Spin-glass phase transition on scale-free networks.

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.