GLYPH<s48>GLYPH<s46> GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s53> GLYPH<s50>GLYPH<s46> GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s50>GLYPH<s53> GLYPH<s48>GLYPH<s46> GLYPH<s53>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s55>GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s50>GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s53>GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s55>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s50>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s53>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s55>GLYPH<s53> GLYPH<s50>GLYPH<s46>GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s55>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s50>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s53>GLYPH<s48> GLYPH<s99>GLYPH<s114>GLYPH<s111> GLYPH<s115>GLYPH<s115> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49>GLYPH<s46>GLYPH<s54>GLYPH<s54>GLYPH<s40>GLYPH<s50>GLYPH<s41> GLYPH<s84> GLYPH<s99>GLYPH<s114>GLYPH<s111> GLYPH<s115>GLYPH<s115> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49>GLYPH<s46>GLYPH<s48>GLYPH<s50>GLYPH<s40>GLYPH<s51>GLYPH<s41> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s107> GLYPH<s66> GLYPH<s84> GLYPH<s47> GLYPH<s74> GLYPH<s32>GLYPH<s47>GLYPH<s32>GLYPH<s74> GLYPH<s32> GLYPH<s84>GLYPH<s114> GLYPH<s105>GLYPH<s99>GLYPH<s114> GLYPH<s105>GLYPH<s116>GLYPH<s105>GLYPH<s99>GLYPH<s97>GLYPH<s108>GLYPH<s32> GLYPH<s32> GLYPH<s80> GLYPH<s117>GLYPH<s114> GLYPH<s101> GLYPH<s32> GLYPH<s82>GLYPH<s97>GLYPH<s110>GLYPH<s100>GLYPH<s111>GLYPH<s109>GLYPH<s32> GLYPH<s40> GLYPH<s114> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s54>GLYPH<s41> FIG. 2: Phase diagrams of the pure 2d BC model and its random version at the disorder strength r = 0 . 75 / 1 . 25. The inset illustrates the crossing of the phase boundaries including an approximate estimate of the crossing point. the crossing point (Δ$_{cross}$ , T$_{cross}$ ) = (1 . 66(2) , 1 . 02(3)), well before the tricritical point. The approximation of this point was obtained by interpolation using the four points of the phase diagram in the range Δ = 1 . 0 − 1 . 9. Table I summarizes the phase diagram points ob- tained in this paper, by the above described traditional FFS method for both the pure and random version of the 2d BC model for disorder strengths r = 0 . 9 / 1 . 1, r = 0 . 75 / 1 . 25, and r = 0 . 6 / 1 . 4, together with corre- sponding phase diagram points given by Beale [48] for the pure model. Note that, in Silva et al. [49] one can find an analogous table for the pure 2d BC model includ- ing the phase diagram points given by Beale [48] and the points produced in their two-parametric WL sam- pling [49]. It can be seen that there is an excellent co- incidence of our points with those of Beale [48]. The points of Beale [48] are based on the very accurate phe- nomenological FSS scheme using a strip geometry [47], whereas our points are obtained via the present simul- taneous FSS analysis, based on lattices with linear sizes L = 20 − 100. The points of Silva et al. [49] are based in much smaller lattices of the two-parameter WL sampling (linear sizes L ≤ 16). However, in the case Δ = 1 . 95 our phase diagram point does not agree, within errors, with that of Beale and for this reason our estimation with a rather generous error bars (shown also on the panel) has been illustrated in Fig. 1(b). The rest of Table I con- tains our estimates for the random-bond versions. One may note from this table that for small values of Δ for instance Δ = 1 . 5 the corresponding critical temperatures decrease as the disorder becomes stronger (compare the three cases of disorder: r = 0 . 9 / 1 . 1, r = 0 . 75 / 1 . 25, and r = 0 . 6 / 1 . 4 at this value of the crystal field). On the other hand, for Δ = 1 . 9 the trend is reversed as can be seen by comparing the corresponding three cases of disorder. Apparently this is a kind of reflection of the phenomenon of the enhancement of ferromagnetic order which appears to influence the geometry of the critical surface for the 2d random-bond BC model. disorder case r = 0 . 9 / 1 . 1 at the value Δ = 1 . 5 and Fig. 1(b) illustrates the simultaneous fitting attempted for the pure model at Δ = 1 . 95, further discussed at the end of Sec. IV B. As noted already, the data fitted for the random version of the model are only those of the pseudocritical temperatures of the peaks of the aver- aged over disorder thermodynamic parameters, as indi- cated in the figure, where the asterisk denotes the peak of the averaged over disorder parameter [ . . . ] ∗ $_{av}$. The alter- native route of using averages of individual sample pa- rameters gives almost identical estimates. However, in cases of strong lack of self-averaging, very large sample- to-sample fluctuations may be present. For simplicity, in all fitting attempts, the whole range L = 20 − 100 has been used. Following this practice for the pure and random version of the 2d BC several phase dia- gram points were produced. Figure 2 shows the result- ing phase diagrams using our phase diagram points for the pure 2d BC model and its random-bond version for the disorder strength r = 0 . 75 / 1 . 25 = 0 . 6. The tricrit- ical point shown is taken from the estimate given by Beale (Δ$_{t}$ , T$_{t}$ )=(1 . 9655(10) , 0 . 610(5)) [48]. For the dis- order strength r = 0 . 75 / 1 . 25 the points of the phase dia- gram were chosen with the intension to be able to approx- imately locate the emergence of the enhancement of fer- romagnetic order observed in our earlier study [37] in the ex-first-order regime at Δ = 1 . 975, where we found a con- siderable increase of the critical temperature by ∼ 9%. A microscopic explanation of this phenomenon, based on the preference s$_{i}$ = ± 1 states to the strong-coupling connectivity sites has been given in Ref. [37] (see also Ref. [52]), as a microsegregation process, due to quenched bond randomness, that evolves continuously within the ferromagnetic and paramagnetic phases. The inset of Fig. 2 shows that phase diagram of the random version crosses the diagram of the pure model at approximately As pointed out in the introduction, in our recent inves- tigation of the 2d random-bond BC model [37] we found dramatically different critical behaviors of the second- order phase transitions emerging from the first- and second-order regimes of the pure model. Namely, dif- ferent sets of critical exponents on two segments of the same critical line appeared to describe the two regimes: the still-second-order and ex-first-order regimes. The study in Ref. [37] was carried out for two values of the crystal-field coupling corresponding to the second- order (Δ = 1) and first-order (Δ = 1 . 975) regimes of A. Strong violation of universality: The ex-first-order regime IV. PHASE TRANSITIONS OF THE PURE AND RANDOM-BOND 2D BLUME-CAPEL MODELS 5