Absence of local order in topologically frustrated spin chains

We show that a wide class of spin chains with topological frustration cannot develop any local order. In particular, we consider translational-invariant one-dimensional chains with frustrated boundary conditions, i.e. periodic boundary conditions and an odd number of sites, which possess a global SU(2) symmetry. This condition implies, even at a finite sizes, an exact degeneracy of the ground state and is quite general in absence of external fields. We directly evaluate the expectation value of operators with support over a finite range of lattice sites and show that, except for some precise conditions, they all decay algebraically, or faster, with the chain length and vanish in the thermodynamic limit. The exceptions that admit a finite order are cases with a higher ground state degeneracy in which the translational symmetry is broken by the ground state choice.