Consistent partial bosonization of the extended Hubbard model

We design an efficient and balanced approach that captures major effects of collective electronic fluctuations in strongly correlated fermionic systems using a simple diagrammatic expansion on a basis of dynamical mean-field theory. For this aim we perform a partial bosonization of collective fermionic fluctuations in leading channels of instability. We show that a simultaneous account for different bosonic channels can be done in a consistent way that allows to avoid the famous Fierz ambiguity problem. The present method significantly improves a description of an effective screened interaction $W$ in both charge and spin channels, and has a great potential for application to realistic $GW$-like calculations for magnetic materials.

Figure 1

Fermion-boson vertex function ${\mathrm{\Lambda }}_{\nu \omega }$ in the charge (left) and spin (right) channels as a function of fermionic ${\nu }_n$ and bosonic ${\omega }_m$ frequencies. The result is obtained for different values of the local Coulomb interaction.

Figure 2

The sketch of the approximation for the full local fermion-fermion vertex function ${\mathrm{\Gamma }}_{\nu {\nu }^{\prime }\omega }$ introduced in Eq. (7) (left). The illustration of the ${\mathrm{TRILEX}}^2$ approximation of the nonlocal self-energy $\tilde{\mathrm{\Sigma }}$ of the ladder dual theory that accounts only for the horizontal (shown in red) contribution to the fermion-fermion vertex function (right).

Figure 3

Charge and spin components of the exact (${\mathrm{\Gamma }}_{\nu ,{\nu }^{\prime }\omega }$) and approximate (${\mathrm{\Gamma }}_{\nu ,{\nu }^{\prime }\omega }^{\prime }$) fermion-fermion vertex functions at zeroth bosonic frequency ${\omega }_0$. The result is obtained for $U=1.0$.

Figure 4

Frequency dependence of charge, spin, and $\uparrow \uparrow$ components of the exact (black triangles) and approximate (lines with diamonds) fermion-fermion vertex function ${\mathrm{\Gamma }}_{\nu ,{\nu }^{\prime }\omega }$ along ${\nu }_0^{\prime }$ (left column) and ${\nu }_n={\nu }_n^{\prime }$ (right column) lines at zeroth bosonic frequency ${\omega }_0$. Results are obtained for $U=0.5$ (top row), $U=1.0$ (middle row), and $U=1.5$ (bottom row).

Figure 5

Real and imaginary parts of the nonlocal self-energy ${\tilde{\mathrm{\Sigma }}}_{\mathbf{k}\nu }$ at the first fermionic Matsubara frequency ${\nu }_0$ obtained for the ladder dual fermion and ${\mathrm{TRILEX}}^2$ (for the “unique” and Ising form of the bare interaction) approaches. Results are calculated for $U=0.5$ (top row), $U=1.0$ (middle row), and $U=1.5$ (bottom row).