Beyond extended dynamical mean-field theory: Dual boson approach to the two-dimensional extended Hubbard model

The dual boson approach [Rubtsov, Katsnelson, and Lichtenstein, Ann. Phys. 327, 1320 (2012)] provides a means to construct a diagrammatic expansion around the extended dynamical mean-field theory (EDMFT). In this paper, we present the numerical implementation of the approach and apply it to the extended Hubbard model with nearest-neighbor interaction $V$. We calculate the EDMFT phase diagram and study the effect of diagrams beyond EDMFT on the transition to the charge-ordered phase. Including diagrammatic corrections to the EDMFT polarization shifts the EDMFT phase boundary to lower values of $V$. The approach interpolates between the random phase approximation in the weak coupling limit and EDMFT for strong coupling. Neglecting vertex corrections leads to results reminiscent of the $\text{EDMFT}+GW$ approximation. We, however, find significant deviations from the dual boson results already for small values of the interaction, emphasizing the crucial importance of fermion-boson vertex corrections.

Figure 1

The building blocks of the dual perturbation theory and frequency conventions. (Top) The bare DF and boson propagators $\tilde{\mathcal{G}}$ (a) and $\tilde{\mathcal{X}}$ (b). (Bottom) Dual fermions interact via the vertex function $\gamma$. $\lambda$ is the fermion-boson interaction. $\nu$ denotes fermionic frequencies; $\omega$ denotes bosonic Matsubara frequencies.

Figure 2

Second-order diagrams contributing to the nonlocal fermionic (a) and bosonic (b) DF self-energy ${\tilde{\Sigma }}_{\mathbf{k}\nu \sigma }$ and ${\tilde{\Pi }}_{\mathbf{q}\omega }$, respectively.

Figure 3

Second-order diagrams with a local DF loop.

Figure 4

Summary of the computational scheme. The loop is started with an initial guess for the hybridization function and retarded interaction. Local observables are calculated in the impurity solver (IS) step. Working with bare dual propagators and neglecting diagrammatic corrections is equivalent to EDMFT (dashed arrow). Corrections beyond EDMFT are taken into account by evaluating diagrams for the dual self-energies, which involve dual propagators and the vertex functions $\lambda$ and $\gamma$. The diagrams are renormalized self-consistently using Dyson equations. From the dual Green's functions one obtains the physical lattice Green's functions and, in turn, an update for the hybridization and retarded interaction.

Figure 5

EDMFT phase diagram for the extended Hubbard model in the plane of on-site interaction $U$ and nearest-neighbor interaction $V$ at temperature $T=0.01$. Phase boundaries marked in red have been obtained starting from a metallic seed, while the boundaries in brown were obtained starting from an insulating solution as the initial guess. CO, FL, and MI denote charge-ordered, Fermi-liquid metallic, and Mott insulating phases, respectively. Colored points indicate positions at which quantities of interest, such as the self-energy, are evaluated.

Figure 6

EDMFT self-energy for fixed on-site interaction $U=2.2$ and different values of the nearest-neighbor interaction $V$. The different curves correspond to the positions marked by circles in the EDMFT phase diagram in Fig. 5. The inset shows the convergence of the self-energy to its high-frequency behavior. Horizontal lines mark the corresponding calculated values of the first moment.

Figure 7

EDMFT charge susceptibility for the same parameters as in Fig. 6.

Figure 8

Three-leg vertex in EDMFT for two bosonic frequencies ${\omega }_m=2m\pi /\beta$ with $m=0$ and $m=4$ and different values of the nearest-neighbor interaction $V$. The parameters are the same as in Fig. 6. With increasing $V$, the transition to the CO phase is approached and the vertex becomes flatter. Note the change of sign.

Figure 9

Three-leg vertex for two different bosonic frequencies as a function of fermionic frequency. The nearest-neighbor interaction is kept fixed at $V=0$ corresponding to a DMFT calculation. The corresponding points are marked by triangles in the phase diagram of Fig. 5. With increasing values of $U$, the Mott transition is approached from the metallic side.

