High-temperature series expansions for random Potts models
We discuss recently generated high-temperature series expansions for the free energy and the susceptibility of random-bond q-state Potts models on hypercubic lattices. Using the star-graph expansion technique, quenched disorder averages can be calculated exactly for arbitrary uncorrelated couplin...
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irk-123456789-1193842017-06-07T03:04:49Z High-temperature series expansions for random Potts models Hellmund, M. Janke, W. We discuss recently generated high-temperature series expansions for the free energy and the susceptibility of random-bond q-state Potts models on hypercubic lattices. Using the star-graph expansion technique, quenched disorder averages can be calculated exactly for arbitrary uncorrelated coupling distributions while keeping the disorder strength p as well as the dimension d as symbolic parameters. We present analyses of the new series for the susceptibility of the Ising (q = 2) and 4-state Potts model in three dimensions up to the order 19 and 18, respectively, and compare our findings with results from field-theoretical renormalization group studies and Monte Carlo simulations. Ми обговорюємо нещодавно генеровані високотемпературні розклади для вільної енергії та сприйнятливості q -станової моделі Потса на гіперкубічній гратці із безладом у формі випадкових зв’язків. Використовуючи техніку розкладу зіркових графів, усереднення за замороженим безладом можна провести точно при довільних розподілах нескорельованих зв’язків, зберігаючи концентрацію безладу p та вимірність гратки d як символічні параметри. Ми представляємо аналіз нових рядів для сприйнятливості тривимірних моделі Ізинга ( q = 2 ) та 4-станової моделі Потса до відповідно 19-го та 18-го порядків, і порівнюємо отримані нами результати із даними досліджень в рамках теоретико-польової ренормалізаційної групи та симуляцій Монте Карло. 2005 Article High-temperature series expansions for random Potts models / M. Hellmund, W. Janke // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 59–74. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 05.50.+q, 64.60.Fr, 75.10.Hk, 75.10.Nr DOI:10.5488/CMP.8.1.59 http://dspace.nbuv.gov.ua/handle/123456789/119384 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
We discuss recently generated high-temperature series expansions for the
free energy and the susceptibility of random-bond q-state Potts models on
hypercubic lattices. Using the star-graph expansion technique, quenched
disorder averages can be calculated exactly for arbitrary uncorrelated coupling
distributions while keeping the disorder strength p as well as the dimension
d as symbolic parameters. We present analyses of the new series
for the susceptibility of the Ising (q = 2) and 4-state Potts model in three
dimensions up to the order 19 and 18, respectively, and compare our findings
with results from field-theoretical renormalization group studies and
Monte Carlo simulations. |
| format |
Article |
| author |
Hellmund, M. Janke, W. |
| spellingShingle |
Hellmund, M. Janke, W. High-temperature series expansions for random Potts models Condensed Matter Physics |
| author_facet |
Hellmund, M. Janke, W. |
| author_sort |
Hellmund, M. |
| title |
High-temperature series expansions for random Potts models |
| title_short |
High-temperature series expansions for random Potts models |
| title_full |
High-temperature series expansions for random Potts models |
| title_fullStr |
High-temperature series expansions for random Potts models |
| title_full_unstemmed |
High-temperature series expansions for random Potts models |
| title_sort |
high-temperature series expansions for random potts models |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119384 |
| citation_txt |
High-temperature series expansions for random Potts models / M. Hellmund, W. Janke // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 59–74. — Бібліогр.: 40 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hellmundm hightemperatureseriesexpansionsforrandompottsmodels AT jankew hightemperatureseriesexpansionsforrandompottsmodels |
| first_indexed |
2025-07-08T15:46:41Z |
| last_indexed |
2025-07-08T15:46:41Z |
| _version_ |
1837094268077867008 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 59–74
High-temperature series expansions for
random Potts models
M.Hellmund ∗1 , W.Janke † 2 ,
1 Mathematisches Institut, Universität Leipzig,
Augustusplatz 10/11, D–04109 Leipzig, Germany
2 Institut für Theoretische Physik, Universität Leipzig,
Augustusplatz 10/11, D–04109 Leipzig, Germany
Received December 3, 2004
We discuss recently generated high-temperature series expansions for the
free energy and the susceptibility of random-bond q-state Potts models on
hypercubic lattices. Using the star-graph expansion technique, quenched
disorder averages can be calculated exactly for arbitrary uncorrelated cou-
pling distributions while keeping the disorder strength p as well as the di-
mension d as symbolic parameters. We present analyses of the new series
for the susceptibility of the Ising (q = 2) and 4-state Potts model in three
dimensions up to the order 19 and 18, respectively, and compare our fin-
dings with results from field-theoretical renormalization group studies and
Monte Carlo simulations.
