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 < 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 < 5 (3 < 4) at the P-F (P-SG) boundary.