Spin glasses in the Bethe-Peierls-Weiss and other mean-field approximations

We obtain the thermodynamic properties of a system of Ising spins interacting with various random potentials in the Bethe-Peierls-Weiss (BPW) approximation. When the effective number of neighbors $z$ approaches infinity, we show that all the magnetic properties arising from the BPW approximation, the mean random field (MRF) and the Sherrington-Kirkpatrick (SK) replica treatment are identical. Also, the internal energy in the BPW method is identical to that obtained by SK, while the MRF neglects correlations and thus gives a different internal energy. Introducing a plausible phenomenological constant of integration we obtain the microscopic free energy derived by Thouless, Anderson, and Palmer (TAP). Using this free energy, we show that the BPW method with a random distribution of fields reproduces all the results of SK including a negative entropy of $-\frac{k}{\left(2\mathrm{}\pi \right)}$ at $T=0\mathrm{}$, and that all probability distributions which do not go to zero at zero field give a negative entropy at $T=0\mathrm{}$ For finite $z$, we obtain the phase diagram for the MRF method as a function of $z$ and find that for $z>\mathrm{}8\mathrm{}$ the phase diagram is already very close to that of the $z\to \infty$ case. We also derive the thermodynamic properties for the Ruderman-Kittel-Kasuya-Yosida system near the spin-glass transition temperature in the BPW method. We find that the method gives a discontinuous slope in the magnetic susceptibility $\chi$ and the specific heat $C_M$ at the spin-glass transition temperature $T_g$, however the maxima in $\chi$ and $C_M$ occur well below $T_g$.