Key words: random Potts models, quenched disorder, high-temperature
series expansions, effective critical exponents
PACS: 05.50.+q, 64.60.Fr, 75.10.Hk, 75.10.Nr
1. Introduction
Systematic series expansions [1] for statistical physics models defined on a lat-
tice provide an useful complement to field-theoretical renormalization group studies
and large-scale numerical Monte Carlo simulations. This is in particular true when
studying phase transitions and critical phenomena of quenched, disordered systems.
In the field-theoretic treatment [2] the necessary average over disorder realizati-
ons at the level of the free energy requires the application of the so-called “replica
trick” which loosely speaking introduces n different, interacting copies of the original
system, with the formal limit n → 0 taken at the end. In the numerical approach
the average over a large but finite number of different disorder realizations can, at
least in principle, be performed explicitly but is very time consuming such that only
∗E-mail: meik.hellmund@math.uni-leipzig.de
†E-mail: wolfhard.janke@itp.uni-leipzig.de
c© M.Hellmund, W.Janke 59
M.Hellmund, W.Janke
few points in the vast parameter space of the systems can be sampled with realistic
effort. Moreover, extrapolations of the data on finite lattices to the infinite-volume
limit are required. On the other hand, using high-temperature series expansions,
one can obtain for many quantities exact results up to a certain order in the inverse
temperature. Here the quenched disorder is treated exactly and the infinite-volume
limit is implicitly implied. Moreover, one can keep the disorder strength p as well
as the dimension d as symbolic parameters and therefore analyse large regions of
the parameter space of disordered systems. The critical part of the series expansi-
on approach lies in the extrapolation techniques which are used in order to obtain
information on the phase transition behaviour from the finite number of known co-
efficients of the high-temperature series. While for pure systems this usually works
quite well, one can question the use of these extrapolation techniques in disordered
systems, where the singularity structure of the free energy or susceptibility may be
very complicated, involving Griffiths-type singularities or logarithmic corrections [3].
Pure Potts models show either first- or second-order phase transitions, depending
on the dimension d and the number of states q. Since in the second-order case the
specific-heat exponent α is non-negative for this class of models, the Harris criterion
[4] suggests for the corresponding disordered systems either the appearance of a new
random fixed point (d = 2, q = 3, 4 and d = 3, q = 2) or logarithmic corrections to
the pure fixed point (d = 2, q = 2). At first-order phase transitions, the randomness
softens the transitions [5]. For d = 2 even infinitesimal disorder induces a continuous
transition [6,7], whereas for d = 3, q > 2 a tricritical point at a finite disorder
strength is expected [8].
In this work we studied these scenarios by means of “star-graph” high-temper-
ature series expansions where the disorder average can be taken at the level of indi-
vidual graphs. Using optimized cluster algorithms for the symbolic, exact calculation
of spin-spin correlators on finite graphs with arbitrary inhomogeneous couplings, we
obtained series expansions for the free energy and susceptibility in the inverse tem-
perature up to the order 19 respectively 18 for bond-diluted Ising and Potts models
in dimensions d 6 5, and up to order 17 in arbitrary dimensions. Here we shall
focus on analyses of these series in three dimensions where a direct comparison with
field-theoretic renormalization group studies and recent Monte Carlo simulations is
possible.
