Multi-orbital dynamical mean-field theory with a complex-time solver

TL;DR

The paper tackles the challenge of obtaining high-resolution real-frequency spectra in DMFT, especially for multi-orbital systems, where imaginary-time analytic continuation or real-frequency solvers are costly. It couples a complex-time tensor-network impurity solver with an ESPRIT-based exponential-fitting continuation to reconstruct $G^{\\mathrm{R}}(\\omega)$ from complex-time data along a contour at fixed angle $\\alpha$. Benchmark results on the single- and multi-orbital DMFT problems show accurate spectral functions and self-energies at reduced bond dimensions and computation time, successfully capturing features such as the metal–insulator transition and Kanamori multiplets without excessive broadening. This approach offers a practical, efficient route for ab initio studies of strongly correlated materials, enabling reliable, high-fidelity spectral calculations across challenging multi-orbital regimes.

Abstract

We present the combination of a complex-time tensor-network impurity solver with an analytic continuation scheme based on exponential fitting as an efficient framework for single and multi-orbital dynamical mean-field calculations. By performing time-evolution along a complex-time contour, the approach balances computational cost with the difficulty of spectral recovery, offering greater flexibility than methods confined to the real or imaginary axis. By complementing the complex-time evolution with an exponential fitting scheme, we faithfully extract real-time information at negligible cost. The resulting method obtains high-resolution spectra at a significantly lower computational cost than real-time evolution, offering a promising tool for ab initio studies of strongly correlated materials.

Figures (11)

• Figure 1: Fixed-angle contours parameterized by $t\geq 0$ and $\alpha \in [0,\pi/2]$.
• Figure 2: Complex and real-time evolution results at $U=2D$ and $N_b=199$. Panels (a) and (b): Real and imaginary parts of the ESPRIT fitted complex-time Green's function with $\alpha=0.2$. Panels (c) and (d): Comparison of the real and imaginary parts of the real-time Green’s function obtained by direct real-time evolution and by analytic continuation from complex-time data. The reference curve is computed by real-time evolution with $\chi=1500$, all other results use $\chi=80$.
• Figure 3: Comparison of spectral functions obtained from different complex angles at $U=2D$, $N_b=199$ and $D t_\text{max}=80$. (a) $\alpha=0.0$, (b) $\alpha=0.2$, (c) $\alpha=0.4$. The $\alpha=0.0$ results are computed by direct Fourier transform, while the others are obtained by ESPRIT fitting. The reference curve is the direct Fourier transform results of $\alpha=0.0$, $N_b=199$, $D t_\text{max}=100$ and $\chi=1500$.
• Figure 4: Self-energies computed with $N_b=399$ and $\chi=80$. Panels (a) and (b): Comparison of results extracted from complex-time evolution with $\alpha=0.4$ and different end times at $U=2D$ and $3D$. Panel (c): Comparison of results obtained from real (dashed lines) and complex (solid lines) time evolutions with $\alpha=0.4$.
• Figure 5: Comparison of real (dashed lines) and complex (solid lines) time single-orbital DMFT simulation results at $U=2D$, $N_b=179$, and $D t_\text{max}=60$. Results are shown for $\mu/D =-1.00$ ($n=1$, blue lines) with $\chi=30$ and $\mu/D=-0.25$ ($n \approx 0.72$, orange lines) with $\chi=40$. Complex-time simulations are performed with $\alpha=0.2$. Panel (a): Spectral functions. Panels (b) and (c): Imaginary part of the self-energy.