Stability of the Griffiths phase in a 2D Potts model with correlated disorder
A Griffiths phase has recently been observed by Monte Carlo simulations in the 2D q-state Potts model with strongly correlated quenched random couplings. In particular, the magnetic susceptibility was shown to diverge algebraically with the lattice size in a broad range of temperatures. However, onl...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1535572019-06-15T01:25:34Z Stability of the Griffiths phase in a 2D Potts model with correlated disorder Chatelain, C. A Griffiths phase has recently been observed by Monte Carlo simulations in the 2D q-state Potts model with strongly correlated quenched random couplings. In particular, the magnetic susceptibility was shown to diverge algebraically with the lattice size in a broad range of temperatures. However, only relatively small lattice sizes could be considered, so one can wonder whether this Griffiths phase will not shrink and collapse into a single point, the critical point, as the lattice size is increased to much larger values. In this paper, the 2D eight-state Potts model is numerically studied for four different disorder correlations. It is shown that the Griffiths phase cannot be explained as a simple spreading of local transition temperatures caused by disorder fluctuations. As a consequence, the vanishing of the latter in the thermodynamic limit does not necessarily imply the collapse of the Griffiths phase into a single point. By contrast, the width of the Griffiths phase is controlled by the disorder strength. However, for disorder correlations decaying slower than 1/r, no cross-over to a more usual critical behavior could be observed as this strength is tuned to weaker values. Використовуючи метод Монте Карло, недавно отримано фазу Грiффiтса у двовимiрний q-становiй моделi Поттса з сильно скорельованими замороженими хаточними зв’язками. Зокрема, показано, що магнiтна сприйнятливiсть розбiгається алгебраїчно з розмiром гратки в широкому iнтервалi температур. Оскiльки тiльки вiдносно малi розмiри граток можуть розглядатися, цiкаво дiзнатися, чи ця фаза Грiффiтса не стягується i колапсує в одну точку, критичну точку, якщо розмiр гратки стає набагато бiльшим. В цiй статтi, двовимiрна восьмистанова модель Поттса вивчається чисельно для чотирьох рiзних кореляцiй. Показано, що фазу Грiффiтса не можна пояснити як просте поширення локальних температур переходу, спричинених флуктуацiями безладу. Як наслiдок, зникнення останнього в термодинамiчний границi не обов’язково означає колапс фази Грiффiтса в одну точку. На вiдмiну вiд цього, ширина фази Грiффiтса контролюється силою безладу. Проте, для кореляцiй безладу, що згасають повiльнiше нiж 1/r , жодний кросовер до бiльш звичайної критичної поведiнки не мав би спостерiгатись, якщо ця сила зменшується до певного значення. 2014 Article Stability of the Griffiths phase in a 2D Potts model with correlated disorder / C. Chatelain // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33601:1-11. — Бібліогр.: 23 назв. — англ. 1607-324X DOI:10.5488/CMP.17.33601 PACS: 64.60.De, 05.50.+q, 05.70.Jk, 05.10.Ln arXiv:1404.6431 http://dspace.nbuv.gov.ua/handle/123456789/153557 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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A Griffiths phase has recently been observed by Monte Carlo simulations in the 2D q-state Potts model with strongly correlated quenched random couplings. In particular, the magnetic susceptibility was shown to diverge algebraically with the lattice size in a broad range of temperatures. However, only relatively small lattice sizes could be considered, so one can wonder whether this Griffiths phase will not shrink and collapse into a single point, the critical point, as the lattice size is increased to much larger values. In this paper, the 2D eight-state Potts model is numerically studied for four different disorder correlations. It is shown that the Griffiths phase cannot be explained as a simple spreading of local transition temperatures caused by disorder fluctuations. As a consequence, the vanishing of the latter in the thermodynamic limit does not necessarily imply the collapse of the Griffiths phase into a single point. By contrast, the width of the Griffiths phase is controlled by the disorder strength. However, for disorder correlations decaying slower than 1/r, no cross-over to a more usual critical behavior could be observed as this strength is tuned to weaker values. |
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Article |
| author |
Chatelain, C. |
| spellingShingle |
Chatelain, C. Stability of the Griffiths phase in a 2D Potts model with correlated disorder Condensed Matter Physics |
| author_facet |
Chatelain, C. |
| author_sort |
Chatelain, C. |
| title |
Stability of the Griffiths phase in a 2D Potts model with correlated disorder |
| title_short |
Stability of the Griffiths phase in a 2D Potts model with correlated disorder |
| title_full |
Stability of the Griffiths phase in a 2D Potts model with correlated disorder |
| title_fullStr |
Stability of the Griffiths phase in a 2D Potts model with correlated disorder |
| title_full_unstemmed |
Stability of the Griffiths phase in a 2D Potts model with correlated disorder |
| title_sort |
stability of the griffiths phase in a 2d potts model with correlated disorder |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/153557 |
| citation_txt |
