Emergence of a non trivial fluctuating phase in the XY model on regular networks

0.3 MB·English

by  Sarah De Nigris

epl draft
Emergence of a non trivial fluctuating phase in the XY model on
regular networks
Sarah De Nigris and Xavier Leoncini
2
Centre de Physique Th´eorique , CNRS - Aix-Marseille Universit´e, Luminy, Case 907, F-13288 Marseille cedex 9,
1
0France
2

l
u
J PACS 05.20.-y–Classical statistical mechanics
7 PACS 05.45.-a–Nonlinear dynamics and chaos
2
Abstract–WestudytheXYmodelondilutednetworks. Consideringtheregularone-dimensional

] latticetopology,wefocusontheinfluenceofthedilutionparameter2≥γ ≥1. Wefindthatforγ <
h 1.5, thesystem does not exhibit aphasetransition, while for γ >1.5 a second ordertransition of
c
themagnetisationarisesanddisplaysidenticalpropertiesasthemeanfield(HMF)regime. Hence
e
m γc =1.5appearstobeacriticalvalue,forwhich,inanidentifiedenergyrange,themagnetisation
shows important fluctuations. We resort to analytical calculations of the magnetisation in the
- low temperatures approximation regime and we show that our analytics breaks down below the
t
a threshold of γc while it gives thecorrect value above, confirmingthe critical valueγc =1.5.
t
s
.
t
a
m
-
d In recent years systems with long-range interactions dition, which is imposing the connectivity per interacting
nhave attracted increasing attention and have been widely unit. We used the paradigmatic 1D-XY model for rotors
o
studied, proving to have a far richer phenomenology than and we will show that we can identify two limit regimes:
c
[the models with short-range potentials. For the latters, a short-rangedone for low connectivity while, in the limit
the rise of equilibrium, in the microcanonicalensemble, is of high connectivity, the system shows global coherence
1
only governed by the conserved momenta of the dynam- via a second order phase transition. The main result of
v
ics and this unique stationary state does not depend on the paper is, however, the emergence of a peculiar new
0
5the initial particle distribution [1]. Moreover the essen- state in between in which the order parameter is affected
5tial property of additivity allows to construct the canoni- by importantfluctuations. Furthermore,we willshow an-
6calensemblefromthemicrocanonical,thetwoapproaches alyticallythatthisstatestemsfromthespecialtopological
.
7resulting equivalent in the thermodynamic limit. This condition on the connectivity we imposed.
0straightforward picture complexifies when dealing with
In generalthe XY model describes a system of N pair-
2
systems interacting via a long-range unscreened poten-
1 wise interacting units. At each unit i is assigned a real
tial: in first instance, the additivity property is no longer
: number θ , which we refer to as the spin i. In the follow-
vpresent and this loss leads to the necessity of a separate i
ing,wewillconsidertheXY modelfromthepointofview
i
Xtreatment of the two ensembles [2–4]. Even more inter- of classical Hamiltonian dynamical systems by adding a
estingly, those systems keep track of the initial config-
r kinetic energy term to the XY Hamiltonian. The total
auration which actually determines the stationary state:
Hamiltonian H takes the form:
for a particular set of initial conditions, long-lasting qua-
sistationary states (QSSs) arise whose duration diverges
with the system size, implying ergodicity breaking [5–7]. N p2 J N
H = i + ǫ (1−cos(θ −θ )). (1)
Furthermore, recently, an oscillating metastable state has 2 2k i,j i j
been observed [8], enriching the already various scenario Xi=1 iX,j=1
oflong-rangesystems. Inthis Letter,weaddressthe issue
of investigating the transition from short-range to long- Because of the periodicity of the cosine function in the
range regime from a quite different point of view than Eq.(1), the phase space for θ is restricted to the interval
i
previous works. Instead of focusing on dynamical con- [0,2π[. Weassociatetoeachspiniacanonicalmomentum
straints, we chose as control parameter a topological con- p whose coupled dynamics with the {θ } will be given by
i i
p-1
Sarah De Nigris Xavier Leoncini
the set of Hamilton equations:
0.5
N=212
J N NN==221146 0
θ˙ =p , p˙ =− ǫ (cosθ sinθ −sinθ cosθ ). -0.5
i i i k i,j j i j i 0.4
-1
iX,j=1 (2) clog()j -1-.25
The coupling constantJ in Eqs.(1) and(2) is chosenpos- 0.3 ǫ=0.2
-2.5 ǫ=0.3
itive in order to obtain a ferromagnetic behaviour and in M -3 ǫ=0.5
0 10 20 30 40 50 60 70
the following it will be set at 1 without loss of generality. 0.2 distancej
Weencodetheinformationaboutthelinksconnectingthe
units in the adjacency matrix ǫ :
i,j 0.1
1 if i,j are connected
ǫi,j = . (3) 0
(0 otherwise 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ǫ
Byconstruction,theadjacencymatrixisasymmetricma-
trixwithnulltrace. InEq. (1)thenormalisationconstant Fig. 1: (colour online) Equilibrium magnetisation for γ =1.25
k ensures the extensivity of the energy, according to the anddifferentsizes; (inset) Correlation function cj for γ =1.25
Kacprescription,anditcorrespondstothenumberoflinks and N =214.
