The scaling window of the 5D Ising model with free boundary conditions

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by  P. H. Lundow

The scaling window of the 5D Ising model with free boundary conditions
P. H. Lundow1,∗ and K. Markstr¨om1,†
1Department of mathematics and mathematical statistics, Ume˚a University, SE-901 87 Ume˚a, Sweden
(Dated: January 29, 2016)
The five-dimensional Ising model with free boundary conditions has recently received a renewed
interest in a debate concerning the finite-size scaling of the susceptibility near the critical temper-
ature. We provide evidence in favour of the conventional scaling picture, where the susceptibility
scales as O(L2) inside a critical scaling window of width O(1/L2). Ourresults are based on Monte
Carlo data gathered on system sizes up to L = 79 (ca. three billion spins) for a wide range of
temperatures near the critical point. We analyse the magnetisation distribution, the susceptibility
and also the scaling and distribution of the size of the Fortuin-Kasteleyn cluster containing the
6 origin. Theprobabilityofthisclusterreachingtheboundarydeterminesthecorrelation length,and
1 itsbehaviouragrees withthemeanfieldcriticalexponentδ=3,thatthescalingwindowhaswidth
0 O(1/L2).
2
Keywords: Isingmodel,finite-sizescaling,boundary, cluster

n
a
J I. INTRODUCTION possible estimates for the scaling behaviour in the cou-

7 pling region discussed by all the previous papers. Apart
2 from the susceptibility we also study properties of the
TheIsingmodelindimensiond=5isofparticularin-

] terest since it is the firstcase where the model is strictly Fortuin-Kasteleyn cluster containing the origin and use
h above its upper critical dimension d = 4. Rigorous re- those to estimate both the susceptibility and the corre-
c
c lation length of the model.
sults [1, 2] establish that the critical exponents of the
e
m model assume their mean field values. Here the specific To concretize, the predictions from [4] are that the
heat exponent α = 0, and the results of [2] also imply location of the maximum for χ will scale as L−2 and the
t- that the specific heat is bounded at the critical point. maximumvalueasL5/2. Themorerecent[7]agreeswith
a Recent simulation results [3] also indicate that, just as these predictions, and also the prediction from [8] that
t
s for the mean fieldcase, the specific heat is discontinuous the locationofthe maximumofthe susceptibility should
t. at the critical point. scale as L2, as does [9], but both are based on smaller
a system sizes than those considered in the present work.
m In contrast to these asymptotic results there has been
In short, our conclusion is that the data is well fitted
a long running debate over the finite size scaling for the
- bytheconventionalscalingpicture,bothforthe location
d model with free boundary conditions. We’ll refer the
and value of the susceptibility, and location of the finite
n readerto [4] fora fuller overviewofthe history andstick
o to the presently most relevant parts here. For d=5 and size critical point for the magnetization.
c cyclic boundary conditions there is agreement that e.g.
[
χ∝L5/2 foralatticeofsideL. Theconventionalpicture
2 forthefreeboundarycaseisthatχ∝L2. However,ithas II. DEFINITIONS AND DETAILS
v alsobeensuggested[4]thatthefreeboundarycaseshould
3 scale in the same way as the cyclic boundary case near For a given graph G on N vertices the Hamiltonian
5
the finite size susceptibility maximum. A computational withinteractionsofunit strengthalongthe edges is H=
0
4 study [5] of the, then, largest lattices possible supported −Pijsisj wherethesumistakenovertheedgesij. Here
0 the conventional picture, but in [4] it was suggested this the graphGis a5-dimensionalgridgraphoflinearorder
. wasdue to anunderestimate ofthe influence ofthe large Lwithfreeboundaryconditions,i.e. acartesianproduct
1
boundaries of the used systems. For systems exactly at of 5 paths on L vertices, so that the number of vertices
0
6 the critical coupling this issue was resolved in [6] where is N =L5 and the number of edges is 5L5(1−1/L). We
1 a study of systems up to L = 160 demonstrated an in- useK =1/k T asthedimensionlessinversetemperature
B
: creasinglygoodagreementwith the conventionalpicture (coupling) and denote the thermal equilibrium mean by
v
i as the system size was increased. But, this left the be- h···i. ThecriticalcouplingKc wasrecentlyestimatedby
X haviour in the rest of the critical scaling window open. us [3] to K = 0.11391498(2). We will define a rescaled
c
r The aimof the currentpaper is to extendthe study of couplingasκ=L2(K−K )whichgivesascalingwindow
a c
largesystemsfrom[6]to the full criticalwindow,includ- of width O(1/L2). The standard definitions apply; the
ing the location of the maximum of χ, and give the best magnetisation is M =PiSi (summing over the vertices
i) and the energy is E = PijSiSj (summing over the
edges ij). We let m=M/N, U =E/N and U =hUi.
