Self-consistent dual boson approach to single-particle and collective excitations in correlated systems

We propose an efficient dual boson scheme, which extends the dynamical mean-field theory paradigm to collective excitations in correlated systems. The theory is fully self-consistent both on the one- and on the two-particle level, thus describing the formation of collective modes as well as the renormalization of electronic and bosonic spectra on equal footing. The method employs an effective impurity model comprising both fermionic and bosonic hybridization functions. Only single- and two-electron Green's functions of the reference problem enter the theory, due to the optimal choice of the self-consistency condition for the effective bosonic bath. We show that the theory is naturally described by a dual Luttinger-Ward functional and obeys the relevant conservation laws.

Figure 1

Sketch of the DB formalism. The original action with parameters $U,V_{\mathbf{q}}$, and $t_{\mathbf{k}}$ is replaced by an auxiliary impurity problem with fields ${\mathrm{\Delta }}_{\nu },{\mathrm{\Lambda }}_{\omega }$. The expectation values of this impurity model ($g_{\nu },{\chi }_{\omega },{\gamma }_{\nu {\nu }^{\prime }\omega }$, and ${\lambda }_{\nu \omega }$) enter a dual theory in terms of $\tilde{G}$ and $\tilde{X}$.

Figure 2

Interaction vertices in the dual perturbation theory.

Figure 3

Diagrammatic representation of Eq. (20). The dual superline $\tilde{S}$ describes two-particle fluctuations that have both pure bosonic and collective fermionic character. Filled lines denote the dual propagators, dashed lines the impurity ones.

Figure 4

Dual Luttinger-Ward functional for the dual boson approach. The generation of diagrams for the dual self-energy and polarization function occurs by cutting one fermionic or bosonic line, respectively.

Figure 5

The renormalized three- and four-point vertex functions in the dual boson approach. The five- and six-point vertices are introduced here with the local dual fermion line. Such diagrams are identically equal to zero because of the self-consistency condition and inserted only to generate the self-energy diagrams from the Luttinger-Ward functional.

Figure 6

Diagrams for the dual self-energy and polarization function. The renormalized three- and four-point vertices have the ladder structure (see Fig. 5) and do not contain any diagrams with the local fermionic line due to the self-consistency condition.

Figure 7

The structure of the irreducible parts of higher-order vertices with respect to the bosonic arguments. The diagrams (a) and (b) give the contribution ${\tilde{\mathrm{\Sigma }}}^{6,0}$ to the self-energy from the six-point vertex, diagrams (c) and (d) are the contributions ${\tilde{\mathrm{\Sigma }}}^{2,2}$ and ${\tilde{\mathrm{\Sigma }}}^{4,1}$ from the vertices ${\gamma }^{2,2}$ and ${\gamma }^{4,1}$, respectively. The purple parts of the diagrams are equal to the dual polarization function $\tilde{\mathrm{\Pi }}$ (see Fig. 6), the dashed wave lines are the impurity susceptibility $\chi$.

Figure 8

A summary of the computational scheme used. The impurity solver determines local impurity quantities based on hybridization functions ${\mathrm{\Delta }}_{\nu },{\mathrm{\Lambda }}_{\omega }$. The fermionic lattice Green's function is determined directly from the impurity quantities. For the lattice susceptibility, we take additional nonlocal corrections into account via the dual polarization $\tilde{\mathrm{\Pi }}$ in the ladder approximation.

Figure 9

(Left) Effective impurity interaction as a function of Matsubara frequency for various nonlocal interaction strengths $V$, at $U=0.5$. The dashed line indicates the high-frequency asymptotic value of ${\mathrm{\Lambda }}_{\omega }$, and this value is shown in the top right as a function of $V$ (the line is a linear fit for $\left|V\right|\le 0.01$.). Center right is the zero frequency ${\mathrm{\Lambda }}_{\omega }$ and bottom right the zero frequency ${\mathrm{\Lambda }}_{\omega }$ in EDMFT.

Figure 10

(Left) Effective impurity interaction as a function of Matsubara frequency for various nonlocal interaction strengths $V$, at $U=10$. The dashed line indicates the high-frequency asymptotic value of ${\mathrm{\Lambda }}_{\omega }$, and this value is shown in the top right as a function of $V$ (the line is a linear fit for $\left|V\right|\le 0.1$.). Center right is the zero frequency ${\mathrm{\Lambda }}_{\omega }$ and bottom right the zero frequency ${\mathrm{\Lambda }}_{\omega }$ in EDMFT.

Figure 11

Quasiparticle renormalization factor $Z$ as a function of the nonlocal interaction strength $V$, at $U=10$. Nearest-neighbor repulsion $\left(V>0\right)$ screens the local interaction and increases the quasiparticle renormalization factor.

Figure 12

Charge conservation requirement $X_{\mathbf{q}=0,\omega >0}=0$ during the self-consistency scheme. Iteration 0 is single-shot DB. These results are at $U=10,V=0.3$.