Stability of the Griffiths phase in a 2D Potts model with correlated disorder / C. Chatelain // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33601:1-11. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT chatelainc stabilityofthegriffithsphaseina2dpottsmodelwithcorrelateddisorder |
| first_indexed |
2025-07-14T04:39:17Z |
| last_indexed |
2025-07-14T04:39:17Z |
| _version_ |
1837595841158709248 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33601: 1–11
DOI: 10.5488/CMP.17.33601
http://www.icmp.lviv.ua/journal
Stability of the Griffiths phase in a 2D Potts model
with correlated disorder
C. Chatelain
Groupe de Physique Statistique, Département P2M, Institut Jean Lamour, CNRS (UMR 7198),
Université de Lorraine, France
Received April 25, 2014, in final form May 28, 2014
A Griffiths phase has recently been observed by Monte Carlo simulations in the 2D q-state Potts model with
strongly correlated quenched random couplings. In particular, the magnetic susceptibility was shown to diverge
algebraically with the lattice size in a broad range of temperatures. However, only relatively small lattice sizes
could be considered, so one can wonder whether this Griffiths phase will not shrink and collapse into a single
point, the critical point, as the lattice size is increased to much larger values. In this paper, the 2D eight-state
Potts model is numerically studied for four different disorder correlations. It is shown that the Griffiths phase
cannot be explained as a simple spreading of local transition temperatures caused by disorder fluctuations. As
a consequence, the vanishing of the latter in the thermodynamic limit does not necessarily imply the collapse of
the Griffiths phase into a single point. By contrast, the width of the Griffiths phase is controlled by the disorder
strength. However, for disorder correlations decaying slower than 1/r , no cross-over to a more usual critical
behavior could be observed as this strength is tuned to weaker values.
Key words: critical phenomena, random systems, Griffiths phase, Potts model, Monte Carlo simulations
PACS: 64.60.De, 05.50.+q, 05.70.Jk, 05.10.Ln
1. Introduction
The effect of disorder on phase transitions and critical phenomena has attracted a considerable in-
terest in the last decades. In the absence of both frustration and disorder correlation, it is now well es-
tablished that a first-order phase transition is smoothed by the introduction of randomness and can be
made continuous at large enough disorder strength [1]. In 2D, an infinitesimal disorder is sufficient to
remove any discontinuity [2–4]. When the pure system undergoes already a continuous phase transition,
the Harris criterion predicts that the universality class of the pure model will be affected by uncorrelated
disorder if the specific heat diverges, i.e., if the critical exponent α is positive [5]. In this context, the q-
state Potts model has been a useful toymodel, because it displays a rich phase diagram involving two lines
of respectively first and second-order phase transition. Along the latter, the universality class depends on
the number of states q . On the practical side, efficient Monte Carlo and transfer matrix algorithms ex-
ist for this model and Conformal Invariance can be used in 2D in combination with Renormalisation
Group (RG).
For comparison, correlated disorder was much less studied. Nevertheless, in some experimental sit-
uations, impurities cannot be considered as uncorrelated. This is in particular the case when they carry
an electric charge or a magnetic moment and are coupled via an electromagnetic interaction. On the the-
oretical side, Weinrib and Halperin studied the φ4 model with a randommass and showed that a new RG
fixed point, distinct from the random and the pure ones, emerges in the phase diagram when the corre-
lations of this mass decay algebraically [6]. For a sufficiently slow decay of these disorder correlations,
the new fixed point becomes stable. Denoting a the exponent of the algebraic decay of disorder corre-
lations, the perturbation is relevant when a < d if the correlation length exponent ν of the pure model
satisfies the inequality ν< 2/a. At a new fixed point, correlated disorder is marginally irrelevant, which
implies that ν= 2/a [7]. The magnetic exponent η remains small compared to ǫ= 4−d . Even though still
© C. Chatelain, 2014 33601-1
http://dx.doi.org/10.5488/CMP.17.33601
http://www.icmp.lviv.ua/journal
C. Chatelain
controversial, these predictions were confirmed by Monte Carlo simulations of the 3D Ising model with
a = 2 [8, 9].
We recently studied the effect of correlated couplings on the 2D Potts model [10, 11] using large-scale
Monte Carlo simulations. Like in the absence of disorder correlation, the first-order phase transition,
occurring for a pure model when q > 4, was shown to be smoothed and replaced by a continuous tran-
sition. However, the new universality class was shown to be q-independent, a feature shared by the
strong-disorder fixed point of the q-state Potts model with a layered McCoy-Wu-like disorder. This result
is remarkably different from the continuous increase of the magnetic scaling dimension xσ observed for
the Potts model with an uncorrelated disorder. More intriguing is the fact that the phase diagram displays
a Griffiths phase, as in the McCoy-Wu model, where the magnetic susceptibility diverges with the lattice
size. Interestingly, such a phase has been predicted by Weinrib and Halperin, but only above the upper
critical dimension dc = 4. Finally, the hyperscaling relation γ/ν= d −2β/ν was observed to be broken in
the Griffiths phase, as a result of large disorder fluctuations.
