Spectral reconstruction from Euclidean lattice correlators through singular value decomposition

Abstract

Reconstructing spectral densities from Euclidean lattice correlators requires an inverse Laplace transform, which is inherently ill-conditioned when applied to numerical data with statistical uncertainties. The maximum amount of information that can be extracted from the imaginary-time dependence of correlators can be characterized by the singular value decomposition (SVD) of the kernel function $\exp(-ωt)$ defined on discrete sets of imaginary times $t$ and energies $ω$. The SVD provides orthogonal basis functions in both the $t$- and $ω$-spaces, while the singular values determine the magnitude of their contributions to the correlators. By retaining only the components associated with the largest singular values, for which the correlator data remain statistically significant, one can reconstruct smeared spectral functions with controlled uncertainties. The systematic error arising from the truncation can also be bounded under reasonable assumptions. In the limit where the ranges of $t$ and $ω$ become infinitely large and continuous, the SVD basis approaches the Mellin transform, allowing a representation of the smeared spectrum that is independent of the details of the lattice parameters.