per spin, called the degree:
1 22−γ(N −1)γ phase transition of the order parameter, this could have
k ≡ ǫi,j = . (4) been inferred since for very low dilutions, the system is
N N
i>j more or less identical to just short range interactions sys-
X
tem and in that case the Mermin-Wagner theorem dis-
In Eq.(4) the dilution γ, γ ∈ [1,2] is introduced as the
provesthe existenceof long-rangeorderin a 1−Dsystem.
parameterof controlto shift continuously from the short-
Stillfinitesizeeffectsareatplay,andresultsaredisplayed
range to the long-range regime [9]. It is straightforward
inFig.1: themagnetisationvanisheswiththesystemsize,
to see that to the case γ = 1 corresponds to the linear
sothatinthethermodynamiclimitweexpecttheresidual
chain with only nearest neighbours coupling and, on the
magnetisation to be zero. Nevertheless, quasi-long-range
other hand, γ = 2 corresponds to the full coupling of all
ordercouldstillariseatfinite temperatures likeinthe 2D
the spins. In this latter case the Hamiltonian in Eq.(1)
short-ranged XY-model which displays the Berezinskij-
reduces to the HMF model [10]. We construct this way a
Kosterlitz-Thouless phase transition [12,13]. This par-
lattice in which each spin is connected to k/2 neighbours
ticular phase transition is characterized by the change in
on each side and the width of this neighbourhood is im-
behaviour of the correlation function, which decays as a
posed by our choice of the dilution. To investigate the
power law at low temperatures and exponentially in the
macroscopic behaviour of the system, we define the mag-
high temperature phase. Hence
netisation M = (m ,m ), where m = N−1 cos(θ )
x y x i i to test the eventual presence of a Kosterlitz-Thouless
and m = N−1 sin(θ ). The modulus M = |M| indi-
y i i P transition, we monitored the correlationfunction:
catesthe degreeofcoherenceofthe spinangulardistribu-
P
tion: the incoherent state will have M = 0, while finite N
1
values are naturally associated to more coherent states. c(j)= cos(θ −θ ). (5)
N i i+j[N]
Having set the structure of the lattice via the dilution, i=1
X
we performedsimulations in the microcanonicalensemble
andwestudiedtheevolutionofthetotalequilibriummag- At equilibrium, the correlation decays exponentially (See
netisation M where the bar denotes the time average (we insertinFig.1)atanytemperatureintheconsideredphys-
assume ergodicity). The system possesses two constants ical range, confirming the absence of the aforementioned
of motion preserved by the dynamics: the energy H = E phase transition. For those values of γ, we can conclude
and the total angular momentum P = p which are that the number of links is still too low to entail a change
i i
setby the initialconditions. We choseto startthe system in the 1-D behaviour and it is interesting to notice that
P
withaGaussiandistributionforbothforthespinsandthe even a configuration with quite a large neighborhood per
momenta, we also impose P = 0. The numerical integra- spin like γ = 1.4 still corresponds to short range interac-
tionofEqs. (2)isperformedusingasymplecticintegrator tions.