2. Model
The ferromagnetic disordered q-state Potts model on hypercubic lattices Z
d is
defined by the partition function
Z =
∑
{si}
exp
β
∑
〈ij〉
Jijδsi,sj
, (1)
where β = 1/kBT is the inverse temperature, Jij are quenched (non-negative)
nearest-neighbour coupling constants, the spins can take the values si = 1, . . . , q,
60
Series expansions for random Potts models
and δ.,. is the Kronecker symbol. In our series expansion the combination
vij =
eβJij − 1
eβJij − 1 + q
(2)
will be the relevant expansion parameter which in the Ising case (q = 2) simplifies
to vij = tanh(βJij/2). In the symmetric high-temperature phase, the susceptibility
associated with the coupling
∑
i hi(qδsi,1 − 1)/(q− 1) to an external field hi is given
for a graph with N spins by summing over all two-point correlations,
χ =
1
N
∑
i
∑
j
[〈
qδsi,sj
− 1
q − 1
〉]
av
. (3)
Here the brackets [. . .]av indicate the quenched disorder average which in our case
is taken over an uncorrelated bimodal distribution of the form
P (Jij) = (1 − p)δ(Jij − J0) + pδ(Jij − RJ0). (4)
Besides bond dilution (R = 0), which will be in the focus of the present work,
this also includes random-bond ferromagnets (0 < R < 1) and the physically very
different class of spin glasses (R = −1) as special cases. Other distributions such as
Gaussian distributions can, in principle, also be considered using our method.
3. Series generation methodology
In this section we briefly review the main technical ingredients necessary for
our high-temperature series study. We begin with a few basic notations from graph
theory. A graph of order E consists of E links connecting N vertices. We consider
only connected, undirected graphs that are simple: no link starts and ends at the
same vertex (no tadpoles) and two vertices are never connected by more than one
link. Subgraphs are defined by the deletion of links. In this process, isolated vertices
can be dropped. Since each link may be present or absent, a graph of order E
has 2E (not necessarily non-isomorphic) subgraphs. These subgraphs may consist of
several connected components and are called clusters. If the deletion of one vertex
renders the graph disconnected, such a vertex is termed articulation point. The
“star graphs” we are considering here are thus just defined by the absence of such
articulation points. A graph is bipartite if the vertices can be separated into red and
black vertices so that no link connects two vertices of the same color. Equivalently,
all closed paths in the graph consist of an even number of links. Hypercubic lattices
are evident examples for bipartite graphs.
There are a couple of well-established methods [1] known for the systematic
generation of high-temperature series expansions which differ in the way the relevant
subgraphs are selected or grouped together. A recently developed alternative method
[9] exploits ideas from so-called finite-lattice methods usually employed before for the
generation of low -temperature series. Using a clever reformulation of the method,
61
M.Hellmund, W.Janke
Arisue et al. [10] succeeded in generating a very impressive 32th order world-record
high-temperature susceptibility series for the pure Ising model in three dimensions.
For the class of classical O(N) spin models without disorder, quite long series
(up to order β25) have also been produced by linked-cluster expansions [11]. This
technique also allows one to obtain series for more involved observables (such as the
second moment of the spin-spin correlation function yielding the correlation length)
which have no star-graph expansion. Furthermore, it works with free embeddings
of graphs into the lattice which can be counted orders of magnitude faster than
the weak embedding numbers needed by the star-graph technique. Nonetheless, the
linked-cluster method has not yet been applied to problems with quenched disorder.
The star-graph method can be adopted to systems involving quenched disorder
[12,13] (as also can the no-free-end method [14]) since it allows one to take the
disorder average on the level of individual graphs. The basic idea is to assemble the
value of some extensive thermodynamic quantity F on a large or even infinite graph
from its values on subgraphs: Graphs constitute a partially ordered set under the
“subgraph” relation. Therefore, for every function F (G) defined on the set of graphs
there exists another function WF (G) such that F (G) =
∑
g⊆G WF (g), for all graphs
G. This function can be calculated recursively via WF (G) = F (G) −
∑
g⊂G WF (g),
resulting for an infinite (e.g. hypercubic) lattice in F (Zd) =
∑
G(G : Z
d) WF (G),
where (G : Z
d) denotes the weak embedding number of the graph G in the given
lattice structure [15].
