Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions

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by  Youjin Deng

Ferromagnetic phase transition for the spanning-forest model
(q → 0 limit of the Potts model) in three or more dimensions
Youjin Deng,1,∗ Timothy M. Garoni,1,† and Alan D. Sokal1,2,‡
1Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA
7 2Department of Mathematics, University College London, London WC1E 6BT, UK
0 (Dated: 6 October 2006; published 17 January 2007)
0
2 WepresentMonteCarlosimulationsofthespanning-forestmodel(q→0limitoftheferromagnetic
Potts model) in spatial dimensions d = 3,4,5. We show that, in contrast to the two-dimensional
n
case, the model has a “ferromagnetic” second-order phase transition at a finite positive value w .
a c
J We present numerical estimates of wc and of the thermal and magnetic critical exponents. We
conjecture that the uppercritical dimension is 6.
7
1
PACSnumbers: 05.50.+q,11.10.Kk,64.60.Cn,64.60.Fr

]
h
c The Potts model [1, 2] plays an important role in the onto an N-vector model [O(N)-invariant σ-model] ana-
e modern theory of phase transitions and critical phenom- lytically continued to N = −1. It follows that, in two
m
ena,andischaracterizedbytwoparameters: thenumber dimensions, the spanning-forest model is perturbatively
- q of Potts spin states, and the nearest-neighborcoupling asymptotically free, in close analogy to (large classes
t
a v =eβJ−1. Initiallyq isapositiveintegerandv isareal of)two-dimensionalσ-modelsandfour-dimensionalnon-
t
s number in the interval −1 ≤ v < +∞, but the Fortuin– abelian gauge theories. In particular, the only ferromag-
.
t Kasteleyn (FK) representation [3] shows that the parti- netic (w > 0) critical point lies at wc = +∞, in agree-
a
tion function Z (q,v) of the q-state Potts model on any ment[12]withtheexactsolutionsonthesquare,triangu-
m G
finite graph G is in fact a polynomial in q and v. This lar and hexagonal lattices [5] showing that v (q) ∝ q1/2
c
-
d allows us to interpret q and v as taking arbitrary real as q ↓0.
n or even complex values, and to study the phase diagram
In this Letter we study the spanning-forest model in
o of the Potts model in the real (q,v)-plane or in complex
c spatial dimensions d ≥ 3, using Monte Carlo methods.
(q,v)-space. In particular, when q,v > 0 the FK rep-
[ We will show that, in contrast to the two-dimensional
resentation has positive weights and hence can be inter-
case,themodelhasa“ferromagnetic”second-orderphase
2 preted probabilistically as a correlated bond-percolation
v model: the FK random-cluster model [4]. Inthis waywe transition at a finite positive value wc, and we will esti-
3 matethethermalandmagneticcriticalexponentsaswell
can study all positive values of q, integer or noninteger,
9 asauniversalamplituderatio. Itfollowsthatv (q)∝qas
1 within a unified framework. c
q ↓0. Indeed, we see the present study of the spanning-
0
In two dimensions, the behavior of the ferromagnetic forestmodelasthefirststepinacomprehensivestudyof
1
6 (v > 0) Potts/random-cluster model is fairly well un- therandom-clustermodelasafunctionof(noninteger)q.
0 derstood, thanks to a combination of exact solutions [5],
For the random-clustermodel with q ≥1, a collective-
/ Coulomb-gas methods [6] and conformal field theory [7].
t mode Monte Carlo algorithm has recently been invented
a But indimension d≥3,many importantaspects remain
m byChayesandMachta[13];itgeneralizesthewell-known
unclear: the location of the crossover between second-
Swendsen–Wang algorithm [14] and reduces to (a slight
- orderandfirst-orderbehavior[8]; the natureofthe criti-
d variant of) it when q is an integer. But for q < 1, the
calexponentsandtheirdependenceonq;thevalueofthe
n only available algorithm seems to be the Sweeny algo-
o upper critical dimension for noninteger q; and the quali-
rithm [15], which is a local bond-update algorithm. Or-
c tative behavior of the critical curve v (q) near q =0.
