Slow relaxation in the Ising model on a small-world network with strong long-range interactions

0.15 MB·English

by  Daun Jeong

Slow relaxation in the Ising model on a small-world network
with strong long-range interactions
Daun Jeong,1 M.Y. Choi,1,2 and Hyunggyu Park2
1Department of Physics, Seoul National University, Seoul 151-747, Korea
2School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
5
0
WeconsidertheIsingmodelonasmall-world network,wherethelong-rangeinteraction strength
0
J2 isingeneraldifferentfromthelocalinteractionstrengthJ1,andexamineitsrelaxationbehaviors
2
as well as phase transitions. As J2/J1 is raised from zero, the critical temperature also increases,

n manifesting contributions of long-range interactions to ordering. However, it becomes saturated
a eventuallyatlargevaluesofJ2/J1 andthesystemisfoundtodisplayveryslowrelaxation,revealing
J that ordering dynamics is inhibited rather than facilitated by strong long-range interactions. To

6 circumvent this problem, we propose a modified updating algorithm in Monte Carlo simulations,
assisting thesystem to reach equilibrium quickly.

]
h
PACSnumbers: 89.75.Hc,89.75.-k,75.10.Hk
c
e
m
When randomlinks are added to a regularlattice, the display long-range order. In the case of uniform interac-
- latter becomes a small-world network, characterized by tion (J = J ), the system is known to undergo a phase
t 1 2
a short path length and high clustering [1, 2]. In such a transition of the mean-field type [3, 7, 8]. It is expected
t
s small-world network, the diameter increases very slowly that the mean-field transition is prevalent for all finite
t. with the system size: l ∼ logN, while a regular one dis- values of J2/J1 as long as shortcuts account for a finite
a
plays l ∼O(N). Also having common neighbors for two fraction of total links. Shortcuts in general assist spins
m
connectednodesishighlyprobable. Withthesefeatures, to order,which is reflected by the increase of the critical
- all elements on the network can exchange information temperature;similartrendshavebeenfoundinanalytical
d
with each other more efficiently than on a regular lat- studiesofslightlydifferentsystemsinequilibrium[9,10].
n
o tice. Accordingly, it is expected that dynamical systems Thisworkfocusesonhowthe systemapproachesequi-
c on small-world networks may display enhanced perfor- librium, and reveals that strong long-range interactions
[ mance; examples include ordering in spin models [3, 4], (J /J ≫1)givesrisetoextremelyslowrelaxation,mak-
2 1
1 synchronization in coupled oscillators [5], and computa- ingMonteCarlo(MC)dynamicsbasedontheMetropolis
v tional performance of neural network [6]. algorithminefficient. Toavoidsuchslowconvergence,we
9 devise a modified updating algorithm, which assists the
A small-world network, constructed from a one-
9 system to reach equilibrium more quickly. In the limit-
dimensionallattice,hastwokindsofconnections: (short-
0
ing case that J = 0, namely, all nearest-neighbor inter-
1 range) local links and (long-range) shortcuts. It is con- 1
actions are deleted, the remaining links (shortcuts) con-
0 ceivable that the two kinds of couplings in a real system
stitute a randomnetworkwith connectivitykP, where k
5 have different origins and thus different strengths; this
0 makes it desirable to examine the general case that in- denotes the range of local interaction in the underlying
/ lattice and P is the probability of adding or re-wiring
t teractions via local links and via shortcuts are different
a shortcuts on each local link.
in strength. It is obvious that in the absence of long-
m Herethesmall-worldnetworkisconstructedinthe fol-
range interactions (via shortcuts), long-range order does
- notemerge. Asasmallamountofweaklong-rangeinter- lowingway: Wefirstconsideraone-dimensional(1D)lat-
d
ticeofN nodes,eachofwhichisconnectedtoits2knear-
n actions is introduced, however, the system undergoes a
est neighbors, with k being the local interaction range.
