Continuum Extrapolation and Finite-Time Windows

A continuum dynamical claim needs a common interval of physical time that is later than cutoff and preparation transients yet earlier than boundary return, numerical truncation, and accumulated evolution error. “Take $a\to0$ and then study late time” is not an operational prescription: at each regulator, the trusted window can move. The correct object is a joint resolution–volume–time window, followed by extrapolation at fixed physical geometry and fixed time.

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.

Build a common trusted window

Let $a$ be the spatial cutoff, $\Delta t$ the timestep, $L$ the volume, $\ell$ a region size, $\tau_{\rm prep}$ the preparation scale, and $\chi$ a local or tensor-network truncation parameter. For each numerical setup $i$, define an accepted interval

$$W_i=[t_{\min,i},t_{\max,i}],$$

with

$$t_{\min,i}\gtrsim \max\{t_{\rm cutoff}(a_i),\tau_{\rm prep},t_{\rm local}\},$$

and

$$t_{\max,i}\lesssim \min\{t_{\rm return}(L_i,\ell), t_{\rm trunc}(\chi_i),t_{\rm num}(\Delta t_i,\epsilon_{\rm num})\}.$$

Each boundary is operational.

• $t_{\rm cutoff}$ is where regulator refinement stops changing the observable beyond tolerance.
• $t_{\rm local}$ is a local-equilibration scale if a hydrodynamic or membrane law is being tested.
• $t_{\rm return}$ is the first geometry-dependent boundary influence or recoherence relevant to the observable.
• $t_{\rm trunc}$ is where local-cutoff or bond-dimension sweeps cease to agree.
• $t_{\rm num}$ is where timestep or solver comparisons exceed the accepted numerical error.

The continuum comparison uses the intersection

$$W_{\rm common}=\bigcap_i W_i.$$

If this intersection is empty, the data support separate finite-regulator statements, not one continuum growth law.

Return, revival, and recurrence are different

The first influence of a finite boundary is not necessarily an exact recurrence. In a periodic free system, counterpropagating wave packets can return coherently; in an interacting system, the first boundary-sensitive time can be followed by complicated quasiperiodic behavior; an exact many-body recurrence can be vastly later.

Write $t_{\rm return}(L,\ell,\text{geometry},\text{dispersion})$ until a specific setup is chosen. For a one-speed conformal quasiparticle picture on a periodic ring, an interval of length $\ell<L/2$ has a leading saturation time

$$t_{\rm sat}=\frac{\ell}{2v}$$

and a complementary-path return scale

$$t_{\rm return}=\frac{L-\ell}{2v}.$$

For $L=40$, $\ell=8$, and $v=1$, the plateau-like interval $4<t<16$ lies between these two geometric scales. A full-ring conformal revival scale is $L/(2v)=20$. Doubling $L$ moves the complementary-path scale to $36$ while leaving $t_{\rm sat}=4$. These numbers belong to this periodic one-speed geometry, not to a universal recurrence formula.

Calabrese and Cardy 2005, § 4 supplies the infinite-system interval crossover. Finite-ring revival and recoherence structure require additional spectral information; Cardy 2014, Eqs. (1)–(4) discusses quantum revivals in conformal field theories.

Fixed-physical-time extrapolation

Choose lattice spacings $a_i$ and scale site counts so that

$$N_i\approx\frac{L}{a_i}, \qquad n_{A,i}\approx\frac{\ell}{a_i}.$$

Declare the integer rounding rule and record the actual physical length error, which should vanish as $O(a_i)$. Evolve every resolution, restrict to $W_{\rm common}$, and interpolate to a common set of physical times. Only then fit a regulator sequence such as

$$O(t;a)=O_{\rm cont}(t)+c_p(t)a^p+c_{p+1}(t)a^{p+1}+\cdots.$$

The leading exponent $p$ is not determined solely by the bulk finite-difference stencil. State preparation, entangling-surface placement, subsystem rounding, subtraction, composite-observable construction, and logarithmic corrections can change the observed leading behavior. Fit competing forms, use temporal covariance, and repeat after dropping the coarsest resolution.

For entropy,

$$\Delta S_A(t;a)=S_A(t;a)-S_A(0;a)$$

cancels the leading common-cut divergence only when the two states use the same regulator, region construction, and short-distance singularity class. State-dependent divergences can remain. The area-law divergence of field entanglement is explicit already in the scalar-lattice calculation of Srednicki 1993.

Finally, fit a growth law to the extrapolated curve $O_{\rm cont}(t)$. Fitting different time intervals at each $a$, extracting a slope, and extrapolating those slopes mixes cutoff dependence with window selection.

