Multicritical Points and Crossover Mediating the Strong Violation of Universality: Wang-Landau Determinations in the Random-Bond $^{d}$= 2 Blume-Capel model arXiv:1001.1240v1 [cond-mat.stat-mech] 8 Jan 2010 I. INTRODUCTION nent ( α = 0) are marginal cases of Harris criterion (since the criterion does not give any information) and their study, upon the introduction of disorder, has been of par- ticular interest. The paradigmatic model of the marginal case is, of course, the general random 2d Ising model (random-site, random-bond, and bond-diluted) and this model has been extensively investigated and debated [8– 24]. Several recent studies, both analytical (renormal- ization group and conformal field theories) and numeri- cal (mainly Monte Carlo (MC) simulations) devoted to this model, have provided very strong evidence in favor of the so-called logarithmic corrections’s scenario. Ac- cording to this, the effect of infinitesimal disorder gives rise to a marginal irrelevance of randomness and besides logarithmic corrections, the critical exponents maintain their 2d Ising values. In particular, the specific heat is expected to slowly diverge with a double-logarithmic de- pendence [11–14]. Here, we should mention that there is not full agreement in the literature and a different sce- nario predicts a negative specific heat exponent α leading to a saturating behavior [16], with a corresponding cor- relation length exponent ν ≥ 2 /d [25]. In general, a unitary and rigorous physical description of critical phenomena in disordered systems still lacks and certainly, lacking such a description, the study of further models for which there is a general agreement in the the behavior of the corresponding pure cases is very important. Historically, such a suitable candidate for testing the above predictions, that has been also quite extensively studied, is the general 2d q -state Potts model [17, 26–30]. This model includes the Ising model ( q = 2), cases of a pure system having continuous transi- tions with a positive exponent α ( q = 3 , 4) and also the The effect of quenched randomness on the equilib- rium and dynamic properties of macroscopic systems is a subject of great theoretical and practical interest. It is well known that quenched bond randomness may produce drastic changes on phase transitions depending on the type of the transition [1–6]. Thus, symmetry- breaking first-order transitions are converted to second- order phase transitions by infinitesimal bond randomness for spatial dimensionality d = 2 [3, 4] and by bond ran- domness beyond a threshold strength in d > 2 [4], as indicated by general arguments [5] and in some cases by rigorous mathematical work [3]. In particular, this rounding effect of first-order transitions has now been rigorously established in a unified way in low dimensions ( d ≤ 2) including a large variety of types of randomness in classical and quantum spin systems [7]. Historically, the effects of disorder on phase transitions have been studied in two extreme cases, i.e. in the lim- its of weak and strong (near the percolation point) dis- order. The first important conjecture, known today as the Harris criterion [1], relates the value of the specific heat exponent α in a continuous transition with the ex- pected effects of uncorrelated weak disorder in ferromag- nets. According to the Harris criterion, for continuous phase transitions with a negative exponent α , the intro- duction of weak randomness is expected to be an irrel- evant field and the disordered system to remain in the same universality class. On the other hand, the weakly disordered system is expected to be in a different univer- sality class in the case of a pure system having a positive exponent α . Pure systems with a zero specific heat expo- The effects of bond randomness on the phase diagram and critical behavior of the square lat- tice ferromagnetic Blume-Capel model are discussed. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method for many values of the crystal field, restricted here in the second-order phase transition regime of the pure model. For the random-bond version several disorder strengths are considered. We present phase diagram points of both pure and random versions and for a particular disorder strength we locate the emergence of the enhancement of ferromagnetic order observed in an earlier study in the ex-first-order regime. The critical properties of the pure model are contrasted and compared to those of the random model. Accepting, for the weak random version, the assumption of the double logarithmic scenario for the specific heat we attempt to estimate the range of universality between the pure and random-bond models. The behavior of the strong disorder regime is also discussed and a rather complex and yet not fully understood behavior is observed. It is pointed out that this complexity is related to the ground-state structure of the random-bond version. PACS numbers: 75.10.Nr, 05.50.+q, 64.60.Cn, 75.10.Hk A. Malakis $^{1}$, A. Nihat Berker2 , $^{3}$, I. A. Hadjiagapiou$^{1}$, N. G. Fytas$^{1}$, and T. Papakonstantinou1 $^{1}$Department of Physics, Section of Solid State Physics, University of Athens, Panepistimiopolis, GR 15784 Zografos, Athens, Greece $^{2}$Faculty of Engineering and Natural Sciences, Sabanc ı University, Orhanl ı , Tuzla 34956, Istanbul, Turkey and $^{3}$Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. (Dated: November 2, 2018)