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

We investigate the effect of a nonuniform deformation applied to one-dimensional (1D) quantum systems, where the local energy scale is proportional to $g_j=[\sin (j\pi /N)]{}^m$ determined by a positive integer $m$, site index $1⩽j⩽N-1$, and system size $N$. This deformation introduces a smooth boundary to systems with open-boundary conditions. When $m⩾2$, 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 as with periodic boundary conditions. In particular, when $m=2$, the value of $e_0^{(N)}$ obtained from the deformed open-boundary system coincides with the uniform system with periodic boundary conditions. We confirm the fact numerically for correlated systems, such as the extended Hubbard model, in addition to 1D free-fermion models.

Figure 1

Convergence of $e_0^{(N)}$ in Eq. (9) with respect to $N$ at half filling. We choose $t$ as the unit of the energy. Data with OBCs, PBCs, and the deformed cases with $m=1$–$5$ are shown.

Figure 2

Asymptotic behavior of $e_0^{(N)}$ for the deformed chains when $1⩽m⩽5$ with respect to $e_0^{(N)}$ of the system with PBCs. We choose $t$ as the unit of the energy. A logarithmic correction is present when $m=1$.

Figure 3

Expectation value of the bond correlation function $\langle c_j^{†}{c}_{j+1}+c_{j+1}^{†}{c}_j\rangle$ with respect to $j$ under sinusoidal deformation.

Figure 4

Comparison of the expectation value of the bond correlation function at half filling under SSD ($m=2$) and a SBC in Refs. [11] and [12]. The number $M$ in the case of the SBC specifies the length of area where the interactions are modified near the system boundary. The bottom graph shows the numerical details when $M=30$ (SBC) with respect to SSD.

Figure 5

Convergence of $e_0^{(N)}$ with respect to $N^{-2}$ at $1/4$ filling and $1/8$ filling. We choose $t$ as the unit of the energy.

Figure 6

Position dependence of the occupation number $\langle n_j\rangle =\langle c_j^{†}{c}_j\rangle$ at $f=1/2$, $1/4$, and $1/8$ when $m=1$, $2$, and $3$ (upper graph). The left-hand and right-hand lower graphs display the occupancy at $f=1/4$ close to the boundaries.

Figure 7

Energy per bond $e_0^{(N)}$ of the spinless fermion model at $V=1$ up to $N=16$. We choose $t$ as the unit of the energy.

Figure 8

Occupation number $\langle n_j\rangle$ and the bond correlation $\langle c_j^{†}{c}_{j+1}+c_{j+1}^{†}{c}_j\rangle$ when $V=1$ at half filling.

Figure 9

System size dependence of $e_0^{(N)}$ of the extended Hubbard model at half filling. We choose $t$ as the unit of the energy. The full and the dashed lines connect the energies $e_0^{(N)}$ obtained by the PBC and SSD, respectively.

Figure 10

Occupation $\langle n_{j\uparrow }\rangle =\langle n_{j\downarrow }\rangle$ and the bond correlation ${\sum }_{\sigma }\langle c_{j\sigma }^{†}{c}_{j+1}+c_{j+1\sigma }^{†}{c}_{j\sigma }\rangle$ of the extended Hubbard model at half filling.

Figure 11

The $\mu$ dependence of $|p+\partial N{e}_0^{(N)}/\partial \mu |$ for the two cases $f\equiv p/2N=1/4$ and $1/6$. We choose $t$ and $\mu$ as the units of the energy.

Figure 12

The upper and lower graphs show the occupancy $\langle n_{i,\sigma }\rangle$ and the bond correlation ${\sum }_{\sigma }\langle c_{j\sigma }^{†}{c}_{j+1\sigma }+c_{j+1\sigma }^{†}{c}_{j\sigma }\rangle$ at $f=1/6$.