o phasetransitiontothe state withlong-rangeorder. This
Then each local edge is visited once and a random long-
c indicates the importance of shortcuts in ordering, and it
v: is of interest to elucidate how much they are important, range connection (shortcut) is added with probability P
(without removing the local edge). Note the difference
i relativetolocallinks. Onthecontrary,withoutneighbor
X from the original Watts and Strogatz (WS) construc-
(local)interactions,the systemcannotpercolatebelow a
r certainvalueofconnectivity. Therefore,weconcludethat tion [2], where local edges are removed and reconnected
a to randomly chosen nodes.
highclusteringduetolocalinteractionsisalsoimportant
The Hamiltonian for the Ising model on a small-world
for achieving long-range order.
network with such two kinds of interactions is given by
Toprobetherolesoflong-andshort-rangeinteractions
in ordering, we consider the Ising model as a prototype k
system exhibiting an order-disorder transition, and ex- H =−J1XXσiσi+j −J2Xσiσj, (1)
amine the transition behavior on a small-worldnetwork, i j=1 hi,ji
varyingthelong-rangeinteractionstrengthJ relativeto
2
the short-range strength J . When J = 0, the system whereσ (=±1)istheIsingspinonnodeiofthenetwork.
1 2 i
reducestothe one-dimensionalIsingmodelanddoesnot The first term is precisely the Hamiltonian for 1D Ising
2
theinteractionstrengthfarlargerthanthelocal(nearest
0.1
neighbor) one. As the temperature is lowered, spins on
suchclustersalignfirstalongeitherthe up- orthe down-
directionwhileotherspinsonthe1Dchainflipeasilybe-
causethermalfluctuationsarestillstrongcomparedwith
local interactions. For the whole spins to be aligned be-
〉m 0.05 low the critical temperature, all clusters should have the
〈 same spin orientation; otherwise some spins (which do
not have long-range interactions) may confuse between
spin clusters of different spin directions. However, it is
not probable for a spin in the cluster to have opposite
directions, due to the strong long-range interactions at
0 suchlowtemperatures. Thisyieldslowacceptanceratios
0 1 2 3 4 in the algorithm, resulting in extremely long relaxation
T/J1 time. Accordingly, the system tends to remain in a dis-
orderedstatewhichdoesnotcorrespondtotheminimum
FIG. 1: Erratic behavior of the order parameter with tem-
of the free energy, even if the temperature is lower than
perature T/J1 (with the Boltzman constant kB ≡ 1) for
the critical temperature. Finally, at very low tempera-
J2/J1 = 10 and P = 0.1. The magnetization m has been
obtained from the average over 5×104 MC steps after the tures, spins seldom flip, so that the value of the order
dataduringinitial5×104 MCstepsdiscarded. Data,labeled parameter depends on the previous history.
byfivesymbols,representtheresultsoffiveMCruns,respec- Weexaminerelaxationoftheorderparameter,starting
tively,onasinglesmall-worldnetwork. Themagnetizationat fromthefullyorderedstatenearthecriticaltemperature
low temperatures varies largely run by run, and the system T andfromthe disorderedstateatlowtemperatures,to
c
persists to remain in the disordered state.
measure the characteristic time scale for the system to
reach equilibrium. Assuming the exponential relaxation
in the form |m−m |∼e−t/τ, we estimate the value of
eq
model with k-nearest neighbors whereas the second one τ, varying J /J and P. Figure 2 shows the relaxation
2 1
describes the contributions of spin pairs connected via timeτ,measuredinunitsoftheMCstep,inthesystemof
long-range connections. size N = 6400, (nominally) at the critical temperature.
