Simulations of an Ising q-state Pods model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for q=2, there are two critical Ising lines meeting at T=0 at the four-state Potts critical point; (ii) for 24, the Ising:critical line intersects a Line of first-order transitions at a critical end point

Kaufman, Miron

Scholten, P. D.

English

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Physics Faculty Publications Physics Department
7-1-1997
Monte Carlo Study of the Square-Lattice Annealed
Ising Model on Percolating Clusters
P. D. Scholten
Miron Kaufman
Cleveland State University, m.kaufman@csuohio.edu
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Publisher's Statement
Copyright 1997 American Physical Society. Available on publisher's site at http://link.aps.org/doi/
10.1103/PhysRevB.56.59.
Repository Citation
Scholten, P. D. and Kaufman, Miron, "Monte Carlo Study of the Square-Lattice Annealed Ising Model on Percolating Clusters" (1997). Physics Faculty
Publications. 49.
https://engagedscholarship.csuohio.edu/sciphysics_facpub/49
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Original Citation
Scholten, P. D. and Miron Kaufman. "Monte Carlo Study of the Square-Lattice Annealed Ising Model on Percolating Clusters." Physical
Review B 56 (1997): 59-62.
Monte Carlo study of the square-lattice annealed Ising model on percolating clusters
P. D. Scholten
Department of Physics, Miami University, Oxford, Ohio 45056
Miron Kaufman
Department of Physics, Cleveland State University, Cleveland, Ohio 44115
~Received 24 January 1997!
Simulations of an Ising q-state Potts model which is equivalent to the Ising model on annealed percolation
clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different
phase diagrams are obtained: ~i! for q52, there are two critical Ising lines meeting at T50 at the four-state
Potts critical point; ~ii! for 2,q<4, the Ising critical line meets the q-state critical line and a line of first-order
transitions at a bicritical point; ~iii! for q.4, the Ising critical line intersects a line of first-order transitions at
a critical end point. @S0163-1829~97!06725-8#
I. INTRODUCTION
The annealed Ising model on percolation clusters was
shown1 to show interesting phase diagrams from a position-
space renormalization-group technique. In particular, that
study has suggested the existence of novel multicritical
points. Though physical systems such as fluids in porous
media and amorphous magnets are better described by
quenched randomness,2 it is useful to have the annealed
model as a proxy for the more realistic but also less tractable
quenched model.
In this paper we present a Monte Carlo simulation of the
model on a square lattice to validate the renormalization-
group predictions.1 The simulations were carried out on an
Ising-q-state Potts model which is equivalent1 to the Ising
model on percolation clusters. There are three thermody-
namic fields: the fugacity q controlling the number of clus-
ters, the probability p for the presence of bonds, and the
temperature or the inverse Ising coupling. Note that the per-
colation process3 considered here allows for correlations be-
tween bonds. The random-bond percolation holds for q51.
We have studied the phase diagrams for q52, 3, 4, 6, and
10. Though the gross features of the renormalization-group
phase diagram, e.g., the three phases of nonpercolating, per-
colating paramagnetic, and percolating ferromagnetic, are
confirmed by the Monte Carlo method, there are important
differences between the results of the two methods. For q
53 and 4, the Monte Carlo method predicts the occurrence
of a bicritical point4 at the intersection of two continuous
transitions lines ~one Ising like and the other Potts like! with
a line of first-order transitions. For higher q ~we looked at
q56 and 10! we find a critical end point5,6 at the intersection
of the Ising transition line with two lines of first-order tran-
sitions. We have also simulated the model for q52. In this
case we seem to confirm the position-space
renormalization-group1 prediction: two Ising critical lines
meeting at T50 at the four-state Potts multicritical point.
The remainder of the paper is organized as follows. In
Sec. II we present the model and review some exact results
which are used to verify the numerical simulations. Section
III contains our Monte Carlo simulations. The conclusions
are included in Sec. IV.
II. THEORY
At each site of the square lattice, there is an Ising spin
si561 and a Potts spin s i51,2,.. . ,q . The Hamiltonian is
