Stability of the Peierls instability for ring-shaped molecules

We investigate the conventional tight-binding model of L π electrons on a ring-shaped molecule of L atoms with nearest-neighbor hopping. The hopping amplitudes t(w) depend on the atomic spacings w with an associated distortion energy V(w). A Hubbard-type on-site interaction as well as nearest-neighbor repulsive potentials can also be included. We prove that when L=4k+2 the minimum energy E occurs either for equal spacing or for alternating spacings (dimerization); nothing more chaotic can occur. In particular, this statement is true for the Peierls-Hubbard Hamiltonian, which is the case of linear t(w) and quadratic V(w), i.e., t(w)=t0-αw and V(w)=k(w-a)2, but our results hold for any choice of couplings or functions t(w) and V(w). When L=4k we prove that more chaotic minima can occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit L→. Our analysis suggests three interesting conjectures about how dimerization stabilizes for large systems. We also treat the spin-Peierls problem and prove that nothing more chaotic than dimerization occurs for L=4k+2 and L=4k.