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arXiv:0911.4028 · Physical Review B

Correlation function for the one-dimensional extended Hubbard model at quarter filling

Satoshi Nishimoto, Masahisa Tsuchiizu

Published Feb 23, 2010• 3 citations
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Abstract

We examine the density-density correlation function in the Tomonaga-Luttinger liquid state for the one-dimensional extended Hubbard model with the on-site Coulomb repulsion $U$ and the intersite repulsion $V$ at quarter filling. By taking into account the effect of the marginally irrelevant umklapp scattering operator based on the bosonization and renormalization-group methods, we obtain the generalized analytical form of the correlation function. We show that, in the proximity to the gapped charge-ordered phase, the correlation function exhibits anomalous crossover between the pure power-law behavior and the power-law behavior with logarithmic corrections, depending on the length scale. Such a crossover is also confirmed by the highly accurate numerical density-matrix renormalization-group method.

DOI
10.1103/physrevb.81.085116

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We examine the density-density correlation function in the Tomonaga-Luttinger liquid state for the one-dimensional extended Hubbard model with the on-site Coulomb repulsion UUU and the intersite repulsion VVV at quarter filling. By taking into account the effect of the marginally irrelevant umklapp scattering operator based on the bosonization and renormalization-group methods, we obtain the generalized analytical form of the correlation function. We show that, in the proximity to the gapped charge-ordered phase, the correlation function exhibits anomalous crossover between the pure power-law behavior and the power-law behavior with logarithmic corrections, depending on the length scale. Such a crossover is also confirmed by the highly accurate numerical density-matrix renormalization-group method.
We study the phase diagram of a half-filled one-dimensional superlattice Hubbard model with an alternating orbital energy ± ϵ/2 and on-site Coulomb repulsion U(A) and U(B) using the density-matrix renormalization group and finite-size scaling analysis. For U(A) >> U(B), we observe an incommensurate charge correlation phase. For U(B) = 0, we find a transition from the band insulating phase to the correlated metallic phase at the critical point ϵ = U(A)/2. For U(B) > 0, we find three insulating phases as in the ionic Hubbard model. In the gapless region, the correlation functions exhibit the asymptotic behavior of power-law decay as in the Tomonaga-Luttinger liquid.
We calculate the charge and spin Drude weight of the one-dimensional extended Hubbard model with on-site repulsion UUU and nearest-neighbor repulsion VVV at quarter filling using the density-matrix renormalization-group method combined with a variational principle. Our numerical results for the Hubbard model (V=0)(V=0)(V=0) agree with exact results obtained from the Bethe ansatz solution. We obtain the contour map for both Drude weights in the UVUVUV-parameter space for repulsive interactions. We find that the charge Drude weight is discontinuous across the Kosterlitz-Thouless transition between the Luttinger liquid and the charge-density-wave insulator, while the spin Drude weight varies smoothly and remains finite in both phases. Our results can be generally understood using bosonization and renormalization-group results. The finite-size scaling of the charge Drude weight is well fitted by a polynomial function of the inverse system size in the metallic region. In the insulating region, we find an exponential decay of the finite-size corrections with the system size and a universal relation between the charge gap ${\ensuremath{\Delta}}_{c}$ and the correlation length $\ensuremath{\xi}$ which controls this exponential decay.
We study the one-dimensional Anderson-Hubbard model using the density-matrix renormalization-group method. The influence of disorder on the Tomonaga- Luttinger liquid behavior is quantitatively discussed. Based on the finite-size scaling analysis of density-density correlation functions, we find the following results: (i) the charge exponent is significantly reduced by disorder at low filling and near half filling, (ii) the localization length decays as $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}2}$, where $\ensuremath{\Delta}$ is the disorder strength, independently of the on-site Coulomb interaction as well as band filling, and (iii) the localization length is strongly suppressed by the on-site Coulomb interaction near half filling in association with the formation of the Mott plateaus.
Phase diagrams of the one-dimensional extended Hubbard model (including nearest-neighbor interaction V)V)V) at half- and quarter-filling are studied by observing level crossings of excitation spectra using the exact diagonalization. This method is based on the Tomonaga-Luttinger liquid theory including logarithmic corrections which stem from the renormalization of the Umklapp- and the backward-scattering effects. Using this approach, the phase boundaries are determined with high accuracy, and then the structure of the phase diagram is clarified. At half-filling, the phase diagram consists of two Berezinskii-Kosterlitz-Thouless (BKT) transition lines and one Gaussian transition line in the charge sector, and one spin-gap transition line. This structure reflects the $\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{SU}(2)$ symmetry of the electron system. Near the U=2VU=2VU=2V line, the Gaussian and the spin-gap transitions take place independently from the weak- to the intermediate-coupling region, but these two transition lines are coupled in the strong-coupling region. This result demonstrates existence of a tricritical point and a bond-charge-density-wave (BCDW) phase between charge- and spin-density-wave (CDW, SDW) phases. To clarify this mechanism of the transition, we also investigate effect of a correlated hopping term, which plays a role to enlarge BCDW and bond-spin-density-wave (BSDW) phases. At quarter-filling, a similar crossover phenomenon also takes place in the large-VVV region involving spin-gap and BKT-type metal-insulator transitions.
The large-distance behavior of the spin-spin correlation function of the one-dimensional repulsive Hubbard model is evaluated analytically in the strong-coupling regime at quarter filling. In this case, its power-law decay is characterized by an exponent \ensuremath{\gamma}=3/2. We have found that this behavior is generally valid at any nonzero doping, although our argument is not mathematically rigorous away from quarter filling. These results strongly suggest that the renormalization-group scaling to the Tomonaga-Luttinger model is exact in the U\ensuremath{\rightarrow}\ensuremath{\infty} Hubbard model.
A one-dimensional electron system at quarter-filling has been examined by applying the renormalization group method to a bosonized model with on-site (U) and nearest-neighbor (V) repulsive interactions. By evaluating both normal scattering and Umklapp scattering perturbatively, we obtain a phase diagram in which a metallic state with a 2k_F spin density wave (k_F is the Fermi wave number) moves into an insulating state with charge disproportionation of a 4k_F charge density wave with an increase in both U and V. The effect of the next-nearest-neighbor repulsion is also discussed.