The effects of bond randomness on the ground-state structure, phase diagram
and critical behavior of the square lattice ferromagnetic Blume-Capel (BC)
model are discussed. The calculation of ground states at strong disorder and
large values of the crystal field is carried out by mapping the system onto a
network and we search for a minimum cut by a maximum flow method. In finite
temperatures the system is studied by an efficient two-stage Wang-Landau (WL)
method for several values of the crystal field, including both the first- and
second-order phase transition regimes of the pure model. We attempt to explain
the enhancement of ferromagnetic order and we discuss the critical behavior of
the random-bond model. Our results provide evidence for a strong violation of
universality along the second-order phase transition line of the random-bond
version.Comment: 6 LATEX pages, 3 EPS figures, Presented by AM at the symposium
  "Trajectories and Friends" in honor of Nihat Berker, MIT, October 200

Berker, A. Nihat

Fytas, N. G.

Hadjiagapiou, I. A.

Malakis, A.

Papakonstantinou, T.

English

arXiv

The effects of bond randomness on the ground-state structure, phase diagram and critical behavior of the square lattice ferromagnetic Blume-Capel (BC) model are discussed. The calculation of ground states at strong disorder and large values of the crystal field is carried out by mapping the system onto a network and we search for a minimum cut by a maximum flow method. At finite temperatures the system is studied by an efficient two-stage Wang-Landau (WL) method for several values of the crystal field, including both the first- and second-order phase transition regimes of the pure model. We attempt to explain the enhancement of ferromagnetic order and we discuss the critical behavior of the random-bond model. Our results provide evidence for a strong violation of universality along the second-order phase transition line of the random-bond version