2H/kBT5(
^i , j&
@Jd~s i ,s j!1Ld~s i ,s j!SiS j# , ~1!
where d is the Kronecker delta and ^i , j& means nearest
neighbors. ~Note that this Hamiltonian differs from the dilute
Potts model of Nienhuis et al.7! This model was shown1 to
be equivalent to the Ising model on annealed percolation
clusters:
Z5(
conf
qcwBZI
conf
, ~2!
where w5p/(12p), p is the probability for a bond to be
present, C is number of clusters, and B is number of bonds.
ZI
conf is the Ising partition function for the particular perco-
lation configuration:
ZI
conf5Tr~Si!expFK(
^i , j&
~SiS j21 !G , ~3!
where ^i , j& means a pair of sites connected by a bond. The
parameters J and L of Eq. ~1! are related to w and K of Eqs.
~2! and ~3! as follows:
J5 12 ln@~11w !~11we22K!# ,
L5 12 ln@~11w !/~11we22K!# . ~4!
Some exact results can be deduced from the Ising-Potts
model. If K50 from Eq. ~4!, we get L50: ~i.e., this is the
q-state Potts model with critical coupling, Jc5ln(11Aq),
which corresponds to pc5Aq/(11Aq).
If K5` , from Eq. ~4! we get J5L . The Hamiltonian
becomes
2H/kBT5(
^i , j&
@2Jd~s i ,s j!d~Si ,S j!# .
PHYSICAL REVIEW B 1 JULY 1997-IVOLUME 56, NUMBER 1
560163-1829/97/56~1!/59~4!/$10.00 59 © 1997 The American Physical Society
This is the 2q-state Potts model with critical coupling Jc
5 12ln(11A2q), which corresponds to pc5A2q/(11A2q).
For p51 the Hamiltonian in Eq. ~1! reduces to the Ising
model
2H/kBT5(
^i , j&
KSi ,S j ,
apart from an additive constant. The critical coupling is Kc
5 12ln(11&).
Finally, the order of the Potts transitions in two dimen-
sions is known.8 It is continuous if q<4 and it is discontinu-
ous if q.4.
III. MONTE CARLO SIMULATIONS
Monte Carlo ~MC! simulations based on the Hamiltonian
of Eq. ~1! were carried out for systems with q52, 3, 4, 6,
and 10 on square lattices of sizes L514, 20, 30, and 40.
Each site was considered to be in one of 2q possible states,
and the Metropolis algorithm was used to determine changes
in the states. Periodic boundary conditions were used for all
lattices, and each MC simulation consisted of 53106 lattice
passes after the first 53104 passes had been discarded to
help ensure equilibration. For the statistical analysis a com-
plete two-dimensional probability distribution of the Ising
and Potts energies, P(EI ,EP), was collected for each simu-
lation. From this the system energy and heat capacity as well
as the individual Ising and Potts heat capacities could be
computed. In order to calculate the Ising and Potts suscepti-
bilities, the average Ising order parameter as a function of
Ising energy, ^MI(EI)& , and the average Potts order param-
eter as a function of Potts energy, ^MP(EP)&, were also re-
corded for each run. Last, the Ising order parameter probabil-
ity distribution P(MI), along with the average Ising and
Potts energies as functions of the Ising order parameter,
^EI(MI)& and ^EP(MI)&, were recorded so that a free energy
functional of the Ising order parameter could be calculated.
We present the phase diagrams in the r ,p plane where r
512e2K, p5w/(11w). The advantage of this particular
choice of thermodynamic fields is that each varies between 0
and 1. Each data set obtained at given values of r and p was
processed using the histogram method of Ferrenberg and
Swendsen9 to obtain thermodynamic quantities for a range of
nearby values of p at constant r . Each transition point was
then determined by locating the value of p for which the heat
capacity was a maximum. The Ising heat capacity was used
for the PP- ~percolating paramagnetic! PF ~percolating ferro-
magnetic! transition, the Potts heat capacity was used for the
NP- ~nonpercolating paramagnetic! PP transition, and the to-
tal system heat capacity was used for the NP-PF transition.
These calculated transition points were in good agreement
with the known results at r51, p51, and r50. For instance,
although singularities prevented the program from running at
r51, the data at r50.98 showed that for L540 the critical
values of p for q52, 3, and 4 were 0.6694, 0.7129, and
0.7407, respectively. These compare to pc5Aq/(11Aq)
50.6667, 0.7101, and 0.7388 for an infinite system at r
51. Due to the negative slope of the phase boundary, pc is
expected to be higher at r50.98 than at r51. Estimates of
the uncertainties in pc for a given q , L, and r were obtained
by doing a second Monte Carlo run at or very near the origi-
nal value of p . In all 12 such cases examined, both computed
values of pc were within 60.0002.
It was found that for every value of q and r , the critical
