State Preparation, Quench, and Regulator Contracts

A quench calculation is reproducible only when it states what was prepared, what changed, how the field was regulated, which physical region was observed, and which limits were taken. “Quench the mass from $m_i$ to $m_f$” is not enough: a sudden switch, a smooth ramp, and a boundary-state preparation have different high-frequency populations even when their infrared scales agree. This page builds a complete contract around a free scalar mass quench and shows how to test it before interpreting any entropy curve.

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

Helpful background. Separating information scales explains which cutoff-sensitive pieces may cancel in a common-cut subtraction.

The regulated scalar quench

On a periodic one-dimensional lattice with $N$ sites, spacing $a$, and physical length $L=Na$, take

$$H(m)=\frac{a}{2}\sum_{n=0}^{N-1} \left[ \pi_n^2+\frac{(\phi_{n+1}-\phi_n)^2}{a^2}+m^2\phi_n^2 \right], \qquad [\phi_n,\pi_{n'}]=\frac{i}{a}\delta_{nn'}.$$

The factors of $a$ are part of the convention. Equivalently, $q_n=\sqrt a\,\phi_n$ and $p_n=\sqrt a\,\pi_n$ obey $[q_n,p_{n'}]=i\delta_{nn'}$. After the discrete Fourier transform, every momentum mode is a harmonic oscillator with

$$\omega_k(m)^2=m^2+\widehat{k}^{\,2}, \qquad \widehat k=\frac{2}{a}\sin\frac{ka}{2}, \qquad k=\frac{2\pi j}{L}.$$

The benchmark preparation is the exact lattice ground state of $H(m_i)$ at $t=0^-$. At $t=0$, the Hamiltonian is switched instantaneously to $H(m_f)$, and the state evolves with $H(m_f)$. Boundary conditions, zero-mode treatment, Fourier normalization, logarithm base, and the convention for $t=0$ are frozen along with $(a,L,m_i,m_f)$. Cotler et al. 2016, §§ 2–3, pp. 9–24 use this Gaussian mass quench to compare exact entanglement dynamics with a quasiparticle description.

This sudden protocol is ultraviolet finite at any nonzero $a$, but it is not automatically a physically smooth continuum preparation. At large momentum,

$$n_k=\frac{\bigl(\omega_{f,k}-\omega_{i,k}\bigr)^2} {4\omega_{f,k}\omega_{i,k}}$$

falls rapidly for a mass quench, yet local composite observables can still require subtraction and the earliest times remain sensitive to the switch. A ramp $m^2(t)$ introduces a duration $\tau_Q$ and a different set of occupations. A conformal boundary-state preparation introduces an extrapolation length $\tau_0$ rather than literally exciting all frequencies with a zero-duration operation Calabrese and Cardy 2005, § 2. These are different contracts, not alternative descriptions of identical microscopic data.

Exact mode and covariance data

The Bogoliubov transformation between the initial and final oscillator bases gives

$$n_k=\langle a_{f,k}^\dagger a_{f,k}\rangle =\frac14\left( \frac{\omega_{f,k}}{\omega_{i,k}} +\frac{\omega_{i,k}}{\omega_{f,k}}-2 \right).$$

For $m_i=2$, $m_f=1$, and $k=0$, $(\omega_i,\omega_f)=(2,1)$, so $n_0=1/8$. The excess final-Hamiltonian energy of that oscillator is $\omega_f n_0=1/8$. These two exact values are useful unit tests: a missing factor of two in the Fourier transform or Bogoliubov coefficient usually changes both.

At $t=\pi/4$ in final-frequency units, the same oscillator gives

$$X=P=\frac58, \qquad R=\frac38, \qquad XP-R^2=\frac14, \qquad E_f=\frac12(P+\omega_f^2X)=\frac58.$$

The last equality leaves $E_f-\omega_f/2=1/8$ above the final vacuum. This single time slice simultaneously checks the canonical normalization, covariance signs, purity, and injected energy.