However, these observations were made for finite systems, so one cannot exclude the possibility that,
at much larger lattice sizes, the Griffiths phase collapses into a single point, the critical point, where the
hyperscaling relation would be restored. Moreover, the estimate of the correlation length exponent ν
is incompatible with Weinrib-Halperin exact result ν = 2/a, which substantiates the idea that the lat-
tice sizes considered could be too small and that a cross-over would be observed at much larger lattice
sizes. On the other hand, no significant evolution of the Griffiths phase could be observed in the range
of lattice sizes studied [11]. Moreover, the conspiracy of two amplitudes that leads to the violation of the
hyperscaling relation is well verified and no sign of deviation at large lattice sizes is observed.
Since larger lattice sizes are not accessible by Monte Carlo simulations, we turn our attention in this
work to larger exponents a of the disorder correlations. The fact that Weinrib-Halperin predictions were
confirmed by Monte Carlo simulations of the 3D Ising model with a = 2 could indicate that finite-size
effects are weaker for larger values of a. In references [10, 11], only small values of a were consid-
ered because disorder configurations were generated by simulating an auxiliary Ashkin-Teller model
on a self-dual line where its critical exponents are known exactly. The polarization density was then
used to construct the couplings Ji j of the Potts model. Disorder correlations correspond, therefore, to
the polarisation-polarisation correlations of the auxiliary Ashkin-Teller model. When moving along the
self-dual line, only exponents in the range a ∈ [1/4;3/4] can be obtained.
In this work, we present new data for disorder correlation exponents a = 1/3 and 2/3 obtained by
using an auxiliary Ashkin-Teller model. In order to investigate the possible existence of a cross-over to-
wards the Weinrib-Halperin fixed point, we also considered the values a ≃ 1.036 and a = 2 obtained
using the 3D and 4D Ising models as auxiliary models to generate disorder configurations. In the first sec-
tion, details regarding the models and the Monte Carlo simulations are given. In the second section, the
behaviour of the magnetic susceptibility χ̄ is discussed. As already observed in reference [11], χ̄ diverge
algebraically with the lattice size in a broad interval of temperatures, identified as a Griffiths phase, when
a is sufficiently small. A simple explanation of this phenomenon is to assume that disorder fluctuations
induce a spreading of local transition temperatures. Since these fluctuations vanish as L−a/2, this would
imply that a single peak would be recovered at large lattice sizes. Moreover, the smaller the exponent a
the larger are the lattice sizes needed to observe a single peak. In the second section, numerical evidence
is given that disorder fluctuations are not sufficient to explain the observed Griffiths phase, and there-
fore, that the latter phase cannot be expected to collapse as L−a/2. In the third section, the possibility of a
cross-over controlled by the amplitude of disorder correlations is considered. These amplitudes are com-
pared for different disorder correlations considered and, then different disorder strengths are studied.
Finally, conclusions follow.
2. Models and simulation
The classical q-state Potts model is the lattice spin model defined by the Hamiltonian [12, 13]
H =−J
∑
(i , j )
δσi ,σ j
, σi = 0,1, . . . , q −1, (2.1)
33601-2
Potts model with correlated disorder
where the spin σi takes q possible values and is located on the i -th node of the lattice. The sum extends
over all pairs (i , j ) of nearest neighbours on the lattice. In what follows, the Potts model will be considered
on a square lattice. As mentioned in the introduction, the phase transition is continuous for q É 4 and of
first-order when q > 4. We will restrict ourselves to the case q = 8, i.e., a point in the regime of the first-
order transition. Disorder is now introduced as bond-dependent random exchange couplings Ji j . The
Hamiltonian becomes
H =−
∑
(i , j )
Ji j δσi ,σ j
. (2.2)
The spatial correlations between these couplings are assumed to decay algebraically with an exponent a
at a large distance:
Ji j Jkl − Ji j Jkl ∼ |~ri −~rk |
−a . (2.3)
For convenience, we will restrict ourselves to a binary coupling distribution, i.e., Ji j = J1 or J2. The pres-
ence of disorder correlations does not affect the self-duality condition of the random Potts model. Impos-
ing J1 and J2 to be equiprobable and self-dual of each other, the self-dual line is given by the condition [14]
(
eβJ1 −1
)(
eβJ2 −1
)
= q. (2.4)
The coupling configurations are generated by independent Monte Carlo simulations of two auxiliary
models: the Ising and Ashkin-Teller models. The former is defined by the Hamiltonian
H =−JIM
∑
(i , j )
σiσ j , σi =±1 (2.5)
and is equivalent to the q = 2 Potts model. It is well known that this model undergoes a second-order
phase transition in any dimension d > 1. We considered hypercubic lattices of dimension d = 3 and d = 4.