The following observation makes this a useful method: Let G be a graph with an
articulation vertex where two star subgraphs G1,2 are glued together. Then WF (G)
vanishes if F (G) = F (G1) + F (G2). An observable F for which this property is
true on arbitrary graphs with articulation points makes a star-graph expansion
possible. All non-star graphs have zero weight WF in the sum for F (Zd). It is easy
to see that the (properly normalized) free energy log Z has this property and it
can be proved [13] that the inverse susceptibility 1/χ has it too, even for arbitrary
inhomogeneous couplings Jij. This restricts the summation for F (Zd) to a sum over
star graphs. The linearity of the recursion relations then enables the calculation of
quenched averages over the coupling distribution on the level of individual graphs.
The resulting recipe for the susceptibility series is:
• graph generation and embedding number counting;
• calculation of Z(G) and the correlation matrix
Mnm(G) = Tr (qδ(Sn, Sm) − 1)e−βH({Jij})
for all graphs as polynomials in E variables vij defined in (2);
• inversion of the Z polynomial as a series up to the desired order;
• averaging over quenched disorder,
Nnm(G) = [Mnm/Z]P (J) ,
resulting in a matrix of polynomials in (p, v);
62
Series expansions for random Potts models
• inversion of the matrix Nnm and subgraph subtraction,
Wχ(G) =
∑
n,m
(N−1)nm −
∑
g⊂G
Wχ(g);
• collecting the results from all graphs,
1/χ =
∑
G
(G : Z
d) Wχ(G).
Algorithmically the most cumbersome part of this recipe is the first step, i.e.,
the generation of star graphs and calculation of their (weak) embedding numbers.
The graph generation is usually done by recursively adding nodes and edges to a
list of smaller graphs. To make sure that no double counting occurs, this requires an
isomorphism test, i.e., the decision whether two given adjacency lists or adjacency
matrices describe the same graph modulo relabelling and reordering of edges and
nodes. We employed the NAUTY package by McKay [16] which permits very fast
Figure 1. Growth behaviour of the number of star graphs with E links that can
be embedded in hypercubic lattices Z
d.
isomorphism tests by calculating a canonical representation of the automorphism
group of the graphs. By this means, we classified for the first time all star graphs
up to order 19 that can be embedded in hypercubic lattices, see table 1. As with
any series expansion, the effort grows exponentially with the maximal order of the
expansion, rendering each new order roughly as “expensive” as all previous orders
taken together. This is illustrated in figure 1 where already the number of star graphs
is seen to grow exponentially as a function of the links E. The exponential fit in the
range E = 13 − 19 suggests that the number of star graphs increases roughly by a
factor of 2.8 in each of the next higher orders, predicting about 65 000 different star
graphs with E = 20 and about 180 000 with E = 21.
63
M.Hellmund, W.Janke
Table 1. Number of star graphs with E > 8 links and non-vanishing embedding
numbers on Z
d. For E = 1, 4, 6, and 7 only a single star graph exists.
order E 8 9 10 11 12 13 14 15 16 17 18 19
# 2 3 8 9 29 51 142 330 951 2561 7688 23078
�
�
�
�
�
�
�
�
�
�
s
s
s s s
s s s
s s s
s s s
12048
(
d
3
)
+ 396672
(
d
4
)
+ 2127360
(
d
5
)
+ 2488320
(
d
6
)
s s s s s s s s s
s s s s s s s s s
7620
(
d
2
)
+ 76851600
(
d
3
)
+ 14650620864
(
d
4
)
+ 404500471680
(d
5
)
+ 3355519311360
(d
6
)
Figure 2. Two star graphs of order 17 and 19 and their weak embedding numbers
up to 6 dimensions.
