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arXiv:cond-mat/9708204 · Physical Review Letters

Renormalization Group Analysis of the Spin-Gap Phase in the One-Dimensional t - J Model

Masaaki Nakamura, K. Nomura, Atsuhiro Kitazawa

Published Oct 27, 1997• 48 citations
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Abstract

We study the spin-gap phase in the one-dimensional $t$- $J$ 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 $k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ SU(2) Wess-Zumino-Witten model.

DOI
10.1103/physrevlett.79.3214

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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 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 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 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.
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.
We study a one-dimensional spin-orbital model using both analytical and numerical methods. Renormalization group calculations are performed in the vicinity of a special integrable point in the phase diagram with SU(4)\mathrm{SU}(4)SU(4) symmetry. These indicate the existence of a gapless phase in an extended region of the phase diagram, missed in previous studies. This phase is SU(4)\mathrm{SU}(4)SU(4) invariant at low energies apart from the presence of different velocities for spin and orbital degrees of freedom. The phase transition into a gapped dimerized phase is in a generalized Kosterlitz-Thouless universality class. The phase diagram of this model is sketched using the density matrix renormalization group technique.