Disordered Potts model on the diamond hierarchical lattice: Numerically exact treatment in the large-$q$ limit

We consider the critical behavior of the random $q$-state Potts model in the large-$q$ limit with different types of disorder leading to either the nonfrustrated random ferromagnet regime or the frustrated spin-glass regime. The model is studied on the diamond hierarchical lattice for which the Migdal-Kadanoff real-space renormalization is exact. It is shown to have a ferromagnetic and a paramagnetic phase and the phase transition is controlled by four different fixed points. The state of the system is characterized by the distribution of the interface free energy $P\left(I\right)$ which is shown to satisfy different integral equations at the fixed points. By numerical integration we have obtained the corresponding stable laws of nonlinear combination of random numbers and obtained numerically exact values for the critical exponents.

Figure 1

(Color online) First steps of the construction of the hierarchical diamond lattice with $b=2$.

Figure 2

(Color online) Phase diagram of the disordered Potts model in the large-$q$ limit on the diamond hierarchical lattice with $b=2$ using the continuous boxlike distribution of Eq. (10). The critical behavior along the transition line between the paramagnetic and the ferromagnetic phases is controlled by four different fixed points: (i) at $p=1$ the pure systems fixed point $\left(P\right)$, (ii) from $p=1$ to $p=p_{\text{MC}}$ (red line), the RF fixed point, (iii) at $p=p_{\text{MC}}$ the MC fixed point, and (iv) for $p_{\text{MC}}<p\le p_Z$ (green line) the zero-temperature $\left(Z\right)$ fixed point. Note that there is a reentrance in the phase diagram.

Figure 3

(Color online) Probability distribution of the scaled interface free energy at the RF fixed point. At $I=0$ there is a delta peak with strength, $p_0=0.1280795$, the function is nonanalytic at $I=1$ and $I=2$.

Figure 4

(Color online) Probability distribution of the scaled interface free energy, $i=I/\overline{I}$, at the $Z$ fixed point, as well as along the green line of Fig. 2 between MC and $Z$. At $i=0$ there is a delta peak with strength, ${\pi }_0=0.000158$, the function is nonanalytic at $i=0$.

Figure 5

(Color online) Distribution of the interface free energy at the MC point as calculated by the numerical pool method. The green (red) symbols are for $n=12$ $\left(n=10\right)$ iterations. A small fraction $n_0\approx 0.0057$ of the samples have zero interface free energy.