Spin Glass Phase Transition on Scale-Free Networks

Abstract

We study the Ising spin glass model on scale-free networks generated by the static model using the replica method. Based on the replica-symmetric solution, we derive the phase diagram consisting of the paramagnetic (P), ferromagnetic (F), and spin glass (SG) phases as well as the Almeida-Thouless line as functions of the degree exponent $\lambda$, the mean degree $K$, and the fraction of ferromagnetic interactions $r$. To reflect the inhomogeneity of vertices, we modify the magnetization $m$ and the spin glass order parameter $q$ with vertex-weights. The transition temperature $T_c$ ($T_g$) between the P-F (P-SG) phases and the critical behaviors of the order parameters are found analytically. When $2 < \lambda < 3$, $T_c$ and $T_g$ are infinite, and the system is in the F phase or the mixed phase for $r > 1/2$, while it is in the SG phase at $r=1/2$. $m$ and $q$ decay as power-laws with increasing temperature with different $\lambda$-dependent exponents. When $\lambda > 3$, the $T_c$ and $T_g$ are finite and related to the percolation threshold. The critical exponents associated with $m$ and $q$ depend on $\lambda$ for $3 < \lambda < 5$ ($3 < \lambda < 4$) at the P-F (P-SG) boundary.