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arXiv:cond-mat/9801263 · Journal of the Physical Society of Japan

Spin-Gap Phase in the One-Dimensionalt-J-J′Model

Masaaki Nakamura

Published Mar 15, 1998• 18 citations
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Abstract

The spin-gap phase of the one-dimensional t-J-J' model is studied by the level-crossing of the singlet and the triplet excitation spectra. The phase boundary obtained between the Tomonaga-Luttinger and the spin-gap phases is remarkably consistent with the analytical results at the $J,J'\to 0$ and the low-density limits discussed by Ogata et al. The spin-gap phase has a single domain in the phase diagram even if the spin gap opens at half-filling. The phase boundary coincides with the $K_ρ=1$ line where the Tomonaga-Luttinger liquid behaves as free electrons, in the low-density region. The relation between our method and the solution of the two-electron problem is also discussed.

DOI
10.1143/jpsj.67.717

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The spin-gap phase of the one-dimensional t-J-J' model is studied by the level-crossing of the singlet and the triplet excitation spectra. The phase boundary obtained between the Tomonaga-Luttinger and the spin-gap phases is remarkably consistent with the analytical results at the J,J′→0J,J'\to 0J,J′→0 and the low-density limits discussed by Ogata et al. The spin-gap phase has a single domain in the phase diagram even if the spin gap opens at half-filling. The phase boundary coincides with the Kρ=1K_ρ=1Kρ​=1 line where the Tomonaga-Luttinger liquid behaves as free electrons, in the low-density region. The relation between our method and the solution of the two-electron problem is also discussed.
We give details of the analysis of critical properties of spin-gap phases in one-dimensional lattice electron models. In Tomonaga-Luttinger (TL) liquid theory, the spin-gap instability occurs when the backward scattering changes from repulsive to attractive. This transition point is shown to be equivalent to that of the level-crossing of the singlet and triplet excitation spectra, using the c=1 conformal field theory and the renormalization group. Based on this notion, the transition point between the TL liquid and the spin-gap phases can be determined with high-accuracy from the numerical data of finite-size clusters. We also discuss the boundary conditions and discrete symmetries to extract these excitation spectra. This technique is applied to the extended Hubbard model, the $t\ensuremath{-}J$ model, and the $t\ensuremath{-}J\ensuremath{-}{J}^{\ensuremath{'}}$ model, and their phase diagrams are obtained. We also discuss the relation between our results and analytical solutions in weak-coupling and low-density limits.
The phase diagram of the one-dimensional t-J model is investigated by analyzing the results of exact diagonalization and the exact solutions at J/t=0 and 2. Phase separation takes place above a critical value of J around Jc{\mathit{J}}_{\mathit{c}}Jc​/t=2.5--3.5 depending on the electron density. In the small-J region, Tomonaga-Luttinger liquid theory holds and its correlation exponents are calculated as a function of J/t and the electron density. Superconducting correlations become dominant in a region between the solvable case (J/t=2) and phase separation. A spin-gap region is also found at low density.
We study the spin-gap phase in the one-dimensional ttt- JJJ model, assuming that it is caused by the backward scattering process. Based on the renormalization group analysis and symmetry, we can determine the transition point between the Tomonaga-Luttinger liquid and the spin-gap phases, by the level crossing of the singlet and the triplet excitations. In contrast to the previous works, the obtained spin-gap region is unexpectedly large. We also determine that the universality class of the transition belongs to the k=1k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1k=1 SU(2) Wess-Zumino-Witten model.
The phase diagram of one-dimensional dimerized t-J model is investigated by exact-diagonalization, quantum tranfer-matrix, and quantum Monte Carlo methods. The spin gap persists near but away from half-filling as well as in a strongly dimerized region, while the charge can be described as a Tomonaga-Luttinger liquid in the entire region. The spin-gap region extends to the weak-dimerization region as well as to the Haldane-gap region. Critical exponents of various correlation functions are discussed. The pairing correlation shows remarkable enhancement and by far dominates over other correlations near half-filling. At quarter filling, charge-gap formation is observed.
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 study the ground state of the one-dimensional ``$t\text{\ensuremath{-}}{J}_{s}\text{\ensuremath{-}}{J}_{\ensuremath{\tau}}$ model,'' which is a variant of the $t\text{\ensuremath{-}}J$ model with an additional channel degree of freedom. The model is not only a generalization of the $t\text{\ensuremath{-}}J$ model but also an effective model of the two-channel Kondo lattice model in the strong-coupling region. The low-energy excitations and correlation functions are systematically calculated by the density matrix renormalization group method, and the ground-state phase diagram at quarter filling consisting of a Tomonaga-Luttinger liquid, spin-gap state, channel-gap state, insulator, and phase separation is determined. We find that weak channel fluctuations stabilize the spin-gap state, while strong channel fluctuations lead to the transition to the insulator.