Finite-size scaling with modified boundary conditions

An efficient scheme is introduced for a fast and smooth convergence to the thermodynamic limit of electronic properties obtained with finite-size calculations on correlated Hamiltonians. This is obtained by modifying the energy levels of the free electron part of the Hamiltonian in a way consistent with the corresponding one-particle density of states in the thermodynamic limit. After this modification, free electron ground state energies, exact in the thermodynamic limit, are obtained with finite-size calculations and for all the particular fillings that satisfy the so called “closed-shell condition.” For those fillings the auxiliary field quantum Monte Carlo technique is particularly efficient and, by combining it with the present method, we provide strong numerical evidence that phase separation occurs in the low doping region and moderate $U\lesssim 4t$ regime of this model.

Figure 1

$U=0$ energy gap for finite clusters with nondegenerate ground state at the closest filling $N/L=0.9$ with ${45}^{\circ }$ tilted PBC or APBC. This gap is multiplied by the number sites, as it should converge for $L\to \infty$ to $g/N\left(\mu \right)$, where $\mu$ is the thermodynamic chemical potential at this filling and $g$ is the multiplicity of the energy level ($g=8$ in 2D and $g=2$ in 1D). (a) Standard 2D clusters. (b) Modified 2D clusters according to this work (see text). (c) Standard 1D clusters. (d) Energy per hole for standard clusters with PBC (filled symbols) and with MBC (empty symbols). The inset shows a ${45}^{\circ }$ tilted lattice of the type considered in this study.

Figure 2

(a) Energy per hole in the 2D Hubbard model. Finite-size effects are rather well behaved by using MBC and at small doping the energy per hole approaches almost exactly a horizontal line. This implies phase separation as discussed in the text. The curve is a fit (see Table 1) of the data for $\delta >6.5\%$. (b) Energy vs $\tau$ convergence starting with several initial left and right wave functions. “Var” (“no Var”) stands for the (non)variational calculation (see text) with different values of the antiferromagnetic parameter ${\mathrm{\Delta }}_{AF}$ in the mean-field determinant.