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The Truncated Hankel Correlator Method

Johann Ostmeyer, Carsten Urbach

TL;DR

The Truncated Hankel Correlator (THC) method presents an algebraic, Hankel-based approach to extract spectral information from Euclidean correlators in the presence of noise. By constructing a maximal Hankel matrix from $C(t)$, applying an Omega-weighted low-rank approximation, and solving a Prony-type generalized eigenproblem, THC yields energies $E_l$ and amplitudes with near-optimal $\ ext{chi}^2$ control and compatibility with matrix-valued correlators. The approach unifies and subsumes many existing techniques (Prony, Lanczos, GEVP) as special cases and provides rigorous optimality statements for the low-rank approximation, while remaining robust to noise and requiring limited human oversight. Demonstrations on synthetic data and lattice QCD (pion, $\omega$ meson, and nucleon) show accurate ground and excited-state spectra, with symmetry properties preserved for time-symmetric data, underscoring THC's practical utility for hadron spectroscopy and related spectral analyses.

Abstract

We introduce a new method to approximate Euclidean correlation functions by exponential sums. The Truncated Hankel Correlator (THC) method builds a Hankel matrix from the full correlator data available and truncates the eigenspectrum of said Hankel matrix. It proceeds by applying the Prony generalised eigenvalue method to the thus obtained low-rank approximation. A large number of algebraic correlator analysis methods including (block) Prony (and equivalently (block) Lanczos) and the generalised eigenvalue problem (GEVP) can be reproduced as sub-optimal special cases of the THC method. Weights, for instance inverse square errors, can be included in the analysis, so that the result has a close to optimal $χ^2$-value. This makes the THC method very similar in spirit to a closed form solution to multi-state fits, naturally including the case of matrix-valued correlators. We show that, in general, finding approximations better than those provided by the THC method is exponentially hard in the number of exponentials. Moreover, the THC method is robust against noise and requires comparably little human oversight. Finally, when applied to symmetric data, the obtained energy spectrum is guaranteed to be symmetric up to machine precision.

The Truncated Hankel Correlator Method

TL;DR

The Truncated Hankel Correlator (THC) method presents an algebraic, Hankel-based approach to extract spectral information from Euclidean correlators in the presence of noise. By constructing a maximal Hankel matrix from , applying an Omega-weighted low-rank approximation, and solving a Prony-type generalized eigenproblem, THC yields energies and amplitudes with near-optimal control and compatibility with matrix-valued correlators. The approach unifies and subsumes many existing techniques (Prony, Lanczos, GEVP) as special cases and provides rigorous optimality statements for the low-rank approximation, while remaining robust to noise and requiring limited human oversight. Demonstrations on synthetic data and lattice QCD (pion, meson, and nucleon) show accurate ground and excited-state spectra, with symmetry properties preserved for time-symmetric data, underscoring THC's practical utility for hadron spectroscopy and related spectral analyses.

Abstract

We introduce a new method to approximate Euclidean correlation functions by exponential sums. The Truncated Hankel Correlator (THC) method builds a Hankel matrix from the full correlator data available and truncates the eigenspectrum of said Hankel matrix. It proceeds by applying the Prony generalised eigenvalue method to the thus obtained low-rank approximation. A large number of algebraic correlator analysis methods including (block) Prony (and equivalently (block) Lanczos) and the generalised eigenvalue problem (GEVP) can be reproduced as sub-optimal special cases of the THC method. Weights, for instance inverse square errors, can be included in the analysis, so that the result has a close to optimal -value. This makes the THC method very similar in spirit to a closed form solution to multi-state fits, naturally including the case of matrix-valued correlators. We show that, in general, finding approximations better than those provided by the THC method is exponentially hard in the number of exponentials. Moreover, the THC method is robust against noise and requires comparably little human oversight. Finally, when applied to symmetric data, the obtained energy spectrum is guaranteed to be symmetric up to machine precision.
Paper Structure (22 sections, 3 theorems, 61 equations, 13 figures)

This paper contains 22 sections, 3 theorems, 61 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Derivation and theory
  3. The Hankel matrix
  4. The truncation
  5. Reconstructing coefficients
  6. Including weights
  7. Optimality and computational complexity
  8. Counting the degrees of freedom
  9. Relations to other methods
  10. Practical considerations
  11. Choosing the truncation $k$
  12. Adjusting the weights $\Omega$
  13. Filtering for physical states
  14. Interpreting the results
  15. Numerical Experiments
  16. ...and 7 more sections

Key Result

theorem 1

Given the matrix-valued correlator data $C(t)$, the weight matrix $\Omega$ and the truncation $k$, then the approximation $\hat{H}_k$ as in eq. eq:omega-approx has minimal residuum with $\hat{H}$ as in eq. eq:maximal_hankel.

Figures (13)

  • Figure 1: Modulus of the eigenvalues $s_i$ of the full Hankel matrix $\hat{H}$ defined by eq. \ref{['eq:maximal_hankel']} built from the synthetic correlator \ref{['eq:syn']} with $T=48$. Negative eigenvalues are marked red.
  • Figure 2: Comparison of Lanczos, PGEVM with $\delta t=1, \Delta=1, t_0=0$, THC using eq. \ref{['eq:get_X_stable']} with uniform weights and $T=48$, and effective mass for artificial data, see Eq. \ref{['eq:syn']}. All but the Lanczos data is shifted slightly in $x$-direction for legibility. Left: convergence of the ground state energy level as a function of $n$, $k$, or $t$, respectively. The exact value is indicated by the dashed line. Right: the difference to the exact ground state energy is plotted on a log-scale as a function of $n$ or $k$. Empty symbols in the right panel indicate negative differences.
  • Figure 3: Comparison of Lanczos, PGEVM with $\delta t=1, \Delta=1, t_0=0$, and THC using eq. \ref{['eq:get_X_sym']} with uniform weights and $T=48$, and cosh-effective mass for symmetric artificial data, see Eq. \ref{['eq:syn_sym']}. All but the Lanczos data is shifted slightly in $x$-direction for legibility. Convergence of the ground state energy level as a function of $n$, $k$, or $t$, respectively. The exact values (positive and negative) are indicated by the dashed lines. Empty symbols indicate negative values.
  • Figure 4: Modulus of the eigenvalues $s_i$ of the full Hankel matrix $\tilde{H}$ defined by eq. \ref{['eq:weighted_full_Hankel']} built from the pion correlator starting at $t_0=1$ and using the default weights \ref{['eq:diag_weight_mat']}. Negative eigenvalues are marked red. Natural candidates for the truncation $k$ are $k=6$ at the largest ratio $\left|\frac{s_{k}}{s_{k+1}}\right|$ (i.e. the largest gap on the log-scale) and $k=9$ right before the first negative eigenvalue $s_{k+1}\le 0$.
  • Figure 5: Pion correlator data together with the reconstruction from the THC method (red lines) using eq. \ref{['eq:get_X_sym']} and the truncation $k=6$. Error bands on the fitting curves are plotted but not visible at this scale. The elements $C^\pi_{21}=C^\pi_{12}$ and $C^\pi_{22}$ are shifted by a factor $3$ and $1/3$, respectively, for better legibility.
  • ...and 8 more figures

Theorems & Definitions (6)

  • theorem 1: Ref. gillard:2023, Theorem 2.2 with $Q=R=\Omega^\dagger \Omega$
  • remark \@thmcounterremark
  • corollary \@thmcountercorollary: Ref. gillard:2023, eq. (6)
  • remark \@thmcounterremark
  • theorem 2: Ref. ottaviani2014exact, Corollary 3.9
  • remark \@thmcounterremark