Multicritical points and topology-induced inverse transition in the random-field Blume-Capel model in a random network

The interplay between quenched disorder provided by a random field (RF) and network connectivity in the Blume-Capel (BC) model is the subject of this paper. The replica method is used to average over the network randomness. It offers an alternative analytic route to both numerical simulations and standard mean field approaches. The results reveal a rich thermodynamic scenario with multicritical points that are strongly dependent on network connectivity. In addition, we also demonstrate that the RF has a deep effect on the inverse melting transition. This highly nontrivial type of phase transition has been proposed to exist in the BC model as a function of network topology. Our results confirm that the topological mechanism can lead to an inverse melting transition. Nevertheless, our results also show that as the RF becomes stronger, the paramagnetic phase is affected in such way that the topological mechanism for inverse melting is disabled.

Figure 1

Local field distribution $W(h,\tilde{h})$ for $c=5, D=0.6$, and $T=0.2$, for (a) $\theta =0.2$ (NM phase), (b) $\theta =0.4$ (F phase), and (c) $\theta =0.6$ (P phase).

Figure 2

(a) Per-site magnetization (top), correlation parameter (middle), and per-site free energy (bottom) versus temperature, with $c=5, \theta /J=0.25$, and $D/J=0.6$ for increasing and decreasing $T$ in solid and dashed lines, respectively. The arrow indicates the discontinuous thermodynamic transition. (b) Per-site magnetization (top), correlation parameter (middle), and per-site free energy (bottom) versus random-field amplitude, with $c=25$ (black lines) or $c=10$ (brown lines), $T/J=0.02$, and $D/J=0.4$ for increasing and decreasing $\theta$ in solid and dashed lines, respectively. The arrow indicates the discontinuous thermodynamic transition.

Figure 3

Phase diagram $D/J$ vs $\theta /J$ at $T=0$ for $c=5$ (black heavy lines), $c=10$ (brown heavy lines), and $c=25$ (black light lines). Continuous and discontinuous phase boundaries are represented, respectively by solid and dashed lines. NM, F, P, and F/NM represent, respectively, nonmagnetic, ferromagnetic, paramagnetic and ferromagnetic-nonmagnetic phases. The asterisk indicates a tricritical point, over the F-P transition, for $c=10$.

Figure 4

(a) Phase diagram $T/J$ vs $\theta /J$ for $c=5$, for $D/J=0.56, D/J=0.6$, and $D/J=0.25$ (in the inset). (b) Phase diagram $T/J$ vs $\theta /J$ for $c=10$, for $D/J=0.25$ and $D/J=0.6$. (c) Phase diagram $T/J$ vs $\theta /J$ for $c=25$, for $D/J=0.4$ and $D/J=0.502$. In the inset, $D/J=0.5$. Solid and dashed lines represent continuous and discontinuous phase boundaries, respectively.

Figure 5

Effective coupling between the neighbors of a node of degree 2 as a function of the temperature. Upper panel for several values of $D$ and constant $\theta =0.3$; lower panel: for several values of $\theta$ and $D=0.6$.