We perform extensive MC simulations at various val- It is observed that τ grows exponentially from 102 to
ues of the addition probability P and the coupling ratio 108 as J2/J1 is increased. For given value of J2/J1, τ
J /J . Specifically, we anneal the system, starting from is shown to depend algebraically on P: τ ∼ P−σ. We
2 1
disordered states at high temperatures, and employ the stressthatthesefeaturesarenotrestrictedmerelytothe
standard Metropolis algorithm with single-spin flip up- region near the critical temperature; they persist at all
dating to compute various quantities including the order temperatures below the critical temperature, as shown
parameter (magnetization). As well known, this method in Fig. 3. In fact they are even more conspicuous at low
is expected to have the system reachefficiently the equi- temperatures; this manifests the sharp contrast with the
librium,characterizedbytheBoltzmandistribution,and conventionalcriticalslowingdown,presentonlynearthe
to give reliable results at all temperatures except in the critical temperature in systems on regular lattices [11].
critical region, where critical slowing down is unavoid- One can understand the exponential growth of τ in
ableduetostrongfluctuations[11]. Obtainedareresults terms of the inverse updating probability. For large val-
which in general support the mean-field transition and ues of J /J , flipping one spin in a pair which interact
2 1
saturation of the critical temperature, unless long-range strongly with each other will give much influence to the
interactions are far stronger than local ones. relaxation process. The probability of this update is
On the other hand, for J much larger than J , the given by e−∆E/T at temperature T, where the energy
2 1
order parameter (magnetization) m turns out to change change ∆E = J2−cJ1 depends on the neighboring spin
erratically around the critical temperature and the or- states through integer c. Since the temperature is mea-
deredphaseishardlyobservedatverylowtemperatures. suredinunitsofJ1,theinverseoftheupdatingprobabil-
Furthermore, at low temperatures it varies largely MC ity leads to the relaxation time in the form τ ∼ eaJ2/J1,
run by run, even though a single small-world network whereaisaconstant. Onthe otherhand,asthelinkad-
configuration is used (see Fig. 1 for J /J = 10 and dition probability P is increased, the characteristic path
2 1
P = 0.1; for convenience, the Boltzman constant k is length l of the system in general reduces in an algebraic
B
set equal to unity throughout this paper). Namely, the way[12];thisallowsinformationtotravelmoreefficiently
result depends upon the randomnumber sequence. This andthus givesrise to the algebraicdecreaseof τ withP.
may be explained in the following way: At high temper-
atures, all spins can flip easily and the system is in the Accordingly, it is concluded that the true equilibrium
fully disordered state. On a small-world network, there state may not be obtained within moderate MC steps
appear clusters which are connected by shortcuts with when long-range interactions are substantially stronger
3
8 8
10 10
(a) (a)
4 4
τ 10 τ 10
1 1
0 5 10 0 2 4 6
J /J J /J
2 1 2 1
8 12
10 10
J /J =5.0
2 1
J /J =2.0
2 1
(b) (b)
4 8
τ 10 τ 10
1 104
0.01 0.1 1 0.01 0.1 1
P P
FIG. 2: Relaxation time τ (in units of the MC step) near FIG. 3: Relaxation time τ (in units of the MC step) at low
the critical temperature T , estimated from the relation temperature T /2, estimated from the relation m−m ∼
c c eq
m−meq ∼ e−t/τ, with J2/J1 and P varied. (a) Exponen- e−t/τ, with J2/J1 and P varied. (a) Exponential increase
tial increase of τ with J2/J1 for P = 0.1, reflecting that the of τ with J2/J1 for P = 0.1, reflecting that the updating
updating probability is an exponentially decreasing function probability is an exponentially decreasing function of J2/J1.
ofJ2/J1. Thesolid linerepresentsthebestfit: τ =τ0eaJ2/J1 The solid line represents the best fit: τ = τ0eaJ2/J1 with
withτ0 =1.7anda=1.6. (b)Algebraicdecreaseofτ withP τ0 = 13 and a = 2.6. (b) Algebraic decrease of τ with P
for two values of J2/J1, which is related with the character- for J2/J1 =5.0. The solid line corresponds to the power-law
istic path length of the small-world network. The solid and decay τ =τ0P−σ with τ0 =0.8 and σ=7.1.
