Operator Bases, Effective Masses, and Excited-State Control

A finite operator basis resolves only those spectral combinations to which it has linearly independent overlap. Effective-energy plateaus are useful diagnostics, but level isolation requires a basis with the correct lattice quantum numbers, a well-conditioned correlation matrix, a generalized eigenvalue or equivalent multi-state analysis, and stability under changes of basis, reference time, fit window, covariance treatment, and spectral model. A level label is credible only when these tests agree.

Required background. Euclidean Correlators and Spectral Information derives the finite-volume spectral matrix and boundary terms used here.

Helpful background. Normal Forms, Spectra, and Projectors reviews matrix spectral structure. Spectral Decomposition of Two-Point Functions supplies the Hilbert-space interpretation.

Operator bases define what can be resolved

For $N$ interpolating operators in one exact lattice-symmetry sector,

$$C_{ij}(t)=\sum_{n=0}^\infty Z_i^{(n)}Z_j^{(n)*}e^{-E_nt}$$

in the zero-temperature forward-propagation limit. Collect the first $r$ overlap vectors into the $N\times r$ matrix $Z$. If $\operatorname{rank}Z<r$, no analysis can recover all $r$ levels from that basis, even with noiseless data. More time slices do not create missing operator-space directions.

Operators should differ in physical structure—spatial profile, derivative content, multi-particle construction, or displacement—not merely by overall normalization. Smearing changes overlaps and sometimes the regulator-level operator definition; its kernel, radius in physical units, iteration count, gauge transport, and ensemble dependence must be declared.

Basis and normalization conventions. Operators are projected into exact lattice irreducible representations and momenta before the matrix is formed. This page absorbs optional $1/(2E_n)$ factors into $Z_i^{(n)}$ and uses the site-wide conventions. Hermitian conjugation, reflection parity, smearing transformations, and vacuum subtraction remain local data.

At each positive time, an exact reflection-positive correlation matrix is Hermitian positive semidefinite. Estimated matrices can acquire small negative eigenvalues from noise. Large or persistent violations require investigation; blindly clipping them changes the likelihood and can create artificial stability.

Generalized eigenvalue method

Choose a reference time $t_0$ for which $C(t_0)$ is positive definite on the retained numerical subspace. Solve

$$C(t)v_n(t,t_0) =\lambda_n(t,t_0)C(t_0)v_n(t,t_0), \qquad v_m^\dagger C(t_0)v_n=\delta_{mn}.$$

In an exactly $N$-state, full-rank model,

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

The correlation-matrix variational construction was introduced into lattice spectroscopy by Michael 1985, pp. 58–76. Higher states generate corrections whose suppression depends on the gaps, $t$, $t_0$, and the particular estimator. Choosing $t_0$ too early leaves large omitted-state effects; choosing it too late makes $C(t_0)$ noisy and ill-conditioned. The asymptotic correction bounds and favorable choices such as $t_0\gtrsim t/2$ are derived under explicit spectral assumptions by Lüscher and Wolff 1990, §5 and refined by Blossier et al. 2009, §§2–3.

Define a principal effective energy by

$$E_n^{\mathrm{eff}}(t,t_0) =\frac1{a_t}\log \frac{\lambda_n(t,t_0)}{\lambda_n(t+a_t,t_0)}.$$

It is a derived diagnostic. Eigenvectors can reorder near degeneracies, so levels should be tracked by overlap continuity, quantum numbers, and multi-time information rather than by sorting eigenvalues independently at each time.

Whitening and conditioning

Diagonalize $C(t_0)=U\operatorname{diag}(s_1,\ldots,s_N)U^\dagger$. In exact arithmetic, whitening gives

$$\widetilde C(t) =C(t_0)^{-1/2}C(t)C(t_0)^{-1/2}.$$

If $s_N/s_1$ is comparable to statistical or floating-point uncertainty, inverse square roots amplify noise. A retained-rank choice must therefore be based on uncertainty-aware singular values and tested across resamples. The discarded directions define a changed operator subspace; the resulting systematic is not eliminated by calling the cut numerical.

A useful exact fixture has three states and three operators,

$$C(t)=Z\operatorname{diag}(e^{-E_0t},e^{-E_1t},e^{-E_2t})Z^\dagger.$$

With full-rank $Z$, the GEVP recovers all three energies exactly. Making two columns of $Z$ proportional leaves the correlator positive but reduces rank; one energy becomes unidentifiable. Injecting this rank loss is a more informative test than checking a well-conditioned example alone.

Competing spectral descriptions

GEVP analysis and direct multi-exponential fitting use the same spectral content differently. A robust calculation compares several controlled descriptions:

• principal correlators fitted to one or more exponentials;
• simultaneous matrix fits with shared energies and overlap vectors;
• Bayesian multi-state fits with explicit prior sensitivity;
• matrix-pencil or Prony-like methods on synthetic and high-precision data; and
• summation or ratio observables only when their altered contamination is derived.

Adding states always improves the maximum likelihood in an unconstrained nested model. It does not prove those states are identified. Examine held-out times, posterior-prior contraction, parameter correlations, predictive residuals, and recovery on mock data whose gaps and overlaps resemble the measured problem.

