WEAKLY PERIODIC STRUCTURES AND EXAMPLE

Overview & Synopsis

Models for non-quantum structures with translationally invariant Hamiltonian in any finite dimension space, have ground-states which are weakly periodic. In other words, for any finite piece of the ground-state structure (block) and some given accuracy ε, then there exists a length R, such that the same block is found again (within this accuracy ε) in any bail with radius R. This property is also true for the ground-state of any translationally invariant pseudo spin models on a lattice (but with ε=0) where the reciprocal property can also be proven. For any weakly periodic structure C of pseudospins on a periodic lattice, there exists a translationally invariant Hamiltonian which has this configuration C, all the translated configurations of C and all their limits as equivalent ground-states and there exists no other ground-state for this Hamiltonian. This property is used for proving the existence of a translationally invariant hamiltonian for which the unique ground-state (apart phase shifts) is a recently studied structure intermediate between quasiperiodic and random. Its structure factor has no Dirac peaks but has "quasi-peaks" with scaling properties and is presumably a singular continuous spectrum. This is the first known example of ground-state which is neither periodic nor quasi periodic although not random.