A few thousand spin configurations are generated at the critical point, corresponding to βJIM ≃ 0.221655
for d = 3 [15] and βJIM ≃ 0.149694 for d = 4 [16]. For each spin configuration, a two-dimensional section
is cut and random couplings for the 2D Potts model are constructed as
Ji j =
J1 + J2
2
+
J1 − J2
2
σi (2.6)
for each pair (i , j ) of nearest neighbours in the 2D section. Note that, at any site i , two couplings, in two
different directions, are identical. By construction, disorder correlation functions Ji j Jkl − Ji j Jkl decay as
the spin-spin correlation functions of the auxiliary Ising model. Therefore, the decay is algebraic at large
distances with an exponent a = 2β/ν≃ 1.036 for the 3D Ising model [15] and a = 2 for the 4D Ising model.
Note that, in the second case, the exponent a is equal to the dimension d = 2 of the Potts model. Therefore,
according to Weinrib and Halperin, disorder correlations are expected to be marginally irrelevant. As
a consequence, the system falls into the same universality class as the Potts model with independent
random couplings but with additional logarithmic corrections.
The second auxiliary model is the 2D Ashkin-Teller model defined by the Hamiltonian [17, 18]
H =−
∑
(i , j )
[
JATσiσ j + JATτiτ j +KATσiσ jτiτ j
]
, σi ,τi =±1 (2.7)
and corresponding to two Ising models coupled by their energy densities. On the square lattice, the model
is self-dual along the line of the phase diagram given by e−2KAT = sinh 2JAT. Due to the mapping onto the
eight-vertex model, the critical exponents are known exactly along this line. The random couplings for
the Potts model are constructed from the polarisation density as
Ji j =
J1 + J2
2
+
J1 − J2
2
σiτi . (2.8)
The disorder correlations, therefore, decay as the polarisation-polarisation correlation functions of the
auxiliary Ashkin-Teller model. In this work, we considered two points on the self-dual line of the Ashkin-
Teller model (y = 0.50 and y = 1.25 in the language of the eight-vertex model) corresponding to exponents
a = 1/3 and a = 2/3.
The above-described spin models were simulated using Monte Carlo cluster algorithms to reduce the
critical slowing-down. For the Ising and Potts models, the Swendsen-Wang algorithm was employed [19].
The Ashkin-Teller was simulated using a cluster algorithm introduced by Wiseman and Domany [20, 21].
33601-3
C. Chatelain
3. Griffiths phase and disorder fluctuations
The magnetic susceptibility χ of a finite system undergoing a continuous phase transition in the ther-
modynamic limit is expected to display a peak whose maximum diverges with the lattice size L as Lγ/ν.
The location of this maximum goes towards the critical temperature Tc in the limit of an infinite system.
A very different situation was observed in the 2D Potts model with a strongly correlated disorder [11]. As
can be seen in figure 1, two peaks are present for a = 1/3 and 2/3. The data show an algebraic increase
of the average magnetic susceptibility for all temperatures between these two peaks. For this reason, this
region was conjectured to be a Griffiths phase, similar to the one observed in the McCoy-Wu model. The
absence of any evolution of the location of the two peaks was reported in the case a = 0.4. By contrast,
figure 1 shows a slow evolution in the case a = 2/3. Since only lattice sizes up to L = 128 were studied,
the possibility of a collapse of the Griffiths phase into a single point in the thermodynamic limit cannot
be excluded. Moreover, such a collapse is even more clearly seen in figure 1 for a ≃ 1.036. Two peaks are
still visible but they tend to come closer when the lattice size is increased. It seems natural in this case
to assume that the two peaks will merge into a single one at larger lattice sizes. For disorder correlations
with a faster decay a = 2, only one peak is observed (figure 1) and its location tends towards the critical
value βc = 1, expected from self-duality arguments.
As mentioned in the introduction, it may be assumed that the width of the Griffiths phase is due
to large disorder fluctuations. It seems indeed natural to assume that the first peak is caused by the
ferromagnetic ordering of large clusters with a high concentration of weak bonds J2 while the second
one corresponds to clusters of strong bonds J1. Such large clusters are more probable when disorder
correlations decay slowly. In what follows, disorder fluctuations will be compared for the different values
of a considered. To be more specific, consider the general case of a lattice model with an energy density
Figure 1. (Color online) Average magnetic susceptibility of the 8-state Potts model with different disorder
correlation exponents (a = 1/3, 2/3, 1.036 and 2 from left to right and top to bottom) for a disorder
strength r = J1/J2 = 7.5. The different curves correspond to different lattice sizes. Note that the scale of
the y-axis is logarithmic.