Sabanci University Research Database

arXiv:0912.4586v1  [cond-mat.stat-mech]  23 Dec 2009Uncovering the secrets of the 2d random-bond Blume-Capel modelA. Malakisa∗, A. Nihat Berkerbcd, I. A. Hadjiagapioua, N. G. Fytasa and T.PapakonstantinouaaDepartment of Physics, Section of Solid State Physics, University of Athens,Panepistimiopolis, GR 15784 Zografos, Athens, GreecebFaculty of Engineering and Natural Sciences, Sabanci University, Orhanli, Tuzla 34956,Istanbul, TurkeycFeza Gu¨rsey Research Institute, TU¨BI˙TAK - Bosphorus University, C¸engelko¨y 34684,Istanbul, TurkeydDepartment of Physics, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139, U.S.A.The effects of bond randomness on the ground-state structure, phase diagram and crit-ical behavior of the square lattice ferromagnetic Blume-Capel (BC) model are discussed.The calculation of ground states at strong disorder and large values of the crystal field iscarried out by mapping the system onto a network and we search for a minimum cut by amaximum flow method. In finite temperatures the system is studied by an efficient two-stage Wang-Landau (WL) method for several values of the crystal field, including boththe first- and second-order phase transition regimes of the pure model. We attempt toexplain the enhancement of ferromagnetic order and we discuss the critical behavior of therandom-bond model. Our results provide evidence for a strong violation of universalityalong the second-order phase transition line of the random-bond version.1. INTRODUCTIONAlthough originally thought to play a rather innocuous role, quenched bond randomnessmay (or may not) modify the critical exponents of second-order phase transitions [ 1],whereas in 2d it always affects first-order phase transitions by conversion to second-orderphase transitions even for infinitesimal strength [ 2, 3, 4]. These predictions have beenconfirmed by various Monte Carlo simulations and they have been also well verified byour recent numerical studies via a two-stage WL method [ 5, 6, 7]. The random-bondversion of the BC model is defined by the HamiltonianH = −∑<ij>Jijsisj +∆∑is2i ; P (Jij) =12[δ(Jij − J1) + δ(Jij − J2)] ; r = J2/J1, (1)∗amalakis@phys.uoa.gr2 A. Malakiswhere the spin variables si take on the values −1, 0, or +1, < ij > indicates summationover all nearest-neighbor pairs of sites and ∆ is the value of the crystal field. The abovebimodal distribution describes our choice for the quenched random interactions corre-sponding to disorder strength r = J2/J1, where both J1 and J2 are taken positive andthe temperature scale may be set by fixing 2kB/(J1 + J2) = 1. The pure square latticemodel exhibits a phase diagram with ordered ferromagnetic and disordered paramagneticphases separated by a transition line that changes from an Ising-like phase transition toa first-order transition at a tricritical point (Tt,∆t) = (0.609(4), 1.965(5)) [ 8, 9, 10, 11].The present paper considers further interesting aspects of the random-bond version ofthe 2d BC model. In particular we will report on the effects of bond randomness on theground-state structure, the phase diagram and the critical behavior of the square latticeferromagnetic BC model. The ground-state problem is briefly discussed in the next Sectionand the order-parameter behavior, as a function of the disorder strength for several valuesof the crystal field, is illustrated. Then, in Section 3, we report on the effects of bondrandomness on the phase diagram of the model and the enhancement of ferromagneticorder is clarified. We continue with a study of the conversion to a second-order phasetransition, due to the introduction of bond-randomness of the first-order transition of thepure model at the value ∆ = 1.975. The more general critical behavior of the second-order phase transition of the random-bond 2d BC model is also briefly discussed. Theimplemented two-stage method of a restricted entropic WL sampling has been describedin our recent studies [ 6, 7]. Finally, our conclusions are summarized in Section 4.2. UNSATURATED FERROMAGNETIC GROUND STATESIn the presence of bond randomness the competition between the ferromagnetic interac-tions with the crystal field may result in destabilization of the ferromagnetic ground state.This phenomenon occurs for strong disorder and sufficiently large values of the crystalfield giving rise to unsaturated ground states. To clarify this possibility, we observe thatthere will be on the average (in a large sample of disorder realizations) a finite portion ofabout 6.25% of lattice sites with all their connections being weak couplings (J2). Thus,at strong disorder (small J2) these lattice sites will be, at T = 0, in the si = 0 stateprovided that ∆ > 4J2. Consequently, for strong disorder and for any ∆ > 4J2, vacantsites (si = 0) will be distributed in the ground state of any given disorder realization.The complexity of this phenomenon may be further illuminated by defining weak clusterssuch that, in T = 0, these weak clusters will be in the all si = 0 state.This subject has not be studied before and the consequences of the unsaturated groundstate on the critical behavior of the 2d (and 3d) random-bond BC model are not known.The calculation of the ground states can be carried out in polynomially bounded com-puting time by mapping the system into a network and searching for a minimum cut byusing a maximum flow algorithm. The results shown in figure 1, obtained via the FORD-FULKERSON method (see for instance [ 12]), illustrate the ground-state behavior of