Exact dispersion and Richardson benchmark

For rescaled canonical variables $[q_j,p_k]=i\delta_{jk}$, take

$$H=\frac12\sum_j\left[ p_j^2+m^2q_j^2+\frac{(q_{j+1}-q_j)^2}{a^2} \right].$$

On a periodic lattice,

$$\omega_a(k)=\sqrt{m^2+\frac4{a^2}\sin^2\frac{ak}{2}}, \qquad v_a(k)=\frac{\sin ak}{a\omega_a(k)}.$$

At fixed physical $k$,

$$\omega_a^2=m^2+k^2-\frac{a^2k^4}{12}+O(a^4),$$

so a smooth fixed-mode frequency has leading $a^2$ error. For $m=1$ and $k=0.5$,

$$\omega_{\rm cont}=\sqrt{1.25}=1.11803398875\ldots.$$

The regulated values are

$$\begin{array}{c|c} a&\omega_a(0.5)\\ \hline 0.4&1.11766174533\\ 0.2&1.11794084642\\ 0.1&1.11801069807 \end{array}$$

and the error decreases by approximately four when $a$ is halved. Second-order Richardson extrapolation,

$$\omega_R(a)=\frac{4\omega_{a/2}-\omega_a}{3},$$

gives $1.11803388012$ from $a=(0.4,0.2)$ and $1.11803398196$ from $a=(0.2,0.1)$. This benchmark checks fixed physical momentum, the sign of the lattice correction, and the refinement ratio.

Do not transfer its $a^2$ convergence blindly to entropy. A correct bulk dispersion can coexist with $O(a)$ surface-placement error or nonanalytic regulator corrections in $\Delta S_A$.

Timestep and truncation control

A timestep horizon is not a function of $\Delta t$ alone. It depends on integrator order, the cutoff-dependent largest frequency, elapsed time, observable, and tolerance. Require, throughout the common window,

$$|O_{\Delta t}(t)-O_{\Delta t/2}(t)|<\epsilon_{\rm num}.$$

For an order-$r$ method in its asymptotic regime, a Richardson estimate can be formed from $\Delta t$ and $\Delta t/2$, but norm growth and high-frequency modes can spoil a simple power fit. Product-formula errors and adaptive Hilbert-space truncation in real-time many-body evolution are analyzed in concrete settings by Suzuki 1990 and Daley et al. 2004.

Vary timestep, local bosonic cutoff, and tensor-network bond dimension independently. Increasing two together can hide compensation. Monitor energy conservation where appropriate, covariance purity in Gaussian evolution, discarded weight, occupation tails, and held-out observables.

Controls that close false windows

Drop one resolution. Recompute the continuum curve and its uncertainty after removing each $a_i$ in turn.

Enlarge the volume. Hold $(a,\ell,t)$ fixed. The accepted result should remain unchanged while $t_{\rm return}$ moves outward.

Move both time endpoints. A growth parameter should remain stable inside a visibly nonempty plateau of fit choices.

Predict a held-out time. Fit only an early subset of $W_{\rm common}$ and predict a later point still inside it.

Cross the boundary deliberately. Extend beyond $t_{\rm return}$ or $t_{\rm trunc}$ and verify that the finite-size or numerical deviation appears where expected.

Reverse the extrapolation order. Compare volume-first and continuum-first sequences. Explain any noncommutation rather than silently merging them.

The chapter orientation map places the validity window alongside the dynamical class. Its failure controls require resolution and time variation, while the diagnostic comparison states the window needed by each observable.

Common pitfalls

Refining at fixed site count. The physical volume and region then shrink. Scale both counts and report rounding error.

Calling the first boundary return a recurrence. Return, recoherence, revival, and exact recurrence can have parametrically different times.

Imposing the stencil order on entropy. Test the observed regulator sequence and surface construction rather than assuming $p=2$.

Exercises

Derive the lattice dispersion expansion through $O(a^2)$.

Solution

Using $\sin(ak/2)=ak/2-(ak)^3/48+O(a^5)$,

$$\frac4{a^2}\sin^2\frac{ak}{2} =k^2-\frac{a^2k^4}{12}+O(a^4).$$

Adding $m^2$ gives the stated expansion of $\omega_a^2$.

Use the table values at $a=0.2$ and $0.1$ to calculate the second-order Richardson estimate.

Solution

$$\omega_R=\frac{4(1.11801069807)-1.11794084642}{3} =1.11803398195\ldots,$$

which agrees with $\sqrt{1.25}$ to about $7\times10^{-9}$.

For the periodic one-speed example $L=40$, $\ell=8$, and $v=1$, compute $t_{\rm sat}$, the complementary-path return scale, and the full-ring revival scale. What changes when $L$ doubles?

Solution

The scales are $t_{\rm sat}=8/2=4$, $t_{\rm return}=(40-8)/2=16$, and $t_{\rm rev}=40/2=20$. At $L=80$, the first stays $4$, while $t_{\rm return}=36$ and $t_{\rm rev}=40$. Volume variation therefore distinguishes the region crossover from finite-ring effects.

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.
• Cardy, John. “Thermalization and Revivals after a Quantum Quench in Conformal Field Theory.” Physical Review Letters 112 (2014): 220401. DOI.
• Daley, A. J., C. Kollath, U. Schollwöck, and G. Vidal. “Time-Dependent Density-Matrix Renormalization-Group Using Adaptive Effective Hilbert Spaces.” Journal of Statistical Mechanics: Theory and Experiment 2004 (2004): P04005. DOI.
• Srednicki, Mark. “Entropy and Area.” Physical Review Letters 71 (1993): 666–669. DOI.
• Suzuki, Masuo. “Fractal Decomposition of Exponential Operators with Applications to Many-Body Theories and Monte Carlo Simulations.” Physics Letters A 146 (1990): 319–323. DOI.