The time-dependent Gaussian covariance provides a second independent route. For one real normal mode, define

$$X_k(t)=\langle q_k(t)q_{-k}(t)\rangle, \quad P_k(t)=\langle p_k(t)p_{-k}(t)\rangle, \quad R_k(t)=\frac12\langle\{q_k(t),p_{-k}(t)\}\rangle.$$

Evolution with $\omega_{f,k}$ from the ground state of $\omega_{i,k}$ gives

$$\begin{aligned} X_k(t)&=\frac{1}{2\omega_{i,k}} \left(\cos^2\omega_{f,k}t+ \frac{\omega_{i,k}^2}{\omega_{f,k}^2}\sin^2\omega_{f,k}t\right),\\ P_k(t)&=\frac{\omega_{i,k}}{2} \left(\cos^2\omega_{f,k}t+ \frac{\omega_{f,k}^2}{\omega_{i,k}^2}\sin^2\omega_{f,k}t\right),\\ R_k(t)&=\frac14\left( \frac{\omega_{i,k}}{\omega_{f,k}}- \frac{\omega_{f,k}}{\omega_{i,k}} \right)\sin(2\omega_{f,k}t). \end{aligned}$$

Fourier transform these matrices to position space, restrict them to the sites in a physical interval $A$, and assemble

$$\Gamma_A= \begin{pmatrix} X_A&R_A\\ R_A^{\mathsf T}&P_A \end{pmatrix}.$$

In the ordering used above, the subsystem symplectic form is

$$\Omega_A= \begin{pmatrix} 0 & \mathbf1_A\\ -\mathbf1_A & 0 \end{pmatrix}.$$

The eigenvalues of $i\Omega_A\Gamma_A$ occur in pairs $\pm\nu_j$. Take each positive eigenvalue once; these are the symplectic eigenvalues $\nu_j\geq1/2$. Taking absolute values without removing the doubled pair would count every mode twice. The interval entropy is

$$S_A=\sum_j\left[ \left(\nu_j+\frac12\right)\log\left(\nu_j+\frac12\right) -\left(\nu_j-\frac12\right)\log\left(\nu_j-\frac12\right) \right].$$

Using only $\sqrt{X_AP_A}$ is safe when the mixed covariance vanishes in the chosen convention; after a quench $R_A$ is generally nonzero, so the full symplectic spectrum is the robust implementation. The covariance method and its relation to free-system entanglement are reviewed by Peschel and Eisler 2009, §§ 2–3.

A complete calculation contract

Record the problem as

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

Here $\mathcal P$ is the preparation protocol, $\mathcal O$ names the observable, $\mathcal R$ is its subtraction or renormalization prescription, and $\mathcal L$ is the order of limits. A useful run record includes:

• the exact or approximate method used to construct $\rho_0$;
• the quench time profile and post-quench Hamiltonian;
• $a$, $N$, boundary conditions, local Hilbert-space truncation, and timestep;
• the physical endpoints of $A$, including how noninteger $\ell/a$ is rounded;
• the entropy or correlation definition and logarithm base;
• every fit window, threshold, and uncertainty estimate;
• code revision, random seeds, and convergence data.

This is not administrative decoration. Each item changes a mathematical input to $\rho_A(t)$ or to the estimator applied to it.

Continuum and thermodynamic limits

For a target interval length $\ell$ and observation time $t$, refine the lattice while holding $(m_i,m_f,\ell,t,L)$ in physical units. Thus $N=L/a$ and $n_A=\ell/a$ increase together. A useful hierarchy is

$$a\ll \{m_i^{-1},m_f^{-1},\ell,t\}, \qquad t\ll \frac{L-\ell}{2v_{\rm front}}, \qquad \Delta t\,\omega_{\max}\ll1,$$

with the last condition adapted to the integrator. The first inequality is shorthand for separate dimensionless convergence tests; time is converted to length using the relativistic speed convention or an explicitly named lattice scale.

The sudden mass-quench benchmark here is scoped to one spatial dimension. In higher dimensions, its high-momentum energy and local observables can have stronger ultraviolet sensitivity; a smooth ramp or explicitly matched high-frequency preparation may be required before a finite-energy continuum claim is available.