The susceptibility is χ=(cid:10)M2(cid:11)/N while we define the
∗ [email protected] modulus susceptibility as χ¯ = var(|M|)/N. The stan-
† [email protected] dard deviation is denoted σ, as is customary. We will
2
refer to the point where the distribution of M changes ΚcHLL
from unimodal to bimodal as K∗(L), or, in its rescaled 0.88
c
form,κ∗(L). Recallalsothatthermodynamicderivatives
c
of e.g. logχ can be obtained through correlation mea- 0.86
surements ∂log(cid:10)|M|ℓ(cid:11)/∂K =(cid:0)hE|M|ℓi/h|M|ℓi(cid:1)−hEi.
LethSidenotetheaveragesizeofaflippedclusterand
0.84
recall[10]thatχ=hSi. Weusethesubscriptotodenote
the origin (i.e. centre) vertex of the grid G so that hS i
o
isthe averagesizeofaclustercontainingthe origin. The 0.82
eventthataclustercontainingtheoriginalsoreachesthe
boundary ∂G of the graph is denoted o↔∂G. 0.80
WehavecollecteddatausingWolff-clusterspinupdat-
ingforL=15,19,23,31,39,47,55,63,79forawideand
0.78 L
dense set of κ-values in 0 ≤ κ ≤ 0.92. The data include 79 63 47 39 31 23 19
e.g. magnetisation, energy (L ≤ 63), size and radius of
the origin cluster. The total number of measurements
FIG. 1. (Colour on-line) Location κc(L) of the maximum
range from about 300000for the small systems (L≤39)
χ¯, ∂logh|m|i/∂K and ∂logχ/∂K plotted against 1/L for
down to 100000for L=63 and 50000 for L=79.
L = 19, 23, 31, 39, 47, 55, 63 and 79 (L = 79 only for χ¯).
Fitted lines are respectively 0.8763−1.21x, 0.8742−1.49x
and 0.8725−1.50x (downwards in figure) where x=1/L.
III. THE WIDTH OF THE SCALING WINDOW
AND THE SHIFT EXPONENTS
is well-fitted by the simple formula f(x)=c exp(c x2+
0 2
Webeginbyestablishingthenaturalwidthofthescal- c4x4). ThelowerinsetofFigure2showsthe distribution
ing window, i.e. the form of a natural rescaled temper- of M/σ for L = 23 at κ = 0.76 (K = 0.11535165) to-
ature. We have already defined the rescaled coupling as gether with the fitted formula. The binning of the data
κ = L2(K −K ) with the expectation that this will be wasdonewithMathematica’sbuilt-inproceduresbutthe
c
therelevantscaling. Inprinciple,thelocationofdifferent results are not very sensitive to this choice. At this κ
effective finite size critical points could scale with differ- (i.e. for L=23) the distribution is very close to shifting
ent exponents and we will compare a number of such from unimodal to bimodal, as is indicated by the very
possibilities. small coefficient c2 = −0.002735. The distribution has
We used a 2nd order interpolation of the data points kurtosis 2.2217 and when c2 = 0 the kurtosis becomes
toestimatethe locationofthe maximumforthreediffer- Γ(1/4)4/8π2 ≈ 2.1884. In fact, it works equally well to
ent parameters: χ¯, ∂logh|m|i/∂K and ∂logχ/∂K. For simply measure the kurtosis and then estimate through
the last two we only have these data for L ≤ 63. The interpolation at which point the kurtosis is 2.1884. All
locations are plotted versus 1/L in Figure 1. The values this suggests that the distribution’s shape is well cap-
clearly point to three different limit κ-values. Based on tured by f(x). See also [3] where we apply this formula
L ≥ 19 we estimate the limit values and an error bar to 5D grids with periodic boundary conditions.
