Model Hamiltonian for strongly correlated systems: Systematic, self-consistent, and unique construction

An interacting lattice model describing the subspace spanned by a set of strongly correlated bands is rigorously coupled to density functional theory to enable ab initio calculations of geometric and topological material properties. The strongly correlated subspace is identified from the occupation number band structure as opposed to a mean-field energy band structure. The self-consistent solution of the many-body model Hamiltonian and a generalized Kohn-Sham equation exactly incorporates momentum-dependent and crystal-symmetric correlations into electronic structure calculations in a way that does not rely on a separation of energy scales. Calculations for a multiorbital Hubbard model demonstrate that the theory accurately reproduces the many-body macroscopic polarization.

Figure 1

Mean-field energy bands of the two-orbital Hubbard model versus $ka$ for $t_{s0}=t_{d0}=3,g_s=g_d=10,{\mathrm{\Delta }}_s={\mathrm{\Delta }}_d=0.5,\xi a=0.0080,t_{sd}=0.8$, and $U=2.0$ (all in eV). The strongly hybridized bands are colored according to their orbital character (blue = s, red = d, purple = hybridized).

Figure 2

Natural occupation number bands of the two-orbital Hubbard model are shown for the series $U/t_{d0}=0.5,2,8,32$ (left to right) for $t_{s0}=t_{d0}=3,g_s=g_d=10,{\mathrm{\Delta }}_s={\mathrm{\Delta }}_d=0.5,\xi a=0.0080$, and $t_{sd}=0.8$ (all in eV). The predominantly $s$-orbital bands (blue) are close to 0 or 1 for all $U$, while the predominantly $d$-orbital bands (red) split off as $U$ increases. The bands are shown on the domain $ka\in (0,2\pi )$ so that zone boundary is positioned at the center.

Figure 3

Lower natural occupation number bands of the two-orbital Hubbard model are shown for the series $t_{sd}=0,0.8,1.6$ (dark to light) for $t_{s0}=t_{d0}=3,g_s=g_d=10,{\mathrm{\Delta }}_s={\mathrm{\Delta }}_d=0.5,\xi a=0.0080$, and $U/t_{d0}=2.4$; all parameters in eV.

Figure 4

The densities $n_d$ (red) and $n_s$ (blue) are plotted vs ${\mathrm{\Delta }}_s$ for $t_{s0}=t_{d0}=3,g_s=g_d=10,{\xi }_a=0.0020,{\mathrm{\Delta }}_d=0.5,U/t_{d0}=2.4$ and for the series $t_{sd}=0.0,0.8,1.6,2.4$ (dark to light); all parameters in eV.

Figure 5

Model parameters $t_{d1}^{\mathrm{model}}$ (top left), $t_{d2}^{\mathrm{model}}$ (bottom left), ${\mathrm{\Delta }}_d^{\mathrm{model}}$ (top right), and $U^{\mathrm{model}}$ (bottom right) are plotted vs ${\mathrm{\Delta }}_s$ for the series $t_{sd}=0.0,0.8,1.6,2.4$ (dark to light) for the same parameters as Fig. 4.

Figure 6

The densities $n_d$ (red) and $n_s$ (blue) are plotted vs ${\mathrm{\Delta }}_d$ for $t_{s0}=t_{d0}=3,g_s=g_d=10,\xi a=0.0020,{\mathrm{\Delta }}_s=0.5,U/t_{d0}=2.4$ and for the series $t_{sd}=0.0,0.8,1.6,2.4$ (dark to light); all parameters in eV.

Figure 7

Model parameters $t_{d1}^{\mathrm{model}}$ (top left), $t_{d2}^{\mathrm{model}}$ (bottom left), ${\mathrm{\Delta }}_d^{\mathrm{model}}$ (top right), and $U^{\mathrm{model}}$ (bottom right) are plotted vs ${\mathrm{\Delta }}_d$ for the series $t_{sd}=0.0,0.8,1.6,2.4$ (dark to light) and the same parameters as Fig. 6.

Figure 8

Many-body geometric phases ${\gamma }_{d\sigma }$ (red) and ${\gamma }_{s\sigma }$ (blue) for the two-orbital Hubbard model are plotted vs ${\mathrm{\Delta }}_d$ for the series $t_{sd}=0,0.8,1.6\mathrm{eV}$ (dark to light) and the same parameters as Fig. 6.

Figure 9

Geometric phase ${\gamma }_{d\sigma }+({\gamma }_{s\sigma }-\pi )$ for the noninteracting two-orbital Hubbard model is plotted vs ${\mathrm{\Delta }}_d$ for the series $t_{sd}=0,0.8,1.6$ (dark to light gray) and the same parameters as Fig. 8. The curves for different values of $t_{sd}$ nearly coincide.

Figure 10

Comparison of ${\gamma }_d$ from the full solution of the two-orbital Hubbard model (black), the model Hamiltonian solution (dashed green) and mean-field approximations (orange and gray) for $t_{s0}=3,t_{d0}=3,\xi =0.0020,{\mathrm{\Delta }}_s=0.5,U/t_{d0}=2.4$, and $t_{sd}=0.8$ as functions of ${\mathrm{\Delta }}_d$; all parameters in eV. The dashed red line connects the paramagnetic (gray) and antiferromagnetic (orange) mean-field results.

Figure 11

Comparison of BALDA (red) and exact (black) results for the geometric phase in the Rice-Mele-Hubbard as a function of $U$ for $t_{d0}=3,g_d=10,{\mathrm{\Delta }}_d=0.5$, and $\xi =0.0020$; all parameters in eV. Dashed lines show $\pi$ and $3\pi /2$.