theorder parameter [M(T = 0)]av as a function of the disorder strength r, for several valuesof the crystal field ∆ = 0.5, 1.5, 1.95, and 1.99 using a square lattice of linear size L = 30.The growing importance of the si = 0 state as we increase the value of the crystal fieldreflects the weak clusters complexity mentioned above and it will be very interesting toUncovering the secrets of the 2d random-bond Blume-Capel model 30,0 0,2 0,4 0,6 0,8 1,00,650,700,750,800,850,900,951,001,05    r = J2 / J1[M(T = 0)]av    L = 30Figure 1. Ground-state behavior of the order parameter of the random-bond 2d BC modelversus r for various values of ∆ averaged over 250 disorder realizations.find any possible interconnections with the critical properties of the random-bond model.3. ENHANCEMENT OF FERROMAGNETIC ORDER AND CRITICALBEHAVIORIn general, the introduction of bond randomness is expected to decrease the phase-transition temperatures which at the percolation limit of randomness (r = 0 and J2 = 0)should tend to zero. However, for weak randomness only a slight decrease is expected,if the average bond strength is maintained, as implemented here. Such a slight decreasein the critical temperature (by 1%) was reported earlier, for ∆ = 1, whereas, in sharpcontrast and for the same disorder strength (r = 0.6), for ∆ = 1.975 (first-order regime ofthe pure model) we have found a considerable increase of the critical temperature, by 9% [7]. A microscopic explanation of this phenomenon was attempted in [ 7] by discussing thebehavior of the connectivity spin densities, Qn = 〈s2i 〉n, where the subscript n = 0, 1, 2, 3, 4denotes the class of lattice sites and is the number of the quenched strong couplings(J1) connecting to each site in this class. For ∆ = 1.975, a pronounced preference wasobserved for the si = 0 state on the low strong-coupling connectivity sites whereas thesi = ±1 states preferentially occurred with strong-coupling connectivity. It was thenpointed out that this naturally leads to a higher transition temperature and effectivelycarries the ordering system to higher non-zero spin densities, the domain of second-orderphase transitions. A similar microsegregation phenomenon, of the si = ±1 states and ofthe si = 0 state, has been seen in the low-temperature second-order transition betweendifferent ordered phases under quenched randomness [ 13].As verified by our subsequent studies (for r = 0.6) this enhancement of the ferromag-netic order takes place in the neighborhood of ∆ = 1.64, where the phase diagrams of thepure and the random-bond models cross each other. Figure 2(a) illustrates the temper-ature behavior of the connectivity densities. The illustration gives the behavior for twovalues of ∆, namely ∆ = 1.5 and ∆ = 1.75 which lie below and above the estimated phase4 A. Malakis0,5 1,0 1,5 2,00,000,250,500,751,001,251,155 1,160 1,165 1,1700,40,50,60,70,80,90,970 0,975 0,980 0,9850,40,50,60,70,80,9upper curves:   QnTlower curves: (a) (b)   TQn  (c)   TQn  Figure 2. Average (10 realizations) behavior of the connectivity spin densities Qn = 〈s2i 〉nfor ∆ = 1.5 and for ∆ = 1.75. The lattice linear size used is L = 40.diagrams crossing. Furthermore, figures 2(b) and (c) illustrate the heights and the thedifferences (between the smallest n = 0 and the largest n = 4) of the connectivity densi-ties in the corresponding pseudocritical region, defined by the specific heat and magneticsusceptibility finite-size peaks. For ∆ = 1.75, the difference is roughly 0.32, whereas for∆ = 1.5 the difference between the smallest and the largest of the connectivity densities isonly 0.239. The middle curves correspond to the leading connectivity density defined onthe lattice sites with two strong and two weak couplings (n = 2). These sites amount onthe average to a relative majority of 37.5% of the lattice sites. The height of the middlecurve falls below the value 2/3 (shown by the dot-line parallel to T -axis) as we move fromthe value ∆ = 1.5 to the value ∆ = 1.75 signaling that the preference on the si = 0 stateis now extended even to the relative majority connectivity sites. This observation maybe related to the enhancement of the ferromagnetic order as we pass through the crossingpoint of the two phase diagrams at ∆ = 1.64. Thus, in agreement with our previousconclusion microsegregation does occur in the first-order regime of the pure model wheremacrosegregation occurs in the absence of bond randomness. However, the enhancementof ferromagnetic order has already appeared before the first-order regime of the puremodel as a kind of a precursor effect induced by weak randomness.We now consider the critical behavior at ∆ = 1.975 [ 7]. Figure 3(a) contrasts thespecific heat results for the pure 2d BC model and two strengths of disorder, r = 17/23 '0.74 and r = 3/5 = 0.6. The saturation of the specific heat is clear and signals theconversion of the first-order transition to a second-order transition with a negative criticalexponent α. Figures 3(b) - (d) summarize our finite-size scaling (FSS) analysis for thedisorder strength r = 3/5 = 0.6. The behavior of five pseudocritical temperatures T[Z]∗av,T[Z]∗av= Tc+bL−1/ν (corresponding to the peaks of susceptibility, derivative of the absoluteorder parameter with respect to the inverse temperature (K = 1/T ) and first-, second-,and fourth-order logarithmic derivatives of the order parameter with respect to the inverseUncovering the secrets of the 2d random-bond Blume-Capel model 510 10010100100010 1000.420.440.460.480.50.5250 60 70 80 90 