Universal Behavior of Crossover Scaling Functions for Continuous Phase Transitions

We consider two different systems exhibiting a continuous phase transition into an absorbing state. Both models belong to the same universality class; i.e., they are characterized by the same scaling functions and the same critical exponents. Varying the range of interactions, we examine the crossover from the mean-field-like to the non-mean-field scaling behavior. A phenomenological scaling form is applied in order to describe the full crossover region, which spans several decades. Our results strongly support the hypothesis that the crossover function is universal.

Figure 1

The order parameter of the CTTP for various values of the interaction range $R$. With increasing interaction range, the critical density tends to the mean-field value ${\rho }_{\mathrm{c},R\to \infty }=1/2$. The inset displays the order parameter which is rescaled according to Eq. (18). The dotted line corresponds to the scaling laws $y=mx$ with $m=(a_{\rho ,R=1}/{\rho }_{\mathrm{c},R=1}{)}^{{\beta }_{D=2}}$.

Figure 2

The rescaled order parameter. The metric factors are given by $c={\mathfrak{a}}_{\rho }{\mathfrak{a}}_R^2$ and $d={\mathfrak{a}}_R^2$. The data of both models display an excellent collapse to the universal crossover scaling function $\tilde{\mathfrak{R}}(x,1)$. The dashed lines correspond to the asymptotic behavior of the two-dimensional system (${\beta }_{D=2}=0.639$ [13]) and of the mean-field behavior (${\beta }_{\mathrm{M}\mathrm{F}}=1$). Neglecting the metric factors, each model is characterized by its own scaling function (see lower right inset). The upper left inset displays the effective exponents ${\beta }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ for both considered models. The data of the CLG model and of the CTTP exhibit an excellent data collapse.

Figure 3

The rescaled fluctuations of the order parameter. The metric factor is given by $c={\mathfrak{a}}_{\rho }{\mathfrak{a}}_R^2$. The data of both models display a good collapse to the universal crossover scaling function $\tilde{\mathfrak{D}}(x,1)$. The dashed lines correspond to the asymptotic behavior of the two-dimensional system (${\gamma }_{D=2}^{\prime }=0.381$ [13] and $m_{\Delta ,\rho }=1.28$) and of the mean-field behavior (${\gamma }_{\mathrm{M}\mathrm{F}}^{\prime }=0$). The inset displays the corresponding effective exponent ${\gamma }_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\prime }$.