[11],whichensurestheconservationofthemomentaEand Symmetrically,theotherimportantrangetoconsideris
P (which were monitored) and the symplectic structure. γ >1.5, when we approach the full coupling of the spins.
The thermodynamic quantities are calculated by averag- AsshowninFig.2a,themeanfieldtransitionoftheorder
ing over time. parameter is recoveredin this dilution regime: it is worth
We first concentrated on low dilution values, γ < 1.5 stressing here that we recover the mean field result even
. For this regime of dilution, the system doesn’t show a for γ significantly lower than 2, e.g. for γ = 1.6, imply-
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Emergence of a non trivial fluctuating phase in the XY model on regular networks
while in the interesting interval of energies it is heavily
1
theory affected by the fluctuations and it is impossible to prop-
γ=1.75
γ=1.5 erly determine its behaviour. We observed these effects
0.8 γ=1.25
on several sizes from N = 212 up to N = 218 and, when
considering the scaling of σ2 with the size (reported in
0.6
the inset in Fig. 2b), it is evident that the variance is not
affected by the increasing system size.
M 0.4
We argue that at γ = 1.5 the number of links is at its
lowervalue to allowthe arisingoflongrangeorderandto
0.2
shedlightonthemechanismunderneath,wederiveanan-
alytical form for the magnetisation which shows that the
0
criticalfactorisembeddedintheadiacencymatrix,viaits
-0.2 spectrum. As first hypothesis, we restrict our analysis to
0 0.2 0.4 0.6 0.8 1
the low temperature regime, hence assuming that the dif-
(a) ǫ
ference θ −θ ∀i,j is small. We thereforeobtaina simple
i j
1 γγ==11.7.55 -2 quadraticHamiltonian: H = i p22i +4Jk i,jǫij(θi−θj)2.
-2.5 This assumption is justified by the simulations, as previ-
0.9 2σlog()10 -3-.35 oseunstlyatidoinscfuosrsethde; {toθip,proi}ceaesdafsuPurmtheorf,rawnedPcoomnspidhearseadrweapvrees-
0.8 -4 [14]:
3.5 4 4.5 5 5.5
M 0.7 log10(N) pθi == lαα˙l((tt))ccooss((22Nππllii ++φφl)),, (6)
i Pl l N l
0.6 where φl are randomPly distributed phases on the circle.
Since we make the hypothesis that the time dependence
0.5 is totally encoded in the amplitudes α , the momenta are
l
simplyrelatedtotheanglesviathefirstHamiltonequation
0.4 θ˙ = p . The basic idea behind this reasoning is that, at
3000 3500 4000 4500 5000 i i
(b) Time equilibrium, the momenta are Gaussian distributed vari-
ables, justifying the representation in Eqs. (6). We also
Fig. 2: (colour online) (a) Equilibrium magnetisation for N = observe that it consists in a linear changing of variable
216 and different γ. (b) Time series for the order parameter since we use N modes for our representation. If we now
withN =218 andǫ=0.6; (inset)Scalingofthemagnetisation consider different sets of phases {φ } labeled by m, we
variance(cid:10)σ2(cid:11) for γ =1.5, ǫ=0.60 (stars) and ǫ=0.74 (dots). have that each one of them corresplonmds to a phase space
trajectory and, hence, it is possible to replace the ensem-
ble average with the average on the random phases [14].
