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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 

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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 

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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. 

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 

