Hereditary structure in Hamiltonians: Information geometry of Ising spin chains

This Letter deals with the problem of approximating the canonical state of an Ising spin chain by a hierarchical series of independent and identically distributed cluster states. Based on information geometry, it is shown that the structure of the effective Hamiltonian for each cluster state is inherited from the total Hamiltonian. This fact partly justifies the methodology of mean field theories. The issue of a phase transition is further analyzed from the point of view of statistical hypothesis testing.