Dual Bethe-Salpeter equation for the multiorbital lattice susceptibility within dynamical mean-field theory

Dynamical mean-field theory describes the impact of strong local correlation effects in many-electron systems. While the single-particle spectral function is directly obtained within the formalism, two-particle susceptibilities can also be obtained by solving the Bethe-Salpeter equation. The solution requires handling infinite matrices in Matsubara frequency space. This is commonly treated using a finite frequency cutoff, resulting in slow linear convergence. A decomposition of the two-particle response in local and nonlocal contributions enables a reformulation of the Bethe-Salpeter equation inspired by the dual boson formalism. The reformulation has a drastically improved cubic convergence with respect to the frequency cutoff, considerably facilitating the calculation of susceptibilities in multi-orbital systems. This improved convergence arises from the fact that local contributions can be measured in the impurity solver. The dual Bethe-Salpeter equation uses the fully reducible vertex which is free from vertex divergences. We benchmark the approach on several systems including the spin susceptibility of strontium ruthenate ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, a strongly correlated Hund's metal with three active orbitals.

Figure 1

Generalized susceptibility $\chi$ (top), schematic scattering experiment (bottom).

Figure 2

Diagrammatic representation of elements of the dual Bethe-Salpeter equation. At the top, the orbital and frequency definition of $L$ is illustrated. In our definition, $L(\omega ,\nu )$ implies that the incoming electron emits energy $\omega$ to the bosonic degree of freedom. At the bottom, the nonlocal contribution to the susceptibility is drawn. In this process, energy $\omega$ and momentum $q$ comes in from the left, is transferred to an electron-hole pair that propagates nonlocally ($\tilde{\chi }$) and is then emitted. In the triangle on the left, since the electron absorbs energy, the corresponding argument of the first $L$ in Eq. (23) is $-\omega$.

Figure 3

Flowchart summarizing the calculation. The blue boxes are local two-particle correlation functions extracted from the impurity solver. For clarity, orbital labels are suppressed here.

Figure 4

Magnetic susceptibility at $\mathbf{q}=0, \omega =0$ in a two-orbital square lattice tight-binding model ($t=1, U=0, \mu \approx -1.84$) as a function of the number of frequencies $N_{\nu }$. The dual Bethe-Salpeter equation (DBSE) converges as $1/N_{\nu }^3$, whereas the usual Bethe-Salpeter equation (BSE) converges only as $1/N_{\nu }$, as is visible in the log-log plot on the right.

Figure 5

Static magnetic susceptibility $\chi (\mathbf{0},0)$ ($\mathbf{q}=0, \omega =0$) in the square lattice Hubbard model as a function of the number of frequencies $N_{\nu }$. The dual BSE result (DBSE) (orange) is compared with the standard BSE (blue) and the linear response result from self-consistent DMFT in an applied field extrapolated to zero field (DMFT $+$ Field) (green). (a) Infinite-frequency extrapolation (open markers) of calculations for several values of $N_{\nu }$ (filled markers) with $1/N_{\nu }^3$ frequency scaling on the $x$ axis showing the DBSE asymptotic $1/N_{\nu }^3$ scaling. (b) Same as panel (a) but with $1/N_{\nu }$ scaling on the $x$ axis showing the BSE asymptotic $1/N_{\nu }$ scaling. (c) Magnetization $M$ curve in applied magnetic field $B$ from self-consistent DMFT, used for zero-field extrapolation (green).

Figure 6

(upper panel) Static spin susceptibility ${\chi }_{S_z{S}_z}\left(\mathbf{q}\right)$ of strontium ruthenate ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ at $T=464$ K along the pseudotetragonal high-symmetry path ($\mathrm{\Gamma }\text{−}X\text{−}M\text{−}\mathrm{\Gamma }$). Results for $N_{\nu }=5$, 10, 20, and 40 fermionic frequencies are shown for the DMFT $+$ BSE method (dashed lines) and our dual BSE method (DMFT $+$ DBSE) (solid lines). (lower panel) Convergence close to the incommensurate peak at $\mathbf{q}=(1/3,1/3,0)$ as a function of $1/N_{\nu }$ for $3\le N_{\nu }\le 40$. Extrapolations to the infinite frequency limit $N_{\nu }\to \infty$ are also shown (lines), using the $N_{\nu }^{-3}$ scaling (solid line) for the DMFT $+$ DBSE results (circles) and the $N_{\nu }^{-1}$ scaling (dashed line) for the DMFT $+$ BSE results (squares). As an independent reference also the zero-field extrapolated self-consistent DMFT result from Ref. [35] is shown (magenta diamond marker).

Figure 7

Magnetic susceptibility of ${\mathrm{SrVO}}_3$. (upper panel) Susceptibility along a path between high-symmetry points, the solid lines are the DBSE and the dashed lines are the BSE. These calculations are performed with the Hamiltonian of Galler et al. [29], where the susceptibility was also calculated using the BSE (cf. their Fig. B.19), although their code is also capable of using the DBSE. (lower panel) Fermionic frequency convergence of the BSE and DBSE.