Free-energy surface of spin-glasses: Thouless-Anderson-Palmer and Bethe-Peierls-Weiss models

Previous numerical studies of the Thouless, Anderson, and Palmer (TAP) infinite-range Ising spin-glass equations have suggested that solutions were difficult to find and ill behaved in temperature. In order to test whether these results are a consequence of inadequate numerical schemes, we have devised and applied an improved approach. A search for minima of the TAP and (finite-range) Bethe-Peierls-Weiss (BPW) free-energy surfaces indicates that, in both cases, the well-behaved field-cooled minimum evolves with decreasing temperature $T$ into a negative-entropy state. All other attempts to obtain minima led, at best, to piecewise continuous (in $T$ physical solutions. The inability to calculate well-behaved temperature-dependent magnetizations for the finite-size TAP and BPW theories appears to be a serious drawback for these approaches.