The free energy of the Ising spin glass on the Bethe lattice

The free energy of the ±J model at the finite temperatures on the Bethe lattice was obtained to be βF = (1 − 1 2z ) ∫ −x x ( 1 2 x) cosech (πx/2β)[S(x)] z d x+ 1 2 z∫ −x x [ cos (J 0 x)/2x] cosech (πx/2β) {S(x)] z−1 dx , where S(x) is the Fourier transform of the distribution function g(h) of the single bond effective field h. A tentative self-consistent S(x) gives the spin glass transition temperature and the energy at T = 0 exactly.