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REVIEW 4 major objections 5 minor 31 references

Non-equilibrium nature of non-linear optical response: Application to the bulk photo voltaic effect

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Pith's one-line read This paper claims that the bulk photovoltaic effect, once phonon scattering and uneven illumination are included, requires an out-of-time-ordered three-operator correlator that equilibrium field theory cannot compute.

desk verdict Strong formalism and a clean strong-drive prediction, but the advertised OTOC necessity for inhomogeneous response rests on a locality premise that is almost certainly false. read the letter →

arxiv 1908.01793 v2 pith:HITVAIDB submitted 2019-08-05 cond-mat.mes-hall

Tamoghna Barik , Jay D. Sau This is my paper · ORCID
classification cond-mat.mes-hall
keywords bulkphotovoltaiceffectshiftcurrentout-of-time-orderedcorrelatornonlinearopticalresponseelectron-phononscatteringFloquetmasterequationsemiclassicalexcitondipolediffusion
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 argues that the bulk photovoltaic effect—a DC current generated by light in non-centrosymmetric materials—cannot in general be described by equilibrium response theory. Expanding the second-order current response with electron-phonon scattering included, the authors find three operator correlation functions, one of which is not time-ordered and therefore cannot be evaluated by equilibrium Feynman diagrams. Using a semiclassical exciton-dipole picture and a Floquet quantum master equation, they show that the homogeneous shift current matches the known noninteracting result, but that the scattering rate enters implicitly and the DC response scales linearly with field amplitude in the weak-scattering limit. For spatially inhomogeneous illumination, the steady state requires a nonlocal diffusion current, and the authors conclude that this nonlocal piece must come from the out-of-time-ordered correlator. The stakes are that standard shift-current calculations miss a genuinely nonequilibrium part of the photovoltaic response.

What carries the argument

The central object is the second-order current response written as four terms: linear response, two time-ordered nonlinear terms $\chi_{n,2}$ and $\chi_{n,3}$, and an out-of-time-ordered term $\chi_{\mathrm{otoc}}$ built from three current operators with a non-time-ordered product. The paper's arguments are carried by three tools: a semiclassical exciton-dipole picture in which phonon scattering interrupts coherent electron-hole pair evolution as a Poisson process, a Floquet quantum master equation that yields steady-state density matrices in the Floquet basis, and a diffusion-equation analysis for inhomogeneous illumination. The out-of-time-ordered term is the load-bearing piece: because $\chi_{n,2}$ and $\chi_{n,3}$ are argued to be local from analytic continuation of imaginary-time correlations, the nonlocal diffusion current is attributed to $\chi_{\mathrm{otoc}}$, and the paper uses the semiclassical solution to constrain its long-range form.

What would settle it

Compute, on the Rice-Mele model with a phonon bath, all three nonlinear response kernels in Eq. (2) using a Keldysh scheme and check whether the nonlocal part of the steady-state DC current is carried by $\chi_{\mathrm{otoc}}$, or by $\chi_{n,2}$ and $\chi_{n,3}$; experimentally, scanning the carrier-density profile under a tightly focused beam and comparing the diffusion current with Eq. (22) would test the same attribution.

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Extended reading notes

Core claim

The central claim is that the nonlinear optical response contains three distinct three-operator correlation functions—$\chi_{n,2}$, $\chi_{n,3}$, and $\chi_{\mathrm{otoc}}$—and that $\chi_{\mathrm{otoc}}$ is an out-of-time-ordered correlator that equilibrium field theory cannot compute. For the bulk photovoltaic effect with electron-phonon interactions, the shift current in a homogeneous excitation profile is reproduced by the dipole moment of optically generated excitons and agrees with the noninteracting shift-current formula in the zero-scattering limit. The master-equation treatment shows a quadratic-to-linear crossover in the vector-potential amplitude, with the electron-phonon scattering rate setting the crossover scale. In an inhomogeneous excitation profile, charge conservation forces a diffusion current that cancels the local shift current in steady state; because the time-ordered correlators are argued to be spatially local, the paper concludes that this strongly nonlocal DC response must be carried by $\chi_{\mathrm{otoc}}$, making the nonlinear response an intrinsically nonequilibrium object rather than an equilibrium Kubo-style response.

