On the thermodynamics of the spin-glass state in infinite dimension

Abstract

Some exact results for the infinitely long-range Ising spin glass are derived at arbitrary temperature and magnetic field below the transition line within Parisi's replica symmetry breaking theory. The distribution of overlaps between pure states, P(q), proves to be non-analytic at q=0 in zero field. The usual relation S=- delta F/ delta T is found to break down, showing that the thermodynamic limit and the temperature derivative do not commute in the ordered phase, at least in non-zero magnetic field. The upper breakpoint of the order parameter function and the Edwards-Anderson order parameter are rigorously derived in the low-temperature limit. The coefficient of the leading low-temperature correction to the latter turns out to be different from that derived from the TAP equations, indicating that Parisi's theory and the TAP equations may not be strictly equivalent.