: c dinarily one would expect such a local algorithm to ex-
v
Interesting special cases of the random-cluster model hibit severecritical slowing-down,atleastwhen the spe-
i
X arise in the limit q →0. In particular, the limit q,v →0 cificheatisdivergent[16]. Buttherandom-clustermodel
r withw =v/q heldfixedgivesrisetoamodelofspanning withq <q0(d)≈2hasanon-divergentspecificheat(i.e.,
a forests, i.e. spanning subgraphs without cycles, in which criticalexponentα<0), whichsuggeststhat the critical
each occupied edge gets a weight w [9]. Very recently, slowing-down might not be so severe after all. Indeed,
it was shown[10] — generalizing Kirchhoff’s matrix-tree our numerical studies of the spanning-forest model (i.e.,
theorem [11] — that this spanning-forest model can be the q → 0 limit) in dimensions d = 2,3,4,5 strongly
mappedontoafermionic(Grassmann)theoryinvolvinga suggest that there is no critical slowing-down, i.e., the
quadratic(Gaussian)termandaspecialnearest-neighbor dynamic critical exponent z associated to the expo-
exp
four-fermion term. Moreover, this fermionic model pos- nential autocorrelation time is zero. Better yet, the ex-
sesses an OSP(1|2) supersymmetry andcan be mapped, ponentz associatedtotheintegrated autocorrelation
int,O
toallordersoftheperturbationtheoryinpowersof1/w, time[17]turnsouttobenegative for“global”observables
2
such as the mean-square cluster size; that is, one “effec-
1
tively independent” sample can be obtained in a time
much less than a single “sweep” — a kind of “critical
speeding-up”. 00..88
Onthe otherhand,the Sweenyalgorithmforq 6=1re-
quires a non-local connectivity check each time one tries 0.6
toupdateasinglebond. Ifdoneinthenaiveway(e.g.,by
R
6
depth-first or breadth-first search), this would require a
CPUtimeoforderthemeanclustersizeχ∝Lγ/ν =L≈2 0.4 8
12
per “hit” of a single bond, leading to a severe “compu-
tational critical slowing-down”. Recent work by com- 0.2 16
24
puter scientists on dynamic connectivity algorithms [18]
32
shows how this can be reduced to (logL)p, but at the 0
expense of fairly complicated algorithmsand data struc- 0.2 0.3 0.4 0.5
w
tures. We therefore adopted an intermediate solution:
a simple “homemade” dynamic connectivity algorithm Figure 1: Coarse plot of R versus w for spanning forests in
that empirically has a slowing-down L≈0.7. The details dimension d=3 and lattice sizes 6≤L≤32.
of this algorithm, along with measurements of the dy-
namic critical behavior of the Sweeny algorithm in the
spanning-forest limit, will be reported separately [19]. 0.861
Wesimulatedthespanning-forestmodelindimensions
d=3,4,5onhypercubic lattices ofsize Ld with periodic
boundary conditions. We measured the cluster-size mo-
mentsS = #(C)k fork =0,2,4. Wefocussed 0.86
k clustersC
attentiononPthe ratioR=hS4i/hS22i, whichtends inthe
R
infinite-volume limit to 0 in a disordered phase and to 1
inanorderedphase,andisthereforediagnosticofaphase 32
0.859
transition. We also studied hS i in order to estimate the 40
2
64
magnetic critical exponent.
80
Ineachdimension, webeganby makinga“coarse”set
120
of runs coveringa wide range of w values, using modest- 0.858
sizedlatticesandmodeststatistics. Iftheplots ofRver- 0.4334 0.4336 0.4338
sus w indicated a likely phase transition, we then made w
a “fine” set of runs coveringa smallneighborhoodof the
Figure2: “Super-fine”plotofRversuswforspanningforests
estimated critical point, using larger lattices and larger
in dimension d=3 and lattice sizes 32≤L≤120.
statistics. Finally, using the results from these latter
runs, we made a “super-fine” set of runs extremely close
totheestimatedcriticalpoint,usingaslargelatticesand 0.43365±0.00002,the critical exponents ν =1.28±0.04
statistics aswe couldmanage,withthe goalofobtaining and γ/ν =2.1675±0.0010, and the universal amplitude
precisequantitativeestimatesofthecriticalpointwc and ratio Rc = 0.8598±0.0003 (68% subjective confidence
thecriticalexponents. Thecompletesetofrunsreported intervals, including both statistical error and estimated
in this Letter used approximately 7 years CPU time on systematic error due to unincluded corrections to scal-
a 3.2 GHz Xeon EM64T processor. ing). A finite-size-scaling plot using these parameters is
The “coarse” plot of R versus w for dimension d = 3 shown in Figure 3. A “coarse” plot of hS i/Lγ/ν using
2
and lattice sizes 6 ≤ L ≤ 32 is shown in Figure 1, and the estimated value of γ/ν is shown in Figure 4.
