Continuum Extrapolation and Finite-Time Windows

A continuum dynamical claim is supported only on times that are simultaneously later than cutoff and preparation transients and earlier than finite-volume return, recurrence, truncation, and accumulated integration error. The useful object is a joint resolution–size–time window, not a formal sequence of limits evaluated at one lattice.

Required background. The quench and regulator contract fixes physical quantities, and entanglement growth identifies the stages to be separated.

Helpful background. Lattice-to-continuum entropy supplies static subtraction and extrapolation methods.

Window inequalities

Let $a$ be the spatial cutoff, $\Delta t$ the timestep, $L$ the volume, $\ell$ a subregion size, $\xi$ a physical correlation length, $v_{\max}$ the fastest relevant lattice group velocity, and $\chi$ a numerical truncation parameter. A candidate window satisfies schematically

$$t_{\rm UV}(a,\Delta t,\xi) \ll t\ll \min\!\left\{ \frac{L-\ell}{2v_{\max}}, t_{\rm rec}(L), t_{\rm trunc}(\chi), t_{\rm err}(\Delta t) \right\}.$$

Each boundary must be measured or bounded. For tensor networks, $t_{\rm trunc}$ can be the time when discarded weight or bond-dimension comparisons cease to agree. For exact finite Gaussian systems, recurrence rather than bond dimension is limiting. For Trotter evolution, compare at least two timesteps in physical units.

Fixed-physical-time extrapolation

At several lattice spacings $a_i$, choose $N_i=L/a_i$ and $n_{A,i}=\ell/a_i$ so $L$ and $\ell$ are fixed. Interpolate every dataset to common physical times within the overlap of all trustworthy windows. Fit, for example,

$$\Delta S_A(t;a) =\Delta S_A^{\rm cont}(t) +c_p(t)a^p+c_{p+1}(t)a^{p+1},$$

where $p$ follows from the discretization rather than a convenient fit. Repeat after dropping the coarsest lattice and vary the fit order. Only then fit a growth law to $\Delta S_A^{\rm cont}(t)$.

This order matters. Fitting a slope separately at each resolution over different time intervals and then extrapolating the slopes mixes cutoff error with window selection.

The validity window is an independent input joining microscopic evolution to every reported diagnostic. A continuum extrapolation is meaningful only on the common overlap of all resolutions. The map is schematic.

Controls that close false windows

• Drop one resolution at a time and compare the continuum curve.
• Double the volume while keeping $a$, $\ell$, and the initial state fixed.
• Halve the timestep and increase local or bond truncation independently.
• Move the fit window inward from both ends.
• Predict one later time inside the claimed window without refitting.
• Extend beyond the first estimated recurrence deliberately and confirm that the result ceases to follow the continuum collapse.

If no common interval survives these controls, the correct output is a finite-regulator result, not a continuum velocity. The finite-size saturation scales in the conformal quench analysis of Calabrese and Cardy 2005, §§ 3–4 provide an analytic benchmark for separating growth from return effects.

Dropping resolution control or extending a fit beyond recurrence can make a discretization artifact appear more precise. Nested-window and held-out-time tests expose the failure. The map is schematic.

Common pitfalls

Extrapolating at fixed site number. The physical geometry then changes with $a$. Scale $N$ and the region together.

Using late time as a synonym for asymptotic. In a finite calculation, sufficiently late time is dominated by saturation or recurrence. An asymptotic continuum regime must occur before those scales.

References

• Calabrese, Pasquale, and John Cardy. “Evolution of Entanglement Entropy in One-Dimensional Systems.” Journal of Statistical Mechanics: Theory and Experiment 2005 (2005): P04010. DOI.