value of p increased with the lattice size. Extrapolations of
graphs of pc vs 1/L indicated that critical values of p for
infinite systems would be only about 0.003 higher than those
for L540.
The phase diagrams obtained from the simulations are
shown in Fig. 1. They are in qualitative agreement with those
obtained from the renormalization-group analysis, but the
multicritical points occur at significantly lower values of p
and r . As q decreases, the location of the multicritical point
FIG. 1. Phase diagram for ~a! q52, ~b! q53 ~j! and q54 ~m!,
and ~c! q56 ~j! and q510 ~m!. The lines are merely guides to the
eye. Solid lines indicate continuous transitions, and dashed lines
indicate first-order transitions.
60 56BRIEF REPORTS
moves to higher values of r and lower values of p .
Of special interest to this study is the nature of the phase
transitions and how they depend on q . The PP-PF transition
is an Ising transition and so is continuous for all q . The
NP-PP transition is a Potts transition and therefore is con-
tinuous for q<4, but first order for q.4. The nature of the
NP-PF transition was not initially obvious and so was inves-
tigated further. For q.2 the energy distributions displayed
two distinct peaks ~see Fig. 2!, suggesting a first-order tran-
sition. To confirm this the Lee-Kosterlitz function10 F(E)
was computed for q53 and 4 at r50.95 and 0.98 for L
514, 20, 30, and 40. Figure 3 shows that DF(E), i.e., the
depth of the two minima relative to the height of the barrier
separating them, increases with L. The effect is much more
pronounced for q54, but nonetheless the data show that the
transition is first order in both cases.
From Fig. 1~a! it first appeared that for q52 a multicriti-
cal point existed for r,1; i.e., three phase boundaries
seemed to intersect at around r50.91. If true, this would be
contrary to the predictions of the renormalization-group
analysis which held that the multicritical point was at r51.
To determine whether there existed one phase boundary
~NP-PF! or two boundaries ~NP-PP and PP-PF! close to-
gether, the values of pc , as determined by the maxima of the
Ising and Potts heat capacities and susceptibilities, were ex-
amined in detail. The precision of each pc for an individual
run is 60.000 05, although, as mentioned above, the repro-
ducibility from run to run is about 60.0002. The computed
differences of 0.0001 or less for the cases of q53 and 4
indicate that the Ising and Potts systems are ordering to-
gether. However, the differences for q52 are significant.
They are, at r50.95, for susceptibility Dpc50.0013 and for
heat capacity Dpc50.0005; at r50.98, for susceptibility
Dpc50.0003 and for heat capacity Dpc50.0002. Further-
more, the data show that the Ising spins order at a higher p
value than do the Potts spins, consistent with the percolating-
paramagnetic to percolating-ferromagnetic boundary being
above the nonpercolating-paramagnetic to percolating-
paramagnetic boundary. Although the numerical simulations
cannot provide absolute proof that the multicritical point
does not occur until r51, the data are consistent with the
picture of two phase boundaries tangentially approaching
each other as r!1 or T!0.
IV. CONCLUSIONS
We have established the phase diagram of the square lat-
tice Ising model on percolating clusters by using Monte
Carlo simulations. The simulations were performed on the
equivalent Ising-Potts model. The Monte Carlo results are
consistent with the renormalization-group result1 that the
nonpercolating-paramagnetic to percolating-ferromagnetic
phase transition for any q.2 is in the universality class of
the 2q-state Potts model. Particularly interesting is the oc-
currence of bicritical points for 2,q<4 and of critical end
points for q.4. Further studies of critical exponents and
amplitudes associated with these multicritical points are
needed.
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p50.7161 for a lattice with L530: ~a! Ising and Potts energies
and ~b! system energy.
FIG. 3. DF(E) as function of lattice size L for q53, r50.95
~j!, q53, r50.98 ~d!, q54, r50.95 ~.!, and q54, r50.98 ~m!.
56 61BRIEF REPORTS
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62 56BRIEF REPORTS


Monte Carlo Study of the Square-Lattice Annealed Ising Model on Percolating Clusters

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Monte Carlo Study of the Square-Lattice Annealed Ising Model on Percolating Clusters

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