From inherent structures to pure states: Some simple remarks and examples

The notions of pure states and inherent structures, i.e. stable configurations against 1-spin flip are discussed. We explain why these different concepts accidentally coincide in mean field models with infinite connectivity and present an exactly solvable one-dimensional model where they do not. At zero temperature pure states are to some extent related to k-spin flip stable configurations with k --> infinity after the thermodynamical limit has been taken. This relationship is supported by an explicit analysis of the TAP equations and calculation of the number of pure states and k-spin flips stable configurations in a mean-field model with finite couplings. Finally, we discuss the relevance of the concepts of pure states and inherent structures in finite-dimensional glassy systems.