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REVIEW 3 major objections 6 minor 34 references

Linear-response theory in Floquet systems

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The photocurrent in a Floquet photoemission experiment is proportional to the imaginary part of the $l=0$ component of the Floquet Green function.

desk verdict Sound wave-function derivation of Floquet linear response and golden rule, but the photoemission identification with -Im G^R_00 is asserted rather than derived, and the numerics are not reproducible. read the letter →

arxiv 2506.07788 v1 pith:2E5YJXUC submitted 2025-06-09 cond-mat.str-el

Ayan Pal , Erik G. C. P. van Loon , Ferdi Aryasetiawan This is my paper
classification cond-mat.str-el
keywords FloquettheorylinearresponseFermigoldenrulephotoemissionspectrumGreenfunctionspectralperiodicallydrivensystemstrARPES
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

This paper aims to put linear response of periodically driven many-electron systems on the same footing as equilibrium Kubo theory. It treats a strong time-periodic drive exactly and a second, weak, non-periodic probe to first order, and derives a Floquet Fermi golden rule: transition rates peak at frequency differences $\omega + n\Omega + \Delta$. Applied to photoemission under the sudden approximation, the result ties the measured photocurrent to the imaginary part of a specific component ($l=0$) of the Floquet Green function, so time-resolved ARPES spectra of driven materials can be read as a positive spectral function. The paper illustrates the formalism on a driven electron gas and shows how drive parameters control the sideband structure.

What carries the argument

The working tool is the Floquet picture, an interaction picture built from the evolution operator of the periodic Hamiltonian alone, so that the weak probe enters as $\hat{\Xi}_F(t) = \hat{U}_0(t_0,t)\hat{\Xi}(t)\hat{U}_0(t,t_0)$. The argument is carried by expanding the periodic Floquet-mode matrix elements $\langle\Phi_\beta(t)|\hat{\Xi}|\Phi_\alpha(t)\rangle$ in Fourier series $\sum_n f^{\beta\alpha}_n e^{in\Omega t}$; the time integrals then collapse to delta functions at $\omega + n\Omega + \Delta_{\beta\alpha}$. The photoemission result additionally uses the sudden approximation, factorizing the final Floquet mode as $\hat{c}^\dagger_k |\Phi^{N-1}_\gamma(t)\rangle$, which reduces the golden rule to Eq. (60). This decomposition is what makes the photocurrent equal to a definite component of the Floquet Green function.

What would settle it

A direct calculation of the full photoemission current for a small driven system, for example a few-site driven Hubbard chain, that keeps the photoelectron coupled to the remaining electrons during escape, compared with Eq. (60), would settle the sudden approximation: any sideband that shifts or gains weight beyond the predicted positions $\omega - \epsilon_k - m\Omega - (E_\gamma^{N-1} - E_\alpha)$ would falsify the reduction of the photocurrent to the $l=0$ Floquet Green-function component.

Watch

Extended reading notes

Core claim

In the Floquet picture, the paper's central result is Eq. (60): the photocurrent $I(\epsilon_k)$ is proportional to $\sum_{\gamma m} |f^{k\alpha\gamma}_m|^2 \delta(\omega - \epsilon_k - m\Omega - (E_\gamma^{N-1} - E_\alpha))$, and this expression is proportional to the imaginary part of the $l=0$ component of the Floquet Green function. Since that component is positive definite, the photoemission spectrum is a genuine spectral function rather than a signed quantity. The Floquet generalization of Fermi's golden rule that precedes it shows transitions occur at $\omega + n\Omega + \Delta_{\beta\alpha}$, with extra cross-term resonances when $2\Delta_{\beta\alpha}/\Omega$ is an integer, which the paper calls Umklapp in frequency space. The application to a periodically driven noninteracting electron gas illustrates how the ratio of the drive phase velocity $\Omega/\kappa$ to the Fermi velocity $v_F$ selects between Floquet sidebands and Bloch gap physics.

Load-bearing premise

The load-bearing premise is the sudden approximation, that the outgoing photoelectron's kinetic energy is large enough that it separates cleanly from the remaining $N-1$-electron Floquet system, together with the assumption that the initial state is a single pure Floquet mode, so mixed states and finite temperature are outside the claimed result.

Editorial extensions

If this is right

  • Time-resolved ARPES on a periodically driven material can be interpreted directly: the measured photocurrent at each momentum is proportional to the positive $l=0$ Floquet spectral function.
  • The Floquet Fermi golden rule places photoemission sidebands at $\omega - \epsilon_k - m\Omega - (E_\gamma^{N-1} - E_\alpha)$; drive frequency and amplitude therefore tune sideband positions and weights, a route to Floquet engineering of spectra.
  • When $2\Delta_{\beta\alpha}/\Omega$ is an integer, interference between absorption and emission amplitudes produces additional Umklapp resonances in frequency space that should be observable as extra lines.
  • In the high-frequency limit the spectrum shows Floquet sidebands, in the low-frequency limit a Bloch gap of size approximately $2V_0$ appears, and in the intermediate regime the traveling-wave drive breaks Kramers degeneracy; the ratio $\Omega/\kappa$ versus $v_F$ controls which regime is realized.