dashed lines correspond to the power-law decay τ = τ0P−σ
with τ0 =1.4; σ=1.56 and τ0 =2.8; σ=3.3, respectively.
the same for every relevant spin, which guarantees the
detailed balance condition. The new algorithm is thus
than local ones. To circumvent this problem and to ob-
expected to help the system to reachthe correctequilib-
tain the equilibrium state efficiently, we propose a mod-
rium quickly, yielding appropriate results efficiently.
ified updating method which is efficient in simulations
of such a system. The slow relaxation originates from Todemonstratetheefficiencyofthenewalgorithm,we
the fact that flipping a spin interacting (strongly) via a employ it to probe the case of strong long-rangeinterac-
shortcut is hardly probable, even though the free energy tions (J2/J1 & 5) where the conventional algorithm is
reduces if accepted. Therefore, when a spin in a cluster practicallyinapplicable. To find the criticaltemperature
linkedvia shortcutsis selectedduring sequentialupdate, at given values of P and J2/J1, we examine the scaling
we also consider, with probability one half, the possi- behaviors of the magnetization m, susceptibility χ, spe-
bility of flipping all the spins in the cluster simultane- cific heat C, and Binder’s cumulant [11]. Typically, we
ously. Note thatthe probabilityofsuchclusterupdating considerthesystemofsizeuptoN =12800andtakethe
ismuchhigherthanthatofusualsingle-spinupdatingbe- averageover100 different networkrealizations as well as
cause the energy difference involves only the short-range the thermal average over 5×104 MC steps after equili-
interactions. Stillsingle-spinupdating is alsoallowed,so bration at each temperature.
that ergodicity of the system remains intact. Further, We write the finite-size scaling forms as m =
theprobabilitytobe selectedasaclusteristakenalways N−β/ν¯h(|t|N1/ν¯), χ = Nγ/ν¯g(|t|N1/ν¯), and C =
4
1 1
P=0.25 P=1.0
P=0.1 P=0.5
P=0.05 P=0.3
0.8 0.8
P=0.1
P=0.05
0.6 0.6
T T
/2 /2
J J
-e -e
0.4 0.4
0.2 0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
-J /T -J /T
e 1 e 1
FIG. 4: Phase diagram of the Ising model on a small-world FIG. 5: Phase diagram of the Ising model on a small-world
network, where the region below each boundary represents networkwithrangek=2. Simulationdataforvariousvalues
theorderedphaseforthecorrespondingvalueoftheaddition of P are depicted by symbols on lines; the latter are merely
probability P. Simulation data for various values of P are guidestoeyes. ForP ≥0.3,thephaseboundaryintersectsthe
depicted by symbols on lines; the latter are merely guides to J1 =0lineatafinitevalueofJ2,manifestingthepresenceof
eyes. AnalyticresultsinRefs.9and10arealsoplotted,with a phase transition. This exhibits that a small-world network
the same kinds of thick and thin lines, respectively, for each withlocallinksdeletedhasathresholdvalueofP belowwhich
value of P. They coincide with numerical results when P is nolong-range order emerges.
small and/or J2 is sufficiently smaller than J1. For J2/J1
large, ourdatalocate between thetwoanalytic resultsin the
phase diagram.
