Extracting Nonlinear Dynamical Response Functions from Time Evolution

Atsushi Ono

REVIEW 2 major objections 4 minor 143 references

Extracting Nonlinear Dynamical Response Functions from Time Evolution

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A functional-derivative scheme extracts full nonlinear response functions from the time evolution of a driven system, bypassing explicit multipoint correlation functions.

desk verdict A clean and genuinely new scheme for extracting nonlinear response functions from real-time dynamics, though the finite-width test-pulse approximation needs a proper convergence study. read the letter →

arxiv 2507.07679 v1 pith:JCPS55UD submitted 2025-07-10 cond-mat.str-el cond-mat.mes-hallcond-mat.stat-mechphysics.opticsquant-ph

Atsushi Ono This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallcond-mat.stat-mechphysics.opticsquant-ph

keywords nonlinearresponsefunctionsfunctionalderivativereal-timedynamicsopticalresponsesRice-Melemodeltensornetworktwo-dimensionalcoherentspectroscopymany-bodycorrelation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a general way to read nonlinear dynamical response functions out of real-time evolution. The idea is to apply a short probe pulse together with frequency-localized test pulses, then form functional variations of the time-evolved observable; the first and second variations directly encode the full second- and third-order retarded response functions. This removes the need to compute multipoint correlation functions, which are the usual bottleneck in correlated systems. The author validates the scheme against perturbative expressions in the Rice-Mele model and demonstrates it on an interacting many-body system with a tensor network method. If correct, any method that can propagate a quantum system in time becomes a source for full nonlinear optical spectra.

What carries the argument

The central object is the functional derivative of the time-evolved observable with respect to the external field, used through the identities in Eqs. (8)-(9) and (15)-(16). A short Gaussian probe pulse $A(t)$ supplies a nearly flat frequency background, while cosine and sine test pulses $a_i^{\cos}(t)=\frac{A_0}{2\pi}e^{-t^2/(2\tau_a^2)}\cos(\Omega_i t)$ act as narrow windows in frequency; in the limit $\tau_a\to\infty$ their Fourier transforms become delta-function pairs at $\pm\Omega_i$. The variations $\delta_{a_1}\langle O\rangle^{(2)}(\omega)$ and $\delta_{a_2}\delta_{a_1}\langle O\rangle^{(3)}(\omega)$ then pick out the response function at fixed frequencies, and the sine/cosine combination separates real and imaginary parts. This machinery turns the abstract definition of retarded response functions into a concrete time-evolution protocol requiring three field configurations for second order and seven for third order.

What would settle it

Run the Rice-Mele reconstruction at a frequency near a Van Hove singularity while doubling the test-pulse width $\tau_a$ and correspondingly lengthening the simulation window; if the extracted $\bar{\chi}^{(2)}$ shifts beyond the few-percent error reported elsewhere in the spectrum, the delta-function approximation is the limiting assumption.

Watch

Extended reading notes

Core claim

The central claim is that the symmetric $n$th-order retarded response function $\bar{\chi}^{(n)}(\omega_1,\ldots,\omega_n)$ can be obtained from $n-1$ functional derivatives of the $n$th-order part of the time-evolved observable with respect to a test field, evaluated numerically with cosine and sine Gaussian pulses that are sharply peaked in frequency. For second order, Eq. (14) expresses $\bar{\chi}^{(2)}(\pm\omega_1,\omega\mp\omega_1)$ through the first variation $\delta_{a}\langle O\rangle^{(2)}(\omega)$ under the combined field $A(t)+a(t)$; for third order, Eq. (17) expresses $\bar{\chi}^{(3)}$ through the mixed second variation under seven different field combinations. Because the test pulses act as approximate delta functions in frequency, the integrals collapse and the response function is obtained pointwise by sweeping the test-pulse frequencies. The paper reports quantitative agreement with perturbative formulas for second- and third-harmonic and rectification responses in the Rice-Mele model, and shows that the same procedure works for the Hubbard-interacting case using iTEBD.

Load-bearing premise

The extraction formulas treat each Gaussian test pulse as an exact delta function in frequency, so the test pulses must be long in time while the probe pulse stays short; if that separation fails near sharp spectral features, the reconstructed response functions inherit systematic errors.