Load-bearing premise

The argument depends on the assertion that the time-ordered correlators $\chi_{n,2}$ and $\chi_{n,3}$ are spatially local because equilibrium analytic continuation says so; if they are not local in the driven, phonon-coupled system, the nonlocal diffusion current need not come from the out-of-time-ordered correlator.

Editorial extensions

If this is right

  • Under inhomogeneous illumination, the bulk photovoltaic response cannot be obtained from equilibrium response functions; a Keldysh-type nonequilibrium computation is required.
  • The steady-state density profile of photoexcited carriers encodes the shift current, so the current can be measured capacitively without electrical contacts.
  • In the weak-scattering regime the DC shift current scales linearly with the vector potential rather than quadratically, with the electron-phonon scattering rate setting the crossover between the two scalings.
  • The homogeneous-excitation shift current reduces to the established shift-current formula in the zero-scattering limit, confirming consistency with earlier noninteracting results.
  • Energy-conservation considerations imply that the DC current in a photovoltaic device is accompanied by phonon-mediated dissipation, so efficiency estimates must include scattering.

Reading between the lines

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

  • A natural extension is to compute the full Keldysh three-operator response for a microscopic model with phonons; if the nonlocal response decomposes differently, the locality assumption on $\chi_{n,2}$ and $\chi_{n,3}$ would be falsified.
  • The same three-correlator structure should appear in other nonlinear probes, such as pump-probe spectroscopy and superconducting nonlinear responses, where equilibrium diagrammatics would similarly miss the out-of-time-ordered part.
  • Because the nonlocal diffusion current is linked to the density profile, spatially resolved imaging of carrier density under a tightly focused beam could provide a contact-free experimental test of the out-of-time-ordered contribution.
  • The linear-in-field DC response at weak scattering suggests that photovoltaic devices in clean, strongly driven materials could behave differently from the perturbative $A^2$ scaling, with implications for power extraction at high intensity.
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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

4 major / 5 minor

Summary. The paper studies the bulk photovoltaic effect (BPVE) in a two-band semiconductor with electron-phonon scattering. It decomposes the second-order current response into linear, time-ordered, and 'out-of-time-ordered' correlation functions, argues that the last cannot be computed from equilibrium field theory, and develops a semiclassical model in which the DC shift current is generated by the dipole moment of photo-excited electron-hole pairs and interrupted by phonon scattering. The semiclassical result is shown to reduce to the known shift-current formula in the homogeneous limit. A Floquet quantum master equation treatment reproduces the small-amplitude shift current and predicts a crossover from quadratic to linear scaling of the DC current with vector-potential amplitude as the scattering rate is decreased. For inhomogeneous illumination, the paper derives a nonlocal diffusion current and claims that this nonlocality requires the out-of-time-ordered correlator χotoc.

Significance. If the central claim about χotoc were established, the paper would point to a qualitative limitation of equilibrium response theory for nonlinear transport in open systems and would be of broad interest in nonlinear optics and photovoltaics. The paper also contains useful independent derivations: the semiclassical exciton-dipole picture is benchmarked against the conventional shift-current expression, the Floquet master equation provides an alternative derivation of the DC current, and the numerical crossover from quadratic to linear scaling in the Rice-Mele model is clearly presented. These parts are internally consistent and give a plausible physical picture. However, the headline claim about the necessity of OTOCs for inhomogeneous excitation is not currently supported, as detailed below.

