Application of the finite-cell-scaling method to the one-dimensional Hubbard model

The finite-cell-scaling method is applied to the one-dimensional Hubbard model. In the half-filled-band case and at zero temperature, the numerical results obtained for the ground-state energy differ from the exact solution by approximately 1%. For the range $1<\mathrm{}\frac{U}{t}<5\mathrm{}$, where $U$ is the Coulomb interaction and $t$ is the transfer integral, the value of the normalized gap $\frac{\Delta }{t}$ agrees well with the exact result while it differs significantly for $0.2<\mathrm{}\frac{U}{t}<0.9\mathrm{}$. For the latter range, we show that the gap varies as ${\left(\frac{U}{t}\right)}^{\frac{1}{2}}\exp \left[{(a+\frac{\mathrm{bU}}{t})}^{-1\mathrm{}}\right]$, where $a$ and $b$ are negative constants which depend on the size of the system, instead of ${\left(\frac{U}{t}\right)}^{\frac{1}{2}}\exp \left(\frac{-2\mathrm{}\pi t}{U}\right)$ as in the exact solution.