Capturing nonlocal interaction effects in the Hubbard model: Optimal mappings and limits of applicability

We investigate the Peierls-Feynman-Bogoliubov variational principle to map Hubbard models with nonlocal interactions to effective models with only local interactions. We study the renormalization of the local interaction induced by nearest-neighbor interaction and assess the quality of the effective Hubbard models in reproducing observables of the corresponding extended Hubbard models. We compare the renormalization of the local interactions as obtained from numerically exact determinant quantum Monte Carlo to approximate but more generally applicable calculations using dual boson, dynamical mean field theory, and the random phase approximation. These more approximate approaches are crucial for any application with real materials in mind. Furthermore, we use the dual boson method to calculate observables of the extended Hubbard models directly and benchmark these against determinant quantum Monte Carlo simulations of the effective Hubbard model.

Figure 1

Local (red) and nearest-neighbor (blue) charge correlation functions of half filled nearest-neighbor hopping Hubbard model on a square lattice obtained from DQMC (full line), DB (diamonds), DMFT (crosses), and RPA (dashed line).

Figure 2

Nearest-neighbor renormalization strength $\alpha \left(\tilde{U}\right)$ of the half-filled nearest-neighbor hopping Hubbard model on a square lattice. (a) $\alpha \left(\tilde{U}\right)$ at $\beta t=2$, as obtained from DQMC (green dots), DB (red diamonds), DMFT (blue crosses), and RPA (dashed line). (b) At $\beta t=2$ (green dots) and $\beta t=5$ (magenta circles) using DQMC.

Figure 3

Observables in the Hubbard model $\left(V=0\right)$ obtained using DQMC (lines) and the DB method (diamonds).

Figure 4

Observables in the extended Hubbard model obtained using the DB method. Every colored square represents a DB calculation; they are separated by $\mathrm{\Delta }U=0.1$ and $\mathrm{\Delta }V=0.1$. The solid lines show isolines where the observable is constant; the dashed lines show lines of constant $\tilde{U}$ according to DQMC. The constant values of the isolines correspond to the tick labels in the color bars.

Figure 5

Local (red) and nearest-neighbor (blue) charge correlation functions of the hole doped $\left(\langle n_0\rangle =0.18\right)$ nearest-neighbor hopping Hubbard model on a square lattice obtained from DQMC (full line), DB (diamonds), DMFT (crosses), and RPA (dashed line).

Figure 6

Nearest-neighbor interaction strength $\alpha \left(\tilde{U}\right)$ of the hole doped $\left(\langle n_0\rangle =0.18\right)$ nearest-neighbor hopping Hubbard model on a square lattice obtained from DQMC (green dots), DB (red error bar), and RPA (dashed line).

Figure 7

Observables in the Hubbard model away from half filling $(〈n_0〉$ and $V=0)$ obtained using DQMC (lines) and the DB method (diamonds), cf. Fig. 3 for the half-filled system.

Figure 8

Observables in the extended Hubbard model, away from half filling at $〈n_0〉=0.18$, obtained using the DB method. See Fig. 8 for the half-filled system.

Figure 9

Observables in the extended Hubbard model obtained using the EDMFT method. Compare to the DB results in Fig. 4. The solid lines show isolines where the observable is constant. The constant values of the isolines correspond to the tick labels in the color bars.

Figure 10

Dynamic interaction $\mathrm{\Lambda }\left(i{\omega }_n\right)=U\left(i{\omega }_n\right)-U$ of the impurity model. Panels (a) and (b) show the dynamic interaction as a function of frequency, the symbols (lines) denote DB (EDMFT) results, respectively. Panel (c) shows how the dynamic interaction depends on the nearest-neighbor interaction $V$. All results are at $U=2$.