tic and necessary for the small-world network to have
an exponential tail in the degree distribution. Further,
one end of each added shortcut is determined sequen-
Nα/ν¯f(|t|N1/ν¯) with appropriate scaling functions and tially, which makes our network have less numbers of
critical exponents γ, α, β, and ν¯, where t≡(T −Tc)/Tc large-degree nodes than the former (superimposed ran-
is the reduced temperature. Finite-size scaling analyses dom) network. Accordingly, the standard small-world
of these quantities obtained for N = 1600,3200,6400, network used in this study lies in between the two types
and12800unanimouslysupportaphasetransitionofthe of network in Refs. 9 and 10. Since spins on those nodes
mean-field type, with exponents γ = 1, α = 0, β = 1/2, which have more links facilitate more spins to order, the
and ν¯ = 2. The critical temperature turns out to agree criticaltemperatureofthesystemonthesmall-worldnet-
well with the value obtained from the unique crossing work should be lower than that in Ref. 10 and higher
point of Binder’s cumulant. It is thus concluded that that that in Ref. 9, and such difference is expected to
the system undergoes a finite-temperature transition of grow as P and J are increased. It is indeed observed
2
mean-field nature for J2/J1 >0 and P 6=0. in Fig. 4 that the phase boundary of the system on the
Hereshortcutinteractionsareessentialforthe 1Dsys- small-worldnetworklocatesinbetweentheboundaryob-
temtodisplaylong-rangeorder. Thecriticaltemperature tained in Ref. 10 and that in Ref. 9, particularly in case
Tc/J1 is expected to increase as J2/J1 is raised. In sim- that J2 is substantially larger than J1 and P is not very
ulations, however, T /J does not keep increasing with small (P &0.05).
c 1
J /J beyonda certainvalue depending on P. InFig. 4, We also consider the system with range k = 2, where
2 1
we present the phase diagram of the system with range local interactions are present between the next nearest
k = 1, for various values of P. In this case of k = 1, neighbors as well as the nearest neighbors, and perform
analytic results have been reported for similar systems: extensive simulations, the results of which are displayed
A replica symmetric solution has been developed on the inFig.5. Asexpected,theregionoftheorderedphasein
networks constructed by superimposing random graphs the phase diagram is increased compared with the case
onto a one-dimensional ring [9]. Subsequently, combi- k = 1. Except for this, when P is small (P < 0.3),
natoricshasbeenusedto treatquencheddisorderonthe the overall features are entirely similar to those of the
networks,whereeachnodeisrestrictedfromhavingmore case k = 1: The critical temperature increases with
thanoneshortcut[10]. Thosenetworkscoincidewithour J /J , eventually saturating to a finite value. In case
2 1
network only in the limit P → 0. For finite P, in con- that P ≥0.3, on the other hand, one observes an order-
trasttothelatter,wealloweachnodetohavemorethan disordertransitionontheJ =0line;thiscorrespondsto
1
one shortcut in the construction, which is more realis- the small-worldnetwork whose locallinks are all deleted
5
sothatthereremainonlyrandomlyaddedshortcutswith conventional regular or disordered lattices, where severe
fraction P. In comparison with the case P < 0.3, where inhomogeneity in the interaction strength is absent and
noorderedphaseexistsonthisline,manifestedistheper- equilibriumisreachedquicklyatalltemperaturesexcept
colationproblemintheresultingrandomgraph. Namely, in the critical region without any erratic behavior. To
the system is percolating only when its connectivity, circumvent this problem, we have developed a modified
given by kP, is higher than 2P ≈ 0.6. It is pleasing updating algorithm, assisting the system to reach equi-
c
that this value agrees with the known expression for the libriumquickly. Any dynamicalsystemona small-world
threshold value Pc = 1−p(k−1)/k [7]. We have also network with strong long-range interactions is expected
performed simulations of the system with k = 3, to ob- to behave similarly, and the modified algorithm devel-
tain fully consistent results. It is of interest that the oped here may be used to obtain (equilibrium) thermo-
threshold value is smaller than that of the Erdo¨s-Renyi dynamic properties efficiently. Finally, it wouldbe of in-
(ER) randomgraph[13], which reflects that our random terest to investigate the case that long- and short-range
graph is still more regular than the ER graph. interactions have opposite signs (J /J < 0). The coex-
2 1
In summary, we have studied via extensive numerical istence of ferromagnetic and antiferromagnetic interac-
simulations the Ising model on a small-world network, tions in general introduces frustration into the system,
where long-range interactions via shortcuts are in gen- which,togetherwiththerandomnessassociatedwiththe
eraldifferentfromlocalinteractions. Ithasbeendemon- long-range connections, may lead to the (truly) glassy
strated that long-range interactions via added shortcuts behavior [14]. The detailed investigation of how such a
help spins to order, raising the critical temperature at glass system relaxes depending on the value J /J and
2 1
first and having it saturated eventually. Of particular comparisonwiththeothercasesareleftforfurtherstudy.
