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Journal of the Physical Society of Japan

Spin Gap of the One-Dimensionalt-J-J′Model

Kazuhiro Sano, Ken'ichi Takano

Published Nov 15, 1993• 11 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.62.3809

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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.
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.
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.
Spin gaps in coupled t-J ladders are investigated by exact diagonalization of small clusters up to 4\ifmmode\times\else\texttimes\fi{}8 sites. At half-filling, the numerical results for the triplet excitation spectrum are in very good agreement with a second-order perturbation expansion in terms of small interladder and intraladder exchange couplings between rungs (J/J'0.25). The band of local triplet excitations moving coherently along the ladder (with momenta close to \ensuremath{\pi}) is split by the interladder coupling. For intermediate couplings, finite-size scaling is used to estimate the spin gap. In the isotropic infinite four-chain system (two coupled ladders) we find a spin gap of 0.245J, roughly half of the single-ladder spin gap. When the system is hole doped, bonding and antibonding bound pairs of holes can propagate coherently along the chains and the spin gap remains finite.
The influence of the staggered magnetic field h on properties of the one-dimensional t-J and Hubbard models with nearly half-filled bands are studied. Using the exact diagonalization method for small systems as well as analytical approaches possible in the anisotropic model and for h\ensuremath{\gg}t, we determine the phase diagram of the t-J-h model at low doping, which displays three different regimes: the phase with repelling holes for JJc{\mathit{J}}_{\mathit{c}}Jc​, the phase with paired holes for Jc{\mathit{J}}_{\mathit{c}}Jc​JJs{\mathit{J}}_{\mathit{s}}Js​, and the phase separation for J>Js{\mathit{J}}_{\mathit{s}}Js​. Employing for h\ensuremath{\ne}0 the analogy with the universal behavior of one-dimensional interacting fermions within the Luther-Emery regime, we calculate the charge exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, which can be ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$>1 in the pairing regime, indicating dominant superconducting fluctuations. An analogous analysis, in particular the perturbation expansion in t/h, performed for the Hubbard t-U-h model shows a phase of stable hole pairs for UUc{\mathit{U}}_{\mathit{c}}Uc​. Again ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ seems to approach a nonuniversal value at half-filling.