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Physical review. B, Condensed matter

Dimer states in the one-dimensionalt-J-J’ model

Ken'ichi Takano, Kazuhiro Sano

Published Oct 1, 1993• 9 citations
Publisher

Abstract

We introduced variational wave functions to describe dimer states in the one-dimensional t-J-J' model, when one hole is doped and J'=J/2. For finite J/\ensuremath{\Vert}t\ensuremath{\Vert} in the first excited state with S=1, there exists a bound state of the hole and an unpaired electron. For the choice of J/\ensuremath{\Vert}t\ensuremath{\Vert}=2, the spin gap is ${\mathrm{\ensuremath{\Delta}}}^{\mathrm{var}}$=0.181J and is smaller than the value of the corresponding spin system, ${\mathrm{\ensuremath{\Delta}}}_{0}^{\mathrm{var}}$=0.25J. The existence of the bound state suggests that the excited state as a function of the electron density is discontinuous at half-filling if J/\ensuremath{\Vert}t\ensuremath{\Vert} is finite.

DOI
10.1103/physrevb.48.9831

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We introduced variational wave functions to describe dimer states in the one-dimensional t-J-J' model, when one hole is doped and J'=J/2. For finite J/\ensuremath{\Vert}t\ensuremath{\Vert} in the first excited state with S=1, there exists a bound state of the hole and an unpaired electron. For the choice of J/\ensuremath{\Vert}t\ensuremath{\Vert}=2, the spin gap is \ensuremathΔvar{\mathrm{\ensuremath{\Delta}}}^{\mathrm{var}}\ensuremathΔvar=0.181J and is smaller than the value of the corresponding spin system, \ensuremathΔ0var{\mathrm{\ensuremath{\Delta}}}_{0}^{\mathrm{var}}\ensuremathΔ0var​=0.25J. The existence of the bound state suggests that the excited state as a function of the electron density is discontinuous at half-filling if J/\ensuremath{\Vert}t\ensuremath{\Vert} is finite.
We investigate the one-dimensional t - J - J ′ model by the numerical diagonalization where a few holes are doped in the half-filled band. By introducing a finite-size scaling, we estimate the spin gap at J ′ / J =0.5 and J / t =2. When one hole is doped, the spin gap decreases to 0.190 J in comparison with 0.236 J which is the spin gap of the half-filled band. The spin gap further reduces to 0.141 J for two-hole doping and to about 0.05 J for three-hole doping. This result shows that the value of the spin gap as a function of the electron density is discontinuous at half-filling.
The one-dimensional t-t'-J-J' model is examined by numerical diagonalization in the case that one hole is doped to the half-filled band. In the presence of t', two types of solitons are found; one is a bound object of the hole and a singlet electron pair in a ferromagnetic background (t'>0, J0), and the other is that of the hole and a triplet electron pair in a singlet background (t'0, J>0). It seems that the solitons remain in the ground state when the system size becomes infinite. The formation of the solitons is ascribed to a kinematic exchange effect due to the t' term.
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 doped and undoped quantum dimer model using exact diagonalization. In the undoped case, previous studies are consistent with the picture that in the region V/J < 0 the ground state is a spin-Peierls state which is four-fold degenerate. We extend the previous calculation and show convincing evidence that a gap exists between the degenerate ground state and the other excitation states. This is further evidence for the spin-Peierls phase. In the doped model, we search for the existence of a stripe phase where the hole-rich region is anisotropic. We introduce a new dimer-hole correlation function, which can indicate the shape of the hole-rich region. Our results show evidence that the holes tend to cluster together with an anisotropic spatial distribution when the holes are not very mobile. But we do not see strong evidence for the existence of a stripe phase.
We study the one-dimensional $t\ensuremath{-}J\ensuremath{-}{J}^{\ensuremath{'}}$-model in the limit of small hole doping x and small J/t,J/t,J/t, ${J}^{\ensuremath{'}}/t.$ Special emphasis is put on the regime ${J}^{\ensuremath{'}}/J\ensuremath{\approx}0.5,$ where a spin gap is present at small doping and the undoped spin chain is strongly dimerized. Using a perturbative approach and Luttinger liquid arguments, we demonstrate for this nonintegrable class of models that the charge degrees of freedom behave as noninteracting spinless solitons in the dilute hole limit. Our approach is also used to evaluate the energy and mass renormalization of a single hole. Interestingly, the corrections of these quantities are in powers of J/t.\sqrt{J/t}.J/t​. At ${J}^{\ensuremath{'}}/J=0.5$ we construct a variational spin-polaron wave function for the hole and find good agreement with our perturbative results.