# Critical behavior of the pure and random-bond two-dimensional triangular Ising ferromagnet 

Department of Physics, Section of Solid State Physics, University of Athens, Panepistimiopolis, GR 15784 Zografos, Athens, Greece 

Nikolaos G. Fytas and Anastasios Malakis 

(Dated: October 28, 2018) 

We investigate the effects of quenched bond randomness on the critical properties of the two- dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat’s logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-known scenarios - logarithmic corrections versus weak universality - strongly support the field-theoretically predicted scenario of logarithmic corrections. A particular interest is paid to the sample-to-sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality. 

Understanding the role played by impurities on the nature of phase transitions is of great importance, both from experimental and theoretical perspectives. First- order phase transitions are known to be dramatically softened under the presence of quenched randomness [1– 10], while continuous transitions may have their expo- nents altered under random fields or random bonds [11, 12]. There are some very useful phenomenological ar- guments and some, perturbative in nature, theoreti- cal results, pertaining to the occurrence and nature of phase transitions under the presence of quenched ran- domness [2, 5, 13, 14]. Historically, the most celebrated criterion is that suggested by Harris [11]. This criterion relates directly the persistence, under random bonds, of the non random behavior to the specific heat exponent α$_{p}$ of the pure system. According to this criterion, if α$_{p}$ > 0, then disorder will be relevant, i.e., under the ef- fect of the disorder, the system will reach a new critical behavior. Otherwise, if α$_{p}$ < 0, disorder is irrelevant and the critical behavior will not change. 

Pure systems with a zero specific heat exponent ( α$_{p}$ = 0) are marginal cases of the Harris criterion and their study, upon the introduction of disorder, has been of particular interest [15]. 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 de- bated [see Ref. [16] and references therein]. Several re- cent studies, both analytical (renormalization group and conformal field theories) and numerical (mainly Monte Carlo (MC) simulations) devoted to this model, have provided very strong evidence in favor of the so-called logarithmic corrections’s scenario [17–21]. According to this, the effect of infinitesimal disorder gives rise to a marginal irrelevance of randomness and besides logarith- mic corrections, the critical exponents maintain their 2d 

PACS numbers: 75.10.Nr, 05.50.+q, 64.60.Cn, 75.10.Hk 

## I. INTRODUCTION 

Ising values. In particular, the specific heat is expected to slowly diverge with a double-logarithmic dependence of the form C ∝ ln(ln t ), where t = | T − T$_{c}$ | /T$_{c}$ is the reduced critical temperature [17–21]. Here, we should mention that there is not full agreement in the literature and a different scenario, the so-called weak universality scenario [22–25], predicts that critical quantities, such as the magnetization, zero-field susceptibility, and cor- relation length display power-law singularities, with the corresponding exponents β , γ , and ν changing continu- ously with the disorder strength; however this variation is such that the ratios β/ν and γ/ν remain constant at the pure system’s value. The specific heat of the disor- dered system is, in this case, expected to saturate, with a corresponding correlation length’s exponent ν ≥ 2 /d [12]. 

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 behavior of the corresponding pure cases is very important. In this sense, the triangular Ising ferromag- net is a further suitable candidate for testing the above predictions that has not been previously investigated in the literature. Thus, our investigation will be related to the extensive relevant literature concerning the critical properties of the disordered 2d square Ising model [15– 23, 26–59]. In particular, our discussion will focus on the main point of dispute over the last two decades, concern- ing the two well-known conflicting scenarios mentioned above and we will provide additional new evidence in fa- vor of the well-established scenario of strong universality. We should note here that, the theoretically predicted sce- nario of strong universality has been confirmed by several MC studies on the square lattice starting from the early 90’s to nowadays [see Ref. [16] for a detailed historical review]. 

As mentioned above, it is always important to consider further models for which the critical properties of the 

