The Stability of the Peierls Instability for Ring-Shaped Molecules

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\/} $L=4k$.

32 pages, Plain TeX, 3 figures appended as postscript files, EHLBN26/June/94

References in corpus (4)

• The Flux-Phase of the Half-Filled Band
• Geometric Aspects of Quantum Spin States
• Metal-insulator and insulator-insulator transitions in the quarter-filled band organic conductors
• Persistent Currents in Mesoscopic Hubbard Rings with Spin-Orbit Interaction

Cited by in corpus (28)

• The Flux-Phase of the Half-Filled Band
• The one-dimensional Hubbard model: A reminiscence
• On the flux phase conjecture at half-filling: an improved proof
• Dynamical effects of phonons on soliton binding in spin-Peierls systems
• Possible Lattice Distortions in the Hubbard Model for Graphene
• A positive density analogue of the Lieb-Thirring inequality
• Proof of phase separation in the binary-alloy problem: the one-dimensional spinless Falicov-Kimball model
• Magneto-elastic instability in Molecular Antiferromagnetic Rings
• Ferromagnetic Ordering of Energy Levels
• Charged and spin-excitation gaps in half-filled strongly correlated electron systems: A rigorous result
• Quantum Griffiths inequalities
• The flux phase problem on the ring
• Symmetry breaking in the periodic Thomas--Fermi--Dirac--von Weizs{ä}cker model
• Parity effect in a small superconducting grain: A rigorous result
• Spin-Peierls Dimerization of a s=1/2 Heisenberg Antiferromagnet on a Square Lattice
• Topology of superconductors beyond mean-field theory
• Variational states for the spin-Peierls system
• Phase transition in the Peierls model for polyacetylene
• Existence of charge-density waves in two-dimensional ionic Hubbard model
• Long-range charge order in the extended Holstein--Hubbard model
• Combined effect of frustration and dimerization in ferrimagnetic chains and square lattice
• Incommensurate Charge Density Waves in the adiabatic Hubbard-Holstein model
• Charge density wave and quantum fluctuations in a molecular crystal
• Ground state of one-dimensional fermion-phonon systems
• Spontaneous mass generation and chiral symmetry breaking in a lattice Nambu-Jona-Lasinio model
• Exponential decay of the critical points in a discrete model of polyacetylene
• Möbius molecules and fragile Mott insulators
• Spin of the ground state and the flux phase problem on the ring