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

Abstract

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, σ, 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 θon σ. A good fit to this dependence is θ=3/4-σ. This implies that the upper critical value of σis 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, σ=5/8, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.

27 pages, 27 figures, 4 tables