Structure of metastable states in a random Ising chain

We investigate the structure of metastable states (states of local-energy minima) in a random Ising chain. It is shown that one can always achieve a unique reduction of a metastable state into a series of irreducible spin clusters. In a certain sense each irreducible spin cluster is equivalent to a two-level system. We present the exact degeneracy for a random Ising chain by a very simple argument. We obtain an analytical expression for the distribution function of the barrier heights $r$ and the excitation energies $\epsilon$ of the clusters $-D(\epsilon ,r)$. This includes contributions from all the possible spin clusters of various shapes and sizes. Even though the numerical results of this paper have been obtained for a Gaussian distribution, all the formulas of the distribution functions obtained are general, independent of the concrete form of any random continuous distribution. To illustrate how these results can be applied, we recover the "logarithmic law" for energy versus relaxation times at different temperatures, a well-known characteristic of spin-glasses.