Dual boson approach with instantaneous interaction

The dual boson approach to strongly correlated systems generally involves a dynamic (frequency-dependent) interaction in the auxiliary impurity model. In this work, we explore the consequences of forcing this interaction to be instantaneous (frequency independent) via the use of a self-consistency condition on the instantaneous susceptibility. The result is a substantial simplification of the impurity model, especially with an eye on realistic multiband implementations, while keeping desirable properties of the dual boson approach, such as the charge conservation law, intact. We show and illustrate numerically that this condition enforces the absence of phase transitions in finite systems, as should be expected from general physical considerations, and respects the Mermin-Wagner theorem. In particular, the theory does not allow the metal-to-insulator phase transition associated with the formation of the magnetic order in a two-dimensional system. At the same time, the metal-to-charge-ordered phase transition is allowed, as it is not associated with the spontaneous breaking of a continuous symmetry, and is accurately captured by the introduced approach.

Figure 1

Top panel: Effective interaction as a function of temperature and interaction strength. In all cases shown here, ${\mathrm{\Lambda }}^{\text{ch}}>0$ and ${\mathrm{\Lambda }}^{\text{sz}}<0$. Bottom panel: Effective interactions in the two-particle self-consistent method.

Figure 2

Inverse antiferromagnetic susceptibility $X_{\mathbf{q}=(\pi ,\pi ),\omega =0}^{\text{AF}}$ in the square lattice Hubbard model at $\beta t=3$. A phase transition to an antiferromagnetically ordered phase occurs when this inverse susceptibility is equal to zero (dashed black line).

Figure 3

Dynamic susceptibilities at $U=5$ and $\beta =3$, for the square lattice. The points correspond to $\chi$, the lines to $X^{\text{loc}}$, both in the magnetic channel.

Figure 4

Self-energy at $U=5$ and $\beta =3$, for the square lattice. The points correspond to the Matsubara frequencies ${\nu }_n$; lines are guides to the eye.

Figure 5

Analysis of the self-consistency condition at $U=4$ and $\beta =2.5$, for a cubic lattice.

Figure 6

Self-energy at $U=4$ and $\beta =2.5$, for a cubic lattice. The points correspond to the Matsubara frequencies ${\nu }_n$; lines are guides to the eye.

Figure 7

Changes in the impurity model obtained for the cubic lattice at $\beta =2.5$. Effective interaction $U^{\prime }$, double occupancy $〈n_{\uparrow }{n}_{\downarrow }〉$ of the impurity model, and average perturbation order $〈k〉$ of the CT-HYB solver. The latter two are related to the energy as $E_{\text{pot}}=U〈n_{\uparrow }{n}_{\downarrow }〉$ and $E_{\text{kin}}=-\frac{2}{\beta }〈k〉$ [68]. The TPSC result for the double occupancy is shown by a black dashed line.

Figure 8

The extended Hubbard model on the cubic lattice, for $U=5$ and $\beta =2.5$. Solid lines are guides to the eye; the dashed line in the bottom panel is a linear fit with intercept at $V\approx 0.99$. The same quantities at $V=0$ are shown in Fig. 7. The bottom panel shows the inverse of the charge density wave susceptibility. Where this quantity reaches zero, the system becomes unstable towards checkerboard charge order.

Figure 9

Comparison of the charge order transition in the square and cubic lattices. The square lattice simulations correspond to $\beta =3$, the cubic lattice with $\beta =2.5$, and all simulations are at $U=5$. The curves labeled DB constant $\mathrm{\Lambda }$ correspond to the method proposed in this work; reference results using the scheme of Ref. [26] are shown as non-sc DB. Arrows indicate $V=U/6$ and $V=U/4$.

Figure 10

Effective interaction as a function of temperature and interaction strength, with the perturbative result of Eq. (A9) as the dashed lines and the DB results of Fig. 1 as the symbols.

Figure 11

Left: Self-energy diagram that contributes to the asymptote. Right: Diagram of trivial contribution to ${\gamma }^{2,2}$.