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

TL;DR

This paper studies the properties of a one-dimensional vector spin glass with long-range interactions, analyzing zero and finite temperature behaviors, defect energies, and finite-size scaling, revealing critical exponents and phase transition characteristics.

Contribution

It introduces a detailed analysis of the infinite-component vector spin glass with power-law decaying interactions, including the relation of critical exponents to the decay parameter , and compares fully connected and diluted models.

Findings

Defect-energy exponent  = 3/4 - , indicating a critical  of 3/4.

The upper critical  is 3/4, matching the lower critical dimension of the short-range model.

Finite-temperature properties show distinct finite-size scaling behaviors below and above  = 5/8.

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, \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 \theta on \sigma. A good fit to this dependence is \theta =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.