On Non-stability of One-Dimensional Non-periodic Ground States

Abstract

We address the problem of stability of one-dimensional non-periodic ground-state configurations in classical lattice-gas models with respect to finite-range perturbations of interactions. We show that a relevant property of ground-state configurations in this context is their homogeneity. The so-called strict boundary condition says that the number of finite patterns of a configuration has bounded fluctuations uniform in any finite subset of the lattice \(\mathbb Z\). We show that if the strict boundary condition is not satisfied and interactions between particles decay at least as fast as \(1/r^{\alpha }\) with \(\alpha >2\), then ground-state configurations are not stable. In the Thue–Morse ground state, the number of finite patterns may fluctuate as much as the logarithm of the length of a lattice subset. We show that the Thue–Morse ground state is unstable for any \(\alpha >1\) with respect to arbitrarily small two-body interactions favoring the presence of molecules consisting of two neighboring up or down spins. We also investigate Sturmian systems defined by irrational rotations on the circle. They satisfy the strict boundary condition but nevertheless they are unstable for \(\alpha >3\).