 94330 Elliott Lieb, Bruno Nachtergaele
 The Stability of the Peierls Instability for RingShaped Molecules
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Oct 26, 94

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Abstract. We investigate the conventional tightbinding model of $L$
$\pi$electrons on a ringshaped 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 onsite interaction as well as nearestneighbor 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 PeierlsHubbard
Hamiltonian which is the case of linear $t(w)$ and quadratic
$V(w)$, i.e., $t(w)=t_0\alpha w$ and $V(w)=k(wa)^2$, 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 spinPeierls problem and prove that
nothing more chaotic than dimerization occurs for $L=4k+2$ {\it and\/}
$L=4k$.
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