by removing each of the data points in turn and fitting TheupperinsetofFigure2showsthemeasuredcoeffi-
a line to the remaining points. Thus we find κ (L) = cientc plottedagainstκforarangeofL. Thevalueofc
c 2 4
0.8763(5) − 1.21(2)/L for the χ¯-maximum, κ (L) = (not shown) at κ∗(L) is somewhat noisy but it seems to
c c
0.8742(5)−1.49(1)/Lfor the maximum ∂h|m|i/∂K and converge to −0.114(3). For a cyclic boundary this value
0.8725(3)−1.50(1)/Lfor the maximum ∂logχ/∂K. We is close to −0.602 [11].
conclude that our rescaledcoupling κ is the correctscal- Figure 2 shows κ∗(L) (where c = 0) versus 1/L.
c 2
ing,andeachoftheeffectivecriticalpointsscalewiththe The fitted 2nd degree polynomial suggests κ∗(L) =
c
same exponent. 0.8561(6)−2.52(3)/L+7.6(4)/L2, with error bars ob-
tained as above, so that κ∗ ≈ 0.856. This point then
c
representswherethetruelow-temperatureregionbegins.
IV. THE LOW AND HIGH-TEMPERATURE The point κ∗(L) constitutes an effective critical tem-
REGIONS c
perature but its scaling rule appears to depend strongly
on the boundary conditions. Note that for periodic
Next we estimate the location of the point where the boundaryconditionsthecorrespondingpointK∗(L)con-
c
magnetisationdistributionswitchesfromunimodaltobi- verges to K with rate 1/L3 [12], i.e. faster than the
c
modal, denoted κ∗(L). This point will be seen as the width of the scaling window, which is 1/L5/2 in that
c
effective finite size separator between the high and low- case. For free boundary conditions we instead see that
temperature regions of the model. K∗(L) → K with the same rate as the width of the
c c
To do this we note that near this point the standard- scaling window, i.e. 1/L2.
ised distribution of M (i.e. of M/σ where σ2 =var(M)) We note that in [7] an estimate based in the kurtosis
3
ΚcHLL Χ(cid:144)L2
10 ç
0.85 1.0c2 æ àì òôáó
0.80 óæà 0P.3rH0M(cid:144)ΣL óæà æà--000000æà......422468 ìòæà ìæà 0.ìçóáòôæíà70ìæàìòôæàæìæàæìæà0à.ìçóáòôæíà7à5ìòæàììòàììçóáòôíàòìòô0òô.çóáòôí8çô0çóáòôíçáíçóáôíóáíçóáíóóí0.85 Κ 68 çô çô çô çô çô çòô òô
0.25
4 ò
0.20 ò ì
0.75 0.15 ò ò
ò ì
0.10 ì
0.05 2 ì ì ì ì à
-3 -2 7-9163 471 39 2 313 Σ 23 19 15 L 0 7æà9 6æà3 æà 4æà7 3æà9 3æà1 2æ3 L
FIG. 2. (Colour on-line) Effectivecritical temperatureκ∗c(L) FIG. 3. (Colour on-line) Normalised susceptibility χ/L2 at
(where c2 = 0) versus 1/L for L = 15, 19, 23, 31, 39, 47, fixed κ versus 1/L for L = 23, 31, 39, 47, 55, 63, 79 at
2
55, 63, 79. The fitted curve is 0.856−2.52x+7.6x where
κ = 0.70, 0.75, 0.80, 0.83, 0.85, 0.86 (upwards). The lines
x=1/L. Lower inset: Probability densityfunction Pr(M/σ)
are fitted on L≥31. Error bars are smaller than the points.