major comments (4)
  1. [Sec. III B, Eqs. (15)-(22)] The assertion that the nonlocal diffusion current 'must involve' χotoc is not established. Equation (19) is obtained from the ordinary linear diffusion equation (15), whose Green's function produces the kernel sgn(x−x') in the long-time limit; this is the standard hydrodynamic diffusion pole of a conserved density, which appears in retarded and time-ordered equilibrium response functions and does not require an out-of-time-ordered correlator. The preceding paragraph argues from Refs. [27,28] that χn,2 and χn,3 are local, but the argument is not derived for the present driven, finite-temperature, zero-frequency situation, and the text explicitly labels the locality as an assumption ('Assuming that χn,2 and χn,3 are local...'). Since the conclusion that χotoc must carry the nonlocal response follows only from this assumption, the central claim is unsupported. Please provide a direct calculation of the nonlocal parts of χn,2 and χn,3, or a controlled argument for their locality in the relevant limit; otherwise the abstract's statement about 'must involve' should be withdrawn.
  2. [Sec. II, Eq. (6)] The correlator in Eq. (6) is a three-point function with a specified operator ordering; it is not an out-of-time-ordered correlator in the sense of nested commutators and Lyapunov growth, and it can be evaluated in the Keldysh formalism. The statement that this term 'cannot be computed from equilibrium field theory' is stronger than what is shown: equilibrium finite-temperature response is not limited to the imaginary-time Matsubara channel, and time-ordered/retarded correlators can be obtained by analytic continuation. If the intended claim is that the zero-frequency DC limit is not accessible from the imaginary-time functional integral, that limitation should be stated precisely and proven. As written, the dichotomy between time-ordered equilibrium correlators (χn,2, χn,3) and the non-equilibrium χotoc is not justified.
  3. [Sec. IV, Eq. (47)] The linear-in-A0 scaling in the weak-scattering limit is a headline result, but the double limit in Eq. (47) is not defined unambiguously: taking A0|J⊥0,k| → 0 while requiring p,q ≪ A0|J⊥0,k| requires a specified ordering of the two small parameters. As written, the linear term could be an artifact of taking p,q to zero first. Please state the limiting procedure precisely (for example, a fixed small ratio p/A0 or p,q → 0 at fixed nonzero A0) and demonstrate that the linear scaling survives a controlled A0 → 0 limit. Without this, the claim that the result 'contradicts Kubo formula' is not fully supported.
  4. [Sec. III B, Eqs. (16) and (21)] There is a sign inconsistency in the diffusion-current derivation. With JDC_shift = αA0^2, Eq. (16) should read f(x) = −∂xJDC_shift = −2αA0∂xA0; the printed expression has the opposite sign. Equation (21) is consistent with the corrected sign after integration by parts, but not with the printed Eq. (16). Please correct the sign and confirm that Eq. (20), JDC_shift = −Jd = D∂xρ, follows. This is a technical error, but it occurs in the derivation that underpins the nonlocal-response conclusion.
minor comments (5)
  1. [References] Ref. [8] lists 'Struman'; the correct spelling is 'Sturman'.
  2. [Sec. III C] The abbreviation 'BVPE' appears in Sec. III C while the abstract and elsewhere use 'BPVE'; please standardize the terminology.
  3. [Eq. (42)] The expression '1−ι + ...' in Eq. (42) appears to contain a typo; the imaginary unit should be written consistently with the rest of the paper.
  4. [Fig. 4] The caption of Fig. 4 should state explicitly that kD is the degeneracy-point momentum defined in the text, rather than leaving this to the body.
  5. [Sec. II, Eq. (6)] The term 'out-of-time-ordered correlator' is used for a three-point function; consider adding a sentence distinguishing it from the standard four-point OTOC used in many-body chaos.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the response decomposition, semiclassical benchmark, and Floquet master-equation derivation are self-contained.

