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 θ on σ. A good fit to this dependence is θ=3/4−σ. This implies that the upper critical value of σ 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, σ=5/8, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.
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Beyer, F., Weigel, M., & Moore, M. A. (2012). One-dimensional infinite-component vector spin glass with long-range interactions. Physical Review B, 86, . https://doi.org/10.1103/PhysRevB.86.014431