Hyperscaling violation in the 2D 8-state Potts model with long-range correlated disorder
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l epl draft Hyperscaling violation in the 2D 8-state Potts model with long-range correlated disorder
C. Chatelain , School of Physics, Indian Institute of Science Education and Research (IISER), Thiruvananthapuram, India Groupe de Physique Statistique, D´epartement P2M, Institut Jean Lamour (CNRS UMR 7198), Universit´e de Lor-raine, France
PACS – Statistical mechanics of phase transitions in model systems
PACS – Potts models in lattice theory and statistics
PACS – Critical phenomena in thermodynamics
PACS – Monte Carlo methods in statistical physics and nonlinear dynamics
Abstract – The first-order phase transition of the two-dimensional eight-state Potts model isshown to be rounded when long-range correlated disorder is coupled to energy density. Criticalexponents are estimated by means of large-scale Monte Carlo simulations. In contrast to uncor-related disorder, a violation of the hyperscaling relation γ/ν = d − x σ is observed. Even thoughthe system is not frustrated, disorder fluctuations are strong enough to cause this violation in thevery same way as in the 3D random-field Ising model. In the thermal sector too, evidence is givenfor such violation in the two hyperscaling relations α/ν = d − x ε and 1 /ν = d − x ε . In contrastto the random field Ising model, at least two hyperscaling violation exponents are needed. Thescaling dimension of energy is conjectured to be x ε = a/
2, where a is the exponent of the algebraicdecay of disorder correlations. Introduction. –
Quenched disorder when coupled tothe energy density, say by dilution or random couplings,is known to soften first-order phase transitions. As ar-gued by Imry and Wortis [1], local fluctuations of impurityconcentration can destabilise the ordered phases in coex-istence at the transition temperature if the surface tensionis sufficiently small. In 2D, it was rigorously proved thatan infinitesimal amount of disorder is sufficient to makeany first-order transition continuous [2, 3]. The completevanishing of the latent heat was first observed numericallyin the case of the 2D 8-state Potts model [4]. The criti-cal behaviour of the disorder-induced second-order phasetransition is governed by a new random fixed point [5].The universality class was later shown to depend on thenumber of states q [6]. In 3D, a finite disorder is requiredto round completely the first-order phase transition. Thephase diagram exhibits a tricritical point separating afirst-order regime from the disorder-induced continuousone, as first observed in the bond-diluted 4-state Pottsmodel [7]. A rounding of the first-order phase transitionof the 2D Potts model was also reported for anisotropicaperiodic sequences of couplings [8], and for layered ran-dom couplings [9, 10]. In both cases, the couplings are infinitely correlated in one direction. In the random case,the critical behaviour was shown to be governed by a q -independent infinite-randomness fixed point. The criticalexponents are therefore those of the layered random Isingmodel, the celebrated McCoy-Wu model [11, 12]. Inter-estingly, the same critical behaviour is observed for thePotts model with homogeneous uncorrelated disorder inthe limit q → + ∞ [13].In this letter, we consider the case of random bond cou-plings J ij > J (0) − ¯ J )( J ( ~r ) − ¯ J ) ∼ r − a . According to the Imry-Wortis criterion, the low-temperature phase is destabilisedwhen the fluctuations of exchange energy inside a ferro-magnetic domain of characteristic length ℓ , increase fasterwith ℓ than the interface free energy σℓ d − . Since thecontribution of correlations to these fluctuations reads s(cid:2) X i,j ( J ij − ¯ J ) (cid:3) ∼ (cid:20) ℓ d Z ℓ d d d ~rr a (cid:21) / ∼ ℓ d − a/ , ( a ≤ d )(1)we expect the first-order phase transition to be softenedfor a ≤ a > d anduncorrelated disorder, the main contribution is due to thep-1. Chatelain1,2fluctuation term qP i,j ( J ij − ¯ J ) ∼ ℓ d/ . The case ofan uncorrelated disorder is therefore equivalent to a = d .Unexpectedly, we observe that the critical exponents atthe randomness-induced second-order phase transition donot satisfy hyperscaling relations. Such a violation hadonly been reported in random systems with frustration,spin glasses or random-field systems, but, to our knowl-edge, never for purely ferromagnetic systems. In the firstsection of this letter, the details of the Monte Carlo sim-ulation are presented. Hyperscaling violation is studiedfirst in the magnetic sector and then in the energy sector. Description of the simulation. –
