Dyson’s equations for the (Ising) spin-glass

A complex problem is solved herej we show how to write Dyson's equations for trie (Ising) spin-glass that relate the propagator G ta trie mass operator M. In other words we are able to reduce trie inversion of an ultrametric matrix M to the solution of a Dyson's equation in ail sectors (for the replicon sectcr trie result had already been derived). It turns oui that what renders the problem tractable is using, instead of trie components of G (or M), an object called here the "kernel" from which one con deduce the components themselves after dressing ii with ultrametric weights and summing over eigenvalue indices with their appropriate multiplicity. Dyson's equations are then established as stationarity equations of tr In M tr GM, where trie kinetic terms are incorporated in M. At each stage we illustrate trie calculation by providing explicit answers for the bore system (mean field in M). In particular trie introduction of the "kernel" allows us to construct trie bore propagator for a Lagrangean where one retains ail quartic invariants. The case of trie system in a magnetic field is also treated. Understanding the properties of the spin glass in trie condensed phase has been outstanding as a major challenge for many years. Field theory from the paramagnetic phase has been thoroughly investigated yielding exponents at the critical temperature and above (Harris et 1288 JOURNAL DE PHYSIQUE I N°9 a1. iii, Elderfield and Mc Kane