33601-4
Potts model with correlated disorder
denoted ǫi j = ǫ(σi ,σ j ) on the edge between the spins on sites i and j . The weak disorder limit of the
partition function can be calculated using the replica trick:
lnZ = lim
n→0
1
n
(
Z n −1
)
. (3.1)
Introducing the interaction energy ǫα
i j
= ǫ(σα
i
,σα
j
) between the two spins σα
i
and σα
j
of the α-th replica,
the partition function of n replicas reads
Z n =
∑
{σα
i
}
e
−β
∑
(i , j ),α Ji j ǫ
α
i j
≃
∑
{σα
i
}
exp
−β
∑
(i , j ),α
Ji j ǫ
α
i j +
β2
2
∑
(i , j ),(k,l ),
α,β
(
Ji j Jkl − Ji j Jkl
)
ǫαi j ǫ
β
kl
+ . . .
. (3.2)
The first contribution of disorder to the partition function involves the correlations Ji j Jkl − Ji j Jkl , and
is obviously a function of a. In order to characterise the disorder strength by a scalar, we considered the
sum of these correlations, which also corresponds to the fluctuations of the couplings:
∆J2 =
[
1
N
∑
(i , j )
(
Ji j − J̄
)
]2
1/2
, (3.3)
where N = 2L2 is the number of bonds of the square lattice. Since the couplings Ji j are constructed
from the polarization density σiτi of the auxiliary Ashkin-Teller model (for a = 1/3 and 2/3), or from
the magnetization density σi of the auxiliary Ising model (for a ≃ 1.036 and 2), ∆J2 is related, up to a
prefactor (J1 − J2)/2, to the fluctuations of the polarization, or magnetization, density. Therefore, ∆J2 is
expected to scale as
∆J2 ∼ L−a/2 (3.4)
for both auxiliary models. This result is obtained by expanding the square in equation (3.3) and inte-
grating out the disorder correlations in the continuum limit. Up to a further factor Ld , (∆J2)2 is also
proportional to the electric or magnetic susceptibility of the Ashkin-Teller and Ising models. The hyper-
scaling relation for these auxiliary models leads to ∆J2 ∼ L−β/ν where the exponents β/ν are equal to a/2
by construction of the random couplings. Equation (3.4) shows that ∆J2 behaves as a shift of the critical
temperature |Tc(L)−Tc(∞)| in a finite system. Indeed, one expects |Tc(L)−Tc(∞)| ∼ L−1/ν and, at the
Weinrib-Halperin fixed point, ν= 2/a.
The variance of the average coupling ∆J2 is plotted in figure 2 versus the lattice size in the four cases
a = 1/3, 2/3, 1.036 and 2. Note that in the last two cases (a ≃ 1.036 and a = 2), only the magnetization
in the two-dimensional section that was used to construct the exchange couplings Ji j is considered. As
expected, an algebraic decay with an exponent compatible with a/2 is observed. In figure 1, the collapse
of the two peaks of the magnetic susceptibility is observed for a ≃ 1.036 for lattice sizes L ∼O(102) when
a ≃ 1.036. According to figure 2, this corresponds to disorder fluctuations of the order ∆J2 ≃ 0.06. For a =
1/3 and 2/3, none of the lattice sizes that were considered correspond to so small disorder fluctuations.
Indeed, ∆J2 = 0.159(4) when a = 1/3 for the largest lattice size L = 128 and ∆J2 = 0.090(3) when a = 2/3.
This strengthens the idea that the collapse will be observed for larger lattice sizes for a = 1/3 and 2/3.
Using the scaling law (3.4), one can even predict these sizes to be of the order of L∗ ≃ 128(0.06/0.16)−2/a ≃
44 000 for a = 1/3 and L∗ ≃ 128(0.06/0.09)−2/a ≃ 432 for a = 2/3. On the other hand, disorder fluctuations
are small for a = 2 (∆J2 = 0.0618(4) already for L = 24), smaller than for a ≃ 1.036 with lattice sizes
L ∼O(102).
These results do not depend on the quantity used to measure disorder fluctuations. The scaling law
(3.4) suggests to use the order parameter, polarization |σiτi | or magnetization |σi |, of the auxiliary mod-
els as an alternative measure of the fluctuations of the couplings. This quantity will be denoted by ∆J1.
