Accelerating dynamical mean-field theory convergence by preconditioning with computationally cheaper quantum embedding methods

TL;DR

The paper addresses the high computational cost of dynamical mean-field theory (DMFT) by proposing a preconditioning strategy: initialize the DMFT self-consistency with self-energies from cheaper quantum embedding methods. The key idea is that accurate low-energy self-energy features, on the scale of the non-interacting bandwidth $W$, are enough to place the DMFT loop near its fixed point, enabling substantial reductions in iteration count, sometimes achieving near-DMFT results after a single DMFT step. Through systematic benchmarks on a paramagnetic Hubbard model, the ghost rotationally invariant slave bosons (g-RISB) emerge as the most effective initializer, with $\mathcal{B}=3$ (and $\mathcal{B}=5$ offering further improvements) delivering rapid convergence across metallic and insulating regimes. The approach holds strong potential for accelerating DMFT in large or complex systems, including DFT+DMFT workflows, multi-orbital and cluster extensions, and high-throughput materials screening, while possibly mitigating sign problems in quantum Monte Carlo solvers.

Abstract

Dynamical mean-field theory (DMFT) is a cornerstone technique for studying strongly correlated electronic systems. However, each DMFT step is computationally demanding, and many iterations can be required to achieve convergence. Here, we accelerate the convergence of DMFT by initializing its self-consistent cycle with solutions from computationally cheaper and more approximate methods. We compare the initialization with the non-interacting solution to a range of quantum embedding compatible approaches: Hartree-Fock, the Hubbard-I approximation, rotationally invariant slave bosons (RISB), and its ghost extension (g-RISB). We find that these initializations can reduce the number of DMFT iterations by up to an order of magnitude, with g-RISB providing the most effective and reliable benefits. In most regimes, initializing with g-RISB and performing a single DMFT iteration suffices to recover the full dynamical structure. The improvement in convergence is controlled by the initial solution's accuracy in the low-energy part of the self-energy, on the scale of the non-interacting bandwidth. This strategy is especially effective at the Mott insulator-metal transition, where an initialization from the non-interacting limit can lead to a breakdown of DMFT due to the sign problem. Our results establish the usage of accurate yet cheaper quantum embedding methods as a powerful means to substantially reduce the computational cost of DMFT, particularly in regimes where convergence is slow or prone to failure.

Figures (5)

• Figure 1: Comparison of the self-energies and Weiss fields obtained from converged solutions. When the low-frequency part of the self-energy $\Sigma$ (rows 1 and 2) is accurate, the impurity model defined through the Weiss field $G_0^{\mathrm{imp}}$ (rows 3 and 4) is very close to the converged DMFT one. In contrast, high-frequency errors in $\Sigma$ largely cancel in $G_0^{\mathrm{imp}}$ (cf. Eq. (\ref{['eq:sigma_cancellation']})), as illustrated by the performance of RISB in the metals. Results shown are at fixed chemical potential $\mu$ for the paramagnetic square-lattice Hubbard model ($t'/t = 0.15$ and $U/W=1.875$); the solid black line denotes converged DMFT, and $W$ is the non-interacting bandwidth.
• Figure 2: DMFT convergence as a function of its initialization in four different regimes of fixed chemical potential. A better initialization markedly improves convergence across metallic and insulating regimes, as well as near the Mott phase transition. Among them, g-RISB performs best because it reproduces the low-energy correlation effects with accuracy comparable to DMFT, cf. Fig. \ref{['fig:weiss_sigma_fixed_mu']}. Here we plot the kinetic energy, $E_{\mathrm{kin}}$, potential energy, $E_{\mathrm{pot}}$, quasiparticle weight, $Z$, impurity filling, $n_{\mathrm{imp}}$, and norm difference of the Weiss field between successive iterations, $\mathcal{C}$.
• Figure 3: DMFT convergence as a function of its initialization at two different electronic fillings. Targeting a fixed electron filling $n$ makes to choice convergence more sensitive to the choice of initial solutions, since enforcing charge conservation introduces an additional nonlinear feedback between the chemical potential and the self-energy. The initialization with g-RISB starts DMFT very near its converged solution, so that remaining iterations only adjust minor spectral features (as $\mathcal{C}W$ continues to decrease), keeping the total cost comparable to the fixed–chemical-potential case. Results are shown for metals at half filling ($n=1$, $U/W=1$) and hole-doped from half filling ($n=0.85$, $U/W=1.5$).
• Figure 4: Demonstration that a single DMFT iteration initialized with a computationally cheaper method can closely reproduce the converged DMFT spectral function. Among the tested approaches, g-RISB gives the closest agreement across all correlated regimes. Near the Mott transition all methods deviate from the converged DMFT spectra because of small low-frequency errors in the self-energy that affect the Weiss field (cf. Fig. \ref{['fig:weiss_sigma_fixed_mu']}). The spectral functions $A(\omega)$ at fixed chemical potential $\mu$ are shown before (top: Iteration 0) and after one DMFT step (bottom: Iteration 1), obtained via maximum-entropy analytic continuation.
• Figure 5: Comparison of the spectra before and after the DMFT iteration for the fixed filling cases. In these cases, one-step DMFT only accurately reproduces the full DMFT spectrum when initialized with a correlated method that captures the essential low-energy physics. Cheaper initial methods such as NI, HF, and HI fail to recover the correct quasiparticle and Hubbard-band structure, while RISB is qualitatively correct but quantitatively poor. Initialization with g-RISB yields spectra nearly indistinguishable from the converged DMFT.

Theorems & Definitions (1)

• Lemma 1: Woodbury inversion identity