Reading between the lines

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

  • Iterating the Floquet-picture perturbation expansion beyond first order would yield nonlinear response tensors for periodically driven systems, connecting this Kubo analogue to pump-probe and harmonic-generation experiments.
  • Because only the $l=0$ component of the Floquet Green function enters the photoemission formula, angle-integrated spectra may hide Floquet physics living in off-diagonal frequency sectors; phase-resolved or two-time measurements would be needed to expose those sectors.
  • A density-matrix version of the Floquet golden rule would extend the result to thermal and mixed-state initial conditions, a natural next step since the paper explicitly starts from a pure Floquet mode.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper develops a linear-response formalism for many-body Floquet systems using an interaction (Floquet) picture in which the time-periodic Hamiltonian H0(t) is treated exactly and a non-periodic probing field Ξ(t) is treated to first order. It derives a retarded density-response function (Eq. 33), a Floquet Fermi golden rule (Eqs. 49-51), and, under the sudden approximation, a formula for the photocurrent from an N-electron Floquet state (Eq. 60). The authors then claim that this photocurrent is proportional to the imaginary part of the l=0 component of the Floquet Green function, and they illustrate the formalism with calculations for a noninteracting electron gas driven by a traveling-wave potential.

Significance. If the central identification between the photocurrent and the Floquet Green function is valid, the paper would provide a useful formal bridge between time-resolved photoemission of driven materials and the Floquet spectral function, whose positivity was established in Ref. 16. The linear-response derivation in Sec. III is internally consistent and parameter-free, and the Floquet golden rule is a clean extension of standard results. However, the key link to the retarded Green function is asserted rather than derived, and this gap is load-bearing for the main conclusion. The numerical application in Sec. IV is also too terse to reproduce. The paper is promising, but the central claim needs to be either proven or explicitly qualified.

major comments (3)
  1. [Sec. III.C, Eq. (60)] The identification of Eq. (60) with the imaginary part of the l=0 component of the Floquet Green function is not established by the preceding derivation. Equation (60) sums over the removal amplitudes f^{kαγ}_m, i.e., transitions from the N-particle Floquet state |Φ_α(t)> to (N−1)-particle states. The retarded Floquet Green function, however, contains both removal (N→N−1) and addition (N→N+1) Lehmann contributions. In a generic pure Floquet state there is no equilibrium-like particle-hole separation, because quasienergies are only defined modulo Ω; an addition resonance at ω = E^{(N+1)}_δ − E^{(N)}_α + nΩ can coincide with a removal resonance at ω = E^{(N)}_α − E^{(N−1)}_γ + mΩ for suitable integers n,m. Therefore, without an additional occupation/filtering argument or an explicit restriction to a state in which the addition terms vanish, the claim that the photocurrent is proportional to −Im G^R_{00} does not follow. The authors should derive the relation to the lesser (particle-removal) Floquet Green function and state under which conditions it equals −Im G^R_{00}, or amend the claim.
  2. [Sec. III.C, Eq. (57)] The sudden-approximation factorization |Φ_β(t)> = c_k^† |Φ_γ^{N−1}(t)> is an extra physical input beyond the perturbation expansion in Sec. III.B. It is valid only when the photoelectron kinetic energy is large compared to the binding energy; the paper acknowledges this in the text but does not state the resulting quantitative limitation on Eq. (60). Because this factorization is what converts the general golden rule into a single-particle removal rate, the scope of the photoemission result — for example, its failure at low photoelectron kinetic energies or for strongly correlated initial states — should be stated explicitly in the conclusion.
  3. [Sec. IV] The illustrative calculation for the traveling-wave driving field V(x,t)=V0 cos(κx−Ωt) is described only verbally. The 'combined Bloch-Floquet formalism' is not presented, and no equations are given for the Green function, the photocurrent, or the numerical procedure used to produce Fig. 2. As a result, the numerical results cannot be reproduced or checked. Please provide the defining equations, the Brillouin-zone parameters, and the discretization/integration scheme, and clarify how Eq. (60) is used to compute I(ε_k).
minor comments (6)
  1. [Sec. III.A, Eq. (34)] There is a typo: 'sates' should be 'states'. Also, the time t0 in Eq. (34) is not explicitly defined; it should be stated that the initial Floquet mode is taken at t0.
  2. [Sec. III.B, Eqs. (40)-(51)] The Fourier index for the b amplitude is written as n in Eqs. (44)-(47), although the conjugate expansion in Eq. (43) carries −m. The reindexing should be made explicit; otherwise the cross-term condition n = −m − 2M in Eq. (51) is difficult to follow.
  3. [Sec. III.B, Eq. (51)] The condition for non-vanishing cross terms should be re-derived. As written, the two delta-function conditions in the long-time limit appear to require simultaneous resonance conditions that are inconsistent with the Fourier indices in Eqs. (40)-(47).
  4. [Sec. IV.B] There is a typo: 'band gand' should be 'band gap'. Also, the statement that the low-frequency limit 'returns to a usual time-independent Bloch system' should be qualified, since the drive amplitude V0 remains nonzero and only the spatial periodicity dominates.
  5. [Sec. IV, Fig. 2] The vertical axis is labeled only as I(a.u.); please specify the normalization of the photocurrent and, if possible, the units of the horizontal axis relative to the Fermi energy.
  6. [Sec. II] The sentence 'exp(−iεαT) can be regarded as a one-dimensional irreducible representation of the element T of the Abelian time translation group' is standard but would benefit from a more precise statement, since the representation is of the cyclic subgroup generated by T rather than the full time-translation group.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Floquet golden-rule and photoemission derivations are self-contained; the Green-function identification uses an external cited result, not the paper's own inputs.

