On non-stability of one-dimensional non-periodic ground states
Damian Głodkowski, Jacek Miȩkisz
TL;DR
The paper investigates stability of one-dimensional non-periodic ground-state configurations in classical lattice-gas models under finite-range perturbations. It introduces the strict boundary condition and shows that, in 1D, non-periodic ground states become unstable under small perturbations unless the decay of interactions is extremely slow, with a threshold around $\alpha>2$. Through detailed analyses of Toeplitz, Thue-Morse, and Sturmian constructions, it demonstrates instability for the Toeplitz case (and $\alpha>1$ for Thue-Morse) and for Sturmian systems when $\alpha>3$, highlighting a fundamental tension between non-periodicity and stability in 1D. The results suggest that stable, uniquely non-periodic ground states in 1D may be exceptional and motivate further exploration of specially structured Sturmian systems at small $\alpha$.
Abstract
We address the problem of stability of one-dimensional non-periodic ground-state configurations with respect to finite-range perturbations of interactions in classical lattice-gas models. We show that a relevant property of non-periodic 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 have bounded fluctuations on any finite subsets of the lattice. We show that if the strict boundary condition is not satisfied, then in order for non-periodic ground-state configurations to be stable, interactions between particles should not decay faster than $1/r^α$ with $α>2$. In the Thue-Morse ground state, number of finite patterns may fluctuate as much as the logarithm of the lenght of a lattice subset. We show that the Thue-Morse ground state is unstable for any $α>1$ with respect to arbitrarily small two-body interactions favoring the presence of molecules consisting of two spins up or down. We also investigate Sturmian systems defined by irrational rotations on the circle. They satisfy the strict boundary condition but nevertheless they are unstable for $α>3$.