One-dimensional infinite-component vector spin glass with long-range interactions

We investigate zero and finite-temperature properties of the one-dimensional spin-glass model for vector spins in the limit of an infinite number m of spin components where the interactions decay with a power, [sigma], of the distance. A diluted version of this model is also studied, but found to deviate significantly from the fully connected model. At zero temperature, defect energies are determined from the difference in ground-state energies between systems with periodic and antiperiodic boundary conditions to determine the dependence of the defect-energy exponent [straighttheta] on [sigma]. A good fit to this dependence is [straighttheta] = 3/4 - [sigma]. This implies that the upper critical value of [sigma] is 3/4, corresponding to the lower critical dimension in the d-dimensional short-range version of the model. For finite temperatures, the large m saddle-point equations are solved self-consistently, which gives access to the correlation function, the order parameter, and the spin-glass susceptibility. Special attention is paid to the different forms of finite-size scaling effects below and above the lower critical value, [sigma] = 5/8, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.