We consider the2D q -state Potts model with Hamiltonian − βH = X ( i,j ) J ij δ σ i ,σ j (2)where σ i ∈ { , , . . . , q − } and the sum extends over pairsof nearest neighbours of the square lattice. We restrictourselves to the case q = 8 for which the correlation lengthof the pure model is ξ ≃
24 at the transition temperature.The order parameter m is defined as m = qρ max − q − , ρ max = max σ N X i δ σ i ,σ (3)where ρ max is the density of spins in the majority state.This definition breaks the Z q symmetry of the Hamilto-nian in the same way as an infinitesimal magnetic field.In the case q = 2 corresponding to the Ising model, theusual order parameter N h| P i σ i |i is recovered.We consider a binary distribution of coupling constants J ij ∈ { J , J } with (cid:0) e J − (cid:1)(cid:0) e J − (cid:1) = q. (4)In the case of uncorrelated disorder, Eq. (4) is the self-duality condition that gives the location of the critical line.The ratio r = J /J is used as a measure of the strengthof disorder. Here, we present results for the case r = 8.To generate correlated coupling configurations { J ij } , wesimulate another spin model, namely the Ashkin-Tellermodel ( σ i , τ i = ± − βH AT = X ( i,j ) (cid:2) J AT σ i σ j + J AT τ i τ j + K AT σ i σ j τ i τ j (cid:3) (5)at different points of its critical line e − K AT = sinh 2 J AT .Two symmetries of the Hamiltonian are spontaneouslybroken at low temperatures: the global reversal of thespins σ i and the reversal of both σ i and τ i . Therefore, twoorder parameters can be defined, magnetisation P i σ i andpolarisation P i σ i τ i , leading to two independent scalingdimensions: β AT σ = 2 − y − y , β AT στ = 112 − y (6) wherein we use the parametrizationcos πy h e K AT − i , ( y ∈ [0; 4 / . (7)The correlation length exponent is ν AT = − y − y . Spin con-figurations of this model are generated by Monte Carlosimulation using a cluster algorithm [14]. For each ofthem, a coupling configuration of the Potts model is con-structed as J ij = J + J J − J σ i τ i , (8)where the site j is either at the right or below the site i . This construction ensures that the constraint (4) im-plies the self-duality of our random Potts model. Sincethe Askin-Teller is considered on its critical line, disor-der fluctuations are self-similar and the coupling constantsdisplay algebraic correlations( J ij − ¯ J )( J kl − ¯ J ) ∼ | ~r i − ~r k | − a (9)at large distances with a = 2 β AT στ /ν AT = 14 − y . (10)We have considered six points on the critical line, y ∈{ , . , . , . , , . } , leading to six correlated disor-der distributions with a ≃ .
25, 0 . . .
4, 0 . . χ AT = L d (cid:2) p − | p | (cid:3) (11)where p = N P i σ i τ i is the polarisation density, is com-puted. According to the fluctuation-dissipation theorem,this quantity is equal to the integral of the correlations (9)over the volume of the system. It is therefore expected toscale as L d − a . The data are plotted on figure 1. A nicepower-law behaviour is observed over the whole range oflattice sizes that are considered. The fitted exponents aregiven in Tab. 1 for all values of y . For y ≥ .
75, they arecompatible with exact exponents. For smaller values of y ,the deviation to the exact result is at most of 5%. Notethat for y = 0, the Ashkin-Teller model is equivalent to the4-state Potts model. The critical behaviour is therefore af-fected by logarithmic corrections. These values have beenobtained with, and only with, the coupling configurationsused during the simulation of the Potts model with corre-lated disorder. The agreement with the expected values,even for small lattice sizes, indicates that the number ofdisorder configurations is sufficient to reproduce correctlythe expected disorder fluctuations.For comparison, simulations for the Potts model withuncorrelated disorder are also performed. The same simu-lation code is used but with an infinite temperature of thep-2yperscaling violation in the 2D 8-state Potts model with long-range correlated disorder Table 1: Critical exponents d − a of the electric susceptibility χ AT of the Ashkin-Teller model, or equivalently of fluctuations ofthe couplings J ij , for different values of the parameter y . The numerical estimate of d − a is denoted by MC (second line) andthe exact value by Th. (third line). y .
25 0 . .
75 1 1 . d − a (MC) 1 . . . . . . d − a (Th.) 1 .
75 1 .
714 1 .
667 1 .
600 1 .
500 1 .
