Unbiased Krylov subspace method for the extraction of ground state from lattice correlators

TL;DR

This work tackles the challenge of extracting ground-state properties from noisy lattice QCD correlators by diagonalizing the transfer matrix within Krylov subspaces while mitigating bias from truncation. It introduces a two-step approach: (1) a low-rank SVD of the correlator matrix to filter spurious eigenvalues, and (2) an eigenvalue-variance extrapolation to remove residual truncation bias, enabling unbiased estimates of ground-state energies $E_0$ and matrix elements $J_{00}$ without relying on exponential fits. The method is validated on noiseless and noisy mock data, where it accurately reproduces $E_0$ and $J_{00}$, and on realistic $K$ and $D_s$ meson data from JLQCD, yielding results consistent with standard plateau and summation analyses. Overall, the approach provides a data-driven, robust framework for assessing ground-state saturation and obtaining reliable, uncertainty-quantified results in lattice QCD analyses.

Abstract

Ground-state energy and matrix element are reconstructed from correlators in lattice QCD by diagonalizing transfer matrix $\hat{T}$ within the Krylov subspace spanned by $\hat{T}^n|χ\rangle$, where $|χ\rangle$ is a state generated by an interpolating field on the lattice. In numerical applications, this strategy is spoiled by statistical noise. To circumvent the problem, we introduce a low-rank approximation based on a singular-value decomposition of a matrix made of the correlators. The associated bias is eliminated by an extrapolation to the limit of vanishing variance of energy eigenvalue. The strategy is tested using a set of mock data as well as real data of $K$ and $D_s$ meson correlators.

Figures (18)

• Figure 1: Normalized singular values $\sigma_r^{(m)}/\sigma_0^{(m)}$ of $C^{+(m)}_{ij}$ for the noiseless two-point correlation functions. They are plotted in decreasing order of singular values, for each choice of $m$ denoted in the label.
• Figure 2: Estimated ground-state mass $E_0'^{(m)}=-\ln\lambda_0'^{(m)}$ for various $m$ and $r$. The horizontal axis represents the matrix size $m+1$. The black dashed line shows the exact mass, which is an input when generating the mock data.
• Figure 3: Largest eigenvalue $\lambda_0'^{(m)}$ plotted against the eigenvalue-variance $\Delta\lambda^{(m)}_0$. The base Krylov subspace is for $m=10$. Each point corresponds to $r=0,\cdots,5$ from right to left. The points for $r=4$ and 5 are nearly degenerate and only visible in the inset. The violet plus represents the exact mass.
• Figure 4: Ground-state matrix element $J_{00}'^{(8)}$ for three types of the three-point correlation function. The plot shows the dependence on the "variance" $\Delta \lambda^{(8)}_0$. From right to left, the points correspond to $=0,\cdots,5$. Results with small $\Delta \lambda^{(m)}_0$ are also plotted in the inset. The violet plus represents the input value $J_{00}=1$.
• Figure 5: Same as Fig. \ref{['fig:svNoiseless']} but with noisy data sets. Each symbol represents a result for a given bootstrap sample; there are 500 of them nearly overlapping in the plot.