1000.640.660.680.7010 1000.1110100[]* av/ = 1.751(5)  L(c) (d)/ = 0.128(5)[M] av(T=Tc)  L(b)  T [z]* avL Z =  Z =  <|M|>/  K Z =  ln <M>/  K Z =  ln <M2>/  K Z =  ln <M4>/  KTc = 0.626(2)= 1.30(6) C*  ; [C]* avL C*: pure [C]*av: r = 17 / 23 = 0.74 [C]*av: r = 3 / 5 = 0.6(a)Figure 3. Behavior of the 2d BC model at ∆ = 1.975: (a) Illustration of the saturation ofthe specific heat for the random-bond model (open symbols). FSS behavior, for r = 0.6,of (b) pseudocritical temperatures, (c) susceptibility peaks, and (d) order parameter atTc. Linear fits are applied for L ≥ 50.temperature) is presented in figure 3(b) and allow us to estimate Tc = 0.626(2) and forthe correlation length exponent ν = 1.30(6). From the peaks of the average susceptibilityand the values of the average order parameter at Tc = 0.626, we have estimated, asshown in Figures 3(c) and (d), that the values of the exponent rations γ/ν = 1.751 andβ/ν = 0.128 are very close to the exact values of the Ising model. This “weak universality”is well obeyed for several pure and disordered models, including the pure and random-bond version of the square Ising model with nearest- and next-nearest-neighbor competinginteractions [ 6].At ∆ = 1 the random version should be comparable with the random Ising model, amodel that has been extensively investigated and debated [ 6, 14, 15, 16, 17, 18, 19, 20].Fitting our data for the corresponding pseudocritical temperatures in the range L =50− 100 to the expected power-law behavior mentioned above, we find that the criticaltemperature is Tc = 1.3812(4) and the shift exponent is 1/ν = 1.011(22). This lastestimate is a strong indication that the random-bond 2d BC at ∆ = 1 and weak disorderhas the same value of the correlation’s length critical exponent as the 2d Ising model.Furthermore, our data for the specific heat maxima averaged over disorder, [C]∗av, showedthat the expected double-logarithmic divergence scenario is well obeyed. Finally, the FSSbehavior of the susceptibility peaks (giving the estimate 1.749(7) for γ/ν) and the order-parameter values at the estimated critical temperature Tc = 1.3812 (giving an estimateβ/ν = 0.126(4)) are in good agreement with the expected 2d Ising universality classbehavior.4. CONCLUSIONSThe effects of bond randomness on the ground-state structure of the 2d BC modelhave been briefly illustrated by mapping the system onto a network and searching for aminimum cut by using a maximum flow algorithm. Furthermore, we clarified some aspects6 A. Malakisof the enhancement of ferromagnetic order, due to bond randomness, and we found thatthis appears before the first-order regime of the pure model. Our study at ∆ = 1.975shows a conversion of the first-order transition of the pure model to a second-order phasetransition, giving a distinctive universality class with ν = 1.30(6) and supporting anextensive but weak universality as in a wide variety of 2d systems without [ 21] andwith [ 6, 22] quenched disorder. At ∆ = 1, our study of the random-bond 2d BC modelindicates that for weak disorder the random system belongs to the same universality classas the random Ising model and the effect of the bond disorder on the specific heat is welldescribed by the double logarithmic scenario. These results amount to a strong violation ofuniversality since the two second-order phase transitions mentioned above, with differentsets of critical exponents, are between the same ferromagnetic and paramagnetic phases. Amore complete presentation of the effects of strong disorder on the ground-state structureand also on the critical behavior of the 2d BC model will appear in a forthcoming paper.ACKNOWLEDGEMENTSThis research was supported by the Special Account for Research Grants of the Uni-versity of Athens under Grant No. 70/4/4071.REFERENCES1. A.B. Harris, J. Phys. C 7 (1974) 1671.2. M. Aizenman and J. Wehr, Phys. Rev. Lett. 62 (1989) 2503;M. Aizenman and J. Wehr, Phys. Rev. Lett. 64 (1990) 1311(E).3. K. Hui and A.N. Berker, Phys. Rev. Lett. 62 (1989) 2507;K. Hui and A.N. Berker, Phys. Rev. Lett. 63 (1989) 2433(E).4. A.N. Berker, Physica A 194 (1993) 72.5. F. Wang and D.P. Landau, Phys. Rev. Lett. 86 (2001) 2050.6. N.G. Fytas, A. Malakis and I.A. Hadjiagapiou, J. Stat. Mech. (2008) P11009.7. A. Malakis, A.N. Berker, I.A. Hadjiagapiou and N.G. Fytas, Phys. Rev. E 79 (2009)011125.8. M. Blume, Phys. Rev. 141 (1966) 517; H.W. Capel, Physica (Utr.) 32 (1966) 966.9. P.D. Beale, Phys. Rev. B 33 (1986) 1717.10. D.P. Landau and R.H. Swendsen, Phys. Rev. B 33 (1986) 7700.11. C.J. Silva, A.A. Caparica and J.A. Plascak, Phys. Rev. E 73 (2006) 036702.12. A.K. Hartmann and H. Rieger, Optimization Problems in Physics (Wiley-VCH, 2002).13. C.N. Kaplan and A.N. Berker, Phys. Rev. Lett. 100 (2008) 027204.14. V.S. Dotsenko and V.S. Dotsenko, Sov. Phys. JETP Lett. 33 (1981) 37.15. B.N. Shalaev, Sov. Phys. Solid State 26 (1984) 1811.16. R. Shankar, Phys. Rev. Lett. 58 (1987) 2466.17. A.W.W. Ludwig, Nucl. Phys. B 285 (1987) 97.18. F.D.A. Aara˜o Reis, S.L.A. de Queiroz and R.R. dos Santos, Phys. Rev. B 54 (1996)R9616.19. H.G. Ballesteros, L.A. Ferna´ndez, V. Mart´ın-Mayor, A. Mun˜oz Sudupe, G. Parisi andJ.J. Ruiz-Lorenzo, J. Phys. A 30 (1997) 8379.20. W. Selke, L.N. Shchur and O.A. Vasilyev, Physica A 259 (1998) 388.Uncovering the secrets of the 2d random-bond Blume-Capel model 721. M. Suzuki, Prog. Theor. Phys. 51 (1974) 1992.22. J.-K. Kim and A. Patrascioiu, Phys. Rev. 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Uncovering the secrets of the 2D random-bond Blume-Capel model