10 100 L χ A T y=0.00y=0.25y=0.50y=0.75y=1.00y=1.25 Fig. 1: Electric susceptibility χ AT of the Ashkin-Teller model,or equivalently fluctuations of the couplings J ij , versus thelattice size L . The different symbols correspond to differentvalues of the parameter y of the Ashkin-Teller model. Thestraight lines are power-law fits to the data. Ashkin-Teller model in order to obtain uncorrelated spinsand therefore uncorrelated couplings J ij .The Potts model is then simulated using the Swendsen-Wang algorithm [15]. Lattice sizes between L = 16 and256 are considered. For each disorder configuration, 1000MCs are performed to thermalise the system and 20 , τ ≃ L = 256). Thermodynamic quantities are averagedover a number of disorder configurations proportional to1 /L . For the largest lattice size L = 256, 2560 disorderconfigurations are generated while for L = 64 for instance,this number is raised up to 40960. Stability of disorderaverages is checked. In the following, we will denote h X i the average of an observable over thermal fluctuations and h X i the average of the latter over disorder.On the critical line, the typical spin configurations of theAshkin-Teller model display a large cluster of polarisation στ = +1 or −
1. As a consequence, our random couplingconfigurations also exhibit large clusters of either strongor weak bonds. The probability distribution of the totalenergy of the Potts model shows two peaks correspondingto these two kinds of bond configurations. This distribu-tion is highly correlated to the probability distribution ofpolarisation of the Ashkin-Teller model. Since the latterundergoes a second-order phase transition, the two peakscome closer as the lattice size is increased. Note that in our model, macroscopic region of strong couplings are notrare: they have a probability 1 /
2. Moreover, they have afractal dimension 1 < d f < Magnetic sector. –
We estimate critical exponentsby Finite-Size Scaling. The exponent β/ν can be ex-tracted from magnetisation h m i and its moments h m n i with n = 2 , ,
4. We observe nice power laws withoutany significant correction to scaling. Our estimates of x σ = β/ν evolve with a and range from 0 . y = 0)to 0 . y = 1 . . χ = L d h m i − h m i . (12)The data display large corrections to scaling (see Fig. 2).A cross-over is observed around L = 48, not far fromthe correlation length ξ = 24 of the pure 8-state Pottsmodel. In the region L ≥
96, power-law fits give stableestimates for γ/ν going from 1 . y = 0 .
00) to 1 . y = 1 . . γ/ν = d − β/ν istherefore not satisfied (see values in Tab 2). We shallidentify disorder fluctuations as the origin of this hyper-scaling violation, like in the 3D Random-Field Ising Model(RFIM). Consider the following decomposition:¯ χ = L d (cid:2) h m i − h m i (cid:3) − L d (cid:2) h m i − h m i (cid:3) . (13)As observed on Fig. 2, the difference of the two terms (13)leaves an average susceptibility which is smaller by a factorrougly equal to 3 in the case of uncorrelated disorder whileit is two orders of magnitude smaller for correlated disor-der. The first term, when computed separately, displaysa power law behaviour with an exponent ( γ/ν ) ∗ incom-patible with γ/ν but in agreement with the hyperscalingrelation (Tab. 2). The second term of (13), the so-calleddisconnected susceptibility, involves the ratio R m = h m i − h m i h m i , (14)which is expected to behave as R m ∼ R m ( ∞ ) + A L − φ ,if magnetisation is not self-averaging [17]. We indeed ob-p-3. Chatelain1,2
10 100 L χ , L d [ < m >−< m > ] y=0.00y=0.25y=0.50y=0.75y=1.00y=1.25Uncorr.10 100 L Fig. 2: Average magnetic susceptibility ¯ χ (bottom) and L d [ h m i − h m i ] (top) versus lattice size L for different val-ues of y and for uncorrelated disorder (Uncorr.). The straightlines correspond to power-law fits. Dashed lines indicate thatthe fit was performed over lattice sizes L ≥
