The Stability of the Peierls Instability for Ring-Shaped Molecules

TL;DR

This paper rigorously analyzes the stability of Peierls dimerization in ring-shaped molecules, proving that for certain sizes only uniform or alternating spacings are minimal energy configurations, with implications for large systems.

Contribution

It provides a rigorous proof that for rings with size $L=4k+2$, only uniform or alternating spacings minimize energy, and explores the stability of dimerization in large systems.

Findings

For $L=4k+2$, only uniform or alternating spacings are energy minima.

For $L=4k$, more complex minima can occur, but dimerization is asymptotically stable.

Dimerization remains stable in the spin-Peierls case for all $L=4k+2$ and $L=4k$.

Abstract

We investigate the conventional tight-binding model of $L$ $\pi$-electrons on a ring-shaped mol\-e\-cule 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., $t(w)=t_0-\alpha w$ 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 {\it can\/} occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit $L\to\infty$. 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 $L=4k+2$ {\it and\/} $L=4k$.