(dots)forL=23atκ=0.76(K =0.11535165). Fitted(red)
curve is 0.3223exp(−0.002735x2 −0.1164x4). Upper inset:
Coefficientc2ofthefitteddensityfunctionf(x)plottedversus
κ for L=15, 19, 23, 31, 39, 47, 55, 63, 79 (left to right).
Χ(cid:144)L2
1.4
was used, but instead of using the universal kurtosis-
valuegivenatthe pointwherec2 =0theyhandpickeda 1.2
ì
valueforthekurtosisandthenusedthefactthatallfixed ì
ssumcahllevralcuoersreschtoiounlds tgoivtehtehsecasalimnge.scaling, with larger or 01..80 òà òà ìòà ìòæà ìæà ìæà æà
0.6 ôæ ôæ ôæ ô òô ò ìò
V. SCALING OF SUSCEPTIBILITY 0.4 ç ç ç ç ç çô çô
0.2
Let us now return to the matter of the finite-size scal-
ing of χ and χ¯. With the onset of the low-temperature
0.0 L
regionat(asymptotically)κ∗ =0.856welookatthescal- 79 63 47 39 31 23
c
ingbehaviourofχatfixedκbelowandabovethispoint.
Consider Figure 3 where we show χ/L2 versus L for six
FIG. 4. (Colour on-line) Normalised modulus susceptibility
different fixed κ-values. Lines are fitted on L ≥ 31 to χ¯/L2 at fixed κ versus 1/L for L=23, 31, 39, 47, 55, 63, 79
demonstrate the presence of correction terms. The be- at κ = 0.82, 0.84, 0.86, 0.88, 0.90 and 0.92. Downwards at
haviourisclearlylinearforlargerL,withthepossibleex- y-axisisκ=0.86,0.88,0.84,0.90,0.92,0.82. The2nddegree
ceptionofκ=0.86,i.e. aboveκ∗ =0.856where we have curves are fitted on L≥31. Error bars are smaller than the
c
entered the low-temperature region and have a bimodal points.
magnetization, but even here the correction term is still
ratherweak. However,higherκ-valuesrenderusstronger
corrections. The normalised susceptibility was also used
in[7], but there the estimates,which leadto scaling pro-
portional to L5/2 were based on much smaller lattices values,basedon2nddegreepolynomialsfittedonL≥31,
L≤36, where finite size effects are much stronger. we now obtain Figure 5.
Movingontothemodulussusceptibilityχ¯/L2 thecor-
rections become clear and present, even for κ ≤ 0.856, For this range of lattice sizes and at least some values
but never more than can be captured by a simple 2nd ofκonecanmakeareasonablefittothedatawithaL5/2
degree polynomial. Figure 4 shows this effect. Note scaling as well, by including sufficient correction terms.
also that once we have gone beyond the maximum at But,aswecanseeinFigure5,theL2scalinggivesvalues
κ = 0.876 the behavior quickly becomes linear again as that decrease to their limit value for a large range of κ,
is clearly seen at κ=0.92, which is where our data end. possibly all κ below κ∗, making an even larger power of
c
Combining all the measuredχ¯/L2 with their asymptotic L seem unlikely here.
4
Χ(cid:144)L2 XSo\(cid:144)L2
1.5 æ æ á
æ 40 á
ç
á
01..50ìçóáòôææíà ìæà ìæà ìòæà ìæà ìçóáòôææíà ìæà ìòôæàæìæàæìæààìçóáòôææíààìòæàììòæàììçóáòôææíàòìòôæàòôìçóáòôææíàçôìçóáòôææíàçáíìçóáòôææíàóáíìçóáòôææíàóìçóáòôææíà ìçóáòôææíà ìçóáòôæíà ìçóáòôæíà ìçóáòôæíà ìçóáòôææíà ìçóáòôææíà ìçòôæà ìçóáòôææíà 123000 ìçáòôæà ìçáòôæà ìçáòôæà ìçáòôæà ìçòôæà ìçòôæà ìòôæà
0.65 0.70 0.75 0.80 0.85 0.90 Κ 0 79 63 47 39 31 23 L
FIG. 5. (Colour on-line) Normalised modulus susceptibility FIG. 6. (Colour on-line) Normalised size of origin cluster
χ¯/L2 versus κ for L=15, 19, 23, 31, 39, 47, 55, 63, 79. The hSoi/L2 at fixed κ versus 1/L for L=23, 31, 39, 47, 55, 63,
thick red curve is the estimated limit found by fitting a 2nd 79atκ=0.70,0.75,0.78,0.80,0.82,0.83and0.84(upwards).