Quantities ∆J1 and ∆J2 provide essentially the same information and, as can be seen in figure 2, take sen-
sibly the same value, but ∆J1 presents the advantage of being more stable numerically. More surprising
33601-5
C. Chatelain
Figure 2. (Color online) Fluctuations∆J (up to a factor (J1−J2)/2) of the average random couplings versus
the lattice size L. Different curves correspond to different disorder correlations, i.e., to different expo-
nents a = 1/3, 2/3, 1.036 and 2 (from top to bottom).
is the fact that the same conclusions can be drawn from the second contribution of disorder to the par-
tition function (3.2). Expanding further, the next term will involve the connected four-point correlation
function Ji J j Jk Jl c
of disorder. This quantity was assumed to be irrelevant by Weinrib and Halperin. We
considered the fourth-order cumulant
∆J4 =
{
3
[ 1
N
∑
i , j
(Ji j − J̄ )
]2
2
−
[ 1
N
∑
i , j
(Ji j − J̄ )
]4
}1/4
. (3.5)
As can be seen in figure 2, no qualitative difference between the three quantities ∆J1, ∆J2 and ∆J4 is
observed.
However, there are small differences between the four cases a = 1/3, 2/3, 1.036 and a which cannot
be explained only in terms of disorder fluctuations. The value of ∆J for the largest lattice size L = 128 at
a = 2/3 is close to the one estimated for L = 48 at a ≃ 1.036. Therefore, the average susceptibility should
look qualitatively the same for a = 2/3 at L = 128 and for a ≃ 1.036 at L = 48. It is not clear whether this is
indeed the case in figure 2. Moreover, a nice collapse of the magnetic susceptibility is observed at large β
for a = 1/3 and 2/3 while this is not the case for a ≃ 1.036 and 2. Stronger statements might be formulated
by comparing thermodynamic quantities displaying universal properties. The natural candidate is the
4th-order Binder cumulant
UM = 1−
〈m4〉
3〈m2〉2
(3.6)
whose value at the intersection of two curves with respect to temperature is expected to be universal in
the thermodynamic limit. A notable difference between different values of a is that the crossing points
occur for inverse temperatures β well below βc = 1 when a = 1/3 and 2/3 and very close to βc = 1 when
a ≃ 1.036 and 2. Unfortunately, the error bars are large and do not permit to be conclusive.
Another quantity displaying universal properties is the ratio [22]
Rm =
〈m〉2 −〈m〉
2
〈m〉
2
(3.7)
that measures the sample-to-sample fluctuations of magnetization. Outside a critical point, all disorder
realizations are expected to lead to the same average magnetization 〈m〉 in the thermodynamic limit.
Therefore, the ratio Rm vanishes as L → +∞ and magnetization is said to be self-averaging. This is no
longer true at a fixed point where disorder is relevant. In this case, Rm goes towards a finite value in
the thermodynamic limit. This limit is expected to be a universal quantity [23]. Numerical data for this
ratio Rm are plotted in figure 3. Two distinct behaviours are observed. For a = 1/3 and a = 2/3, Rm
displays a peak in the paramagnetic phase (small β= 1/kBT ), followed by a broad shouldering. The latter
33601-6
Potts model with correlated disorder
Figure 3. (Color online) Ratio 〈m〉2/〈m〉
2
−1 of the 8-state Potts model with different disorder correlation
exponents (a = 1/3, 2/3, 1.036 and 2 from left to right and top to bottom) for a disorder strength r =
J1/J2 = 7.5. Different curves correspond to different lattice sizes.
extends over a range of temperatures which roughly corresponds to the range between the two peaks of
the average magnetic susceptibility (see figure 1). Interestingly, the estimates of Rm at any temperature
in this shouldering are compatible, within error bars, for all lattice sizes L ∈ [32;128]. Unless a sudden
decay of Rm occurs at much larger lattice sizes, we are led to the conclusion that magnetization is a non-
self-averaging quantity in the whole range of temperatures between the two peaks of susceptibility. This
conclusion is consistent with the assumption of the existence of a Griffiths phase. On the other hand, for
a ≃ 1.036 and a = 2, the peak in the paramagnetic phase is softer and is not followed by a shouldering but
by amonotonous decay. More interesting is the fact that the curves corresponding to different lattice sizes
cross each other at a single point, close to the self-dual point βc = 1. This is consistent with the existence of
a unique critical point at βc = 1. Would it be possible that, in the case a = 2/3, the shouldering disappears
at large lattice sizes to be replaced by amonotonous decay with a single crossing point for different lattice
sizes? If the coupling fluctuations ∆J provides a measure of the width of the Griffiths phase as discussed
above, it should also determine the range of temperatures around βc for which Rm is finite and size-
independent. Then, the ratio Rm should look similar for a = 2/3 at L = 128 and for a ≃ 1.036 at L = 48.
This is definitely not the case in figure 3. Therefore, the Griffiths phase is not solely the consequence of
disorder fluctuations and there is no reason to expect the Griffiths phase to collapse into a single point as
L−a/2.