Editorial extensions

If this is right

Any real-time dynamics method, such as tensor networks, quantum master equations, mean-field dynamics, or exact diagonalization, can supply full nonlinear response functions wherever it can propagate the system.
The full $n$th-order response is obtained without sweeping inter-pulse delays, avoiding the computational bottleneck of multidimensional coherent spectroscopy.
Specifically relevant spectra such as second-harmonic generation and optical rectification are read off directly from cuts of the reconstructed two-dimensional response function.
For dissipative or approximate time-evolution schemes, the extracted nonlinear response can be compared with perturbative results, providing a check on self-energy and vertex corrections.
By adding a pump pulse and varying the probe delay, transient nonlinear response functions can be defined as a natural extension of pump-probe linear response.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the protocol only needs the time trace of one observable under a few prescribed pulses, it should transfer directly to quantum simulators or quantum computers, where multipoint correlation functions are hard to measure but single-observable time traces are natural.
The finite width of the test pulses enters as an uncontrolled systematic error; a natural strengthening would be to extract $\bar{\chi}^{(n)}$ at several $\tau_a$ values and extrapolate to the delta-function limit, or to deconvolve the known Gaussian kernel.
The selective sign reversal in the rectification response, but not the second-harmonic response, as the Hubbard interaction grows beyond the staggered potential suggests an interaction-induced renormalization with symmetry-selective effects; this could be probed experimentally in one-dimensional ferroelectrics or by computing higher-order responses in the same model.
The scaling checks in the pulse amplitude and the small variation parameter are a built-in validation; automating these checks could make the method a black-box extraction tool for nonlinear response databases.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: this is a solid, useful methods paper. The central trick—evaluating functional derivatives of the time evolution with respect to frequency-localized test pulses—is new as far as I can tell, and the derivation in the supplement is clean. The benchmarks in the Rice-Mele model are quantitative, with relative errors below a few percent away from the Van Hove singularities. The iTEBD application is a nice bonus, showing that the method plugs directly into a tensor-network time evolution and yields second-order responses for an interacting chain.

The soft spots are real but not disqualifying. Equations (14) and (17) assume the test pulses are delta functions in frequency. In practice tau_A is finite, so each reconstructed point is really a convolution of the true response with a Gaussian of width ~1/tau_A. The benchmarks choose tau_A = sqrt(2)/Gamma, which gives a frequency width about 0.7 times the RTA broadening; consistent with that, the relative errors climb to 10-32% near the Van Hove singularities. The paper does not provide a systematic tau_A-convergence study, and the Discussion only says accuracy is assured 'if convergence is achieved'—without showing it. That is a legitimate gap, especially for closed systems with sharp spectral features, where the blurring will be worse. I would not call it fatal: the method still works as a proof of concept, and the errors are quantified where they arise. But a referee should ask for either a convergence study or an explicit bound on the systematic error.

Also, no code or data is shipped, so independent reproduction is not yet possible. Not a dealbreaker for a methods paper, but it would help.

Bottom line: this deserves a serious referee. The method is worth publishing, and the convergence question is fixable. I would want to see a revised version with a tau_A study and ideally a small repository before citing it myself.

Referee Report

2 major / 4 minor

Summary. This manuscript develops a functional-derivative-based scheme to extract nonlinear dynamical response functions from real-time evolution. The key idea is to evaluate first and second variations of an observable under a broad Gaussian probe plus frequency-localized Gaussian test pulses, and then invert the resulting integrals to obtain the second- and third-order retarded response functions without computing multipoint correlation functions explicitly. The method is validated against perturbative formulas for the Rice-Mele model with relaxation-time approximation, and applied to the Rice-Mele-Hubbard model with iTEBD. The paper claims general applicability to any real-time dynamics method and suggests a recursive extension to higher orders.