degreepolynomialin1/LtothedatapointsforL≥31ateach Lines fitted on L≥23.
fixed κ. The left vertical line at κ=0.856 (orange) indicates
wLh→ere∞th.eTmheagrnigehtitsavteirotnicdalisltirnibeuattioκn=be0c.o8m77es(bbliumeo)diaslwwhheerne lnHPrHSoLL 1 2 3 4 5 6 lnHSoL
2
themaximum χ¯/L is located when L→∞.
-2
VI. THE FK-CLUSTER AT THE ORIGIN -4
OurnextobjectofstudyistheFortuin-Kasteleynclus- -6
ter containing the origin. This is the cluster which the
-8
Wolff random cluster algorithm builds when started at
the central vertex of our lattice, and is one of the clus-
-10
tersintheFK-randomclusterrepresentationoftheIsing
model. From the FK-model the usual susceptibility can
-12
be computed as the expected cluster size, and with free
boundary the expected size of the origin cluster will be
larger than the average cluster size, thereby giving an FIG. 7. (Colour on-line) Log-log plot of the size distribution
upper bound on the susceptibility. This observation was of the origin cluster for L = 63 and κ = 0.85. The red line
used in [6] to give a susceptibility estimate which was has slope −4/3.
hopedtobelessaffectedbytheeffectsofthefreebound-
ary, in part as a response to the truncation method of
[4]. here having the mean field value δ =3.
As described we expect hS i to place an upper bound In Figure 7 we show a log-log plot of the distribution
o
on χ, but we do not expect them to be of different scal- density (for small x = So) together with a line having
ing orders. Plotting hS i/L2 versus 1/L demonstrates a slope −4/3, consistent with δ =3. The line is clearly an
o
similar behaviour to that of χ/L2. In Figure 6 we show excellent fit. The plot shows the distribution for L=63
the scaling at a number of different κ-values below κ∗. at κ = 0.85 but for this range of cluster sizes the plot is
c
In fact, a strong correction enters the picture already at indistinguishablefromthatofotherκ-values,andindeed
κ=0.85(notshowninthefigure). Thisistobeexpected of other L (except the smallest L).
since above κ∗ there are additional terms from the FK-
c
modelaffectingtheIsingsusceptibility,andthosearenot
VII. CORRELATION LENGTH
included in our sampled data.
The distribution of the origincluster size S is also an
o
interesting object. It is a Pareto-like distribution with The origin cluster can also be used to estimate the
density proportional to x−1−1/δ, or, by definition [13], correlationlengthofthe model. Inthe hightemperature
region the correlation length ξ is given, see [13], by the
Pr(So ≤x)=1−x−1/δ (1) limit
whereδisthecriticalexponentreflectinghowanexternal 1 −logPr(o↔∂G)
= lim (2)
fieldaffectsthemagnetisationatthecriticaltemperature, ξ L→∞ L
5
aHΚL random spanning tree for the lattice.
PrHoǦGL
0.21 For the random spanning tree Pemantle [14] proved
ô
0.8 ò æ that for large enough d the distance between two points
0.20 0.6 ô òô ì in the tree scales as L2 with free boundary and as Ld/2
ò ì à æ with cyclic boundary. For percolation, Aizenmann [15]
00..1189 00..24 ìòôæà ìòôæàìòôæà ìòôæà ìòôæà ìòôæà ìæà æà æ ææ cclounsjteecrtushreodultdhastcafoleralasrgLe4ewniotuhgfhredetbhoeusnizdeaoryftahnedlaLrg2de/s3t
with cyclic boundary. This conjecture was later proven
æ
0.17 0.0 L ææææ in[16,17]. So,forbothq =0andq =1wehaverigorous
7963 47 39 31 23 19 15 æ æ
results showing that free andcyclic boundary conditions
0.16 æ æ
æ lead to distinct scalings.
