Sturmian Ground States in Classical Lattice–Gas Models

Abstract

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Ground state criteria in one-dimensional antiferromagnetic Ising model with long range interaction

“…The contradiction between inequalities ( 5) and (6) shows that φ is nearly uniform. Now, suppose that the configuration φ ∈ Φ per p,q is nearly uniform.…”

“…5 Ground states satisfying Hubbard's condition in the context of non-periodic long-range order, instead of periodic long-range order, are recently investigated in Ref. 6. It is well known that the models with potentials satisfying ∑ x∈Z + xU(x) < ∞ have a unique Gibbs state.…”

Typical Ground States for Large Sets of Interactions

“…But this is no longer true if one allows even fast decaying interactions in one dimension, or nearest-neighbour interactions based on tilings in higher dimensions, see e.g. [1,22,52,53,69]. For example, it is not known if it is the case that the quasicrystalline order such as occurs in Devil's Staircases [2][3][4]6,33,34], or in more general Sturmian (balanced) ground states [69], is stable under some class of short-range perturbations.…”