4. Griffiths phase and disorder strength
All data presented in the previous section correspond to a disorder strength r = J1/J2 = 7.5. Since
the two peaks of susceptibility were interpreted as the ordering of macroscopic clusters with a majority
33601-7
C. Chatelain
of strong, or weak, couplings, the disorder strength r controls the width of the Griffiths phase. One can,
therefore, wonder whether disorder is not too strong in the cases a = 1/3 and 2/3 which implies that
a cross-over to the Weinrib-Halperin fixed point would be observed at larger lattice sizes. For the Potts
model with uncorrelated disorder, strong scaling corrections depending on r = J1/J2 were indeed ob-
served. Accurate estimates of the critical exponents became accessible only after an appropriate disorder
strength was determined. The by-far most efficient technique was, in this case, to compute an effective
central charge ceff by transfer matrix techniques and search for the maximum of ceff. The central charge
is unfortunately difficult to measure by Monte Carlo simulations. Consequently, we will restrict ourselves
to the observation of the effect of a variation of the disorder strength r .
Weinrib and Halperin considered disorder correlations of the form
C (rik ) = Ji j Jkl − Ji j Jkl = vδ(~ri −~rk )+
w
|rik |
a
. (4.1)
When simulating a finite system with an amplitude w much larger (or much weaker) than the value w∗
taken at the fixed point, the critical behaviour may be affected by strong scaling corrections. In a simpler
case of a critical system with a perturbation controlled by a parameter w , the magnetic susceptibility
χ=−∂2 f /∂h2 is indeed expected to scale at the critical temperature as
χ(1/L, w) = Lγ/ν
F
(
Lyw (w −w∗)
)
, (4.2)
where yw is the scaling dimension associated to the perturbation. The scaling function F involves a
cross-over length ℓ∼ (w−w∗)−1/yw . The dominantfinite-size scaling behaviour χ∼ Lγ/ν will be hidden by
scaling corrections if L ≪ ℓ. Similar scaling corrections should also be expected at the long-range random
fixed point, even though they are certainly more complex. In particular, Weinrib and Halperin showed
that the scaling dimensions yw can have an imaginary part, leading to oscillating scaling corrections.
In the previous section, the fluctuations of the average coupling have been compared for different ex-
ponents a. To compare now the amplitudes w of disorder correlations, note that integrating out disorder
correlations leads, on the one hand, to
1
N 2
∑
i j ,kl
[
Ji j Jkl − Ji j Jkl
]
≃
1
L2
∫
L2
C (~r )d2
~r ≃
w
L2
L
∫
0
r dr
r a
= w
L−a
2−a
(4.3)
while, on the other hand, the same quantity is equal to (∆J2)2 according to equation (3.3). The amplitude
w can, therefore, be recomputed as w ≃ (2−a)(∆J2)2La . This estimate is plotted in figure 4. Note that the
Figure 4. (Color online) Amplitude w of disorder correlations (up to a factor (J1− J2)2/4) versus the lattice
size L. Different curves correspond to different disorder correlations, i.e., to different exponents a = 1/3,
2/3 and 1.036.
33601-8
Potts model with correlated disorder
amplitude w is not plotted for a = 2 because the definition is inappropriate in this case (the integration
of the correlations involves a logarithm) and leads to w = 0. As can be seen in figure 4, the amplitude
w does not evolve monotonously with a. This should not be a surprise because the couplings have been
generated from different auxiliary models. The amplitude w for a ≃ 1.036 lies in between the amplitudes
for a = 1/3 and a = 2/3. Therefore, the Griffiths phase and the small ν exponents reported in [11] for
a = 1/3 and a = 2/3 cannot be explained as the result of strong scaling corrections. Indeed, if one assumes
that the amplitude w is close to w∗ in the case a ≃ 1.036, which would explain why the collapse of the two
peaks of χ̄ is observed for reachable lattice sizes, one can conceive that the Griffiths phase is the result of
too strong disorder correlations, i.e., w > w∗, in the case a = 1/3. However, it is hard to understand how
weak correlations, i.e., w < w∗, would lead to a similar result for a = 2/3. The explanation in terms of a
cross-over is, therefore, not supported by the numerical data.