96 only. serve that R m goes to a non-vanishing constant in thelimit L → + ∞ (Fig. 3). The second term of Eq. (13)therefore behaves as L d h m i ∼ L d − β/ν , i.e. with an ex-ponent satisfying the hyperscaling relation. Since the dataindicate that it is also the case for the first term of Eq.(13), one can imagine that, if their amplitudes are equal,the dominant terms will cancel. To test this hypothesis,we compute the ratio h m i − h m i h m i − h m i . (15)We observe a plateau at 1 . χ ∼ L d − β/ν − ω of any of the two termsof Eq. (13) and the hyperscaling violation exponent θ isthe exponent ω of this correction. Note that this is alsothe mechanism of hyperscaling violation invoked in thecontext of the RFIM [18] where our exponent ( γ/ν ) ∗ isdenoted 4 − ¯ η [19, 20]. In the case of uncorrelated disor-der, the ratio (15) goes to a value significantly differentfrom 1 in the large size limit (Fig. 3). The dominant con-tribution of the two terms of (13) do not cancel in thiscase and therefore hyperscaling is not violated. Energy sector. –
The divergence of specific heat iscompletely washed out by the introduction of disorder,which means that the specific heat exponent α/ν is eitherzero or negative (Fig 4). The average specific heat¯ C = L d < e > − < e > (16)can be decomposed in the same way as ¯ χ :¯ C = L d (cid:2) h e i − h e i (cid:3) − L d (cid:2) h e i − h e i (cid:3) . (17) < m > / < m > − ( < m >−< m > ) / ( < m > −< m > ) y=0.00y=0.25y=0.50y=0.75y=1.00y=1.25Uncorr. Fig. 3: On the left, ratio defined by Eq. (14) for differentvalues of y and for uncorrelated disorder (Uncorr.) versus theinverse of the lattice size. The non-vanishing asymptotic limit,i.e. the extrapolated value on the y -axis, indicates that magne-tization is not self-averaging for both correlated and uncorre-lated disorder. On the right, ratio (15) of the two terms whosedifference gives the average susceptibility. Asymptotically, thisratio displays a plateau at a value compatible with 1 for corre-lated disorder. For uncorrelated disorder, the asymptotic valueis incompatible with 1. Like in the case of susceptibility, the two terms are severalorders of magnitude larger than their difference for corre-lated disorder. We observe a nice power-law behaviour ofthe first term with an exponent in good agreement with( α/ν ) ∗ = ( γ/ν ) AT = d − a , which means that the fluctu-ations of energy are dominated by the fluctuations of thecouplings and therefore of the polarisation density in theoriginal Ashkin-Teller model. The second term involvesthe ratio R e , constructed in the same way as R m (14).Our numerical data show that energy is not self-averaging( R e ( ∞ ) = 0) and the ratio < e > − < e > < e > − < e > (18)exhibits a plateau at the value 1 . C so that a violation of the hyperscaling re-lation α/ν = d − x ε is expected. Even though we cannotmeasure x ε from the scaling behaviour of energy, we in-fer that it can be extracted from the hyperscaling relation( α/ν ) ∗ = d − x ε which implies x ε = a/
2. In the case ofuncorrelated disorder, R e is compatible with zero whichmeans that energy is self-averaging and therefore the twodominant contributions of Eq. (17) do not cancel. Asobserved, hyperscaling is not violated in this case.In pure systems, a good estimator for the determinationof the correlation length exponent ν is − d ln h m i dβ = L d h me i − h m ih e ih m i . (19)p-4yperscaling violation in the 2D 8-state Potts model with long-range correlated disorder
10 100 L C , L d [ < e >−< e > ] y=0.00y=0.25y=0.50y=0.75y=1.00y=1.25Uncorr.10 100 L Fig. 4: Average specific heat ¯ C (bottom) and L d [ h e i − h e i ](top) versus lattice size L for different values of y and uncor-related disorder (Uncorr.). The straight lines correspond topower-law fits. For clarity, error bars of ¯ C in the uncorrelatedcase have been drawn as dashed line when they overlap withother points. This is generalised to random systems as: − d ln h m i dβ = L d h me i − h m ih e ih m i . (20)and is expected to scale as d − x ε . A power-law fit toour data yields exponents 1 /ν close to zero but with largeerror bars. Consider again the decomposition − d ln h m i dβ = L d (cid:2) h me i − h m i h e i (cid:3) h m i − L d (cid:2) h m ih e i − h m i h e i (cid:3) h m i . (21)The first term displays a power-law behaviour with expo-nents 1 /ν ∗ close to, though slightly above, d − a/
2, whichimplies x ε ≃ a/ R me = h m ih e i − h m i h e ih m i h e i , (22)which behaves as R me ( L ) ∼ R me ( ∞ ) + aL − φ ′ with φ ′ ≃ .
