{"id":"7a4eaf52-1bdf-4c6f-9bef-2ada7712b7f9","arxiv_id":"1908.01793","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The nonlinear optical response that drives the bulk photovoltaic effect contains an out-of-time-ordered correlator, and in the strong-driving limit the DC photocurrent scales linearly with the optical field amplitude.","lead":"This paper argues that the bulk photovoltaic effect, a light-induced DC current in non-centrosymmetric materials, cannot be fully described by equilibrium field theory once electron-phonon scattering is included. It derives new terms involving out-of-time-ordered correlators and predicts a linear-in-field photocurrent in the strong-driving limit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The necessity of χotoc for the nonlocal response rests on the unproven and likely false claim that χn,2 and χn,3 are local at DC; equilibrium diffusive response already yields nonlocal kernels of the form Eq. 19.","rationale":"The reader correctly identified the Sec. III B locality assumption as the weakest point, but treated it as a clarification issue that does not invalidate the central formalism. I disagree with that severity assessment: the locality claim is not merely unproven; it is inconsistent with standard equilibrium response in a diffusive plasma. Conservation laws produce hydrodynamic poles in retarded correlation functions, and these poles are present in Matsubara/analytic-continuation calculations. They generate exactly the nonlocal sgn kernel the paper attributes to χotoc. The semiclassical diffusion calculation in Sec. III B is itself a hydrodynamic calculation, so it is circular to use it to infer the presence of OTOC: the same diffusion equation can be derived from the density-density response of the finite-temperature electron-hole plasma. Therefore the abstract's central claim that the inhomogeneous response 'must involve OTOC' and 'cannot be computed by equilibrium field theory' is not established and is likely false. The master-equation and semiclassical results for the shift current itself may remain valuable, but the headline conceptual claim would need to be withdrawn or substantially qualified. I therefore recommend moving the verdict from CONDITIONAL to REJECT. The proposed Kubo computation with e-ph ladder corrections provides a concrete test: if it reproduces the diffusion kernel without χotoc, the locality premise is refuted and the central claim is invalidated.","tokens_in":18900,"tokens_out":20597,"duration_ms":242572,"concrete_test":"Compute the second-order DC current of the Rice-Mele model with electron-phonon coupling (parameters of Sec. V) from the standard Kubo double-commutator formula, using the time-ordered three-current correlation function with e-ph self-energy and ladder vertex corrections and no χotoc term, for a spatially inhomogeneous envelope A0(x)=A0 θ(L/2-|x|). If the resulting steady-state current reproduces the nonlocal diffusion kernel of Eq. 19 (or its finite-diffusion-length version), then the Sec. III B locality premise is false and χotoc is not required. A quicker analytic check is to evaluate the retarded density-density response χ(q,ω=0) at finite temperature in this model; a 1/(Dq^2) diffusion pole for q→0 would directly contradict the claimed DC locality of equilibrium time-ordered correlations.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim that inhomogeneous BPVE response 'must involve' χotoc and 'cannot be computed by equilibrium field theory' depends entirely on the Sec. III B assertion that χn,2 and χn,3 are local in space and time because they are related by analytic continuation to imaginary-time correlation functions that decay on correlation-length scales. This premise is load-bearing: if χn,2 or χn,3 can be nonlocal, the diffusion current of Eq. 19 can be described without χotoc. The premise is not derived, and it is in tension with standard equilibrium linear response in a system with a conserved density at finite temperature. In a diffusive electron-hole plasma, the retarded density-density response contains a hydrodynamic pole ~1/(Dq^2 - iω); the associated current response to a local source is nonlocal, with a kernel ~sgn(x-x') in one dimension after the long-time limit, which is precisely the form assigned to χotoc in Eq. 22. This pole is present in ordinary time-ordered/Matsubara correlation functions and in Kubo linear response; it does not require an out-of-time-ordered correlator. The paper's additional qualifier that time-ordered correlations are local 'at frequencies larger than temperature' is inapplicable here, because the diffusion current is a zero-frequency response and the semiclassical model explicitly invokes finite-temperature carriers (Eqs. 14-19). Moreover, Sec. II defines the response functions with ground-state expectation values, while Sec. III B switches to a finite-temperature plasma; for that plasma, equilibrium response functions are not local at DC. Thus the identification of χotoc as the sole carrier of nonlocal response is unsupported; if χn,3 (or the retarded combination of the χn,i terms) contains the diffusion pole, the paper's advertised conclusion fails.