full rationale

The paper derives the Floquet interaction picture, the first-order linear response, the Floquet generalization of Fermi's golden rule, and the photoemission current directly from the time-dependent Schrödinger equation and the Floquet-mode expansion of the unperturbed evolution operator in Eq. (16). No parameter is fitted from the quantity being predicted, and no equation is defined in terms of the result it is used to derive. The one load-bearing external input is the identification of the right-hand side of Eq. (60) with the l=0 component of the imaginary part of the Floquet Green function, attributed to Ref. [16]; that is an independent cited result rather than a self-citation, and it is invoked as supporting identification, not as an input that constructs the derivation. The concern that a general pure Floquet state may lack the occupation/addition filtering needed for the removal-only photocurrent to equal -Im G^R is a potential rigor or correctness issue, not a circularity: it does not make Eq. (60) equivalent to its inputs by construction. The derivation chain is therefore not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation is parameter-free in its formal part; no free parameters are fitted to data. The illustrative calculation uses arbitrary physical parameters (V0, kappa, Omega) but these are not part of the central claim. Five assumptions are identified: completeness of many-body Floquet modes, the sudden approximation, first-order light-matter coupling, a pure Floquet initial state, and the standard asymptotic identity for the long-time limit. No new entities are introduced.

assumptions (5)
  • domain assumption Completeness and orthonormality of many-body Floquet modes (Eq. 16)
    The time-evolution operator U0 is expanded in a complete set of Floquet modes |Phi_alpha(t)> for the many-body Floquet Hamiltonian. This is assumed without proof; for interacting many-body systems, completeness and the absence of resonances or heating are non-trivial.
  • domain assumption Sudden approximation for photoemission (Eq. 57)
    The final Floquet state is factorized as c_k^dagger |Phi_gamma^{N-1}(t)>, decoupling the outgoing electron. This is valid only for high photoelectron kinetic energies; the paper states this but does not quantify the regime.
  • domain assumption First-order perturbation and neglect of A^2 term (Eq. 52)
    The light-matter coupling is truncated at linear order in the vector potential; the A^2 term is neglected as a two-photon process. This is standard but restricts the result to weak fields.
  • domain assumption Pure Floquet state initial condition (Sec. III.A)
    The response is computed from |Psi_F(t0)> = |Phi_alpha(t0)>. Real driven systems may be in mixed or prethermal states; the paper does not address statistical averaging.
  • standard math Long-time limit sin^2(xt)/x^2 -> pi t delta(x)
    Used to obtain the Floquet golden rule Eqs. (49)-(51). This is standard asymptotic distribution theory, but requires t large compared to inverse level spacings.

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Cite this review

Pith. "Pith review of Linear-response theory in Floquet systems." pith.science (2026). https://pith.science/paper/2E5YJXUC

@misc{pith2026250607788,
  author       = {Pith},
  title        = {Pith review of: Linear-response theory in Floquet systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2E5YJXUC}},
  note         = {Machine review of arXiv:2506.07788}
}
read the original abstract

Nonequilibrium quantum physics greatly simplifies in the case of time-periodic Hamiltonians, since Floquet theory provides an analogue to Bloch's theorem in the time domain. Still, the formal properties of Floquet many-body theory remain underexplored. Here, we develop linear response theory for Floquet systems, in the sense that we have a time-periodic potential of arbitrary strength and a perturbatively small but non-periodic probing field. As an application, we derive the analogy of Fermi's Golden Rule and the photoemission spectrum of a many-electron system. As in the equilibrium case, the latter is related to the spectral function which is positive definite. We also analyze the parameter dependence of the controllable photoemission spectra by virtue of Floquet engineering.

Figures

Figures reproduced from arXiv: 2506.07788 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of the formalism followed in this [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Spectral representation of photoelectric current of a noninteracting electron gas driven by an external periodic field [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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