For each of these graphs we calculated their (weak) embedding numbers for d-
dimensional hypercubic lattices (up to order 17 for arbitrary d and up to order 19 for
dimensions d 6 5). Two typical results are depicted in figure 2. For the embedding
count we implemented a refined version of the backtracing algorithm by Martin [15],
making use of a couple of simplifications for bipartite hypercubic lattices Z
d. After
extensive tests to find the optimal algorithm for the “innermost” loop, the test for
collisions in the embedding, we ended up using optimized hash tables.
The second step of the series generation requires an exact calculation of the par-
tition function and the matrix of correlations Mnm for each star graph with arbitrary
symbolic couplings Jij defined on the E 6 19 edges. The crucial observation is that
this can be done most efficiently by using the cluster representation
Z ∝ Z = q−NTr
∏
〈ij〉
[
1 − vij + vijqδsi,sj
]
=
∑
C
qe+c−N
∏
〈ij〉∈C
vij
∏
〈ij〉/∈C
(1 − vij)
, (5)
where the sum goes over all clusters C ⊆ G, e is the number of links of the cluster and
c is the number of connected components of C. The reduced partition function Z ≡
ZqE−N/
∏
〈ij〉(e
βJij −1+q) is normalized such that logZ has a star-graph expansion.
64
Series expansions for random Potts models
Similarly, the calculation of susceptibility involves the matrix of correlations
Mnm ∝
∑
Cnm
qe+c−N
∏
〈ij〉∈C
vij
∏
〈ij〉/∈C
(1 − vij)
, (6)
where the sum is restricted to all clusters Cnm ⊆ G in which the vertices n and m
are connected.
This representation essentially reduces the summation over qN states to a sum
over 2E clusters which, compared with previous implementations, results in a huge
saving factor in computing time (of the order of 106). Further improvements take
place if the 2E clusters belonging to a graph are enumerated by Gray codes [17]
such that two consecutive clusters in the sum (5) differ by exactly one (added or
deleted) link. In the Ising case q = 2 another huge simplification takes place since
only clusters in which all vertices are of even degree contribute to the cluster sum.
Since general purpose software for symbolic manipulations turned out to be too
slow for our purposes, we developed a C++ template library using an expanded
degree-sparse representation of polynomials and series in many variables. For arbi-
trary-precision arithmetics the open source library GMP was used. Finally, for the
case of bond dilution (R = 0 in (4)) considered here, we made use of the fact that
the disorder average is most easily calculated via
[vn1
1 . . . vnk
k ]av = (1 − p)kvn1+...+nk
0 . (7)
4. Series analysis: techniques and results
4.1. Bond-diluted 3D Ising model
Disordered magnetic systems belonging to the 3D Ising model universality class
have been studied extensively in experiments [18–20] and also by field theoretical and
numerical methods. A comprehensive compilation of recent results can be found in
[21], showing a wide scatter in the critical exponents of different groups, presumably
due to large crossover effects.
Our high-temperature series expansion for the susceptibility up to order 19 is
given with coefficients as polynomials in p, χ(v) =
∑
n an(p)vn [22]. Therefore it
should be well-suited for the method of partial differential approximants [23] which
was successfully used to analyse the series with an anisotropy parameter describing
the crossover between 3D Ising, XY and Heisenberg behaviour [24]. But this method
was not capable of giving conclusive results. Therefore, we confined ourselves to a
single-parameter series for selected values of p.
The ratio method assumes that the expected singularity of the form
χ(v) = A(vc − v)−γ + · · · (8)
is the closest to the origin. Then the consecutive ratios of series coefficients behave
asymptotically as
rn =
an
an−1
= v−1
c
(
1 +
γ − 1
n
)
. (9)
65
M.Hellmund, W.Janke
Figure 3. Ratio approximants for different dilutions p vs. 1/n. In order to make
them visually comparable, they are (except for p = 0.75) normalized by their
respective critical couplings vc.
