Annealed Ising model with site dilution on self-similar structures

We consider an Ising model on the triangular Apollonian network (AN), with a thermalized distribution of vacant sites. The statistical problem is formulated in a grand canonical ensemble, in terms of the temperature $T$ and a chemical potential $\mu$ associated with the concentration of active magnetic sites. We use a well-known transfer-matrix method, with a number of adaptations, to write recursion relations between successive generations of this hierarchical structure. We also investigate the analogous model on the diamond hierarchical lattice (DHL). From the numerical analysis of the recursion relations, we obtain various thermodynamic quantities. In the $\mu \to \infty$ limit, we reproduce the results for the uniform models: in the AN, the system is magnetically ordered at all temperatures, while in the DHL there is a ferromagnetic-paramagnetic transition at a finite value of $T$. Magnetic ordering, however, is shown to disappear for sufficiently large negative values of the chemical potential.

Figure 1

Geometrical construction of the first three generations ($g=0,1,$ and 2) of the AN.

Figure 2

Specific heat $\mathcal{C}$ of the diluted system on the AN as a function of temperature $T$ for various values of the chemical potential $\mu$.

Figure 3

Concentration $\rho$ of the magnetic sites of the dilute model on the AN as a function of temperature for various values of chemical potential $\mu$. At zero temperature, there is a discontinuous behavior of $\rho$ as the chemical potential changes through the characteristic value $\mu =-3$.

Figure 4

Entropy $s$ on the AN lattice as a function of temperature for various values of chemical potential $\mu$. There is a residual entropy for $\mu \le -3$. At the critical value $\mu =-3$, the residual entropy is $s(T=0)=0.8432....$This value changes to $s(T=0)=ln2$ for $\mu <-3$.

Figure 5

Magnetization in zero field on the AN as a function of temperature for various values of the chemical potential $\mu$. As in Fig. 3, at zero temperature, there is a discontinuous behavior of the magnetization at $\mu =-3$.

Figure 6

Specific heat $\mathcal{C}$ of the diluted model on the DHL as a function of temperature for several ranges of values of the chemical potential $\mu$. (a) explores large intervals of $\mu$, illustrating phase transitions at $T_c\left(\mu \right)$ indicated by cusps for $\mu =30$, $3,$ and 0. No cusp appears for $\mu \le -3$. (b) illustrates patterns consisting of cusps with a broad maximum superposed for $\mu =-2.0$, $-2.3$, and $-2.6$. At $\mu =-2.8$ the cusp to the left of the broad maximum reaches a very large value $\simeq 29.15$. (c) shows the huge increase of $\mathcal{C}\left(T_c\right)$ as $\mu$ decreases, which is related to the presence of a tricritical point; $\mu =-2.95$ is the lowest value of $\mu$ where the cusp can be well characterized, as $\mathcal{C}\left(T_c\right)$ $\sim {10}^{13}$.

Figure 7

Spontaneous magnetization $m$ on the DHL as a function of temperature for various values of the chemical potential $\mu$. Note that $m\to 0$ at $T=T_c\left(\mu \right)$ for $\mu >-3$, and that $m\equiv 0$ at all temperatures for $\mu <-3$.