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We investigate the effect of a non-uniform deformation applied toone-dimensional (1D) quantum systems, where the local energy scale isproportional to $g_j = [\sin (j \pi / N)]^m$ determined by a positive integer$m$, site index $1 \leq j \leq N-1$, and the system size N. This deformationintroduces a smooth boundary to systems with open boundary conditions. When $m\geq 2$, the leading $1/N$ correction to the ground state energy per bond$e_0^{(N)}$ vanishes and one is left with a $1/N^2$ correction, the same aswith periodic boundary conditions. In particular, when $m = 2$, the value of$e_0^{(N)}$ obtained from the deformed open-boundary system coincides with theuniform system with periodic boundary conditions. We confirm the factnumerically for correlated systems, such as the extended Hubbard model, inaddition to 1D free-Fermion models.

Suppression of finite-size effects in one-dimensional correlated systems