First order phase transitions of the Potts model in fractal dimensions

Effects of random and deterministic discrete scale invariance on the critical behavior of the Potts model

“…1; the calculations of noninteger dimension points have been obtained by means of numerical simulation presented in Ref. [10][11][12]. Such a diagram deserves two comments: (i) A border separating a first order from a second order region in qualitative agreement with the analytical approximation of Ref.…”

“…Error bars have been calculated according to a jackknife resampling analysis. In one of our previous studies of the q-state Potts model on deterministic fractals SP a (5 2 ,24,k) with the help of the Wang-Landau algorithm [12], it has been shown that the transition is a first order one for q 5. As already recalled above, the Wang-Landau algorithm, which calculates the density of states over the whole energy range, is very time consuming and does not enable us to simulate very large sizes; we decided to carry out simulations for the value q = 7 with the help of the canonical Wolff algorithm in order (6) 487 (15) to check that the two approaches agree.…”

“…This paper is devoted to the questions brought up at the end of the conclusion in Ref. [8]: Let us recall that, in translationally invariant systems with a space dimension d, the order of the phase transition of the q-state Potts model changes from the second to the first one as q is increased; thus, two regions can be distinguished in the (d,q) plane; the question of the extrapolation of this phase diagram to noninteger dimensions where the symmetry changes from translation invariance to discrete scale invariance without disorder has been studied for deterministic fractals [10][11][12]; moreover an approximate analytical result has been provided by Andelman and Berker [13]. Furthermore, the introduction of disorder in translationally invariant systems can, under some conditions linked to the Harris criterion, induce a second order transition from a pure system exhibiting a first order one.…”