content_image/422585.jpg content_image/422584.jpg
<img src='content_image/422585.jpg'>

<img src='content_image/422584.jpg'>

In Fig. 3 the corresponding behaviour of the suscepti- bility χ is presented. 

In Fig. 4 the fourth-order Binder cumulant is shown as a function of the temperature for several values of L . Taking two largest lattices (for L = 64 and L = 128) we have T$_{C}$ = 1 . 590(3) and T$_{C}$ = 2 . 100(3) for (3 , 4 , 6 , 4) and (3 $^{4}$, 6) AL, respectively. 

In order to go further in our analysis we also com- puted the modulus of the magnetisation at the inflection M ∗ = M ( T$_{C}$ ). The estimated exponents β/ν values are 0 . 180(20) and 0 . 103(7) for (3 , 4 , 6 , 4) and (3 $^{4}$, 6) AL, re- spectively. 

Basing on the dependence ln χ on ln L we estimated γ/ν = 1 . 46(8) and γ/ν = 1 . 44(8) for (3 , 4 , 6 , 4) and (3 $^{4}$, 6) AL, respectively. 

To obtain the critical exponent 1 /ν , we used the scaling relation (4). The calculated values of the exponents 1 /ν are 0 . 83(5) for (3 , 4 , 6 , 4) and 1 /ν = 0 . 94(5) for (3 $^{4}$, 6). Eq. (5) yields effective dimensionality of the systems 

D$_{eff}$ = 1 . 82(4) for (3 , 4 , 6 , 4) and D$_{eff}$ = 1 . 64(5) for (3 $^{4}$, 6). 

The above results, indicate that the Ising S = 1 model on (3 , 4 , 6 , 4) and (3 $^{4}$, 6) AL does not fall in the same universality class as the square lattice Ising model, for which the critical exponents are known analytically i.e. β = 1 8 = 0 . 125, γ = 7 4 = 1 . 75 and ν = 1. We have checked numerically, that Ising S = 1 2 model repro- duces these critical exponents with reasonable accuracy for both studied lattices [25]. We improved the value of the critical temperature T$_{C}$ for these two lattices and S = 1 2 as well, with respect to Ref. [21]. 

The results are collected in Tab. I. 

Except the exponent ν , all critical exponents for S = 1 differ for more than three numerically estimated uncer- tainties from those given analytically. 