full rationale

The paper's central results are derived from an explicit second-order Dyson expansion (Appendix A) rather than assumed. The decomposition into χlin, χn,2, χn,3, and χotoc follows from expanding the time-evolution operator and its Hermitian conjugate; χotoc is not defined as a nonlocal remainder but as the specific three-operator correlator in Eq. 6. The semiclassical shift current in Eq. 12 is benchmarked in Appendix B against the established shift-current expression, and the Floquet master-equation treatment in Secs. IV–V is an independent derivation that reproduces the semiclassical result in the weak-field limit and predicts the scaling crossover. No parameter is fitted to data and no prior result by the same authors is invoked. The claim that χn,2 and χn,3 are local is an explicit assumption ('Assuming that χn,2 and χn,3 are local for the reasons outlined in the previous paragraph...'), used to attribute nonlocality to χotoc; this is a substantive premise whose correctness may be questioned, but it is not circular because the form of χotoc in Eq. 22 is a consistency constraint derived from the diffusion model, not an input to it. Therefore no circularity is found.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on a few well-flagged physical approximations: Markovian phonons, symmetric e-ph coupling, and a two-band model. The most fragile input is the assumed locality of χn,2 and χn,3, which is not derived and is used to identify χotoc as the source of non-local response.

free parameters (2)
  • p and q (electron and hole phonon scattering rates)
    Effective scattering rates introduced in the Floquet master equation (Eq. 39). They are physical parameters, not fitted to data; the central scaling results hold for any values satisfying the stated limits.
  • η (semiclassical scattering rate)
    Rate of phonon-scattering Poisson process in the semiclassical model (Eq. 10). It is a model parameter that sets the broadening of the shift-current resonance; the final result is independent of its exact value after the η→0 limit.
assumptions (4)
  • domain assumption Electron-phonon interaction is symmetric and does not directly contribute to the shift current.
    Stated in Sec. I as the setup for a 'minimally interacting' system; ensures the photocurrent comes from the intrinsic shift mechanism rather than extrinsic asymmetric scattering.
  • domain assumption The phonon bath is Markovian, at zero temperature, and has a constant density of states G.
    Used in Sec. IV and Appendix D to derive the master equation (Eq. 32). These are standard but nontrivial approximations for driven-dissipative systems.
  • ad hoc to paper The time-ordered response functions χn,2 and χn,3 are local in space and time for the driven system.
    Assumed in Sec. III B (after Eq. 21) to conclude that the non-local diffusion current must be carried by χotoc. The paper argues from equilibrium analytic-continuation intuition, but the system is out of equilibrium, so this is an unverified assumption.
  • domain assumption A0|J⊥| is small compared to the electronic bandwidth, so Floquet perturbation theory applies.
    Stated in Sec. IV A to linearize the master equation in the occupancy deviations; required for the analytic expressions in Appendix F.

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Cite this review

Pith. "Pith review of Non-equilibrium nature of non-linear optical response: Application to the bulk photo voltaic effect." pith.science (2026). https://pith.science/paper/HITVAIDB

@misc{pith2026190801793,
  author       = {Pith},
  title        = {Pith review of: Non-equilibrium nature of non-linear optical response: Application to the bulk photo voltaic effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HITVAIDB}},
  note         = {Machine review of arXiv:1908.01793}
}
read the original abstract

The bulk photovoltaic effect is an example of a non-linear optical response that leads to a DC current that is relevant for photo-voltaic applications. In this work, we theoretically study this effect in the presence of electron-phonon interactions. Using the response function formalism we find that the non-linear optical response, in general, contains three operator correlation functions, one of which is not ordered in time. This latter correlator cannot be computed from equilibrium field theory. Using a semiclassical approach instead, we show that the bulk photovoltaic effect can be attributed to the dipole moment of the generated excitons. We then confirm the validity of the semiclassical result (which agrees with the non-interacting result) for non-linear DC response from a quantum master equation approach. From this formalism we find that, in contrast to usual linear response, the scattering rate has a strong implicit effect on the non-linear DC response. Most interestingly, the semiclassical treatment shows that the non-linear DC response for spatially inhomogeneous excitation profiles is strongly non-local and must involve the aforementioned out-of-time-ordered correlators that cannot be computed by equilibrium field theory.

Figures

Figures reproduced from arXiv: 1908.01793 by the authors.

Figure 1
Figure 1. FIG. 1: Two step process determining DC response: [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: In the optically excited region, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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