the SG overlap parameter q2. (d) Same as in (c) but for
x = 0.86. Direct comparison of curves shown in panels (b)
and (d) for x = 0.86 indicate a coupling between m2 and q2.
This coupling does not occur for x = 0.7 (see panels (a) and
(c)).
0
0.5
1
T
1.5
0.1
1
10
ξx,L/L
1
1.5
2
T
0.5
1
1.5
T
0.5
1
gm
0
0.5
1
1.5
T
0.5
1
gm
(a) x=0.8 (b) x=0.86
(m)
FIG. 4: (Color online) (a) Semilog plots of ξ(m)
for x = 0.8, and L = 24 (◦), L = 20 (?), L = 16 (⋄), and
L = 8 (△). In the inset, kurtosis of the m distribution versus
T for the same values of x and system sizes. (b) Same as in
(a) but for x = 0.86.
x,L/L versus T
III.RESULTS
A.The AF phase
The phase diagram shown in Fig. 2 summarizes our
main results for the diluted 2D PAD model. For x > xc
we find a thermally driven second order transition be-
tween the paramagnetic and AF phases at a N´ eel tem-
perature TN(x) that vanishes as x → xc from above.
The phase boundary meets the T = 0 line at xc≃ 0.79.
For concentrations well below xcthe paramagnetic phase
covers the whole range T ? 0. We do not find evidence of
a SG phase at finite temperature. However, our results
are consistent with a SG correlation length that diverges
algebraically near or at TSG = 0. In the following we
report the numerical evidence on which these qualitative
results are based.
First we focus our attention on the paramagnetic–AF
transition.34The AF phase is defined by the staggered
magnetization (3). We illustrate in Fig. 3a how the mo-
ment of staggered magnetization m2behaves with tem-
perature for x = 0.7. Note that m2appears to decrease
as L increases even at low T. Plots of m2versus L (not
shown) indicate a faster than algebraic decay in L, as one
expects for a non AF phase. This is in sharp contrast to
the behavior of m2for x = 0.86 (see Fig. 3b). Curves for
different L cross at TN ≃ 1.15. Below this temperature
m2 increases with L indicating the existence of an or-
dered AF phase. Similar results are obtained for higher
values of x. In the inset of Fig. 2, plots of m1versus x
for different system sizes at low temperature show that
the system exhibits AF order for x ? 0.8. Similar graphs
were obtained for m2. These are our first pieces of evi-
dence for the existence of an AF phase above xc∼ 0.8.
It is instructive to compare the behavior of m2with that
of q2shown for x = 0.86 in Figs. 3b and 3d. m2and q2
are not qualitatively different. This is not so for x = 0.7
where there is no AF order (compare Figs. 3a and 3c).
From Fig. 3c it is not obvious whether q2vanishes or not
as L increases at very low temperatures. We will return
to this point in the discussion of Fig. 5.
For further information about the extent of the AF
phase, we also examine how the cumulant–like quantity
gmand the finite–size AF correlation length behave for
several pairs of values x, T. Let us first outline how gm
is expected to behave in the various magnetic phases. It
clearly follows from its definition that gm→ 1 as L →
∞ in the case of long–range AF order. From the law
of large numbers it also follows that gm → 0 as L →
∞ in the paramagnetic phase.
imply that curves of gmvs T for various values of L cross
at the phase boundary between the paramagnetic and
AF phases. We make use of this fact to quantitatively
determine the paramagnetic–AF phase boundary. Plots
of gmvs T are shown in the insets of Figs. 4a and 4b for
x = 0.8 and 0.86, respectively. The signature of an AF
phase below T ≃ 1.2 is clear for x = 0.86. The inset of
Fig. 4a shows that within errors curves of gmvs T for x =
These two statements