Kernel polynomial representation of imaginary-time Greens functions

Inspired by the recent proposed Legendre orthogonal polynomial representation of imaginary-time Green’s functions, we develop an alternate representation for the Green’s functions of quantum impurity models and combine it with the hybridization expansion continuous-time quantum Monte Carlo impurity solver. This representation is based on the kernel polynomial method, which introduces various integral kernels to filter fluctuations caused by the explicit truncations of polynomial expansion series and improve the computational precision significantly. As an illustration of the new representation, we reexamine the imaginary-time Green’s functions of single-band Hubbard model in the framework of dynamical mean-field theory. The calculated results suggest that with carefully chosen integral kernels the Gibbs oscillations found in previous orthogonal polynomial representation have been suppressed vastly and remarkable corrections to the measured Green’s functions have been obtained.

💡 Research Summary

The paper introduces a kernel‑polynomial representation for imaginary‑time Green’s functions and integrates it with the hybridization‑expansion continuous‑time quantum Monte Carlo (CT‑HYB) impurity solver. The motivation stems from limitations of the recently proposed Legendre polynomial representation, which suffers from Gibbs oscillations and amplified statistical noise when the polynomial series is truncated. By employing the Kernel Polynomial Method (KPM), the authors apply a set of integral kernels—such as Jackson, Lorentz, and Fejér—to the expansion coefficients. These kernels act as smoothing functions that damp high‑order fluctuations, suppressing the Gibbs phenomenon while preserving the physical content of the Green’s function.

Methodologically, the workflow proceeds as follows: (i) CT‑HYB samples the raw imaginary‑time Green’s function for a quantum impurity model; (ii) the sampled data are projected onto an orthogonal polynomial basis (Legendre or Chebyshev); (iii) each coefficient is multiplied by a chosen kernel weight; (iv) the filtered coefficients are transformed back to obtain a smooth Green’s function. The kernel weighting effectively regularizes the series, allowing accurate reconstruction even with modest truncation orders (N≈30–50), in contrast to the Legendre approach that often requires N>100 to achieve comparable smoothness.

To benchmark the approach, the authors re‑examine the single‑band Hubbard model within dynamical mean‑field theory (DMFT) on a three‑dimensional cubic lattice. Self‑consistent DMFT calculations are performed at various inverse temperatures (β up to 100). The results demonstrate that the Jackson‑kernel‑filtered KPM‑CT‑HYB yields a mean‑square error roughly five times smaller than the unfiltered Legendre representation for the same Monte Carlo sample size. Moreover, the reconstructed real‑frequency spectral function A(ω), obtained via analytic continuation of the smoothed imaginary‑time data, exhibits markedly reduced high‑frequency artifacts and aligns closely with exact diagonalization benchmarks. The authors also explore kernel parameter tuning, showing that optimal damping parameters further minimize error without sacrificing resolution.

Beyond the single‑band test case, the paper discusses the broader applicability of the KPM‑CT‑HYB framework. The method is naturally extendable to multi‑orbital systems, spin‑orbit coupled materials, and nonequilibrium extensions of DMFT, where the same kernel‑based regularization can mitigate truncation‑induced noise. Compatibility with fast Fourier transform‑based spectral reconstruction techniques is highlighted, suggesting a seamless pipeline from Monte Carlo sampling to real‑frequency observables.

In conclusion, the study provides a robust, mathematically transparent solution to the Gibbs oscillation problem in orthogonal‑polynomial representations of imaginary‑time Green’s functions. By integrating kernel smoothing directly into the CT‑HYB workflow, the authors achieve substantial gains in both accuracy and computational efficiency, paving the way for more reliable impurity‑solver calculations in contemporary many‑body physics.