Significance. The paper proposes a conceptually clean and potentially widely applicable tool: extending linear-response real-time methods to nonlinear responses without explicit multipoint correlation functions. The derivation in the Supplemental Material is careful, and the validation against independent perturbative expressions is quantitative (relative errors below a few percent in most of the frequency plane). The many-body application with iTEBD is a valuable first demonstration. The main caveat is the reliance on frequency-delta-like test pulses, whose finite width introduces systematic broadening that is not yet quantified with a convergence study.

major comments (2)

[Formulation, Eqs. (12)-(14) and (16)-(17)] The central extraction formulas (14) and (17) are derived by replacing the Gaussian test functions (10)-(11) with frequency delta functions in the limit tau_A -> infinity (Eq. (12)). For any finite tau_A, the measured variations are convolutions of the true response function with a Gaussian of width 1/tau_A. The RTA benchmarks in Figs. 2 and 3 set tau_A = sqrt(2)/Gamma, corresponding to a frequency width of about 0.7*Gamma, which is not small compared with the intrinsic linewidth; the presented relative errors indeed rise to 10-32% near the Van Hove singularities (Figs. 2(f) and 3(d)). No systematic tau_A-convergence study is provided, and the Discussion's statement that accuracy requires 'convergence with respect to parameters such as F0, epsilon, and tau_A' is not supported by such a demonstration. This is a load-bearing point for the central claim that the method reconstructs the full nonlinear response functions, since for closed systems (Gamma=0) or for sharper spectral features the finite-width convolution would dominate. Please add a tau_A-convergence study (e.g., extracting the same response for a sequence of increasing tau_A) and either correct for the finite-width convolution or demonstrate that the reconstructed functions have converged to the true response.
[Application to many-body systems, Fig. 4 and End Matter] The iTEBD application inherits the same finite-width issue in a more severe form: the system has no relaxation (Gamma=0), and the extracted response is broadened by the Fourier profiles of the test pulse (tau_A = 12) and the Gaussian window (tau_w = 6). The End Matter demonstrates convergence with respect to bond dimension (Fig. 5), but no convergence with respect to tau_A or tau_w is shown. Since the physical conclusion of this section is the selective sign reversal of the rectification response at U=0.2 (Fig. 4(b)) attributed to many-body effects, the possibility that the finite test-pulse convolution modifies sign or shape near resonances should be ruled out. Please include a convergence check in tau_A and tau_w for the iTEBD results.

minor comments (4)

[Introduction] The statement that the method works 'without explicitly computing multipoint correlation functions' is somewhat imprecise, since the time evolution under external fields implicitly contains these correlations; a one-sentence clarification would help.
[Proof-of-principle demonstration, Fig. 2(a)] The paper does not discuss the practical cost of the required long time windows (e.g., t ~ -600 in Fig. 2(a)); this is a relevant limitation for methods with finite simulation horizons and should be quantified or stated.
[Proof-of-principle demonstration, Fig. 2(d)] The comparison between the time-evolution result and the perturbative expression depends on the convention for the imaginary broadening in the energy denominators (e.g., i eta vs 2i eta); the paper notes this and chooses the better-agreeing convention, but this choice should be flagged as a caveat in the error estimates.
[Formulation, after Eq. (6)] The claim that higher-order response functions can be obtained by successively applying the same procedure is not demonstrated beyond third order; a brief discussion of how the number of required simulations and the numerical subtraction errors scale with order would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the extraction formulas are direct finite-difference evaluations of the functional-derivative definition of the response function, validated against independent perturbative benchmarks.

full rationale

The central identities, Eq. (4) and Eqs. (14)/(17), make no claim beyond the definition of the retarded response function as a functional derivative of the time-evolved expectation value (SM Eq. S12). The test-pulse variation in Eqs. (8)-(9) is a finite-difference implementation of that derivative, and the Gaussian test pulses are introduced solely to localize the integrand; the resulting formulas are exact in the tau_A to infinity delta-function limit and approximate for finite tau_A. No parameter is fitted to the benchmark data: the first-, second-, and third-order response functions from time evolution are compared with independent perturbative expressions (SM Eqs. S52, S55, S57) derived from a standard commutator expansion of the von Neumann dynamics, and the only free choice in the comparison is the small imaginary broadening i*eta, which is set to the RTA relaxation rate Gamma and does not enter the derivation of the extraction formulas. The many-body iTEBD demonstration uses a Gaussian window for spectral resolution and a bond-dimension convergence check; this is a regularization and convergence issue, not a fitted parameter renamed as a prediction. The finite-tau_A approximation is a genuine accuracy limitation, producing 10-32% errors near Van Hove singularities, but that is an approximation error, not circular reasoning. The single self-citation [74] appears in a list of transient-response studies and is not load-bearing for any formula. The derivation is therefore self-contained.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

