Spectral extraction and variational methods

Prerequisites: glueball spectroscopy and operator overlap.

You will learn to derive an ideal generalized eigenvalue problem, explain the role of the operator basis, and recognize fit-window, finite-volume and excited-state limitations.

Several operators probe one spectrum

Choose operators $O_i$ with the same momentum and symmetry channel, and subtract vacuum expectations when appropriate. Under a positive spectral representation, their connected correlator matrix is

$$C_{ij}(t)=\sum_n Z_{in}\overline{Z_{jn}}e^{-E_nt}, \qquad Z_{in}=\langle\Omega|O_i|n\rangle.$$

This notation assumes a discrete finite-volume spectrum for the teaching example, vacuum-subtracted energies, and negligible effects from finite temporal extent. With periodic temporal extent $T$, backward propagation can matter; a single forward exponential is then not automatically an adequate fit.

Changing operators changes $Z$, not the underlying energies. A useful basis is one that distinguishes the relevant states and has manageable statistical noise. A larger but nearly linearly dependent basis can create an ill-conditioned matrix.

Work the ideal two-state problem exactly

Let the two overlap vectors be the columns of

$$Z=\begin{pmatrix}1&1\\1&-1\end{pmatrix}, \qquad D(t)=\operatorname{diag}(e^{-E_1t},e^{-E_2t}), \qquad C(t)=ZD(t)Z^\dagger.$$

Explicitly, the diagonal entries are $e^{-E_1t}+e^{-E_2t}$ and the off-diagonal entries are $e^{-E_1t}-e^{-E_2t}$. The combinations $(1,1)$ and $(1,-1)$ separate the two energies.

More generally, if a square overlap matrix $Z$ is invertible and exactly $N$ states contribute to $N$ operators, solve

$$C(t)v_n=\lambda_n(t,t_0)C(t_0)v_n,\qquad t>t_0.$$

Multiply by $Z^{-1}$ and set $u_n=Z^\dagger v_n$. Then

$$D(t)u_n=\lambda_nD(t_0)u_n, \qquad \lambda_n=e^{-E_n(t-t_0)}.$$

The result is independent of the mixing in $Z$ because the basis spans the assumed finite state space. This algebra is the reason the method can separate states that contaminate a single correlator.

What changes in an actual calculation

A finite operator basis generally overlaps with more than $N$ states. Its eigenvalues are no longer exact single exponentials. Time choices, overlap structure, the next omitted energy and statistical errors matter. Blossier et al., arXiv:0902.1265v2, §2.1, equations (2.3)–(2.10) sets out the ideal problem; §2.2 studies corrections under its stated assumptions.

A practical effective energy uses

$$E_n^{\mathrm{eff}}(t,t_0)= \frac1a\log\frac{\lambda_n(t,t_0)}{\lambda_n(t+a,t_0)}.$$

If $t$ is recorded as an integer step, report the result first as $aE_n^{\mathrm{eff}}$. Do not remove $a^{-1}$ while still labelling the axis in physical energy units.

At early times omitted high-energy contributions can matter. At late times noise may dominate. Selecting a plateau requires a fitting analysis, not only a visually flat line. Correlated data, resampling, stability across admissible windows and sensitivity to the basis belong in the uncertainty assessment.

A channel contains more than a particle name

The chosen symmetry channel may contain multiparticle states or finite-volume winding states. Enlarging the basis to include relevant operators can expose states absent from a narrow single-loop basis. Volume dependence and continuum symmetry restoration help interpret the levels, but neither is replaced by numerical stability of the eigenvalue solver.

The glueball quantum-number page explains the relation between cubic irreducible representations and continuum spin. Athenodorou–Teper, arXiv:2106.00364v3, §§2.2–2.3 and 5.8 is a concrete source for spectroscopy methods and scattering-state caveats in pure $SU(N)$ calculations.

Exercise: discover a rank failure

Let $O_2=2O_1$ exactly. What happens to the two-by-two correlator matrix and the proposed two-state extraction? Would adding more measurements fix the mathematical problem?

Solution. The second row is twice the first and the second column twice the first, so the matrix has rank at most one. $C(t_0)$ is singular and cannot resolve two independent directions. More measurements can reduce noise but cannot create a missing direction in the operator space. One needs an independent operator or an explicit restriction to the non-null subspace.

For a second diagnostic, suppose all operators have zero overlap with a lighter state. Even a perfectly solved matrix problem returns only visible states. It cannot certify that the physical spectrum has no lower level elsewhere.

Use the method with its scope intact

The calculation above is an exact finite-state identity. Real variational spectra are numerical evidence with a finite basis, finite statistics, regulator and extrapolation choices. Continue to continuum scaling and the research benchmarks to see what must accompany an exported mass estimate.