Density-matrix renormalization-group study of the spin gap in a one-dimensional Hubbard model: Effect of the distant transfer and exchange coupling

The spin gap of a one-dimensional repulsive Hubbard model is numerically calculated with the density-matrix renormalization group, with a special emphasis on the effect of a next-nearest-neighbor hopping ${(t}^{\prime })$ and the nearest-neighbor ferromagnetic exchange $\mathrm{}\left(J\right)\mathrm{}$ interaction. At half filling, a significant spin gap opens if $|t^{\prime }|\simeq |t|$ and $J=0\mathrm{}$, in agreement with the weak-coupling theory, while the gap is strongly suppressed by the introduction of $J$. On the other hand, the quarter-filled system has very small spin gaps regardless of the values of $t^{\prime }$ and $J$. Implications for the ${\mathrm{CuO}}_2$ chain in ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ and related materials are discussed.