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Physical review. B, Condensed matter

Dimerization in the two-dimensional Hubbard model

Fu‐Chun Zhang, Peter Prelovšek

Published Feb 1, 1988• 29 citations
Publisher

Abstract

We have used both an exact diagonalization method for finite-size systems and a perturbation technique to study the lattice instability due to dimerization in a two-dimensional Hubbard model in the large-U limit. We show that a critical electron-phonon coupling strength is required to form a dimerized state, in contrast to the 1-dimensional system. Our calculations suggest the phase transition to the dimerized state is of first order. The relevance to high-${T}_{c}$ superconductor materials is discussed.

DOI
10.1103/physrevb.37.1569

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We have used both an exact diagonalization method for finite-size systems and a perturbation technique to study the lattice instability due to dimerization in a two-dimensional Hubbard model in the large-U limit. We show that a critical electron-phonon coupling strength is required to form a dimerized state, in contrast to the 1-dimensional system. Our calculations suggest the phase transition to the dimerized state is of first order. The relevance to high-Tc{T}_{c}Tc​ superconductor materials is discussed.
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 show that the zero-temperature susceptibility of the one-dimensional, dimerized Hubbard model at quarter-filling can be accurately determined on the basis of exact diagonalization of small clusters. The best procedure is to perform a finite-size scaling of the spin velocity ${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$, and to calculate the susceptibility from the Luttinger-liquid relation \ensuremath{\chi}=2/\ensuremath{\pi}${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$. We show that these results are reliable by comparing them with the analytical results that can be obtained in the weak- and strong-coupling limits. We have also used quantum Monte Carlo simulations to calculate the temperature dependence of the susceptibility for parameters that should be relevant to the Bechgaard salts. This shows that, used together, these numerical techniques are able to give precise estimates of the low-temperature susceptibility of realistic one-dimensional models of correlated electrons.
To gain insight into the behavior of the Hubbard model, we define a SU(n)\mathrm{SU}(n)SU(n) invariant generalization of the Hubbard-Heisenberg model and, in the large-nnn limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional SU(n)\mathrm{SU}(n)SU(n) Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term ttt, the doping $\ensuremath{\delta}$, and the biquadratic spin interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. In particular, the undimerized "flux" or "s+ids+ids+id" phase is the ground state at half filling for sufficiently large ttt or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large-nnn problem to the high-Tc{T}_{c}Tc​ superconductors.
We apply an exact diagonalization method to the the infinite-D two-band Hubbard model. The method is essentially exact for the calculation of thermodynamic properties for all but the smallest frequencies and yields a resolution unavailable in Monte Carlo calculations. We establish the instability of the normal state with respect to singlet superconductivity at small frequencies at small doping, the regime of relevance for High-TcT_cTc​ superconductors. We also present evidence for the existence of an instability towards triplet superconductivity in the large doping regime n∼2n \sim 2n∼2.
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.