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arXiv:cond-mat/9410100 · Physical review. B, Condensed matter

Stability of the Peierls instability for ring-shaped molecules

Élliott H. Lieb, Bruno Nachtergaele

Published Feb 15, 1995• 60 citations

Abstract

We investigate the conventional tight-binding model of L \ensuremath{\pi} 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)=${\mathit{t}}_{0}$-\ensuremath{\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 can occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit L\ensuremath{\rightarrow}\ensuremath{\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 and L=4k.

DOI: 10.1103/physrevb.51.4777

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Sanyee Tang, J. E. Hirsch

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Physical review. B, Condensed matter•Jun 1, 1988•85 citations

We investigate the conventional tight-binding model of L \ensuremath{\pi} 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)=

{\mathit{t}}_{0}

-\ensuremath{\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 can occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit L\ensuremath{\rightarrow}\ensuremath{\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 and L=4k.

A Green-function formalism is presented to study a Peierls-Hubbard Hamiltonian in two dimensions. The lattice consists of parallel dimerized chains with alternating nearest-neighbor hoppings t and ${\mathit{t}}_{\mathrm{\ensuremath{\parallel}}}$ and another hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ between different chains. The method treats the interdimer hopping as a perturbation and yields exact results in the uncorrelated case and for isolated dimers. The calculated spectral functions exhibit a number of narrow subbands with typical low-dimensional singularities. The dependence of the gap at the Fermi level on the electron-electron interaction U agrees qualitatively with the exact result in the known one-dimensional nondimerized limit. The paramagnetic susceptibility shows a maximum structure at low temperatures that is enhanced by U and by dimerization and a Curie-Weiss behavior at high temperatures.

Fr\"ohlich and Peierls showed that a one-dimensional system with a half-filled band can lower its ground-state energy by a dimerization from period 1 to period 2. It was an open question whether or not this dimerization was exact, i.e., whether additional symmetry breaking would further lower the energy. We prove that the dimerization is exact for a periodic chain of infinitely massive, harmonically bound atoms with nearest-neighbor electron hopping matrix elements that vary linearly with the nearest-neighbor distance.

We investigate the general applicability of the standard Peierls-Hubbard model, involving both on-site (

U

) and nearest-neighbor (

V

) Coulomb repulsions, by adding nearest-neighbor off-diagonal terms representing the bond (

W

) and mixed bond-site (

X

) electron-electron repulsive interactions. For a broad range of the site-diagonal Hubbard parameters (

U,V

), we find that the off-diagonal terms (

W,X

) initially enhance dimerization, thereby stabilizing the dimerized ground state.

The Peierls–Hubbard model applied to N-membered ring-shaped molecules is investigated numerically and analytically. The bond configurations minimizing the total energy at half-filling of the electronic states are determined by a self-consistent method for N⩽8 in a wide range of the electron–lattice and electron–electron coupling parameters. In the even-N case, only dimerized and homogeneous configurations have been found to be stable. Odd-N rings show three types of bond configurations depending on the coupling parameters. Two of these configurations have reflection symmetry whereas the third one is irregular.

We study extended Hubbard and Peierls-Hubbard models with up to N=10 sites and half-filled bands with modified boundary conditions. For the extended Hubbard model, the finite-size dependence of the transition from the charge-density-wave regime (CDW) to the spin-density-wave (SDW) regime and of the condensation transition is critically examined, clarifying some previous results. For the Peierls-Hubbard model and N\ensuremath{\le}10 with modified periodic boundary conditions, we always see a decrease of dimerization with U. Extrapolating these results to infinite N, we find that dimerization is enhanced by U for small electron-phonon coupling \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}0.75, whereas it decreases with U for 0.75\ensuremath{\le}1. By extending this model by a nearest-neighbor repulsion V, the amplitude of dimerization reaches a maximum at the same value U\ensuremath{\le}2V, where the CDW-to-SDW transition occurs in the extended Hubbard model.

We have studied the tendency of the two-dimensional half-filled Hubbard model towards dimerization as a function of the on-site repulsion U. We have found that the phonon with (\ensuremath{\pi},\ensuremath{\pi}) wave vector is the most favorable vibration mode when U is less than a critical value

{U}_{c}

, where

{U}_{c}

is a function of the dimerization. When U is greater than

{U}_{c}

, the (\ensuremath{\pi},0) phonon becomes preferred. In leading order, the energy gain under dimerization appears to be proportional to ${\ensuremath{\delta}}^{2}$ln\ensuremath{\delta} if the (\ensuremath{\pi},\ensuremath{\pi}) phonon is favorable, so that the system displays a Peierls instability in this case. For large U the energy gain to leading order appears to be proportional to ${\ensuremath{\delta}}^{2}$ for both the (\ensuremath{\pi},\ensuremath{\pi}) and (\ensuremath{\pi},0) phonons, and the latter one is preferred; in that case the system is stable unless the electron-phonon coupling exceeds a critical value. Unlike the effect in the one-dimensional case, the on-site repulsion U seems to have essentially no effect on the tendency towards dimerization for small U, and suppresses it for large U. This difference may be related to the fact that there exists long-range antiferromagnetic order in the two-dimensional half-filled Hubbard model when U is present.