A wrong approximation can conserve energy perfectly
The full resonant oscillation of the four-site example is the exact solution of its two-state matrix. Raising the electric cutoff from one to two restores four more configurations. Their couplings can change the curve. Both calculations may conserve norm and energy to excellent precision.
Ask two separate questions. Did the program solve its matrix correctly? Did that matrix retain the states needed for this physical question? Analytic limits and an independent construction address the first. Cutoff and size comparisons address the second.
Reading: Sandbox Physics FR04 finite Hamiltonian · Sandbox Physics FR04 original numerical references and reproduction scripts
Check probability at the electric boundary
For each link, add the probabilities of configurations with . This is its electric-edge weight. Large weight can signal that flux-raising or lowering processes are being removed at the boundary.
The Lab displays , the probability that at least one internal link reaches the edge. A configuration is counted once even if several links touch it. This global weight differs from the weight defined for one link.
Edge weight is a diagnostic, not an error bound. A unit-flux initial string already sits entirely at the edge when the cutoff is one. Whether that matters depends on where the Hamiltonian can send its amplitude. Small edge weight at one instant also cannot rule out effects of paths removed earlier.
Raise the cutoff and compare the same local fields and pair counts over the same time window. Their change measures the actual effect on the chosen observable. If the two retained ranges still give appreciably different answers, report that result as unconverged.
Match the geometry before increasing size
Keeping sources at the endpoints makes a four-to-six-site comparison change the string length and its initial electric energy. A difference can come from a longer string or from more distant boundaries. Those are different questions.
To isolate a boundary effect, put the same source pair inside larger chains. Keep source separation, local sublattice parity, couplings and initial local configuration fixed. Compare the same sites and links near the sources. The staggered pattern repeats every two sites, so a translation must preserve that convention.
Added source-free regions can themselves quench from the bare product reference. A whole-chain pair count therefore does not compare the same local problem. Local fields and densities show more directly when the extra region starts affecting the center.
Identify a boundary return by changing the boundary
Plot a matched local observable at two chain lengths and locate the time when the curves separate. Then test whether a return feature shifts later when the boundary is moved away. If the source separation changes too, that test no longer isolates the boundary.
A small chain has no automatically applicable classical propagation speed. Interference among a few energies can produce complicated oscillations. Retaining the size scan makes local differences reproducible over the chosen time window and connects a return feature to a specified boundary position.
Keep the sources of uncertainty traceable
Save Hamiltonian parameters, basis dimension, sample times and raw local observables. Report norm and energy drift, squared Gauss residuals, independent-solver differences, cutoff changes and matched-geometry size changes separately. A deterministic calculation needs no random seed. Synthetic shot noise, if added later, needs its own seed and sample count.
These diagnostics answer different questions. Adding them does not automatically produce a confidence interval. A quantitative interval for one observable requires a stated rule for turning those comparisons into an uncertainty estimate.
Reading: ITensorMPS.jl