Figure 10

Three-leg vertex for different bosonic frequencies as a function of the fermionic frequency. The nearest-neighbor interaction is fixed at $V=0.6$. Parameters are otherwise the same as in Fig. 9. The corresponding points are marked by diamonds in Fig. 5.

Figure 11

Diagrammatic approximations to the bosonic self-energy ${\tilde{\Pi }}_{\mathbf{q}\omega }$.

Figure 12

The renormalized triangular vertex ${\Lambda }_{\mathbf{q}\nu \omega }$ in the ladder approximation.

Figure 13

Diagrammatic representation of the Bethe-Salpeter equation for the renormalized vertex function.

Figure 14

$U\text{−}V$ phase diagram in EDMFT and dual boson (DB) with polarization corrections only, at $T=0.02$. The blue line is for the second-order diagrammatic correction [Fig. 11] to the polarization and the green line includes the ladder diagrams [Fig. 11]. Finite-temperature RPA data (dash-dotted line) is shown for comparison. The dashed line corresponds to $V_c=U/z$, where $z=4$ is the coordination number. The value of the bandwidth is marked by the vertical dotted line. Arrows bound the location of the transition according to lattice Monte Carlo results for $U=1$ and $T=0.125$ (see text for details).

Figure 15

Comparison of the momentum dependence (top panels) and frequency dependence (bottom panels) of the polarization in EDMFT (red circles), second-order DB (blue triangles), ladder DB (green diamonds), and RPA (black crosses) for $U=0$ and $V=0.045$ in the immediate vicinity of the charge-ordering transition.

Figure 16

Frequency dependence of the physical polarization ${\Pi }_{\mathbf{q}\omega }$ for fixed momentum $\mathbf{q}=(0,0)$ (left column) and $\mathbf{q}=(\pi ,\pi )$ (right column) in the ladder approximation. We show results for EDMFT (red circles), second-order DB (blue triangles) and ladder DB (green diamonds). The colors are the same as in the phase diagram Fig. 14. For $U=0.5$ and $\mathbf{q}=(\pi ,\pi )$, the polarization in second-order DB turns negative at $\omega =0$.

Figure 17

Momentum dependence of the physical polarization ${\Pi }_{\mathbf{q}\omega }$ for fixed Matsubara frequencies $\omega =0$ (left column) and $\omega =6\pi /\beta$ (right column) in the ladder approximation. In EDMFT, this quantity is a constant.

Figure 18

The diagrammatic correction ${\chi }_{\omega }{\tilde{\Pi }}_{\omega ,\mathbf{q}=(\pi ,\pi )}$ (see text) in second-order approximation, evaluated at the wave vector associated with charge order and shown as a function of Matsubara frequency. With increasing values of $U$, the diagrammatic correction is reduced. Its value is only weakly dependent on $V$.

Figure 19

Second-order DB approximation for the fermionic (b) and bosonic (c) self-energies defined in terms of a common dual functional (a). The open triangles stand for the impurity triangular vertices, straight and wiggly lines denote fully dressed fermionic and bosonic propagators, respectively.

Figure 20

Dual boson ladder approximation constructed from a functional involving the renormalized triangular vertex shown in Fig. 12 (shaded triangle).

Figure 21

Phase diagram for the extended Hubbard model in the plane of on-site interaction $U$ and nearest-neighbor interaction $V$ including fermionic self-energy corrections computed within the approximations depicted in Figs. 19 and 20. The EDMFT and DB data with polarization corrections only (labeled “DB, $\Pi$”) are the same as in Fig. 14 and included for comparison. Black crosses mark ladder DB results with full outer self-consistency.

Figure 22

Momentum dependence of the nonlocal part of physical self-energy $\mathrm{Im}{\Sigma }_{\mathbf{k}\nu }-\mathrm{Im}{\Sigma }_{\nu }^{\text{EDMFT}}$ for fixed Matsubara frequencies $\nu =\pi /\beta$ (left column) and $\nu =7\pi /\beta$ (right column) in the ladder approximation. In EDMFT, this quantity is a constant (equal to zero).