Figure 3 shows these ratios for different values of p. For small p they show the
typical oscillations related to the existence of an antiferromagnetic singularity at
−vc. Near the percolation threshold at pc = 0.751 188 [25] (where Tc goes to 0, vc
to 1) the series is clearly ill-behaved, related to the exp(1/T ) singularity expected
there. Besides that, the slope (related to γ) is increasing with p.
Table 2. Transition points vc = tanh(βcJ0/2) and critical exponents γ for different
dilutions p as obtained from DLog-Padé approximants.
p vc γ
0 0.21813(1) 1.2493(7)
0.075 0.23633(1) 1.2589(8)
0.15 0.25788(1) 1.2714(8)
0.225 0.28382(1) 1.2873(10)
0.3 0.31566(2) 1.305(4)
0.375 0.35557(5) 1.329(4)
0.45 0.40743(10) 1.365(6)
0.525 0.4772(2) 1.400(10)
0.6 0.576(1) 1.435(60)
The widely used DLog-Padé method consists in calculating Padé approximants
to the logarithmic derivative of χ(v),
d ln χ(v)
dv
=
γ
vc − v
+ · · · . (10)
66
Series expansions for random Potts models
The smallest real pole of the approximant is an estimation of vc and its residue
gives γ. The results presented in table 2 are the averages of 45 – 55 different Padé
approximants for each value of p, with the error in parentheses indicating the stan-
dard deviation. The scattering of the Padé approximants increases with p, getting
again inconclusive near the percolation threshold. Nevertheless, up to about p = 0.6
the series estimates for vc respectively Tc are in perfect agreement1 with the Monte
Carlo (MC) results of [26]. This is demonstrated in figure 4 where also the (prop-
erly normalized) mean-field and effective-medium approximation [27] are shown for
comparison.
Figure 4. Transition temperatures of the bond-diluted Ising model for different
dilutions p as obtained from our DLog-Padé high-temperature series (HTS) anal-
ysis and from Monte Carlo (MC) simulations [26]. For comparison the (properly
normalized) mean-field and effective-medium approximations are shown as well.
The critical exponent γ, as provided by this method, apparently varies with
the disorder strength. More sophisticated analysis methods, such as inhomogeneous
differential approximants [28,29], the Baker-Hunter method [30] or the methods M1
and M2 [31], especially tailored to deal with confluent singularities as one would
expect in a crossover situation, give improved results in the pure (p = 0) case but
do not essentially change the results in the presence of disorder.
Thus, while for theoretical reasons we still find it likely that the variation of γ
with the disorder strength can be attributed to neglected or insufficiently treated cor-
rection terms, it proved clearly impossible to verify this effect in the series analysis.
In fact, a plot of γ vs. p does not even show an indication of a plateau. In the central
disorder regime, p = 0.3−0.5, the high-temperature series estimates given in table 2
are at least compatible with Monte Carlo results for site and bond dilution [26,32,33]
which cluster quite sharply around γMC = 1.34(1). Field-theoretic renormalization
1Notice that “p” in the present notation corresponds to “1 − p” in [26].
67
M.Hellmund, W.Janke
group estimates [21,34] favor slightly smaller exponents of γRG = 1.32 − 1.33, while
experiments [18–20] report values between γexp = 1.31 − 1.44, cp., e.g., the table in
[35].
4.2. Bond-diluted 4-state Potts model
In three dimensions the 4-state Potts model exhibits in the pure case a strong
first-order transition [36] which is expected to stay first order up to some finite
disorder strength, before it gets softened to a second-order transition governed by a
disorder fixed point.
Figure 5. Ratio approximants for different dilutions p vs. 1/n (normalized by vc
as in figure 3).
