Application of Laplace filters to the analysis of lattice time correlators

TL;DR

The paper introduces regulated Laplace filters as invertible high-pass transforms to tackle two persistent lattice QCD data challenges: ill-conditioned time covariance matrices and excited-state contamination in spectral analyses. By applying a regulator $\lambda$, these filters decorrelate time data and weight the spectrum via $(\lambda^2-E^2)$, enabling both conditioning improvements and targeted suppression of unwanted states. Empirical results across representative RBC/UKQCD datasets show large reductions in the correlation dynamic range, especially when combined with downsampling, and demonstrate practical benefits for extracting ground-state energies and matrix elements in mesonic two-point and three-point functions, as well as neutral-m meson mixing. The approach is simple to implement, complementary to existing regularization strategies, and holds promise for robust spectrum extraction and expanded application to finite-temperature and multi-scale lattice analyses.

Abstract

The analysis of lattice simulation correlation function data is notoriously hindered by the ill-conditioning of the Euclidean time covariance matrix. Additionally, the isolation of a single physical state in such functions is generally affected by systematic contamination from unwanted states. In this paper, we present a new methodology based on regulated Laplace filters and demonstrate that it can be used to address both issues using state-of-the-art simulation data. Regulated Laplace filters are invertible high-pass filters that suppress local correlations in the data, and we show that they can reduce the condition number of covariance matrices by several orders of magnitude. Furthermore, Laplace filters can annihilate functions that decay exponentially with time, which can be used to alter the spectrum of a lattice correlation function. We show that this property can be exploited to significantly reduce excited-state contamination in the determination of matrix elements. The same property can also be used to constrain the spectral content of a correlation function and has the potential to form the basis of new methods to extract physical information from lattice data.

Figures (12)

• Figure 1: CDR of the filtered correlators as a function of the regulator $\lambda$. The left and right panels show the results for the M0M-mes-ll and F1M-mes-lh correlators, respectively. In each plot, the solid blue and orange curves represent the CDR for the full time range and for the restricted range $\{N_t/8,\ldots,3N_t/8-1\}$, respectively. Additionally, the colored dashed lines represent the CDRs of the unfiltered data, and the black dashed line represents the CDR of the restricted range evaluated at the optimal value of the regulator, with values of approximately $\lambda\simeq0.11$ for M0M-mes-ll and $\lambda\simeq0.88$ for F1M-mes-lh. In the F1M-mes-lh plot, the dotted line indicates the critical 156.54 threshold above which the CDR cannot be resolved using double-precision arithmetic (cf. \ref{['sec:cdr']}).
• Figure 2: Correlation matrix of the M0M-mes-ll time correlator, in both the unfiltered (left panel) and filtered (right panel) cases. For the filtered case, the optimal value $\lambda\simeq0.11$ is used (cf. \ref{['fig:optim']}). The titles of the plots provide the CDRs of the full and restricted matrices, using the same time range as in \ref{['fig:optim']}. Finally, the restricted range is represented as a black square in the figure.
• Figure 3: Representation of the correlation matrix of the F1M-mes-lh time correlator, in both the unfiltered (left panel) and filtered (right panel) cases. In the filtered case, the optimal value $\lambda\simeq0.88$ is used (cf. \ref{['fig:optim']}). Other details are as in \ref{['fig:M0ll-corr']}.
• Figure 4: Laplace filter analysis and optimization when using an additional downsampling step after filtering, as explained in \ref{['sec:ds']}. The two rows of plots are for the analysis of the M0M-mes-ll and F1M-mes-lh correlators, respectively. Plots on the left represent the dependence of the total and restricted CDR on the filter regulator $\lambda$, using identical conventions to \ref{['fig:optim']}. Plots on the right represent the correlation matrix $\Gamma^{\downarrow}_{\lambda}$, for the optimal value $\lambda=\lambda_0$, which minimizes the restricted downsampled CDR (i.e., the orange curve on the associated plot on the left). The variables $t_1$ and $t_2$ are time indices of the downsampled correlator; that is, they correspond to $2t_1$ and $2t_2$ for the original data, respectively. As in \ref{['fig:M0ll-corr', 'fig:F1Mhl-corr']}, the sub-matrix within the black square is the correlation matrix on the restricted range.
• Figure 5: Behavior of the eigenvalues $\sigma_i$ of the correlation matrix of the M0M-mes-ll (upper half) and F1M-mes-lh (lower half) correlation functions as a function of Laplace filtering regulator $\lambda$. In each half, the top and bottom rows show the spectrum for the full and restricted time ranges, respectively. In the right column, an additional downsampling step is applied. The faint horizontal lines correspond to the spectrum when no filtering is applied.