3. The constant R me ( ∞ ) is clearly finite, except maybefor y = 1 .
25. Like in the magnetic case, the two terms ofEq. (21) have the same dominant scaling behaviour. Theratio h me i − h m i h e ih m ih e i − h m i h e i (23)displays a plateau at 1 . /ν = d − x ε is expected to be violated. In the case ofuncorrelated disorder, R me is compatible with zero so wedo not expect any cancellation of the two dominant con-tributions of Eq. (21) and, consequently, no hyperscalingviolation. < e > / < e > − ( < e >−< e > ) / ( < e > −< e > ) y=0.00y=0.25y=0.50y=0.75y=1.00y=1.25Uncorr. Fig. 5: On the left, ratio defined in the same way as Eq.(14), but for energy instead of magnetisation, versus the in-verse of the lattice size L . A non-vanishing asymptotic limitis observed for correlated disorder, indicating that energy isnot self-averaging in this case. On the right, ratio of the twoterms whose difference gives the average specific heat. Errorbars for uncorrelated disorder are large (they overlap other barsfor large lattice sizes) and have not been represented for clar-ity. For correlated disorder, a plateau is observed at a valuecompatible with 1. Conclusions. –
Numerical evidence has been givenof the violation of hyperscaling relations in both the mag-netic and energy sectors of the 2D 8-state Potts modelwith long-range correlated disorder. Even though thismodel is not frustrated, the mechanism causing these vi-olations was shown to be the same as in the 3D RFIM,namely the cancellation of the two dominant contributionsto the magnetic susceptibility, specific heat or derivativeof the logarithm of magnetisation. However, there aretwo important differences between our Potts model withcorrelated disorder and the RFIM: disorder is coupled tothe energy density and not magnetisation, and the ran-dom fixed point does not lie at zero temperature but ata finite temperature. The latter may explain why hyper-scaling violation is observed in both magnetic and energysectors. In the magnetic sector, the hyperscaling violationexponent is estimated to be θ m = ( γ/ν ) ∗ − γ/ν ≃ . θ e = ( α/ν ) ∗ − α/ν & d − a and θ ′ e = (1 /ν ) ∗ − /ν ≃ d − a/
2. For uncorrelatedRFIM, the hyperscaling violation exponent is compatiblewith θ ≃ d/
2. Since in the absence of correlation a = d ,this value is identical to our estimate θ ′ e = d − a/ ∗ ∗ ∗ The author gratefully thanks Sreedhar Dutta and theIndian Institute for Science Education and Research(IISER) of Thiruvananthapuram for their warm hospital-ity and a stimulating environment.p-5. Chatelain1,2
Table 2: Critical exponents measured by Monte Carlo simulations (MC), or computed from them (from MC), and conjecturedvalues (Conj.). y .
25 0 . .
75 1 1 . a .
25 0 .
286 0 .
333 0 . . . β/ν (MC) 0 . . . . . . d − β/ν (from MC) 1 . . . . . . γ/ν (MC) 1 . . . . . . γ/ν ) ∗ (MC) 1 . . . . . . α/ν ) ∗ (MC) 1 . . . . . . d − a (conj.) 1 .
75 1 .
714 1 .
667 1 .
600 1 .
500 1 . /ν ∗ (MC) 1 . . . . . . d − a/ .
875 1 .
857 1 .
835 1 . .
75 1 .
10 100 L d l n < m > / d β , L d ( < m e >−< m >< e > ) / < m > y=0.00y=0.25y=0.50y=0.75y=1.00y=1.25Uncorr.10 100 L Fig. 6: Quantity − d ln h m i dβ (bottom) and L d h me i−h m i h e ih m i (top)versus lattice size L for different values of y and for uncorre-lated disorder (Uncorr.). < m >< e > / < m >< e >− ( < m e >−< m >< e > ) / ( < m >< e >−< m >< e > ) y=0.00y=0.25y=0.50y=0.75y=1.00y=1.25Uncorr. Fig. 7: On the left, ratio defined by Eq. (22) versus 1 /L . .The non-vanishing extrapolation in the limit L → + ∞ indi-cates that h m i and h e i are correlated in the case of correlateddisorder. On the right, ratio (23) of the two terms whose dif-ference gives − d ln h m i dβ . Error bars for uncorrelated disorderare large and have not been represented for clarity. Again, aplateau compatible with 1 is observed for correlated disorder. REFERENCES[1] Imry Y. and Wortis M. Phys. Rev. B Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett. Comm. Math. Phys.
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