ing that global coherence is still reachable with a weaker Consequently, injecting Eqs. (6) in the linearised Hamil-
condition than the full coupling. Naturally, in Fig. 2a, a tonian and averaging on the random phases, we obtain:
shift exists between the simulations, performed at finite
size, and the theoretical curve which is the one obtained hHi 1 N
= α˙2+α2(1−λ ), (7)
for the mean field in the thermodynamic limit. Neverthe- N 2 l l l
less this interval shrinks with the increasing size and, in Xl=1
that sense, it would be interesting to further investigate
where
the finite size scaling of the criticalenergy. In both cases, k/2
2 2πml
γ < 1.5 and γ > 1.5 , the variance of the magnetisation λ = cos( ) (8)
l
k N
σ2 = (M −M)2 vanishes linearly with the system size, m=1
X
ensuring the reaching of equilibrium in our simulations. aretheeigenvaluesoftheadiacencymatrix. Usingthesec-
Thetransitionbetweenthe1-Dbehaviourandthemean ond Hamilton equation d(∂hHi) = −∂hHi, we can thus
dt ∂α˙l ∂αl
field phase appears to be critical for γ =1.5: for low en- derive from Eq. (7) a dispersion relation for the waves
c
ergies0.45≤ǫ≤0.75themagnetisationisaffectedbyim- amplitudes that embeds two levels of information: at the
portantfluctuations anditis notclearif itdoes notreach microscopicallevel, the structure of the links, via the adi-
an equilibrium state (Fig. 2b) on the time scales consid- acency matrix spectrum and, from a more macroscopical
ered,or if these fluctuations are the mere reflectionof the point of view, Eq. (7) results from averaging on the ran-
critical behaviour and will persist forever. Moreover the domphaseswhich,asexplained,accountsfortheensemble
correlation function in Eq. (5) does not prove helpful in averaging. Imposing the equipartition of energy at equi-
characterizingthispeculiarstate: itacquirestheexponen- libriumfortheobtainedcollectionofharmonicoscillators,
tial behaviour only for densities of energy above ǫ = 0.7, givesanadditionalrelationbetweenthefrequenciesω and
l
p-3
Sarah De Nigris Xavier Leoncini
1 1
γ=1.25
γ=1.5
0.8 γ=1.75
0.8
> 0.6
1) 0.6
.=0 0.4
MT( 0.4 λl 0.2
<
N=222
0.2 N=223 0
N=224
N=225
-0.2
theory
0
1 1.2 1.4 1.6 1.8 2
-0.4
γ 0 20000 40000 60000 80000 100000 120000
l
Fig. 3: (colour online) Approximated magnetisation hMi =
exp(−2TN Pl 1−1λl) for T = 0.1 versus the dilution parameter Fig. 4: (colour online) Spectra λl for N =218 and different γ
γ. values.
the amplitudes α: α2ω2 = 2T/N. We evaluate now the
l l l how to quantify this difference is still objectof a more re-
magnetisation in the low temperature regime using the
fined analysis to precisely relate the spectrum properties
same approach: we inject the representation (6) and we
to the magnetisation behaviour.
averageon the phases, obtaining [15]:
In this Letter we first introduced our model for the in-
teraction,theXY modelandfocusedontheregularlattice
hMi= J (α)(cosθ ,sinθ ), (9)
0 l 0 0 topologyinwhichwecontrolledthedegreeofeachspinvia
l
Y the dilution parameter γ. We showed that three different
where θ is the average of the {θ } which is a constant regimesexisted,alowdilutionregime(γ <1.5),wherethe
0 i
becauseoftheconservationofthetotalmomentumP =0. long-rangeorder is absent, a high dilution phase in which
TheabsolutevalueofthemagnetisationhMiwillhencebe, the global coherence and mean field behaviour is recov-
from Eq. (9), the product over the l modes of the Bessel ered (γ > 1.5) and a peculiar behaviour at the threshold
functions. To evaluate the logarithm of hMi , we observe ofγ =1.5. Interestingly,weshowthatthemeanfieldtran-
that, at equilibrium and in the limit of large system size, sition does not necessitate the full coupling of the spins,
weexpecttohavesmallα2. We canthusapproximatethe like in the HMF model or in a random diluted network,
l
Besselfunctions in the limit of small amplitudes α which and it still arises for a regular topology even for γ =1.6 ,
l
is, therefore, the low temperatures regime. This finally quite far hence from the extremal configuration of γ = 2.
