Dual boson diagrammatic Monte Carlo approach applied to the extended Hubbard model

In this work we introduce the dual boson diagrammatic Monte Carlo technique for strongly interacting electronic systems. This method combines the strength of dynamical mean-filed theory for nonperturbative description of local correlations with the systematic account of nonlocal corrections in the dual boson theory by the diagrammatic Monte Carlo approach. It allows us to get a numerically exact solution of the dual boson theory at the two-particle local vertex level for the extended Hubbard model. We show that it can be efficiently applied to description of single-particle observables in a wide range of interaction strengths. We compare our exact results for the self-energy with the ladder dual boson approach and determine a physical regime, where the description of collective electronic effects requires more accurate consideration beyond the ladder approximation. Additionally, we find that the order-by-order analysis of the perturbative diagrammatic series for the single-particle Green's function allows to estimate the transition point to the charge density wave phase.

Figure 1

Schematic diagrammatic interpretation of Eq. (12). The full antisymmetrized fermion-fermion interaction $\overline{\mathrm{\Gamma }}$ (gray box) is a combination of the impurity vertex $\mathrm{\Gamma }$ (white box) and processes involving a boson exchange (wiggly line). The full vertex acquires a momentum dependence due to the presence of the bosonic lines. White and black dots represent incoming and outgoing particles, respectively. Triangles represent ${\mathrm{\Lambda }}_{\nu \omega }$ vertices. The exact dependence on the channel indices and prefactors is shown in Eq. (12).

Figure 2

Convergence of the real (top panel) and imaginary (bottom panel) parts of the dual self-energy ${\tilde{\mathrm{\Sigma }}}_{\mathbf{k},\nu }$ obtained for the zeroth Matsubara frequency ${\nu }_0$. The result is plotted along the high-symmetry path in momentum space $\mathbf{k}$ as a function of the expansion order $n$. The parameters are $U=5, V=1.25$, and $\beta =2$ in the units of the hopping amplitude. The inset shows the convergence of the real part around the $M=\{\pi ,\pi \}$ point.

Figure 3

Most important self-energy diagrams. Top row shows the only nonzero contribution to the first order diagram, taking into account that we cannot connect two slots of the same local vertex with a propagator line due to DMFT self-consistency condition. The middle row shows the second-order diagrams ${\tilde{\mathrm{\Sigma }}}^{\left(2\right)}$. If $V$ is small compared to $U/4$, it can be approximated by the second-order dual fermion diagram on the right-hand side. The last term ${\tilde{\mathrm{\Sigma }}}^{\mathrm{corr}}$ shows few diagrams that enter $\tilde{\mathrm{\Sigma }}$ in our calculations, but are not included in the ladder DB.

Figure 4

Comparison between DiagMC@DB (solid lines with error bars) and ladder DB (dots) results for the real (top panel) and imaginary (bottom panel) parts of the lattice self-energy ${\mathrm{\Sigma }}_{\mathbf{k}{\nu }_0}$. The result is obtained for $U=2, \beta =4$, and different values of the nonlocal Coulomb interaction $V$ specified in the legend of the bottom panel.

Figure 5

Comparison between DiagMC@DB (solid lines with errorbars) and ladder DB (dots) results for the real (top panel) and imaginary (bottom panel) parts of the lattice self-energy ${\mathrm{\Sigma }}_{\mathbf{k}{\nu }_0}$. The result is obtained for $U=4, \beta =4$, and different values of the nonlocal Coulomb interaction $V$ specified in the legend of the bottom panel.

Figure 6

Comparison between DiagMC@DB (solid lines with errorbars) and ladder DB (dots) results for the real (top panel) and imaginary (bottom panel) parts of the lattice self-energy ${\mathrm{\Sigma }}_{\mathbf{k}{\nu }_0}$. The result is obtained for $U=6, \beta =2$, and different values of the nonlocal Coulomb interaction $V$ specified in the legend of the bottom panel.

Figure 7

Comparison between DiagMC@DB (solid lines with errorbars) and ladder DB (dots) results for the real (top panel) and imaginary (bottom panel) parts of the lattice self-energy ${\mathrm{\Sigma }}_{\mathbf{k}{\nu }_0}$. The result is obtained for $U=8, \beta =2$, and different values of the nonlocal Coulomb interaction $V$ specified in the legend of the bottom panel.

Figure 8

Summary of the results of our calculations as a function of the local $U$ and nonlocal $V$ Coulomb interactions. Results for $U\le 4$ were obtained at $\beta =4$, while for $U>4$ they were calculated at $\beta =2$. The mismatch parameter ${\delta }_M$ is depicted by color. Points correspond to physical parameters for which calculations are performed. The red area highlights the region where the mismatch parameter is larger. Transition points between the normal and CDW phases obtained in DiagMC@DB and ladder DB calculations are depicted by an orange circle and cross, respectively. The dashed black line $V=U/4$ represents an estimation of the phase boundary predicted by mean-field arguments.

Figure 9

Left panel: $G_{\text{loc}}(\tau =\beta /2)$ as a function of nonlocal interaction $V$ for the various perturbation orders in the bare DiagMC scheme at $U=2.5$ and $\beta =4$. In the inset it is shown the estimated $V$ of the CDW transition as a function of the order. An additional fitting of the curve with the function $f\left(n\right)=C_0+C_1{n}^{-C_2}$, where $C_0, C_1$, and $C_2$ are fitting parameters, allows to extrapolate the value at infinite order $V_{\mathrm{CDW}}=C_0=0.77\left(2\right)$. Upper right panel: The inverse of the charge susceptibility at the $M=\{\pi ,\pi \}$ point obtained by ladder dual boson calculations as a function of $V$ for the same $U$ and $\beta$. The value $V_{\mathrm{CDW}}$ can be obtained by a linear fitting of the data and checking where the fitting line crosses zero. Lower right panel: Position ${\xi }_s$ of the singularity on the real axis obtained within the method presented in Ref. [66] for the bare and semi-bold DiagMC schemes. The phase transition occurs when ${\xi }_s$ crosses $\xi =1$. This analysis predicts the transition at $V_{\mathrm{CDW}}=0.82\left(1\right)$ for the bare series and $V_{\mathrm{CDW}}=0.81\left(1\right)$ for the semi-bold series.