Figure 23

Frequency dependence of the physical self-energy $\mathrm{Im}{\Sigma }_{\mathbf{k}\nu }$ for fixed momentum $\mathbf{q}=(\pi ,0)$ (left column) and $\mathbf{q}=(\pi ,\pi )$ (right column) in the ladder approximation. We show results for EDMFT (red circles), second-order DB (blue triangles), and ladder DB (green diamonds). The colors are the same as in the phase diagram of Fig. 14.

Figure 24

$U\text{−}V$ phase diagram in the DB approximation with and without vertex corrections and from $\text{EDMFT}+GW$ in the $V$-decoupling scheme. The $\text{EDMFT}+GW$ data obtained in the $V$-decoupling scheme are taken from Ref. [34] (for $T=0.01$). Approximations that neglect the fermionic frequency structure of the three-leg vertex deviate strongly from those with vertex corrections and from the $V=U/4$ line, close to where the true phase boundary is expected. For details, see text.

Figure 25

Momentum dependence of the physical polarization ${\Pi }_{\mathbf{q}\omega }$ in s-DB, for fixed Matsubara frequencies $\omega =0$ (left) and $\omega =6\pi /\beta$ (right). In EDMFT, this quantity is a constant. The same quantity in DB is shown in Fig. 17. In the left panel, the plot range is restricted to improve contrast.

Figure 26

Frequency dependence of the physical polarization ${\Pi }_{\mathbf{q}\omega }$ for fixed momentum $\mathbf{q}=(0,0)$ (left) and $\mathbf{q}=(\pi ,\pi )$ (right) for s-DB (blue triangles) and EDMFT (red circles). The colors correspond to those in the phase diagram of Fig. 24. The same quantity in the DB scheme with vertex corrections is shown in Fig. 16.

Figure 27

Momentum dependence of the nonlocal part of physical self-energy $\mathrm{Im}{\Sigma }_{\mathbf{k}\nu }-\mathrm{Im}{\Sigma }_{\nu }^{\text{EDMFT}}$ for fixed Matsubara frequencies $\nu =\pi /\beta$ (left) and $\nu =7\pi /\beta$ (right) in s-DB. In EDMFT, this quantity is a constant. The corresponding plot with DB data is shown in Fig. 22.

Figure 28

Frequency dependence of the physical self-energy ${\Sigma }_{\mathbf{k}\nu }$ for fixed momentum $\mathbf{k}=(\pi ,0)$ (left) and $\mathbf{k}=(\pi ,\pi )$ (right) in the DB approximation without vertex corrections (blue triangles). For comparison we show the same quantity in EDMFT (red circles). The colors correspond to the ones used in the phase diagram of Fig. 24. The corresponding DB data are shown in Fig. 23.

Figure 29

Illustration of how to determine the sign of a diagram. The vertices (left) in a given diagram should symbolically be replaced by interaction lines as indicated on the right. Each closed fermion loop in the resulting diagram contributes a factor $-1$. See Appendix pp2 for details.

Figure 30

Second-order diagram for the dual fermionic self-energy ${\tilde{\Sigma }}_{\mathbf{k}\nu \sigma }$ (a) and second-order (b) and third-order (c) diagrams contributing to the DB self-energy ${\tilde{\Pi }}_{\mathbf{q}\omega }$.

Figure 31

Illustration of how to count the number of closed fermion loops to determine the sign of the diagram in Fig. 30. The symbolic rules of Fig. 29 yield the diagram depicted here. It contains two closed fermion loops, which contribute a factor ${(-1)}^2$.

Figure 32

Diagrammatic representation of the three-leg $g^{\left(3\right)}$ (top panel) and four-leg correlation function $g^{\left(4\right)}$ (bottom panel) and the definition of the vertex functions in terms of them. The figure illustrates the relation between the three-leg and four-leg vertices ${\gamma }^{\left(4\right)}$ and $\lambda$. Wavy lines represent the charge susceptibility and straight lines with arrows denote fully dressed single-particle Green's functions.