One-Dimensional Continuum Electronic Structure with the Density-Matrix Renormalization Group and Its Implications for Density-Functional Theory

We extend the density matrix renormalization group to compute exact ground states of continuum many-electron systems in one dimension with long-range interactions. We find the exact ground state of a chain of 100 strongly correlated artificial hydrogen atoms. The method can be used to simulate 1D cold atom systems and to study density-functional theory in an exact setting. To illustrate, we find an interacting, extended system which is an insulator but whose Kohn-Sham system is metallic.

Figure 1

The exact ground state density of a chain of 100 widely separated (strongly correlated) artificial atoms. The total length of the system is $L=420$ in atomic units (4200 grid sites with a spacing of 0.1). The upper panel shows the electron density of a central region superimposed with the density at the left edge (the dashed blue curve with corresponding $x$ above). The lower panel compares the exact electron density to DFT predictions within the local spin density approximation.

Figure 2

Spontaneous dimerization of the density for a chain of 10 soft hydrogen atoms with interatomic spacing $b=4$ (dashed lines are a guide to the eye). The upper panel compares the densities predicted by DFT within the LSDA; the lower panel shows the spin densities for unrestricted LSDA. Also shown is the expectation value of the kinetic energy $⟨c_j^{†}{c}_{j+1}+c_{j+1}^{†}{c}_j⟩$ for a Hubbard model with $U/t=4$ and the exchange energy $⟨{\mathbf{S}}_j·{\mathbf{S}}_{j+1}⟩$ for the Heisenberg model on 10 lattice sites. The thickness of the lines indicates the magnitude of these quantities on each bond.

Figure 3

Exact gaps for chains of $N$ soft hydrogen atoms with atomic separation $b=4$ (error bars are less than symbol sizes). The upper curve is a quadratic fit of exact gaps of the largest six systems and extrapolates to a finite value $E_g\simeq 0.33$. The exact Kohn-Sham gaps, in contrast, extrapolate to zero showing that for $N\to \infty$ the true KS system is metallic (lower curve is a linear fit of exat KS gaps of the largest six systems).

Figure 4

Entanglement spectrum at the center of interacting 4-atom chains with various interatomic separations $b$. $N=(N_L,N_R)$ refers to the number of electrons to the left and right of the cut for each density matrix eigenstate. The states with $N_L=3$, 1 primarily correspond to charge fluctuations while those with $N_L=2$ to spin fluctuations.