Optimal Hubbard Models for Materials with Nonlocal Coulomb Interactions: Graphene, Silicene, and Benzene

To understand how nonlocal Coulomb interactions affect the phase diagram of correlated electron materials, we report on a method to approximate a correlated lattice model with nonlocal interactions by an effective Hubbard model with on-site interactions $U^{*}$ only. The effective model is defined by the Peierls-Feynman-Bogoliubov variational principle. We find that the local part of the interaction $U$ is reduced according to $U^{*}=U-\overline{V}$, where $\overline{V}$ is a weighted average of nonlocal interactions. For graphene, silicene, and benzene we show that the nonlocal Coulomb interaction can decrease the effective local interaction by more than a factor of 2 in a wide doping range.

Figure 1

Illustration of the physical process underlying Eq. (5). (a) Half-filled system: an electron hops from a doubly occupied site to an empty one, gaining an energy ($U-V$) in the original model and $U^{*}$ in the effective model. (b) Nearly empty or almost full system: wave packets of spin up and down electrons or holes [$\rho (x)=|{\psi }_{\uparrow ,\downarrow }(x){|}^2$] are separated to the farthest possible position. If the packets are much wider than the lattice spacing, the approximation for the initial energy $U\int \rho (x{)}^{2}dx+V\int \rho (x)\rho (x+1)dx\approx (U+V)\int \rho (x{)}^{2}dx$ holds and the energy gained in the original model is $\sim (U+V)$ and $\sim U$ in the effective model.

Figure 2

(a) Derivatives of the correlation functions $⟨n_{0\uparrow }{n}_{j\sigma }⟩$ with respect to $U^{*}$ at $U^{*}=2t$ for half-filled graphene ($16\times 16$ unit cells). Each circle corresponds to one carbon atom. The thick drawn circles indicate the lattice site with index $i=0$. (b) Correlation function ${\partial }_{U^{*}}⟨n_{0\uparrow }{n}_{j\downarrow }{⟩}^{*}$ for nearly empty honeycomb lattice (two electrons in a $5\times 5$ super cell). In addition, we find ${\partial }_{U^{*}}⟨n_{0\uparrow }{n}_{j\uparrow }{⟩}^{*}=0$ as it must be for a singlet ground state (not shown here).

Figure 3

Effective local Coulomb interaction $U^{*}/t$ color coded for (a) benzene with $U=3.96/t$ and (b) benzene with $U=7.92/t$, for various screenings of $V_{ij}(\epsilon )$ and all fillings. Due to particle hole symmetry of the model only $N\le 6$ is shown.

Figure 4

Correlation as functions of the screening for the extended Hubbard model (continuous lines) and the effective Hubbard model (broken lines) for benzene. The left panel shows spin correlation $⟨S_z^{ij}⟩$ and the right panel shows density correlation $⟨{\rho }_z^{ij}⟩$. $⟨S_z^{01}⟩$ is virtually the same for the effective and original model. The parameters for the original model are $U=10.06\mathrm{eV}$ and $t_{\mathrm{orig}}=2.539\mathrm{eV}$, and $V(\epsilon )$ is calculated by Eq. (8). $U^{*}(\epsilon )$ is calculated by Eq. (5), while $t_{\mathrm{eff}}=t_{\mathrm{orig}}$.