At fixed $N$, sending $a\to0$ shrinks both $L$ and a fixed-site region. At fixed $a$, sending $N\to\infty$ removes boundary return but leaves the lattice dispersion. At fixed finite $L$, sending $t\to\infty$ probes recurrence rather than continuum late-time relaxation. Reverse the continuum and thermodynamic extrapolations as an adversarial test. If they disagree within the claimed window, report the noncommutation instead of combining the data.

Controls that expose a wrong problem

Reverse a limit order. Compare continuum-first and volume-first extrapolations at the same physical $(\ell,t)$. A difference that shrinks only after both refinements is a finite-size–cutoff coupling; a persistent difference defines distinct limits.

Change the preparation bandwidth. Replace the sudden switch by ramps with decreasing $a$ at fixed physical $\tau_Q$. If an “entanglement-production rate” follows the cutoff rather than $\tau_Q$, it is a preparation transient.

Check conserved quantities and purity. Mode evolution must preserve the symplectic eigenvalues of the full-system covariance and the expectation of $H_f$ after the switch. Drift localizes timestep or truncation error before a subsystem entropy is interpreted.

Keep the shared diagnostics visible. The chapter orientation map shows where the contract enters the calculation. Its failure controls separate regulator changes from observable and dynamical-class changes, while the diagnostic comparison records which limit is required by each observable.

Exercises

Derive the post-quench occupation

$$n=\frac14\left(\frac{\omega_f}{\omega_i}+\frac{\omega_i}{\omega_f}-2\right)$$

by writing $a_f$ in terms of $a_i$ and $a_i^\dagger$. Evaluate it for $(\omega_i,\omega_f)=(2,1)$.

Solution

For one oscillator,

$$a_f=\frac12\left(\sqrt{\frac{\omega_f}{\omega_i}}+ \sqrt{\frac{\omega_i}{\omega_f}}\right)a_i +\frac12\left(\sqrt{\frac{\omega_f}{\omega_i}}- \sqrt{\frac{\omega_i}{\omega_f}}\right)a_i^\dagger.$$

The initial vacuum obeys $a_i|0_i\rangle=0$, so $n=|\beta|^2$, which yields the stated expression. At $(2,1)$,

$$n=\frac14\left(\frac12+2-2\right)=\frac18.$$

Suppose $L=24$, $\ell=6$, and $a=0.2$, so $N=120$ and $n_A=30$. What site counts are required after refinement to $a=0.1$? What physical problem would be computed if the site counts were left unchanged?

Solution

Fixed physical lengths require $N=24/0.1=240$ and $n_A=6/0.1=60$. Leaving the counts unchanged would give $L=12$ and $\ell=3$. That is a simultaneous change of volume and observable geometry, not a refinement of the original problem.

Show directly from the covariance formulas that

$$X_k(t)P_k(t)-R_k(t)^2=\frac14.$$

Why is this a valuable numerical check?

Solution

Let $c=\cos\omega_ft$, $s=\sin\omega_ft$, and $r=\omega_i/\omega_f$. Then

$$X P=\frac14(c^2+r^2s^2)(c^2+r^{-2}s^2), \qquad R^2=\frac14(r-r^{-1})^2c^2s^2.$$

Expanding and subtracting leaves $(c^2+s^2)^2/4=1/4$. The equality says that the full one-mode state remains pure under symplectic time evolution. Its violation exposes a normalization, timestep, or matrix-assembly error before subsystem quantities are trusted.

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.
• Cotler, Jordan S., Mark P. Hertzberg, Márk Mezei, and Mark T. Mueller. “Entanglement Growth after a Global Quench in Free Scalar Field Theory.” Journal of High Energy Physics 2016, no. 11 (2016): 166. DOI.
• Peschel, Ingo, and Viktor Eisler. “Reduced Density Matrices and Entanglement Entropy in Free Lattice Models.” Journal of Physics A: Mathematical and Theoretical 42 (2009): 504003. DOI.