Free-energy landscapes, dynamics, and the edge of chaos in mean-field models of spin glasses

Metastable states in Ising spin-glass models are investigated numerically by finding iterative solutions of mean-field equations for the local magnetizations m(i). A number of iterative schemes are employed, and two different mean-field equations are studied: the Thouless-Anderson-Palmer (TAP) equations that are exact for the Sherrington-Kirkpatrick model, and the simpler "naive mean-field" (NMF) equations, in which the Onsager reaction term of the TAP equations is omitted and which are exact for the Wallace model. The free-energy landscapes that emerge are very different for the two systems. For the TAP equations, the numerical studies confirm the analytical results of Aspelmeier , which predict that TAP states consist of close pairs of minima and index-one (one unstable direction) saddle points, while for the NMF equations the corresponding free-energy landscape contains saddle points with large numbers of unstable directions. For the TAP equations the free-energy difference between a minimum and its adjacent saddle point (the "barrier height") appears to scale as 1/(f-f(0))(1/3) where f is the free energy per spin of the solution and f(0) is the equilibrium free energy per spin. This means that for pure states, which are those states for which f-f(0) is of order 1/N (N is the number of spins in the system), the barriers between them would apparently scale as N-1/3, but between states for which f-f(0) is of order one, the barriers are finite and also small so such metastable states will be of limited physical significance. For the NMF equations there are saddles of index K and we can demonstrate that their complexity Sigma(K) scales as a function of K/N. We have also employed an iterative technique with a free parameter that can be adjusted to bring the system of equations close to the "edge of chaos." Both for the TAP and NME equations it is possible with this approach to find metastable states whose free energy per spin is close to f(0). As N (the number of spins) is increased, it becomes harder and harder to find solutions near to the edge of chaos, but nevertheless the results that can be obtained are competitive with those achieved by more time-consuming computing methods and suggest that this method may be of general utility.