The effective-energy curve can exhibit a false plateau when two excited-state terms partially cancel over a finite window. For

$$C(t)=A_0e^{-E_0t} \left[1+r_1e^{-\Delta_1t}+r_2e^{-\Delta_2t}\right],$$

off-diagonal or subtracted observables may have $r_1r_2<0$. The derivative of the bracket can be small even though neither term is. Moving the fit window modestly may not reveal the cancellation; changing the operator basis changes $r_1,r_2$ and is the decisive test.

Level identity in finite volume

Finite-volume eigenstates are classified by total momentum and lattice irreducible representation, not by continuum spin alone. Multi-hadron operators can be essential even when the desired state resembles a single particle, because the spectrum contains every state with the same exact quantum numbers. Omitting them can leave a stable eigenvalue that is nevertheless a distorted mixture.

Volume dependence, overlap patterns, and inclusion of operators with different expected state content help identify levels. Converting the resulting spectrum into scattering amplitudes is a separate step governed by Spectra from Euclidean Correlation Matrices and the rest of Chapter 7.

Adversarial failure cases

Rank-one basis for two nearby states. A precise single correlator can determine an effective combination but cannot establish two independently normalized levels.

GEVP reference time selected after viewing the answer. Scanning $t_0$ is a legitimate stability test; selecting one value without propagating the choice is not.

Eigenvalues sorted at every time. Near a crossing, this can splice two physical levels into one smooth-looking curve. Track eigenvectors and overlaps.

Covariance regularization hidden. Dropping small covariance eigenmodes changes the metric in data space. Report the retained modes and repeat the fit under justified variations.

Plateau agreement across operators that share the same smearing. Their excited-state overlaps may be strongly correlated. A structurally different operator gives a more independent test.

The shared inference map shows exactly where operator-basis rank and spectral model choice enter. Follow the discrete-level branch to see that a GEVP or multi-state fit produces finite-volume energies and overlaps, not yet a renormalized continuum observable.

Euclidean correlators are measured finite-regulator observables. Energies, matrix elements, and continuous spectral features enter through different inference problems and only later reach matched continuum quantities. The diagram is schematic and not to scale.

Observable-level validation checklist

Before assigning a finite-volume level, require:

• exact lattice quantum numbers and momentum for every operator;
• a declared smearing or displacement transformation with physical scale;
• Hermiticity and positive-semidefinite checks on $C(t)$;
• uncertainty-aware rank and condition-number studies;
• variation of operator subsets, $t_0$, fit windows, and state counts;
• level tracking by eigenvectors or overlaps, not time-local sorting;
• covariance treatment repeated through every resample;
• mock-data recovery including near-degeneracy and rank-loss failures; and
• held-out times or observables that were not used to select the model.

A useful exact fixture for these positive and negative tests uses three states and three operators.

Exercises

1. Exact GEVP. Let $C(t)=Z D(t)Z^\dagger$ with square invertible $Z$ and $D_{nn}(t)=e^{-E_nt}$. Show that the generalized eigenvalues of $(C(t),C(t_0))$ are $e^{-E_n(t-t_0)}$.

Solution

Multiply the GEVP by $Z^{-1}$ on the left and write $w=Z^\dagger v$. It becomes $D(t)w=\lambda D(t_0)w$, which is diagonal. Each basis vector gives $\lambda_n=e^{-E_n(t-t_0)}$.

2. Rank-loss diagnostic. For two operators and two states, let the overlap columns be $z_1$ and $z_2=cz_1$. Show that $C(t)$ has rank one for every $t$.

Solution

$$C(t)=z_1z_1^\dagger \left(e^{-E_1t}+\lvert c\rvert^2e^{-E_2t}\right),$$

which is one outer product times a scalar. No choice of $t_0$ can create a second generalized eigenvector. A second independent operator structure is required.

What you can now do

You should now be able to construct and condition a two-point correlation matrix, solve and test a GEVP, and reject a level claim that depends on one basis, window, or spectral model. Continue with Three-Point Functions, Matrix Elements, and Disconnected Contributions to extract a bare operator insertion while carrying the state-isolation problem into both source and sink channels.

References

• Blossier, Benoît, Michele Della Morte, Georg von Hippel, Tereza Mendes, and Rainer Sommer. “On the Generalized Eigenvalue Method for Energies and Matrix Elements in Lattice Field Theory.” Journal of High Energy Physics 2009, no. 04 (2009): 094. doi:10.1088/1126-6708/2009/04/094.
• Lüscher, Martin, and Ulli Wolff. “How to Calculate the Elastic Scattering Matrix in Two-Dimensional Quantum Field Theories by Numerical Simulation.” Nuclear Physics B 339, no. 1 (1990): 222–252. doi:10.1016/0550-3213(90)90540-T.
• Michael, Christopher. “Adjoint Sources in Lattice Gauge Theory.” Nuclear Physics B 259, no. 1 (1985): 58–76. doi:10.1016/0550-3213(85)90297-4.