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 $T_c$ $\left(T_g\right)$ 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.