Generalized Scaling Theory for Critical Phenomena Including Essential Singularities and Infinite Dimensionality

We propose a generic scaling theory for critical phenomena that includes power-law and essential singularities in finite and infinite dimensional systems. In addition, we clarify its validity by analyzing the Potts model in a simple hierarchical network, where a saddle-node bifurcation of the renormalization-group fixed point governs the essential singularity.

Figure 1

Recursive construction of a shortcut network. The number of generation, $n$, equals 1, 2, 3, and 4 from left to right. The vertical lines and solid arcs indicate backbone edges and shortcut edges, respectively. A periodic boundary condition is imposed in the vertical direction. The graph with $n=1$ is a single node with a self-connecting edge.

Figure 2

Phase diagram for $q=8$. The shaded region indicates the critical phase. The (blue) solid line and the (red) dashed line denote the stable and unstable fixed lines, respectively. The circle symbol denotes the SNB point, $(j_{\mathrm{sn}},k_{\mathrm{sn}})$.

Figure 3

Scaling plot of the order parameter for $1/N\approx 0$ (a) and for $t=0$ (b). In the latter case, we show the data for $n=4096$ and 16 384 to confirm the convergence to the large size limit.