GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s49>GLYPH<s54>GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s51>GLYPH<s46>GLYPH<s54> GLYPH<s51>GLYPH<s46>GLYPH<s55> GLYPH<s51>GLYPH<s46>GLYPH<s56> GLYPH<s51>GLYPH<s46>GLYPH<s57> GLYPH<s52>GLYPH<s46>GLYPH<s48> GLYPH<s52>GLYPH<s46>GLYPH<s49> GLYPH<s52>GLYPH<s46>GLYPH<s50> GLYPH<s40>GLYPH<s98> GLYPH<s41> GLYPH<s40> GLYPH<s32> GLYPH<s108> GLYPH<s110> GLYPH<s32> GLYPH<s60> GLYPH<s77> GLYPH<s110> GLYPH<s62> GLYPH<s32> GLYPH<s47> GLYPH<s32> GLYPH<s32> GLYPH<s75> GLYPH<s41> GLYPH<s75> GLYPH<s61> GLYPH<s75> GLYPH<s99> GLYPH<s76> GLYPH<s32>GLYPH<s110> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s58>GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48> GLYPH<s46>GLYPH<s57> GLYPH<s57> GLYPH<s56> GLYPH<s40>GLYPH<s52> GLYPH<s41> GLYPH<s32>GLYPH<s110> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s50> GLYPH<s58>GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48> GLYPH<s46>GLYPH<s57> GLYPH<s57> GLYPH<s57> GLYPH<s40>GLYPH<s53> GLYPH<s41> GLYPH<s32>GLYPH<s110> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s52> GLYPH<s58>GLYPH<s32> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s48> GLYPH<s46>GLYPH<s57> GLYPH<s57> GLYPH<s57> GLYPH<s40>GLYPH<s51> GLYPH<s41> GLYPH<s84> GLYPH<s42> GLYPH<s76> GLYPH<s59> GLYPH<s32>GLYPH<s90> GLYPH<s76> GLYPH<s32>GLYPH<s90>GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s67> GLYPH<s32>GLYPH<s90>GLYPH<s32>GLYPH<s61>GLYPH<s32> GLYPH<s32>GLYPH<s90>GLYPH<s32>GLYPH<s61>GLYPH<s32> GLYPH<s32>GLYPH<s60>GLYPH<s124>GLYPH<s77>GLYPH<s124>GLYPH<s62>GLYPH<s32>GLYPH<s47> GLYPH<s32>GLYPH<s75> GLYPH<s32>GLYPH<s90>GLYPH<s32>GLYPH<s61>GLYPH<s32> GLYPH<s32>GLYPH<s108>GLYPH<s110> GLYPH<s32>GLYPH<s60>GLYPH<s77>GLYPH<s62>GLYPH<s32>GLYPH<s47>GLYPH<s32> GLYPH<s32>GLYPH<s75> GLYPH<s32>GLYPH<s90>GLYPH<s32>GLYPH<s61>GLYPH<s32> GLYPH<s32>GLYPH<s108>GLYPH<s110> GLYPH<s32>GLYPH<s60>GLYPH<s77> GLYPH<s50> GLYPH<s62>GLYPH<s32>GLYPH<s47>GLYPH<s32> GLYPH<s32>GLYPH<s75> GLYPH<s32>GLYPH<s90>GLYPH<s32>GLYPH<s61>GLYPH<s32> GLYPH<s32>GLYPH<s108>GLYPH<s110> GLYPH<s32>GLYPH<s60>GLYPH<s77> GLYPH<s52> GLYPH<s62>GLYPH<s32>GLYPH<s47>GLYPH<s32> GLYPH<s32>GLYPH<s75> GLYPH<s84> GLYPH<s99> GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s51> GLYPH<s46>GLYPH<s54> GLYPH<s52> GLYPH<s49> GLYPH<s50> GLYPH<s40>GLYPH<s53> GLYPH<s41>GLYPH<s32> GLYPH<s84> GLYPH<s99> GLYPH<s40>GLYPH<s101>GLYPH<s120> GLYPH<s97>GLYPH<s99>GLYPH<s116>GLYPH<s41>GLYPH<s32>GLYPH<s61>GLYPH<s32>GLYPH<s51> GLYPH<s46>GLYPH<s54> GLYPH<s52> GLYPH<s48> GLYPH<s57> GLYPH<s61>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s52> GLYPH<s40>GLYPH<s53> GLYPH<s41> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s49>GLYPH<s54>GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s53> GLYPH<s50>GLYPH<s46> GLYPH<s48> GLYPH<s50>GLYPH<s46> GLYPH<s53> GLYPH<s51>GLYPH<s46> GLYPH<s48> GLYPH<s65> GLYPH<s48> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s52>GLYPH<s57>GLYPH<s57>GLYPH<s48>GLYPH<s40> GLYPH<s57>GLYPH<s41> GLYPH<s66>GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s49>GLYPH<s51>GLYPH<s56>GLYPH<s40> GLYPH<s53>GLYPH<s41> GLYPH<s67> GLYPH<s99> GLYPH<s32> GLYPH<s61>GLYPH<s32> GLYPH<s65> GLYPH<s48> GLYPH<s32> GLYPH<s108>GLYPH<s110>GLYPH<s32> GLYPH<s76> GLYPH<s32> GLYPH<s43>GLYPH<s32> GLYPH<s66> GLYPH<s67> GLYPH<s99> GLYPH<s76> FIG. 2: (a) Simultaneous fitting of the form (6) of 6 pseudo- critical temperatures defined in the text. (b) FSS of several powers of the logarithmic derivatives [Eq. (5)] of the order parameter at the critical temperature in a log-log scale. The solid lines are linear fittings giving the value ν = 1 for the correlation length’s exponent. lowing six quantities: specific heat C , magnetic suscepti- bility χ , derivative of the absolute order parameter with respect to inverse temperature K = 1 /T [75] and logarithmic derivatives of the first ( n = 1), second ( n = 2), and fourth ( n = 4) powers of the order parame- ter with respect to inverse temperature [75] Fitting our data for the whole lattice range to the ex- pected power-law behavior where Z stands for the different thermodynamic quan- tities mentioned above, we find the critical temperature to be T$_{c}$ = 3 . 6412(5) which is in excellent agreement with the exact value 3 . 6409 · · · . Additionally, our esti- mate of the critical exponent ν of the correlation length is ν = 1 . 004(5), also in excellent agreement with the value ν = 1. Additional estimates for the critical exponent ν may be obtained from the scaling behavior of the loga- rithmic derivatives (5), which scale as ∼ L 1 /ν with the The universality of the above expansion has been al- ready pointed out and discussed by Fisher [60]. For the three most common 2d lattices, i.e. the square, plane triangular, and honeycomb, a unified approach has been presented by Wu et al. [79], from which one can find also for the plane triangular lattice the first B -coefficients and the amplitude A$_{0}$ at the critical temperature. As shown by Wu et al. [79] for all the three lattices, the coeffi- cient B$_{1}$ is zero at the critical temperature and as in the square lattice, also for the triangular lattice, the crit- ical amplitude of the expansion is A$_{0}$ = 0 . 499069 · · · , equal to the amplitude of the temperature expansion close to the critical point obtained by Houtappel [63]. From the work of Wu et al. [79] we find, using their closed form expressions, for the plane triangular lat- tice (with periodic boundary conditions and aspect ratio where the value of the critical amplitude A$_{0}$ is 0 . 494358 · · · . As pointed out in Ref. [62], this is the same with the amplitude A$_{0}$ in the temperature expansion of the specific heat close to the critical point and this was already known from the original paper of Onsager [76]. The first B coefficients are given in Ref. [62] and further details have been presented in Refs. [68, 77–79]. In par- ticular, at the critical temperature the constant term B is 0 . 138149 · · · and as it is also well-known [62, 78] the coefficient B$_{1}$ is zero. lattice size [75]. The FSS of these logarithmic derivatives at the critical temperature is shown in Fig. 2(b) and in all three cases a value ν ≈ 1 is obtained consistent with the estimate from panel (a) of Fig. 2 and with the exact value ν = 1. We know turn to the most interesting issue in the study of the pure model, which is the specific heat’s logarith- mic expansion, as mentioned in the introduction. For the square lattice, it was shown in 1969 in the pioneer- ing work of Ferdinand and Fisher [62] that close to the critical point the specific heat obeys the following FSS expansion FIG. 3: FSS of the specific heat data at the exact critical temperature. 4 C$_{L}$ ( T ) = A$_{0}$ ln L + B ( T ) + B$_{1}$ ( T ) ln L L + B$_{2}$ ( T ) 1 L + · · · , (7) T ∗ L ; Z = T$_{c}$ + bL − 1 $^{/ν}$, (6) ∂ ln 〈 M $^{n}$〉 ∂K = 〈 M $^{n}$H 〉 〈 M $^{n}$〉 − 〈 H 〉 . (5) ∂ 〈| M |〉 ∂K = 〈| M | H 〉 − 〈| M |〉〈 H 〉 , (4)