TY - JOUR

T1 - Stability of the Peierls instability for ring-shaped molecules

AU - Lieb, Elliott H.

AU - Nachtergaele, Bruno

N1 - Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 1995

Y1 - 1995

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevB.51.4777

DO - 10.1103/PhysRevB.51.4777

M3 - Article

AN - SCOPUS:11644265405

VL - 51

SP - 4777

EP - 4791

JO - Physical Review B

JF - Physical Review B

SN - 0163-1829

IS - 8

ER -