Elliott Lieb, Bruno Nachtergaele
The Stability of the Peierls Instability for Ring-Shaped Molecules
(187K, Plain TeX, 3 postscript figures appended)
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\/}
$L=4k$.