Random field Ising model on networks with inhomogeneous connections

We study a zero-temperature phase transition in the random field Ising model on scale-free networks with the degree exponent $\gamma$. Using an analytic mean-field theory, we find that the spins are always in the ordered phase for $\gamma <3$. On the other hand, the spins undergo a phase transition from an ordered phase to a disordered phase as the dispersion of the random fields increases for $\gamma >3$. The phase transition may be either continuous or discontinuous depending on the shape of the random field distribution. We derive the condition for the nature of the phase transition. Numerical simulations are performed to confirm the results.

Figure 1

Schematic plots of $f\left(m\right)$ (solid line), which has the infinite slope at $m=0$ (a), is convex at $m=0$ (b) and is concave at $m=0$ (c). The dashed line represents the graph of $m$, and the solid circle represents the solution for the SC equation $m=f\left(m\right)$.

Figure 2

$⟨m⟩$ versus $\Delta$ with the magnetic field distribution $p\left(h\right)=p_{+}(h∕\Delta )∕\Delta$ and $\gamma =2.5$ (a), 4.0 (b), and 6.0 (c). The dashed line in (a) has a slope $-1$.

Figure 3

The histogram on an arbitrary scale of the magnetization at several values of $\Delta$ near the transition on the SF networks of $N=64000$ nodes with $\gamma =4.0$ (left column) and 6.0 (right column).

Figure 4

$⟨m⟩$ versus $\Delta$ with $p\left(h\right)=p_{-}(h∕\Delta )∕\Delta$ and $\gamma =2.5$ (a), 4.0 (b), and 6.0 (c). The dashed line in (a) has a slope $-1$.

Figure 5

$U$ versus $\Delta$ for various system sizes with the degree exponent (a) $\gamma =2.5$, (b) $\gamma =4$, and (c) $\gamma =6$.

Figure 6

Scaling plot of $⟨m⟩N^{\beta ∕{\nu }^{\prime }}$ versus ${\mid {\Delta }_c-\Delta \mid }^{{\nu }^{\prime }}N$ for $\gamma =4.0$ in (a) and $\gamma =6.0$ in (b).

Figure 7

The critical exponent $\beta$ as a function of the degree exponent $\gamma$. The solid line corresponds to the mean-field result $\beta =1∕(\gamma -3)$ for $3<\gamma <5$ and $\beta =1∕2$ for $\gamma >5$. The numerical simulation is based on the finite-size-scaling method.