Efficiency of free auxiliary models in describing interacting fermions: From the Kohn-Sham model to the optimal entanglement model

Density functional theory maps an interacting Hamiltonian onto the Kohn-Sham Hamiltonian, an explicitly free model with identical local fermion densities. Using the interaction distance, the minimum distance between the ground state of the interacting system and a generic free-fermion state, we quantify the applicability and limitations of the exact Kohn-Sham model in capturing the various properties of the interacting system. As a by-product, this distance determines the optimal free state that reproduces the entanglement properties of the interacting system as faithfully as possible. The parent Hamiltonian of the optimal free state identifies a system that can determine the expectation value of any observable with controlled accuracy. This optimal entanglement model opens up the possibility of extending the systematic applicability of auxiliary free models into the nonperturbative, strongly correlated regimes.

Figure 1

A schematic illustration of the distances between interacting ${\rho }^{\text{int}}$, optimal ${\rho }^{\text{opt}}$, and Kohn-Sham ${\rho }^{\text{KS}}$ reduced density matrices. Two free manifolds of Gaussian states, $\mathcal{F}$ and ${\mathcal{F}}^{\prime }$, are depicted with possibly different number of degrees of freedom. For small interaction coupling $U$, ${\rho }^{\text{int}}$ is close to $\mathcal{F}$, while as $U$ increases ${{\rho }^{\prime }}^{\text{int}}$ can be close to another manifold ${\mathcal{F}}^{\prime }$. The direction of equal local fermion densities identifies the KS model on $\mathcal{F}$. In the perturbative regime (small $U$), when $D_{\mathcal{F}}\approx 0$, then $D_{\text{tr}}({\rho }^{\text{int}},{\rho }^{\text{KS}})$ and $D_{\text{tr}}({\rho }^{\text{KS}},{\rho }^{\text{opt}})$ are also small, as dictated by (19). When $U$ is large, then the state ${{\rho }^{\prime }}^{\text{int}}$ can effectively admit an optimal free description ${{\rho }^{\prime }}^{\text{opt}}$, with a different number of fermions than the one from the perturbative regime. This change makes the Kohn-Sham model ${{\rho }^{\prime }}^{\text{KS}}$ unsuitable for reproducing the entanglement properties of ${{\rho }^{\prime }}^{\text{int}}$.

Figure 2

Behavior of the chemical potential $\mu$, for the $L=2$ optimal entanglement mode at half-filling, against interaction strength of the interacting model $U$. This chemical potential imitates the affect of interactions in the interacting model and its explicit form is a function of the optimal free-state entanglement spectrum. In the strong interaction regime we find $\mu \approx JU$ to a very good approximation. Inset: a sketch of the auxiliary model described by Hamiltonian (23). The system is built from two noninteracting chains, each with a single spinless fermion. The dashed line shows the partitioning of the system into subsystems $A$ and $B$.

Figure 3

(Top) Trace distance, (middle) natural metric, and (bottom) entanglement entropy for the interacting ${\rho }^{\mathrm{int}}$, optimal ${\rho }^{\mathrm{opt}}$, KS ${\rho }^{\mathrm{KS}}$, and auxiliary ${\rho }^{\mathrm{aux}}$ reduced density matrices, as a function of the interaction coupling $U$, for $L=2$, $J=1$, total spin $S_z=0$, and ${\nu }_1-{\nu }_2=0.5$. In the perturbative limit the KS is a good approximation to the optimal entanglement model which describes spin-$\frac{1}{2}$ free fermions. In the large-$U$ limit the KS model fails to reproduce entanglement, while both the optimal and auxiliary states that describe spinless free fermions provide faithful representations of the local densities (middle) and the entanglement entropy (bottom) of the interacting system. For large $U$ the entanglement entropy of the KS model tends to $S=ln4$ corresponding to the maximally entangled state $\left|\psi \right.〉=(\left|\uparrow \downarrow ,0\right.〉+\left|\uparrow ,\downarrow \right.〉+\left|\downarrow ,\uparrow \right.〉+\left|0,\uparrow \downarrow \right.〉)/2$ while the interacting, optimal, and auxiliary systems tend to $S\approx ln2$ that correspond to $\left|\psi \right.〉=(\left|\uparrow ,\downarrow \right.〉+\left|\downarrow ,\uparrow \right.〉)/\sqrt{2}$, signaling the freezing of double occupations due to interactions.

Figure 4

Free parameter values $b_1,b_2$ that produce boundary curves ${\rho }_j^{\mathrm{opt}}={\rho }_j^{\mathrm{int}}$ (solid lines) for $\left\{{\rho }_j^{\mathrm{int}}\right\}=\{\frac{1}{3},\frac{1}{3},\frac{1}{3},0\}$. The dashed lines enclose the normalized and ordered regions for the free spectra. There are two points of intersection within the normalized and ordered region. The intersection that matches better the low-level entanglement spectrum will give the most faithful representation of the interacting system. Thus, it is the $b_1,b_2$ pair at the intersection between ${\rho }_1^{\mathrm{opt}}={\rho }_1^{\mathrm{int}}$ and ${\rho }_2^{\mathrm{opt}}={\rho }_2^{\mathrm{int}}$ that give the interaction distance $D_{\mathcal{F}}=\frac{1}{6}$.