The effects of bond randomness on the ground-state structure, phase diagram and critical behavior of the square lattice ferromagnetic Blume-Capel (BC) model are discussed. The calculation of ground states at strong disorder and large values of the crystal field is carried out by mapping the system onto a network and we search for a minimum cut by a maximum flow method. In finite temperatures the system is studied by an efficient two-stage Wang-Landau (WL) method for several values of the crystal field, including both the first- and second-order phase transition regimes of the pure model. We attempt to explain the enhancement of ferromagnetic order and we discuss the critical behavior of the random-bond model. Our results provide evidence for a strong violation of universality along the second-order phase transition line of the random-bond version

Uncovering the secrets of the 2d random-bond Blume-Capel model

The effects of bond randomness on the ground-state structure, phase diagram and critical behavior of the square lattice ferromagnetic Blume–Capel (BC) model are discussed. The calculation of ground states at strong disorder and large values of the crystal field is carried out by mapping the system onto a network and we search for a minimum cut by a maximum flow method. At finite temperatures the system is studied by an efficient two-stage Wang–Landau (WL) method for several values of the crystal field, including both the first- and second-order phase transition regimes of the pure model. We attempt to explain the enhancement of ferromagnetic order and we discuss the critical behavior of the random-bond model. Our results provide evidence for a strong violation of universality along the second-order phase transition line of the random-bond version

Uncovering the secrets of the 2d random-bond Blume-Capel model

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