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arXiv:cond-mat/0612050 · Physical Review B

Charge gap in the one-dimensional extended Hubbard model at quarter filling

Kazuhiro Sano, Yūichi Ono

Published Mar 12, 2007• 10 citations
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Abstract

We propose a different combined approach, which consists of the exact diagonalization, renormalization group, and Bethe ansatz methods, for precise estimates of the charge gap $\ensuremath{\Delta}$ in the one-dimensional extended Hubbard model, with on-site and nearest-neighbor interactions $U$ and $V$ at quarter filling. This approach enables us to obtain the absolute value of $\ensuremath{\Delta}$, including the prefactor, without ambiguity even in the critical regime of the metal-insulator transition (MIT) where $\ensuremath{\Delta}$ is exponentially small, beyond usual renormalization group methods and/or finite-size scaling approaches. The detailed results of $\ensuremath{\Delta}$, down to the order of ${10}^{\ensuremath{-}10}$ near the MIT, are shown as contour lines on the $U\text{\ensuremath{-}}V$ plane.

DOI
10.1103/physrevb.75.113103

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We propose a different combined approach, which consists of the exact diagonalization, renormalization group, and Bethe ansatz methods, for precise estimates of the charge gap \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ in the one-dimensional extended Hubbard model, with on-site and nearest-neighbor interactions UUU and VVV at quarter filling. This approach enables us to obtain the absolute value of \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ, including the prefactor, without ambiguity even in the critical regime of the metal-insulator transition (MIT) where \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is exponentially small, beyond usual renormalization group methods and/or finite-size scaling approaches. The detailed results of \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ, down to the order of 10\ensuremath−10{10}^{\ensuremath{-}10}10\ensuremath−10 near the MIT, are shown as contour lines on the U\ensuremath-VU\text{\ensuremath{-}}VU\ensuremath-V plane.
We examine the critical behavior near the metal-insulator transition (MIT) in the one-dimensional extended Hubbard model with the on-site and the nearest-neighbor interactions UUU and VVV at quarter filling using a combined method of the numerical diagonalization and the renormalization group (RG). The Luttinger-liquid parameter ${K}_{\ensuremath{\rho}}$ is calculated with the exact diagonalization for finite size systems and is substituted into the RG equation as an initial condition to obtain ${K}_{\ensuremath{\rho}}$ in the infinite size system. This approach also yields the charge gap $\ensuremath{\Delta}$ in the insulating state near the MIT. The results agree very well with the available exact results for $U=\ensuremath{\infty}$ even in the critical regime of the MIT where the characteristic energy becomes exponentially small and the usual finite size scaling is not applicable. When the system approaches the MIT critical point $V\ensuremath{\rightarrow}{V}_{c}$ for a fixed UUU, ${K}_{\ensuremath{\rho}}$ and $\ensuremath{\Delta}$ behave as ${\ensuremath{\mid}\mathrm{l}\mathrm{n}\ensuremath{\Delta}\ensuremath{\mid}}^{\ensuremath{-}2}={c}_{\ensuremath{\Delta}}(V∕{V}_{c}\ensuremath{-}1)$ and ${({K}_{\ensuremath{\rho}}\ensuremath{-}1∕4)}^{2}={c}_{K}(1\ensuremath{-}V∕{V}_{c})$, where the critical value Vc{V}_{c}Vc​ and the coefficients ${c}_{\ensuremath{\Delta}}$ and cK{c}_{K}cK​ are functions of UUU. These critical properties, which are known to be exact for $U=\ensuremath{\infty}$, are observed also for finite UUU case. We also observe the same critical behavior in the limit of the MIT critical point $U\ensuremath{\rightarrow}{U}_{c}$ when UUU is varied for a fixed VVV.
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 examine the ground-state properties of the one-dimensional Hubbard model at quarter filling with Coulomb interactions between nearest-neighbors V1{V}_{1}V1​ and next-nearest neighbors V2{V}_{2}V2​. Using the density-matrix renormalization group and exact diagonalization methods, we obtain for U=10tU=10tU=10t three different phases in the ${V}_{1}\text{\ensuremath{-}}{V}_{2}$ plane: 2kF2{k}_{\mathrm{F}}2kF​- and 4kF4{k}_{\mathrm{F}}4kF​-charge-density-wave (CDW) and a broad metallic phase in between. Assuming that the metal is a Tomonaga-Luttinger liquid (TLL), we calculate the TLL parameter ${K}_{\ensuremath{\rho}}$. It is largest when V1{V}_{1}V1​ and V2{V}_{2}V2​ are frustrated, and ${K}_{\ensuremath{\rho}}=0.25$ at the boundaries between the metallic phase and each of the two CDW phases.
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.
By utilizing the twisted boundary conditions in the exact diagonalization method, we investigate the single-particle spectral function of the extended Peierls-Hubbard model at both half-filling and quarter-filling. In one-dimensional (1D) interacting systems, the spin-charge separation can typically be identified in the single-particle spectral function by observing the distinct spinon and holon bands. At half-filling, starting from the pure 1D Hubbard model with the on-site interaction U=10U=10U=10, we observe that the band structure indicative of the spin-charge separation gradually transitions to four individual bands as the Peierls instability $\ensuremath{\delta}$ increases. At U=10U=10U=10 and $\ensuremath{\delta}=0.2$ where the spin-charge separation is still observable, increasing the nearest-neighbor interaction VVV can drive the system to a charge-density-wave state when $V\ensuremath{\gtrsim}U/2$. At quarter-filling, on the other hand, the ground state of Peierls-Hubbard model manifests an antiferromagnetic Mott insulator in units of dimers. Increasing UUU results in only a very small gap in the single-particle spectrum because even for $U=+\ensuremath{\infty}$, with the model transforming into a noninteracting half-filled dimerized tight-binding model, its gap determined by the Peierls instability $\ensuremath{\delta}$ remains small. Conversely, increasing VVV can effectively open the single-particle gap and make the spinon and holon bands more prominent.
We consider two aspects of the ground state of the $U=\ensuremath{\infty}$ Hubbard ladder: ferromagnetism and the metal-insulator transition at quarter-filling. First, we present rigorous results for the $U=\ensuremath{\infty}$ Hubbard ladder in the limit of the large interchain hopping ${(t}_{\ensuremath{\perp}}/t\ensuremath{\rightarrow}\ensuremath{\infty})$. In this limit, the total spin SSS of the ground state is shown to be zero for the electron density n< 0.5n<~0.5n< 0.5 and its maximum (S=Smax){(S=S}_{\mathrm{max}})(S=Smax​) for n>0.5n>0.5n>0.5. The charge gap at quarter-filling is ${2t}_{\ensuremath{\perp}}.$ We extend these results to finite ${t}_{\ensuremath{\perp}}/t$ by means of the density-matrix renormalization group method. We estimate the phase boundaries with respect to spontaneous magnetization and the charge gap at quarter-filling for finite ${t}_{\ensuremath{\perp}}/t$. Applying the extended Aharonov-Bohm method, we give numerical evidence that the critical ratio ${t}_{\ensuremath{\perp}}/t$, above which the charge gap opens, is less than 0.001. Ferromagnetism in the ttt-JJJ ladder is briefly discussed.