Absence of phase transitions in certain one-dimensional long-range random systems
An Ising chain is considered with a potential of the form \(J(i,j)/| i-j|^{\alpha}\), where the J(i,j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if \(\alpha >1\). This is to be contrasted with a ferromagnetic interaction which requires \(\alpha >2\). The basic idea is to prove that the ``surface'' free energy between two half-lines is finite, although the ``surface'' energy may be unbounded. For d-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if \(\alpha >(1/2)d\).
scientific article; zbMATH DE number 4046872
scientific article