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arXiv:cond-mat/9904381 · Physical review. B, Condensed matter

Spin-gap phases in Tomonaga-Luttinger liquids

Masaaki Nakamura, Atsuhiro Kitazawa, K. Nomura

Published Sep 15, 1999• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.60.7850

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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−Jt\ensuremath{-}Jt\ensuremath−J model, and the t\ensuremath−J\ensuremath−J\ensuremath′t\ensuremath{-}J\ensuremath{-}{J}^{\ensuremath{'}}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 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 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.
We consider instabilities of two-component Tomonaga–Luttinger (TL) liquids for the one-dimensional extended Hubbard model in a magnetic field. When the density of electrons with spin σ is 1/2, electrons with spin σ tend to occupy every other site, and do not conduct. The instability stems from the umklapp scattering between same-spin electrons. But for the remaining electrons with spin −σ, an energy gap does not appear. We also consider the instability relating to the phase separation. This relates to the intrinsic stability condition of the TL liquid.
We investigate the phase diagram of the half-filled one-dimensional t–U–J model by the level-spectroscopy method. Due to the competition between the Coulomb repulsion U and the antiferromagnetic exchange J , the backward scattering may change sign from repulsive to attractive, leading to spin-gap instability. From the excitation spectra of finite-size clusters, the transition line U c ( J ) can be accurately determined in the weak-coupling and intermediate-coupling regimes. With increasing J , U c ( J ) deviates from that given by the weak-coupling theory based on the spin–charge separation hypothesis. Moreover, we find that the spin gap vanishes robustly when , irrespective of J .
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.
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.