Ground states of one-dimensional antiferromagnetic models with long-range interaction

In the classical lattice antiferromagnetic model on the lattice $Z^1$ with Hamiltonian
$$ H(\varphi)=\sum\limits_{x,y\in Z^1;x>y}U(x-y)\varphi(x)\varphi(y)+\mu\sum\limits_{x\in Z^1}\varphi(x), $$
where $U(x)$ is a strictly convex function $\sum\limits_{x\in Z^1,x>0}U(x)<\infty, \mu$ is the chemical potential, and the spin variables $\varphi(x)$ take the values $0$ and $1$, periodic ground states, i.e., periodic configurations with minimal specific energy, were constructed earlier for rational values of the density by means of the theory of continued fractions. In the present paper, it is shown that other periodic ground states do not exist.