leads to: However we consider that the main result of our analysis
is the evidence of a unsteady almost turbulent like state
ln(hMi)=− α2l =− T 1 . (10) when γ = 1.5 : the important fluctuations affecting the
4 2N 1−λl orderparameterand the invarianceof these effects onthe
l l
X X system size in a whole interval of energies are in total
We calculated numerically Eq. (10) for increasing N and contrast with what observed in the other regimes, where
in Fig. 3 we show how it correctly grasps the behaviour with the same initial conditions the convergence to equi-
for the magnetisation: in the low temperature regime, it libriumisrapid. Wepresentedaanalyticalcalculationfor
retrieves the theoretical value for γ > 1.5 and it vanishes the magnetisation, based on the method in [14], which is
when γ < 1.5. Moreover, with the increasing size, the able to catch the appropriate behaviour in the two lim-
difference between the two regimes becomes sharper con- its discussed before. This result points out that γ = 1.5
firming the critical nature of γ = 1.5. The key for this is indeed the critical value for this passage from the 1-D
c
peculiareffectatγ =1.5appearsthus tobefully encoded topologytothe meanfieldframe. Moreoveritprovesthat
inthe spectrumofthe adiacencymatrix,whichdrivesthe the spectrum of the adiacency matrix, which carries the
system to the mean field regime or to the shortrange one informationonthelinks,iscrucialtounderstandthisshift.
according to the dilution parameter γ. Nevertheless, by Asanticipatedbefore,thisunstablestatestemsfromtopo-
a rapid inspection of Eq. (8), it appears not trivial to logical features of the lattice, instead of from a particular
isolate the dependence of the eigenvalues on the dilution choiceoftheinitialconditionsasin[16–18]. Thisintrinsic
andonthesize,eacheigenvalueconsistinginasumofk/2 differenceindicates the importantroleofthis statethatit
contributions. In Fig. 4, we show the behaviour of the isobserved,atourknowledge,forthefirsttime. Weantic-
spectrum for three representative values of γ: clearly the ipate that the same kind of phenomenon can be observed
spectra qualitatively differ according to the dilution, but withdifferenttopologiesandprobablylowerdilutions,and
p-4
Emergence of a non trivial fluctuating phase in the XY model on regular networks
believe that if we should find an efficient way to modify
the dilution parameters, these systems could prove to be
useful on-off switches for a somewhat large temperature
range.
∗∗∗
TheauthorsaregratefultoW.Ettoumifordiscussions.
S.d.N. has been supported by DGA/DS/MRIS.
REFERENCES
[1] G.Gallavotti, eprint chao-dyn, 9403004 (1994) .
[2] J.Barre, D.Mukamel S., Phys Rev Lett, 87 (2001)
030601.
[3] Leyvraz F. and S.Ruffo, J.Phys A, 35 (2002) 285.
[4] Torcini A. and Antoni M., Phys. Rev. E, 59 (1999)
2746.
[5] D.Mukamel, S.Ruffo N., Phys Rev Lett, 95 (2005)
240604.
[6] Campa A., Dauxois T. and Ruffo S., Phys. Rep., 480
(2009) 57.
[7] P.H. ChavanisJ. V.andBouchetF.,Eur. Phys. J. B,
46 (2005) 61.
[8] H.Morita K., Phys Rev Lett, 96 (2006) 050602.
[9] CianiA.,RuffoS.andFanelliD.,Long-rangeInterac-
tion, Stochasticity and Fractional Dynamics- Dedication
to George M. Zaslavsky (1935-2008) (HEPand Springer)
2010.
[10] AntoniM.andRuffoS.,Phys.Rev.E,52(1995)3261.
[11] McLachlan R. I.andAtela P.,Nonlinearity,5(1992)
541.
[12] V.L.Berezinskij,Sov. Phys. JETP, 32 (1971) 494.
[13] Kosterlitz J. M. and Thouless D. J., Journal of
Physics C: Solid State Physics, 6(1973) 1181.
[14] Leoncini X. and Verga A., Phys. Rev. E, 64 (2001)
066101.
[15] LeonciniX.,VergaA.andRuffoS.,Phys.Rev.E,57
(1998) 6377.
[16] AntoniazziA., Fanelli D., Ruffo S.andYamaguchi
Y. Y., Phys. Rev. Lett., 99 (2007) 040601.
[17] A.Antoniazzi, F.Califano, D.Fanelli and S.Ruffo,
Phys Rev Lett, 98 (2007) 150602.
[18] Antoniazzi A., Fanelli D., Barre J., Chavanis P.-
H., Dauxois T. and Ruffo S., Phys. Rev. E, 75 (2007)
011112.
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