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arXiv:0710.2274 · Physical Review B

Phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping

Satoshi Nishimoto, Kazuhiro Sano, Y. Ohta

Published Feb 26, 2008• 21 citations
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Abstract

We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.

DOI
10.1103/physrevb.77.085119

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We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.
We determine the ground-state phase diagram of the one-dimensional half-filled Hubbard model with on-site (nearest-neighbor) repulsive interaction U (V) and nearest-neighbor hopping t using the density-matrix renormalization group technique. Based on the results of the excitation gaps, Luttinger-liquid exponents, and bond-order-wave (BOW) order parameter, we confirm that the BOW phase appears in a substantial region between the charge-density-wave (CDW) and spin-density-wave phases. Each phase boundary is determined by multiple means and it allows us to make a cross-check on the validity of our estimations. We also find that the BOW-CDW transition changes from continuous to first order at the tricritical point (U(t),V(t)) approximately (5.89 t,3.10 t) and the BOW phase shrinks to zero at the critical end point (U(c),V(c)) approximately (9.25 t,4.76 t).
We study the infinite-UUU Hubbard model on ladders of two, four, and six legs with nearest- (t)(t)(t) and next-nearest- $({t}^{\ensuremath{'}})$ neighbor hoppings by means of the density-matrix renormalization group algorithm. In particular, we analyze the stability of the Nagaoka state for several values of ${t}^{\ensuremath{'}}$ when we vary the electron density $\ensuremath{\rho}$ from half filling to the low-density limit. We build the two-dimensional phase diagram, where the fully spin polarized and paramagnetic states prevail. We find that the inclusion of a nonfrustrating next-nearest-neighbor hopping stabilizes the fully spin polarized phase up until $|{t}^{\ensuremath{'}}/t|=0.5$. Surprisingly, for this value of ${t}^{\ensuremath{'}}$, the ground state is fully spin polarized for almost any electron density $1\ensuremath{\gtrsim}\ensuremath{\rho}\ensuremath{\gtrsim}0$, connecting the Nagaoka state to itinerant ferromagnetism at low density. Also, we find that the previously found checkerboard insulator phase at ${t}^{\ensuremath{'}}=0$ and $\ensuremath{\rho}=0.75$ is unstable against ${t}^{\ensuremath{'}}$.
We study the phase diagram of spinless fermions with nearest- and next-nearest-neighbor interactions in one dimension utilizing the (finite-size) density-matrix renormalization group method. The competition between nearest- and next-nearest-neighbor interactions and nearest-neighbor hopping generates four phases in this model: two charge-density-wave insulators, a Luttinger-liquid phase, and a bond-order phase. We use finite-size scaling of the gap and various structure factors to determine the phase diagram.
We calculate spin and charge excitation spectra of the one-dimensional (1D) quarter-filled Hubbard model with nearest-neighbor ttt and next-nearest-neighbor ${t}^{\ensuremath{'}}$ hopping integrals using the dynamical density-matrix renormalization group technique. We consider a case where ttt (>0)(>0)(>0) is much smaller than ${t}^{\ensuremath{'}}$ (>0)(>0)(>0). First, we examine the accuracy of our method based on comparison between our result and exact noninteracting spectrum. Next, we investigate the spectra with the on-site Coulomb interaction. We find that the spin and charge excitation spectra are essentially the same as those of the 1D quarter-filled Hubbard (and $t\text{\ensuremath{-}}J$) model for the two 1D chains along the hopping integral ${t}^{\ensuremath{'}}$. However, the hopping integral ttt $(<{t}^{\ensuremath{'}})$ plays a crucial role in the short-range correlations and low-energy excitaions; ferromagnetic correlation between electrons on neighboring sites is enhanced and pairing correlation between the electrons is induced. Consequently, a spin-triplet superconducting state may be derived.
Ferromagnetism in the one-dimensional Hubbard model with the next-nearest-neighbor hopping is explored by using the exact-diagonalization method in a small cluster and the equation-of-motion method in the thermodynamic limit with electron density n = 3/2. With these two complementary methods, it is found that an intermediate value of the next-nearest-neighbor hopping amplitude t1 tends to stabilize the fully polarized ferromagnetic state under the condition that the on-site coulomb interaction U is sufficiently large in our model. The ground-state phase diagram of the model is presented in the t1—U plane.
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.