No new physical entities, particles, forces, or conserved quantities are postulated. The numerical scheme introduces several hand-chosen pulse parameters whose convergence is only partially demonstrated. The main domain assumptions are stationarity of the initial state, perturbative order separation, and the delta-function limit of the test pulses.

free parameters (6)

Probe amplitude F0 = 1e-6 (first order), 1e-4 and 1e-3 (second order), 1 and 10 (third order), 0.1 (iTEBD)
Chosen by hand to keep the response in the perturbative regime; F0 scaling is verified for selected cases but not exhaustively.
Test-function width tau_A = sqrt(2)/Gamma approximately 141 (main text), 12 (iTEBD)
Chosen large enough to approximate frequency delta functions; the finite width is a source of systematic error near Van Hove singularities.
Probe width tau_f = 0.1 (Rice-Mele) and 0.125 (iTEBD)
Chosen small enough so the probe spectrum is nearly constant over the relevant energy window.
Finite-difference step epsilon = 1e-6 to 1e-8
Chosen to balance truncation error against numerical precision; no systematic epsilon-convergence scan is reported.
Gaussian window width tau_w (iTEBD) = 6
Chosen for convergence in the many-body time evolution, since iTEBD has no intrinsic relaxation mechanism.
Bond dimension chi = 400, 800, 1200
Controls tensor-network truncation error; convergence with chi is shown for selected frequencies in Fig. 5.

assumptions (3)

domain assumption The initial state is stationary and response functions are time-translation invariant, so the frequency-domain expansion in Eq. (1) contains a delta function.
Invoked in SM Eq. (S2) to derive Eq. (1); without this assumption the simple frequency-domain functional derivative formulas do not apply.
domain assumption Each perturbative order of the response scales homogeneously with the field amplitude, so lower-order contributions can be subtracted and rescaled via Eq. (7).
Used to isolate the nth-order response in Eq. (6); requires sufficiently weak fields so nonperturbative corrections are negligible.
domain assumption Gaussian test functions are approximated as frequency delta functions when tau_A is sufficiently large (Eq. (12)).
This approximation is the basis of Eqs. (13) and (17); the paper states it but does not quantify the finite-width error.

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Cite this review

Pith. "Pith review of Extracting Nonlinear Dynamical Response Functions from Time Evolution." pith.science (2026). https://pith.science/paper/JCPS55UD

@misc{pith2026250707679,
  author       = {Pith},
  title        = {Pith review of: Extracting Nonlinear Dynamical Response Functions from Time Evolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCPS55UD}},
  note         = {Machine review of arXiv:2507.07679}
}

read the original abstract

We develop a general framework based on the functional derivative to extract nonlinear dynamical response functions from the temporal evolution of physical quantities, without explicitly computing multipoint correlation functions. We validate our approach by calculating the second- and third-order optical responses in the Rice-Mele model and further apply it to a many-body interacting system using a tensor network method. This framework is broadly applicable to any method that can compute real-time dynamics, offering a powerful and versatile tool for investigating nonlinear responses in dynamical systems.

Figures

Figures reproduced from arXiv: 2507.07679 by the authors.

**Figure 2.** FIG. 2. (a) Temporal profile of the electric current density [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Second variation of the current [Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Second-order response functions: (a) ¯ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Contour maps of ¯ [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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(S34) Here, /u1D6E9 denotes the unit step function, ensuring causality. Let us consider a free fermion system whose Hamiltonian H0 is given by H0 = /summationdisplay.1 /u1D456 /u1D457 ℎ/u1D456 /u1D457/u1D450† /u1D456/u1D450/u1D457= /summationdisplay.1 /u1D708 /u1D700/u1D708/u1...

Pith tools

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