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arXiv:cond-mat/0504078 · Journal of Physics Condensed Matter

The Peierls–Hubbard model at weak coupling

Michael Dzierzawa, Carmen Luminiţa Mocanu

Published Apr 15, 2005• 8 citations
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Abstract

We investigate the Peierls transition in the one-dimensional Peierls-Hubbard model at half filling in the adiabatic approximation for the lattice. Depending on the value of the electron-lattice coupling constant g the equilibrium dimerization can be either enhanced or suppressed by the Hubbard interaction U. Applying second order perturbation theory we determine the critical value g_c = 0.689348 below which the Hubbard interaction enhances the dimerization.

DOI
10.1088/0953-8984/17/17/014

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We investigate the Peierls transition in the one-dimensional Peierls-Hubbard model at half filling in the adiabatic approximation for the lattice. Depending on the value of the electron-lattice coupling constant g the equilibrium dimerization can be either enhanced or suppressed by the Hubbard interaction U. Applying second order perturbation theory we determine the critical value g_c = 0.689348 below which the Hubbard interaction enhances the dimerization.
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 Uc{U}_{c}Uc​, where Uc{U}_{c}Uc​ is a function of the dimerization. When U is greater than Uc{U}_{c}Uc​, 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.
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.
The effect of charge ordering on lattice dimerization has been investigated in the one-dimensional extended Peierls–Hubbard model at quarter filling. We estimate the strength of the lattice dimerization by applying the bosonization method and the perturbative renormalization-group method. In the ground state, it is shown that a first-order phase transition takes place between the dimerized insulator and the charge-ordered insulator. The lattice dimerization disappears in the charge-ordered state, while the lattice dimerization is appreciably suppressed even in the dimerized insulating state due to charge fluctuations. The phase diagram at finite temperature is also discussed.
We investigate two kinds of quantum phase transitions observed in the one-dimensional half-filled Peierls-Hubbard model with the next-nearest-neighbor hopping integral in the strong-coupling region $U\ensuremath{\gg}t$, ${t}^{\ensuremath{'}}$ [ttt ${(t}^{\ensuremath{'}})$, nearest- (next-nearest-) neighbor hopping; UUU, on-site Coulomb repulsion]. In the uniform case, with the help of the conformal field theory prediction, we numerically determine a phase boundary ${t}_{\mathrm{c}}^{\ensuremath{'}}(U/t)$ between the spin-fluid and the dimer states, where a bare coupling of the marginal operator vanishes and the low-energy and long-distance behaviors of the spin part are described by a free-boson model. To exhibit the conformal invariance of the systems on the phase boundary, a multiplet structure of the excitation spectrum of finite-size systems and a value of the central charge are also examined. The critical phenomenological aspect of the spin-Peierls transitions accompanied by the lattice dimerization is then argued for the systems on the phase boundary; the existence of logarithmic corrections to the power-law behaviors of the energy gain and the spin gap (i.e., the Cross-Fisher scaling law) are discussed.
We investigate critical aspects of the spin-Peierls transition in the one-dimensional half-filled Peierls-Hubbard model. The bosonized Hamiltonian shows that low-energy excitations of the spin and the charge degrees of freedom are coupled through a cross term arising from the transfer-integral alternation. According to the renormalization-group theory, the cross term becomes relevant, which brings about the energy gain and the singlet-triplet gap described by the so-called Cross-Fisher scaling law. To confirm the prediction numerically, the density-matrix renormalization group method is employed. We find that the transition obeys the scaling law; the critical exponents satisfy the hyperscaling relation, whereas a logarithmic correction to the power-law behaviors is suggested. Based on these results, we give a universality argument among the dimerization transition and the criticalities of the two-dimensional qqq-state Potts models (i.e., q=2q=2q=2 and 4) in connection with a role of the on-site Coulomb interaction. Some related experiments on CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ are also discussed.