Double occupancy in dynamical mean-field theory and the dual boson approach

We discuss the calculation of the double occupancy using dynamical mean-field theory in finite dimensions. The double occupancy can be determined from the susceptibility of the auxiliary impurity model or from the lattice susceptibility. The former method typically overestimates, whereas the latter underestimates the double occupancy. We illustrate this for the square-lattice Hubbard model. We propose an approach for which both methods lead to identical results by construction and which resolves this ambiguity. This self-consistent dual boson scheme results in a double occupancy that is numerically close to benchmarks available in the literature.

Figure 1

Visual illustration of Eq. (3): Shown are example diagrams contributing to the local part of the DMFT susceptibility. The impurity susceptibility ${\chi }^{\text{imp}}$ contains the local bubble and local vertex corrections built from local propagators. $X^{\prime }$ contains all the remaining diagrams. These describe processes that start and end at a site $a$ but involve intermediate scattering on different sites $b$ and nonlocal propagators.

Figure 2

Double occupancy. The lines show the DMFT double occupancy according to Eq. (2) (blue triangles) and the impurity double occupancy in DMFT (red squares). The purple crosses show the result of self-consistent dual boson. The double occupancy from DCA is shown [4] as a benchmark (green circles).

Figure 3

Screening of the impurity interaction in sc-DB. The different values of $U$ correspond to the point in Fig. 2. The difference between the screened and bare interactions is shown both in the charge and in the spin channel, normalized by the bare interaction. The retarded interactions appear as $n{\mathrm{\Lambda }}^{\text{ch}}n+S_z{\mathrm{\Lambda }}^{\text{sz}}{S}_z$ in the impurity action, so both the positive sign in the charge channel and the negative sign in the spin channel suppress the double occupancy.

Figure 4

Renormalization of the impurity double occupancy by nonlocal processes in two and three dimensions. Shown is the ratio of the sc-DB and DMFT impurity susceptibilities as a function of the quasiparticle weight $Z$. The results in 2D at $\beta =2$ correspond to Fig. 2. In 2D, the renormalization effect is stronger than in 3D.

Figure 5

Double occupancy away from half-filling at $U=2, \beta =8$, in the same color scheme as Fig. 2. The double occupancy is normalized by ${〈n〉}^2$ for clarity.