interest is the case of strong long-rangeinteractions, rel-
ative to the local ones, where each cluster may play the
role of temporarily quenched randomness. The system
Acknowledgments
then tends to be trapped in a local minimum, inhibited
fromrelaxationtotheglobalminimum(i.e.,equilibrium);
this results in very slow relaxation, making simulations ThisworkwassupportedinpartbytheKOSEFGrant
inefficient. This is in contrast with the Ising model on No. R01-2002-000-00285-0and by the BK21 Program.
[1] Forreviews,see,e.g.,Science284,79-109(1999);M.E.J. A. Vespignani, and R. Zecchina, Eur. Phys. J. B 28,
Newman, J. Stat. Phys. 101, 819 (2000); D.J. Watts, 191 (2002); A. Ramezanpour, Phys. Rev. E 69, 066114
Small Worlds (Princeton University Press, Princeton, (2004).
1999); S.H. Strogatz, Nature 410, 268 (2001); R. Albert [9] T.Nikoletopoulos,A.C.C.Coolen,I.P´erezCastillo,N.S.
andA.-L.Barab´asi,Rev.Mod.Phys.74,47(2002),S.N. Skantzos,J.P.L.Hatchett,andB.Wemmenhove,J.Phys.
Dorogovtsev and J.F.F. Mendes, Adv. Phys. 51, 1079 A: Math. Gen. 37, 6455 (2004).
(2002). [10] J.V. Lopes, Y.G. Pogorelov, J.M.B. Lopes dos Santos,
[2] D.J. Watts and S.H.Strogatz, Nature393, 440 (1998). and R.Toral, Phys. Rev.E 70, 026112 (2004).
[3] M. Gitterman, J. Phys. A: Math. Gen. 33, 8373 (2000); [11] See, e.g., K. Binder (eds.), Applications of the Monte
A. P¸ekalski, Phys. Rev. E 64, 057104 (2001); H. Hong, Carlo Method in Statistical Physics (Springer-Verlag,
B.J. Kim, M.Y. Choi, ibid.66, 018101 (2002). Berlin, 1987); K. Binder and D.W. Heermann, Monte
[4] B.J. Kim,H.Hong, P.Holme, G.S.Jeon, P.Minnhagen, Carlo Simulationin Statistical Physics(Springer-Verlag,
M.Y. Choi, Phys.Rev.E 64, 056135 (2001). Berlin, 1992).
[5] H. Hong, M.Y. Choi, and B.J. Kim, Phys. Rev. E 65, [12] M.E.J. Newman, D.J. Watts, Phys. Lett. A 263, 341
026139 (2002). (1999).
[6] L.G. Morelli, G. Abramson, and M.N. Kuperman, Eur. [13] B. Bollob´as, Random Graphs (Academic Press, London,
Phys. J. B 38, 495 (2004); P.N. McGraw and M. Men- 1985).
zinger, Phys. Rev.E 68, 047102 (2003). [14] Note the difference from the usual case that the long-
[7] A.BarratandM.Weigt,Eur.Phys.J.B13,547(2000). rangeinteractionJ2itselfisrandomlydistributed,taking
[8] The Ising model has also been examined analytically to values ±J; the resulting spin-glass phase has been
and numerically on other typesof complex network. See examined in equilibrium.See Ref. 9.
S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes,
Phys.Rev. E 66, 016104 (2002); M. Leone, A.V´azquez,