Low-rank Green's function representations applied to dynamical mean-field theory

Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well as efficient interpolation grids used to recover the representations. In particular, the intermediate representation with sparse sampling and the discrete Lehmann representation (DLR) make use of low rank compression techniques to obtain optimal approximations with controllable accuracy. We consider the use of the DLR in dynamical mean-field theory (DMFT) calculations, and in particular show that the standard full Matsubara frequency grid can be replaced by the compact grid of DLR Matsubara frequency nodes. We test the performance of the method for a DMFT calculation of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ at temperature $50\mathrm{K}$ using a continuous-time quantum Monte Carlo impurity solver, and demonstrate that Matsubara frequency quantities can be represented on a grid of only 36 nodes with no reduction in accuracy, or increase in the number of self-consistent iterations, despite the presence of significant Monte Carlo noise.

Figure 1

The steps of the DMFT loop. The arrows around the formula for $G_{\mathrm{loc}}$ indicate that this quantity is computed self-consistently with the chemical potential to maintain the correct particle density. Our approach improves the efficiency of the DMFT loop by making two simple changes compared with the standard algorithm: (1) All operations in the Matsubara frequency domain are carried out only at the DLR nodes ${\nu }_n={\nu }_{n_k}$, rather than the full Matsubara frequency grid, and (2) the imaginary time hybridization function $\mathrm{\Delta }\left(\tau \right)$ is obtained from the computed values $\mathrm{\Delta }\left(i{\nu }_{n_k}\right)$ by forming a DLR expansion and obtaining its Fourier transform analytically.

Figure 2

Orbital-resolved hybridization function $\mathrm{\Delta }$ and self-energy ${\mathrm{\Sigma }}_{\mathrm{imp}}$ of the three $t_{2g}$ orbitals during the first iteration of the DMFT loop for the ${\text{Sr}}_2{\text{RuO}}_4$ example, demonstrating the use of the DLR procedure. (a) $\mathrm{\Delta }\left(i{\nu }_n\right)$ (from initial guess with zero self-energy) given by Eq. (7). (b) $\mathrm{\Delta }\left(\tau \right)$ obtained using the standard method, i.e., asymptotic expansion and discrete Fourier transform, and DLR interpolation from the values $\mathrm{\Delta }\left(i{\nu }_{n_k}\right)$. (c) ${\mathrm{\Sigma }}_{\mathrm{imp}}\left(i{\nu }_n\right)$ calculated via the Dyson equation after the impurity problem is solved, with the hybridization functions obtained using both methods. To suppress QMC noise, an asymptotic expansion is fit to the tail above ${\nu }_n=2\mathrm{eV}$. In the DLR approach, all Matsubara frequency quantities are obtained only at the DLR nodes $i{\nu }_{n_k}$ (with the parameters ${\omega }_{max}=12\mathrm{eV}$ and $\epsilon ={{10}^{-6}\mathrm{eV}}^{-1}$), which are indicated in (a) and (c).

Figure 3

Convergence of DMFT self-consistency for the ${\text{Sr}}_2{\text{RuO}}_4$ example using standard and DLR procedures, measured via the norm of the difference between $G_{\mathrm{imp}}$ and $G_{\mathrm{loc}}$ in $\tau$ [see Eq. (14)].

Figure 4

Imaginary part of the orbitally-resolved (a) local Green's function, and (b) impurity self-energy in a low-frequency window for the ${\text{Sr}}_2{\text{RuO}}_4$ example, obtained using both the standard and DLR procedures. We show the converged results, obtained independently using the two methods. In the DLR procedure, Matsubara frequency quantities are obtained only at the indicated DLR nodes (with the parameters ${\omega }_{max}=12\text{eV}$ and $\epsilon ={10}^{-6}{\text{eV}}^{-1}$).