Oscillatory behavior of critical amplitudes of the Gaussian model on a hierarchical structure

Abstract

We studied oscillatory behavior of critical amplitudes for the Gaussian model on a hierarchical structure presented by a modified Sierpinski gasket lattice. This model is known to display non-standard critical behavior on the lattice under study. The leading singular behavior of the correlation length ξ near the critical coupling K = K c is modulated by a function which is periodic in ln | ln(K c − K)|. We have also shown that the common finite-size scaling hypothesis, according to which for a finite system at criticality ξ should be of the order of the size of system, is not applicable in this case. As a consequence of this, the exact form of the leading singular behavior of ξ differs from the one described earlier (which was based on the finite-size scaling assumption).64.60. Ak, 05.50.+q, 05.40.Fb

The Yang–Lee edge singularity for the Ising model on two Sierpinski fractal lattices

“…Let us note here that this result is in certain contrast to the usual finite-size scaling assumption, according to which the correlation length of a finite system at criticality should be of the order of the system's size (which would be ∼3 r 0 in our case, instead of ( 12)). This is not so surprising, because a similar discrepancy with finite-size scaling hypothesis has been noticed earlier in our studies of a Gaussian model on the same structure [19]. Using (11) and (12), one finds that the YL correlation length, instead of a standard power-law, follows the logarithmic behavior…”

“…Using such an approach we determined with four correct digits, which is much better than an estimate that we can obtain directly, i.e. from the slope of the curve shown in Let us note here that the YL correlation length (13) follows the asymptotic behavior of the Gaussian correlation length ξ G on the same structure [19]: ξ G ∼ ln (δK), where is given by (13), and δK = K c − K > 0 provides the distance from the critical coupling K c . Besides, one can demonstrate the coincidence between the period τ YL of the oscillations of the YL critical amplitude, see figure 2(b), and period τ G = ln(2) of the oscillations of the corresponding Gaussian critical amplitude [19].…”

“…from the slope of the curve shown in Let us note here that the YL correlation length (13) follows the asymptotic behavior of the Gaussian correlation length ξ G on the same structure [19]: ξ G ∼ ln (δK), where is given by (13), and δK = K c − K > 0 provides the distance from the critical coupling K c . Besides, one can demonstrate the coincidence between the period τ YL of the oscillations of the YL critical amplitude, see figure 2(b), and period τ G = ln(2) of the oscillations of the corresponding Gaussian critical amplitude [19]. Indeed, one can regard relation (13) as a 'pure' power-law, with the scaling variable ln(δh), rather than δh, and a critical exponent .…”

“…The critical behavior of the Gaussian model is known. In particular, it has been shown that the Gaussian correlation length exhibits a logarithmic singularity in the first case [18,19], while it follows a usual power-law behavior for the model situated on a d = 3 Sierpinski gasket of figure 1(b) (see e.g. [20,21]).…”

“…Here we use the notation of reference [19] to emphasize that the density of zeros of the ferromagnetic Ising model follows the asymptotic behavior of the average Gaussian energy per site, with the same value of . In this sense, one can say that these two models, on the b = 3 Sierpinski gasket embedded in the Euclidean space of dimension d = 2, belong to the same class of universality.…”