Spin-glass theory in the Bethe approximation: Insights and problems

Abstract

We set up and solve the Bethe approximation for a spin-glass with finite-range interactions. We obtain the finite-$z$ analog of the Thouless-Anderson-Palmer equations and the free-energy functional at all $T$. In the configuration-averaged theory the entropy $\overline{S}\mathrm{}\left(T\right)\mathrm{}$ and all other properties are well behaved on a Cayley tree. For real lattices $\overline{S}\mathrm{}\left(0\right)=0\mathrm{}$, but the specific heat is negative near $T=0\mathrm{}$ for all but small values of $z$.