In the latter regime we are interested in locating power-law divergences of the
form (8) from our susceptibility series up to order 18 [37,38]. To localize a first-order
transition point, however, a high-temperature series alone is not sufficient since there
the correlation length remains finite and no critical singularity occurs. In analysing
the series by ratio, Padé or differential approximants, the approximant will provide
an analytic continuation of the thermodynamic quantities beyond the transition
point into a metastable region on a pseudo-spinodal line with a singularity T ∗
c < Tc
and effective “critical exponents” at T ∗
c . Again we first employed the ratio method
which is the least sophisticated method of series analysis, but usually it is quite
robust and gives a good first estimate of the series behaviour. Figure 5 shows these
ratios for different values of p. They behave qualitatively similar to the Ising model
case (oscillations caused by the antiferromagnetic singularity at −vc, strong effect
of the percolation point at pc ≈ 0.75). Notice that the slope (∝ γ − 1) is increasing
with p, changing from γ < 1 to γ > 1 around p = 0.5.
68
Series expansions for random Potts models
Figure 6. Transition temperatures of the bond-diluted 4-state Potts model for
different dilution p as obtained from Monte Carlo (MC) simulations [39] and
DLog-Padé series analysis. The inset shows the difference between the two esti-
mates.
Figure 6 compares the critical temperature, estimated from an average of 25−30
Padé approximants for each value of p,2 with the results of recent Monte Carlo
simulations [39]. For small p, in the first-order region, the series underestimates the
critical temperature. As explained above, this is an estimate not of Tc but of T ∗
c .
Between p = 0.3 and p = 0.5, the estimates confirm, within errors, the Monte Carlo
results, indicating that now both methods see the same second-order transition.
Beyond p = 0.5, the scatter of different Padé approximants increases rapidly, related
to the crossover to the percolation point.
The situation is more complicated with respect to the critical exponent γ. The
DLog-Padé analysis gives inconclusive results due to a large scattering between
different Padé approximants, as shown in figure 7. One possible reason for this
failure is the existence of confluent singularities. The dots in equation (8) indicate
correction terms which can be parametrized as follows:
χ(v) = A(vc − v)−γ
[
1 + A1(vc − v)∆1 + A2(vc − v)∆2 + · · ·
]
, (11)
where ∆i are the confluent correction exponents. Among the various sophisticated
analysis methods (inhomogeneous differential approximants [28,29] and the methods
M1 and M2 [31]), in the case at hand, the Baker-Hunter method [30] appeared to be
the most successful, giving consistent results at larger dilutions p > 0.35 where the
leading-term DLog-Padé analysis failed. The Baker-Hunter method assumes that
2Again, “p” in the present notation corresponds to “1− p” in [39].
69
M.Hellmund, W.Janke
Figure 7. Scattering of different Padé approximants at dilution p = 0.4: critical
exponent γ against critical coupling vc.
the function under investigation has confluent singularities
F (z) =
N
∑
i=1
Ai
(
1 − z
zc
)−λi
=
∑
n=0
anzn, (12)
which can be transformed into an auxiliary function g(t) that is meromorphic and
therefore suitable for Padé approximation. After the substitution z = zc(1− e−t) we
expand F (z(t)) =
∑
n cntn and construct the new series
g(t) =
∑
n=0
n! cn tn =
N
∑
i=1
Ai
1 − λit
, (13)
such that Padé approximants to g(t) exhibit poles at t = 1/λi with residues −Ai/λi.
This method is applied by plotting these poles and residues for different Padé ap-
proximants to g(t) as functions of zc. The optimal set of values for the parameters
is determined visually from the best clustering of different Padé approximants, as
demonstrated in figure 8.
Using this method, our results for the critical exponent γ are plotted in figure 9.
They show an effective exponent monotonically increasing with p but reaching a
plateau at γ = 1 for dilutions between p = 0.42 and p = 0.46. The following sharp
increase is to be interpreted as due to the crossover to the percolation fixed point
at pc ≈ 0.75, Tc = 0, where a χ ∼ exp(1/T ) behaviour is expected.