showsaclearorder-disordertransitionatwc ≈0.43. The The “coarse” plots of R versus w for dimensions d =
corresponding “super-fine” plot, for lattice sizes 32 ≤ 4,5 are shown in Figures 5 and 6, respectively. Once
L≤120,isshowninFigure2. WefitthedatatoAnsa¨tze again they show a clear order-disorder transition. For
obtained from lackof space,we refrainfromshowingthe corresponding
R = R + a (w−w )L1/ν + a (w−w )2L2/ν “super-fine” plots (which use lattice sizes up to 644 and
c 1 c 2 c
205) and simply give the results of fits to Ansa¨tze of the
+b L−ω1 + b L−ω2 + ... (1)
1 2 general type (1). In dimension d = 4, we estimate w =
c
by omitting various subsets of terms, and we systemat- 0.210302±0.000010, ν = 0.80±0.01, γ/ν = 2.1603±
ically varied L (the smallest L value included in the 0.0010and R =0.73907±0.00010. In dimension d=5,
min c
fit). We also made analogous fits for hS i/Lγ/ν. Com- we estimate w = 0.14036±0.00002, ν = 0.59±0.02,
2 c
paring all these fits, we estimate the critical point w = γ/ν =2.08±0.02 and R =0.625±0.015.
c c
3
0.95 1
0.9
0.8
0.85
8
0.6
12
R 0.8 16 R
24 0.4 4
0.75
32 6
0.7 40 0.2 8
64 10
120 12
0.65 0
-0.4 -0.2 0 0.2 0.4 0.12 0.13 0.14 0.15 0.16
1/ν
(w-w )L w
c
Figure 3: Finite-size-scaling plot of R versus (w−w )L1/ν, Figure 6: Coarse plot of R versus w for spanning forests in
c
with w = 0.43365 and ν = 1.28, for spanning forests in dimension d=5 and lattice sizes 4≤L≤12.
c
dimension d=3 and lattice sizes 8≤L≤120.
3
γ/ν=2.1675 q=0 q=1 q=2
6
8 ν =∞ ν =4/3 ν =1
d=2
2 12 γ/ν =2 γ/ν =43/24 γ/ν =7/4
γ/ν 16 d=3 ν =1.28(4) ν =0.874(2) ν =0.6301(5)
-L 24 γ/ν =2.1675(10) γ/ν =2.0455(6) γ/ν =1.9634(5)
2 32
S ν =0.80(1) ν =0.689(10) ν =1/2(log)
1 d=4
γ/ν =2.1603(10) γ/ν =2.094(3) γ/ν =2(log)
ν =0.59(2) ν =0.57(1) ν =1/2
d=5
γ/ν =2.08(2) γ/ν =2.08(2) γ/ν =2
0
TableI: Critical exponentsν andγ/ν versusq andd. d=2:
0.2 0.3 0.4 0.5
w presumed exact values [20]. d = 3,4,5, q = 0: this work.
d=3, q =1: [21]. d=3, q =2: [22]. d=4, q =1: [23, 24].
Figure4: PlotofhS2i/Lγ/ν versusw,withγ/ν =2.1675,for d=5,q=1: [24,25]. d=4,5,q=2: presumedexactvalues.
spanningforestsindimensiond=3andlatticesizes6≤L≤
32.
In Table I we summarize the estimated critical ex-
ponents for ferromagnetic Potts models with q = 0
1
(this work), 1 (percolation) and 2 (Ising) in dimensions
d = 2,3,4,5. It is evident that ν varies quite sharply as
0.8
afunctionofq andd,whileγ/ν variesmuchmoreslowly.
The dimension-dependences of ν and γ/ν for q = 0 are
0.6 consistent with the conjecture that they are tending to
R the mean-field values 1/2 and 2 in dimension d=6, just
4
0.4 6 as they do for q = 1. This in turn supports the more
general conjecture that the upper critical dimension is 6
8
0.2 12 for all random-cluster models with 0 ≤ q < 2, and is 4
16 only when q =2.
20 This conjecture is supported by a field-theoretic
0 renormalization-groupcalculationindimensiond=6−ǫ
0.16 0.18 0.2 0.22 0.24 0.26 throughorderǫ3 [26]inwhichq =2playsadistinguished
w
role (all the correction terms vanish there). Specializing
Figure 5: Coarse plot of R versus w for spanning forests in to q =0, we have
dimension d=4 and lattice sizes 4≤L≤20.
ǫ 7ǫ2 26ζ(3) 269
γ/ν = 2+ + − − ǫ3+O(ǫ4)
15 225 (cid:18) 625 16875(cid:19)
4
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