Abstract
An Ising chain is considered with a potential of the formJ(i, j)/|i−j|α, where theJ(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 ifα>1. This is to be contrasted with a ferromagnetic interaction which requiresα>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. Ford-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions ifα>(1/2)d.
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van Enter, A.C.D., van Hemmen, J.L. Absence of phase transitions in certain one-dimensional long-range random systems. J Stat Phys 39, 1–13 (1985). https://doi.org/10.1007/BF01007972
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DOI: https://doi.org/10.1007/BF01007972