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

Abstract

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.

Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

“…This network is very useful because rigorous real-space renormalization is possible for various models in the simplest way. Furthermore various types of phase transition are observed depending on the model used, e.g., a discontinuous transition of the bond percolation model [12], equivalent to the one-state Potts model [14,15], and continuous transitions with a power-law singularity (PLS) or an essential singularity (ES) for the q-state Potts model with q 3 [16]. These are observed in other graphs [3][4][5]17,18].…”

“…For q 3, the saddle-node (SN) bifurcation of the FP is observed [16]. Consequently, two kinds of singularity appear depending on k = e −K : ES corresponding to SN BP for k k SN and PLS corresponding to USFP for k < k SN .…”

“…If ln N is a proper cutoff length, μ should be unity. Equations (26) Table I, where we also show the Potts model with q = 1 [12] and q 3 [16]. See also the inset of Fig.…”

“…Thus the generalized scaling theory in Ref. [16] assuming instability of a FP cannot be applied to the latter. We emphasize that the stability or instability of the FP of the transition point is the most fundamental criterion of phase transitions.…”

“…(6), for q 3 in Ref. [16], we obtain the PF BP (j,k) = (1/2,0) for q 2 and the recursion relation at j = 1/2: k n+1 − k n = − 2−q 4 k 2 − q−1 4 k 3 + O(k 4 ). Thus the FP of q = 2 is marginal and has weaker stability.…”