0.15 æ Aizenman’s conjectured scaling L2d/3 came from the
æ
fact that the cyclic boundary case was expected to be-
Κ
0.0 0.2 0.4 0.6 0.8 have like the critical Erdo˝s-Renyi random graph, where
themaximumclusterhassizeN2/3,whereN isthenum-
berofvertices,whichwouldbe Ld forthe lattice. Again,
FIG.8. (Colouron-line) Correlation coefficient a(κ)versusκ
therearedetailedrigorousresults[18]concerningtheran-
basedon fitted2nddegreepolynomialsforL≥15(seetext).
dom cluster model on the complete graph and in [19] it
Note the sudden growth that sets in near κ = 0.85. Inset:
was demonstrated that Monte Carlo data for the FK-
ξ/Lplottedagainst 1/LforL=23, 31,39, 47, 55,63 and79
at at κ = 0, 0.40, 0.70, 0.80, 0.84, 0.88 and 0.92 (upwards). model with q = 2 and d = 5 with cyclic boundary gives
Curves are 2nd degree polynomials fitted to 15≤L≤79. a detailed agreement with the scaling for the complete
graph. Inparticular,thesizeofthelargestclusterhasthe
samescalingforbothgraphsforseveraldifferentrescaled
For a fixed κ we expect not just ξ →∞, but rather that coupling ranges.
the correlation length is comparable to L, i.e. ξ/L → Now,the L2 scaling observedin this paper for the ori-
a(κ) for some function a. Thus we propose to estimate gin cluster can be seen as an indicator of the correct
a(κ) by first defining ξ/L = −1/logPr(o ↔ ∂G) and scaling of the largest cluster as well. This would give a
then use a simple projection rule to estimate the limit scalingwhichisdifferentforthefreeandcyclicboundary
function a(κ). cases, just as one would expect from the known results
TheinsetofFigure8showsξ/Lversus1/Lforarange for lower q. In [5] the authors conjectured that for ev-
ofLatdifferentκ-values. Figure8alsoshowsaroughes- ery q there is a d(q) such that above this dimension the
timate of a(κ) basedupon fitted 2nd degreepolynomials FK-model with cyclic boundary behaves like the model
(on L≥15) for each κ. We expect the errors in the plot on a complete graph, giving a partial generalisation of
to be significant although the general behaviour as such Aizenmann’s conjecture for all q. In fact, we also expect
is correct. In the low-temperature region the identity in the model with free boundary to follow a simpler mean-
Equation2isnolongervalid,andtheprobabilitythatthe field scaling, similar to that seen on an infinite tree, or
origin cluster reaches the boundary tends to 1, and this Bethe lattice. The fact that there are several distinct
agreeswell with the steep growthbeginning at κ≈0.85, basic model systems, e.g. the complete graphs and the
the start of the effective finite size low-temperature re- Bethelattices,whichononehandhavemeanfieldcritical
gion. exponents but on the other hand differ on more detailed
propertiesis probablyunder-appreciatedinthe olderlit-
erature.
VIII. CONCLUSIONS
As we have seen here our data fits excellently with
ACKNOWLEDGMENTS
the conventional scaling picture for the free boundary
case. Now, apart from this fact we believe that there
are good reasons for expecting distinct behaviour from Thesimulationswereperformedonresourcesprovided
the free and cyclic boundary cases. As is well known by the Swedish National Infrastructure for Computing
the Ising model is equivalent to the Fortuin-Kasteleyn (SNIC) at High Performance Computing Center North
random cluster model for q = 2. The case q = 1 cor- (HPC2N) and at Chalmers Centre for Computational
responds to ordinary percolation and q → 0 gives the Science and Engineering (C3SE).
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