Impact of partially bosonized collective fluctuations on electronic degrees of freedom

In this work we present a comprehensive analysis of collective electronic fluctuations and their effect on single-particle properties of the Hubbard model. Our approach is based on a standard dual fermion and boson scheme with the interaction truncated at the two-particle level. Within this framework we compare various approximations that differ in the set of diagrams (ladder vs exact diagrammatic Monte Carlo), and/or in the form of the four-point interaction vertex (exact vs partially bosonized). This allows to evaluate the effect of all components of the four-point vertex function on the electronic self-energy. In particular, we observe that contributions that are not accounted for by the partially bosonized approximation for the vertex have only a minor effect on electronic degrees of freedom in a broad range of model parameters. In addition, we find that in the regime, where the ladder dual fermion approximation provides an accurate solution of the problem, the leading contribution to the self-energy is given by the longitudinal bosonic modes. This can be explained by the fact that contributions of transverse particle-hole and particle-particle modes partially cancel each other. Our results justify the applicability of the recently introduced dual triply irreducible local expansion (D-TRILEX) method that represents one of the simplest consistent diagrammatic extensions of the dynamical mean-field theory. We find that the self-consistent D-TRILEX approach is reasonably accurate also in challenging regimes of the Hubbard model, even where the dynamical mean-field theory does not provide the optimal local reference point (impurity problem) for the diagrammatic expansion.

Figure 1

Top row shows the ladder dual fermion self-energy ${\tilde{\mathrm{\Sigma }}}^{\mathrm{LDF}}$, which consists of the first-order Hartree-type term, second-order diagram ${\tilde{\mathrm{\Sigma }}}^{\left(2\right)}$, and the rest of the ladder ${\tilde{\mathrm{\Sigma }}}^{(3+)}$. Keeping only longitudinal modes of the partially bosonized representation for the four-point vertex $\mathrm{\Gamma }$ (black squares), the LDF self-energy reduces to the self-energy of the D-TRILEX approach (red diagram at the end of the top row). For ${\tilde{\mathrm{\Sigma }}}^{\left(2\right)}$ and ${\tilde{\mathrm{\Sigma }}}^{(3+)}$ the result of this approximation is explicitly shown in the middle and bottom rows, respectively. Red parts of these diagrams that consist of two triangles (three-point vertices) connected by two solid lines (dual Green's functions) is the polarization operator of the D-TRILEX theory. Wiggly line corresponds to the renormalized interaction.

Figure 2

The lattice self-energy ${\mathrm{\Sigma }}_{\mathbf{k},\nu }^{\mathrm{latt}}$ obtained for the first Matsubara frequency ${\nu }_0=\pi /\beta$ for the inverse temperature $\beta =2$ along the high-symmetry path in momentum space $\mathbf{k}$. The value of the onsite Coulomb potential $U$ for which the calculation was performed is specified in panels. Upper and lower parts of each panel correspond to real and imaginary parts of the self-energy, respectively. Results are obtained using D-TRILEX (red line), LDF (dark blue line), DiagMC@DF (light blue line), DiagMC@PBDB (purple crosses), and DiagMC@PBDB-s (green crosses) methods.

Figure 3

The normalized deviation $\delta$ calculated for LDF (blue), D-TRILEX (red), DiagMC@PBDT (purple), and DiagMC@PBDT-s (green) approximations with respect to the reference Diag@DF result. The black line shows the leading eigenvalue (l.e.) of AFM fluctuations. The vertical left axis shows the scale for the normalized deviation, while the vertical right axis displays values for the leading eigenvalue. The inset compares $\delta$ obtained for D-TRILEX and second-order DF $({\mathrm{DF}}^{\left(2\right)}$, orange) approaches for different inverse temperatures $\beta =2$ (circles), $\beta =4$ (triangles), and $\beta =10$ (squares).

Figure 4

Imaginary part of the lattice self-energy as the function of Matsubara frequency $\nu$ obtained for $U=2$ at the antinodal $\text{AN}=(0,\pi )$ (top row) and nodal $\text{N}=(\pi /2,\pi /2)$ (bottom row) points. Results are calculated at the inverse temperature $\beta =10$ (left column) and $\beta =15$ (right column) using D-TRILEX (red line), scD-TRILEX (red stars), and LDF (dark blue line) approaches. The DiagMC results (light blue line) are provided by the authors of Ref. [4].

Figure 5

Imaginary part of the D-TRILEX self-energy as the function of Matsubara frequency $\nu$ obtained for $U=2$ at the nodal $\text{N}=(\pi /2,\pi /2)$ (left panel) and antinodal $\text{AN}=(0,\pi )$ (right panel) points for different inverse temperatures $\beta =10$ (red line), $\beta =15$ (dark blue line), and $\beta =20$ (light blue line).

Figure 6

Imaginary part of the lattice self-energy as the function of Matsubara frequency $\nu$ obtained for $U=4$ at the AN (top row) and N (bottom row) points. Results are calculated at the inverse temperature $\beta =4$ (left column) and $\beta =6$ (right column) using D-TRILEX (red line), scD-TRILEX (red stars), and LDF (dark blue line) approaches. The reference DiagMC@DF result (light blue line) is presented only for $\beta =4$ and is reproduced by the scD-TRILEX method.

Figure 7

Imaginary part of the nonlocal self-energy obtained for the zeroth Matsubara frequency ${\nu }_0=\pi /\beta$ along the high-symmetry path in momentum space $\mathbf{k}$. Calculations are performed for $U=5.6$, $t^{\prime }=-0.3$, and $\beta =5$ using scD-TRILEX (red line) and DiagMC@DF (light blue line) methods for $3.4\%$ hole doping. The DiagMC result (dark blue line) for $4\%$ doping is provided by the authors of Ref. [93].

Figure 8

Top row: ladder ${\tilde{\mathrm{\Sigma }}}_{k\sigma }^{\mathrm{ladd}}$ (C3) and mixed ${\tilde{\mathrm{\Sigma }}}_{k\sigma }^{\mathrm{mix}}$ (C5) contributions to the LDB self-energy, and the LDB polarization operator ${\tilde{\mathrm{\Pi }}}_q^{\varsigma }$ (C8). Bottom row: Screened three-point $L_{\nu q}^{\vartheta }$ (C9) and four-point ${\mathrm{P}}_{\nu {\nu }^{\prime }q}^{\vartheta }$ (C10) vertex functions in the LDB approximation.