It is well known (see, e.g., [40]) that series analysis in crossover situations is
extremely difficult. If the parameter p interpolates between regions governed by
different fixed points, the exponent obtained from a finite number of terms of a
series expansion should cross somehow between its universal values, and usually does
70
Series expansions for random Potts models
Figure 8. Values for the critical exponent γ and amplitude A at p = 0.4 as
function of trial vc estimates from the Baker-Hunter analysis. From the clustering
of different Padé approximants in both pictures we estimate vc = 0.3217, γ =
0.966, and A = 1.21.
Figure 9. Effective critical exponent γ as function of the dilution p from Baker-
Hunter analysis.
this quite slowly. Therefore, it does not come as a surprise that the Monte Carlo
simulations quoted above see the onset of a second-order phase transition already for
smaller values of the disorder strength p. The mere existence of a plateau in γeff(p),
however, is an indication that here truly critical behaviour is seen. It is governed by
a fixed point for which we obtain γ = 1.00(3). Here, as always in series analyses, the
error estimates the scattering of different approximants.
5. Discussion
We have implemented a comprehensive toolbox for generating and enumerating
star graphs as required for high-temperature series expansions of quenched, disor-
dered systems. Monte Carlo simulations of systems with quenched disorder require
an enormous amount of computing time because many realizations have to be simu-
lated for the quenched average. For this reason it is hardly possible to scan a whole
71
M.Hellmund, W.Janke
parameter range. Using high-temperature series expansions, on the other hand, one
can obtain this average exactly. Since the relevant parameters (degree of disorder p,
spatial dimension d, number of states q, etc.) can be kept as symbolic variables, the
number of potential applications is very large.
Here we presented an analysis of the susceptibility series for the three-dimensional
bond-diluted Ising and 4-state Potts model. The resulting phase diagrams in the p-
T -plane are in very good agreement with recent Monte Carlo results. As far as the
critical exponent γ is concerned, however, large crossover effects render a reliable
determination from series expansions up to order 19 respectively 18 very difficult.
In the Ising case we estimate values that are clearly different from the pure case but
exhibit a pronounced dependence on the degree of dilution. For the 4-state Potts
model with its strong first-order phase transition in the pure case, the singularity
structure of the disordered model is even more involved. Still, by comparing the
series expansions with numerical data we can identify signals for the onset of a
softening to a second-order transition at a finite disorder strength.
Acknowledgements
It is a great pleasure to thank Yurko Holovatch for giving us the opportunity
to contribute to the Festschrift dedicated to the 60th birthday of Reinhard Folk.
Support by DFG grant No. JA 483/17–3 and partial support from the German-
Israel-Foundation under grant No. I–653–181.14/1999 is gratefully acknowledged.
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Високотемпературні розклади для випадкової
моделі Потса
М.Гельмунд 1 , В.Янке 2
1 Інститут математики, Університет Ляйпцігу
Августуспляц 10/11, D–04109 Ляйпціг, Німеччина
2 Інститут теоретичної фізики, Університет Ляйпцігу
Августуспляц 10/11, D–04109 Ляйпціг, Німеччина
Отримано 3 грудня 2004 р.
Ми обговорюємо нещодавно генеровані високотемпературні роз-
клади для вільної енергії та сприйнятливості q -станової моделі Пот-
са на гіперкубічній гратці із безладом у формі випадкових зв’язків.
Використовуючи техніку розкладу зіркових графів, усереднення
за замороженим безладом можна провести точно при довіль-
них розподілах нескорельованих зв’язків, зберігаючи концентрацію
безладу p та вимірність гратки d як символічні параметри. Ми
представляємо аналіз нових рядів для сприйнятливості тривимірних
моделі Ізинга ( q = 2 ) та 4-станової моделі Потса до відповідно 19-го
та 18-го порядків, і порівнюємо отримані нами результати із даними
досліджень в рамках теоретико-польової ренормалізаційної групи
та симуляцій Монте Карло.
Ключові слова: випадкова модель Потса, заморожений безлад,
високотемпературні розклади, ефективні критичні показники
PACS: 05.50.+q, 64.60.Fr, 75.10.Hk, 75.10.Nr
74
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