State Preparation, Quench, and Regulator Contracts

A QFT quench is defined by more than two Hamiltonian parameters. The initial state, preparation protocol, ultraviolet regulator, volume, timestep, observable subtraction, and limit order determine the problem. Freezing this contract before evolution prevents a lattice transient from being reinterpreted as continuum entanglement production.

Required background. Regulated subregion entropy supplies the cutoff quantity, and equilibration and dephasing distinguishes unitary relaxation from thermal state preparation.

Helpful background. Separating information scales explains which cutoff-sensitive pieces may be subtracted.

A complete quench specification

Write the regulated problem as

$$\mathcal C= \{\rho_0^{(a,L)},\mathcal P,H_i^{(a,L)},H_f^{(a,L)}, A_{\rm phys},\Delta t,\mathcal O,\mathcal R,\mathop{\rm lim}\}.$$

Here $\mathcal P$ is the preparation map, $\mathcal R$ is the renormalization or subtraction rule, and $\mathop{\rm lim}$ records the order of continuum, volume, truncation, and observation-time limits. A sudden quench sets the Hamiltonian from $H_i$ to $H_f$ at a specified time. A ramp adds a time-dependent protocol and a new duration scale.

The initial state should be described operationally. “Ground-state quench” can mean the exact ground state of the regulated $H_i$, an approximate variational state, or a continuum Gaussian covariance sampled on the lattice. These differ in short-distance correlations and early-time entropy.

Massive free-field example

For a one-dimensional lattice scalar,

$$H(m)=\frac12\sum_{n=1}^{N} \left[\pi_n^2+\frac{(\phi_{n+1}-\phi_n)^2}{a^2} +m^2\phi_n^2\right]a ,$$

prepare the ground state at mass $m_i$ and evolve with mass $m_f$. Freeze boundary conditions and the dispersion

$$\omega_k(m)^2=m^2+\frac{4}{a^2}\sin^2\frac{ka}{2}.$$

For a physical interval of length $\ell$, choose its site count $n_A=\ell/a$ at each refinement. Compute the Gaussian covariance, symplectic eigenvalues, and $\Delta S_A(t)=S_A(t)-S_A(0)$. Hold $(m_i,m_f,\ell,t)$ fixed while taking $a\to0$; increase $L=Na$ separately to suppress boundary returns.

Calabrese and Cardy 2005, §§ 2–4 use a boundary-state quench construction to show how a preparation length scale controls universal late behavior. It should not be identified with a literal zero-duration quench at arbitrarily high frequencies.

State preparation and limit order sit upstream of every effective picture and diagnostic. Changing either defines a new dynamical problem even when the nominal pre- and post-quench masses agree. The map is schematic.

Order-of-limits checks

At fixed $N$, $a\to0$ sends both $L$ and $\ell$ to zero. At fixed $a$, $L\to\infty$ leaves lattice dispersion at high momentum. At fixed finite $L$, taking $t\to\infty$ encounters saturation and recurrence. A continuum finite-time claim therefore needs a joint window such as

$$a\ll\{m_f^{-1},\ell,t\} \ll L,\qquad \Delta t\ll\min(a,m_f^{-1}),$$

adapted to the numerical method. Reverse the continuum and thermodynamic limits as a stress test and explain any noncommutation.

The characteristic preparation failure is to change the physical region or time while refining the lattice. The resulting curve can converge numerically while approaching the wrong continuum observable. The map is schematic.

Exercises

If $L=Na$ and $\ell=n_Aa$, how must $N$ and $n_A$ scale when $a$ is halved at fixed physical $L$ and $\ell$?

Solution

Both $N$ and $n_A$ must double. Holding either site count fixed would halve the corresponding physical length.

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.