Decomposing imaginary time Feynman diagrams using separable basis functions: Anderson impurity model strong coupling expansion

TL;DR

A self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca$_2$RuO$_4, demonstrating the promise of the method for modeling realistic strongly correlated multi-band materials.

Abstract

We present a deterministic algorithm for the efficient evaluation of imaginary time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary time Green's functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an $M$th-order diagram at inverse temperature $β$ and spectral width $ω_{\max}$ from $\mathcal{O}((βω_{\max})^{2M-1})$ for a direct quadrature to $\mathcal{O}(M (\log (βω_{\max}))^{M+1})$, with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multi-band impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca$_2$RuO$_4$, demonstrating the promise of the method for modeling realistic strongly correlated multi-band materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust black-box evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams.

Figures (5)

• Figure 1: Single-particle Green's function for the spinless fermion dimer model \ref{['eq:fermion_dimer']}, at inverse temperatures $\beta t = 2$, $16$, $128$, $1024$ (columns), and increasing expansion orders. The exact diagonal (first row) and off-diagonal (second row) Green's functions obtained from exact diagonalization (ED) are quantitatively described by the first-order approximation (O1). The diagonal (third row) and off-diagonal (last row) Green's function error decreases when increasing to second (O2) and third (O3) order.
• Figure 2: Single-particle Green's function for the two-band $e_g$ model with a discrete bath. The left column shows the diagonal Green's function $G_{00}(\tau)$, and the right column shows the off-diagonal component $G_{01}(\tau)$, with decreasing temperatures $\beta t = 2$, $16$, $128$, and $1024$ along the rows. The strong coupling results converge to the exact diagonalization (ED) solution as the expansion order increases.
• Figure 3: Single-particle Green's function for the two-band $e_g$ model with a metallic bath at inverse temperature $\beta t = 8$. The diagonal part G$_{00}$ is shown in the left panel (a), and the off-diagonal part G$_{01}$ in the right panel (b). The strong coupling expansion converges towards the result produced by the continuous time hybridization expansion method (CT-HYB) and the inchworm quantum Monte Carlo (IW) results from Ref. eidelstein2020, but in this case diagrams beyond third-order contribute significantly.
• Figure 4: (a) Diagonal component of the single-particle Green's function, with increasing expansion orders, for the three-band effective model of Ca$_2$RuO$_4$. G$_{xz \, \sigma, xz \, \sigma }(\tau)$ and $G_{yz \, \sigma, yz \, \sigma}(\tau)$ are degenerate and close to half-filling (solid lines), while $G_{xy \, \sigma, xy \, \sigma}(\tau)$ is close to unity-filling (dashed lines). (b) Imaginary part of the off-diagonal component $G_{xz \, \sigma, yz \, \sigma}(\tau)$. (c) Real part of the off-diagonal component $G_{xz \, \sigma, xy \bar{\sigma}}(\tau)$ generated by the local spin-orbit coupling. Calculations were performed at inverse temperature $\beta=10\,$eV$^{-1}$.
• Figure 5: Fourth-order self-energy diagrams for the weak coupling expansion of the Anderson impurity problem assuming a paramagnetic phase at half filling. Lines represent the electronic propagator $\mathcal{G}_0~(G)$ for the bare (bold) expansion, and dots correspond to the interaction vertices $U$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4