However, as predicted byWeinrib and Halperin for the φ4 model, the amplitude w∗ at the fixed point
can be a function of a. In what follows, the effect of a change of the disorder strength, and, therefore, of
w −w∗, is studied in the case a = 2/3. Since F depends on the scaling variable Lyw (w −w∗), tuning the
amplitude w to come closer to w∗ is expected to be equivalent to increasing the lattice size L. Therefore,
the cross-over, if any, should be observed when w is decreased. In figure 5, the magnetic susceptibility
is plotted for two different disorder strengths r = J1/J2 = 3.5 and r = 2. Compared with figure 1 where
the case r = 7.5 was plotted, it is clear that the width of the Griffiths phase is directly proportional to
the disorder strength r , and, therefore, to the amplitude w . The two peaks come closer as the disorder
strength is reduced. However, even for r = 2, the magnetic susceptibility is still very different from what
is observed in figure 1 for a ≃ 1.036 and r = 7.5. One can doubt that the two curves will look similar for
an even weaker disorder in the case a = 2/3. Very probably, the two peaks of the magnetic susceptibility
Figure 5. (Color online) On the left, average magnetic susceptibility of the 8-state Potts model with a dis-
order correlation exponent a = 2/3 and a disorder strengths r = J1/J2 = 3.5 (top) and r = 2 (bottom).
Different curves correspond to different lattice sizes. On the right, ratio Rm = 〈m〉2/〈m〉
2
−1 for the same
models. Note that the horizontal scales are not the same.
33601-9
C. Chatelain
for a = 2/3 will collapse but only when approaching r = 1, i.e., the pure model.
The ratio Rm = 〈m〉2/〈m〉
2
−1 provides a stronger evidence that what is observed for a ≃ 1.036 is not
what should be expected for a = 2/3 at a weaker disorder. As discussed when figure 1 was commented,
the signature of the Griffiths phase is a size-independent ratio Rm over a finite range of temperatures.
By contrast, for a ≃ 1.036, the ratios Rm computed at two different lattice sizes display a single crossing
point at a temperature evolving towards the self-dual pointβc = 1. As observed in figure 5, the diminution
of the disorder strength induces a reduction of the width of the Griffiths phase, and, as expected, of the
range of temperatures where the ratio Rm appears to be size-independent. However, the ratio Rm looks
surprisingly similar, up to a temperature rescaling, at different disorder strengths. Even at a disorder
strength r = 2, the behaviour is still very different from what is observed for a ≃ 1.036.
5. Conclusions
New Monte Carlo simulations of the 2D 8-state Potts model with a disorder involving algebraically
decaying correlations C (r ) ∼ r−a with a = 1/3, 2/3, 1.036 and 2 are presented. While the analysis of
the magnetic susceptibility does not permit to exclude the possibility of a collapse of the Griffiths phase
into a single critical point, the study of the self-averaging ratio Rm = 〈m〉2/〈m〉
2
−1 permits to be more
conclusive. Two different behaviours are indeed observed for a = 1/3 and 2/3 on the one hand and
a ≃ 1.036 and 2 on the other hand. The first case is compatible with the assumption of the existence
of a Griffiths phase while in the second case, the signature of a single critical point is observed. These
difference cannot be explained by larger disorder fluctuations in the first case. Moreover, if the width
of the Griffiths phase depends on the disorder strength for a = 2/3, no cross-over towards the Weinrib-
Halperin critical behaviour is observed at a weak disorder. These numerical results call for a theoretical
understanding of the precise mechanism behind this Griffiths phase.
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33601-10
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Potts model with correlated disorder
Стiйкiсть фази Грiффiтса в 2D моделi Поттса
зi скорельованим безладом
К. Шателєн
Група статистичної фiзики, вiддiл P2M, Iнститут Жана Лямура, CNRS (UMR 7198),
Унiверситет Лотарингiї, Францiя
Використовуючи метод Монте Карло, недавно отримано фазу Грiффiтса у двовимiрний q-становiй моделi
Поттса з сильно скорельованими замороженими хаточними зв’язками. Зокрема, показано, що магнiтна
сприйнятливiсть розбiгається алгебраїчно з розмiром гратки в широкому iнтервалi температур. Оскiль-
ки тiльки вiдносно малi розмiри граток можуть розглядатися, цiкаво дiзнатися, чи ця фаза Грiффiтса не
стягується i колапсує в одну точку, критичну точку, якщо розмiр гратки стає набагато бiльшим. В цiй
статтi, двовимiрна восьмистанова модель Поттса вивчається чисельно для чотирьох рiзних кореляцiй.
Показано, що фазу Грiффiтса не можна пояснити як просте поширення локальних температур переходу,
спричинених флуктуацiями безладу. Як наслiдок, зникнення останнього в термодинамiчний границi не
обов’язково означає колапс фази Грiффiтса в одну точку. На вiдмiну вiд цього, ширина фази Грiффiтса
контролюється силою безладу. Проте, для кореляцiй безладу, що згасають повiльнiше нiж 1/r , жодний
кросовер до бiльш звичайної критичної поведiнки не мав би спостерiгатись, якщо ця сила зменшується
до певного значення.
Ключовi слова: критичнi явища, випадковi системи, фаза Грiффiтса, модель Поттса,
симуляцiї Монте Карло
33601-11
Introduction
Models and simulation
Griffiths phase and disorder fluctuations
Griffiths phase and disorder strength
Conclusions
|