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19250,"tokens_out":12333,"duration_ms":129811,"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":[{"comment":"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.","section":"Sec. III B, Eqs. (15)-(22)"},{"comment":"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.","section":"Sec. II, Eq. (6)"},{"comment":"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.","section":"Sec. IV, Eq. (47)"},{"comment":"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.","section":"Sec. III B, Eqs. (16) and (21)"}],"minor_comments":[{"comment":"Ref. [8] lists 'Struman'; the correct spelling is 'Sturman'.","section":"References"},{"comment":"The abbreviation 'BVPE' appears in Sec. III C while the abstract and elsewhere use 'BPVE'; please standardize the terminology.","section":"Sec. III C"},{"comment":"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.","section":"Eq. (42)"},{"comment":"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.","section":"Fig. 4"},{"comment":"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.","section":"Sec. II, Eq. (6)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper contains a useful semiclassical and master-equation treatment of the shift current, but the central claim about the necessity of OTOCs is explicitly assumption-based and is in tension with standard diffusive linear response. I would ask the authors to either derive the locality of χn,2 and χn,3 or substantially soften the central claim; the paper may be publishable after such a revision, but not in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nYou should read this paper because it says something concrete about nonlinear optical response in driven semiconductors, not because its headline claim is right. The headline claim—that inhomogeneous BPVE must involve an out-of-time-ordered correlator and cannot be computed by equilibrium field theory—is probably wrong, and the stress-test note explains why.\n\nWhat is genuinely good: the decomposition of the second-order current into χn,2, χn,3 and χotoc is a useful organizing principle, and the OTOC term is not in the standard shift-current literature. The semiclassical exciton-dipole derivation is clean and correctly reproduces the known shift-current formula in the homogeneous limit. The Floquet master-equation calculation is independent, internally consistent, and produces a sharp, testable prediction: the DC current crosses from quadratic to linear scaling in the vector potential as the drive amplitude exceeds the phonon scattering rate. The Rice-Mele numerics support that crossover. That alone makes the paper worth referee time.\n\nThe soft spot is load-bearing. In Sec. III B the paper needs χn,2 and χn,3 to be local in space and time so that only χotoc can carry the nonlocal diffusion current (Eq. 22). The argument offered, citing Refs. 27-28, is that these correlators are related by analytic continuation to imaginary-time functions that decay on correlation-length scales. But the diffusion current is a zero-frequency response of a finite-temperature electron-hole plasma, and ordinary time-ordered/retarded response in a conserved-density system has a hydrodynamic diffusion pole. That pole produces exactly the sgn(x−x') kernel in Eq. 19. So the locality assertion fails precisely in the regime the paper needs it. There is also a mismatch: Sec. II defines the correlation functions in the ground state, while Sec. III B explicitly invokes finite-temperature carriers; the finite-temperature response functions are not local at DC.\n\nThe 'cannot be computed from equilibrium field theory' statement is too strong even setting the locality issue aside: Keldysh methods are standard, and the diffusion contribution is ordinary response, not an OTOC. The photon-counting argument in Sec. II is hand-wavy but not central.\n\nIn short: the formalism and the strong-drive crossover are valuable; the nonlocal-response/OTOC conclusion is not established. I would send it to review with a request to focus on Sec. III B, and I would not let the authors keep the 'must involve χotoc' claim without a much better argument.","headline":"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.","tokens_in":19797,"tokens_out":3172,"would_cite":true,"duration_ms":32220,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["bulk photovoltaic effect","shift current","out-of-time-ordered correlator","nonlinear optical response","electron-phonon scattering","Floquet master equation","semiclassical exciton dipole","diffusion current"],"falsifier":"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.","tokens_in":1855,"feed_emoji":"⚡","tokens_out":3608,"duration_ms":76499,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bulk photovoltaic effect demands out-of-time-ordered correlators","feed_subtitle":"With phonon scattering and uneven light, the DC shift current becomes nonlocal and escapes equilibrium response theory.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the quantized circular photogalvanic result whose shift-current form the semiclassical result is matched against.","marker":"[4]"},{"why":"Defines the shift-current formalism for the bulk photovoltaic effect that the homogeneous semiclassical result reproduces.","marker":"[12]"},{"why":"Established the intrinsic nonlinear-response origin of the bulk photovoltaic effect, extended here by the master-equation analysis.","marker":"[13]"},{"why":"Gives the Berry-curvature expression for shift current used to connect the response to resonant optical transitions.","marker":"[14]"},{"why":"Introduced Keldysh and out-of-time-ordered techniques that the paper invokes for correlators beyond equilibrium diagrams.","marker":"[24]"},{"why":"Provides the Rice-Mele tight-binding model used for the numerical crossover study.","marker":"[25]"},{"why":"Supplies the analytic-continuation argument that imaginary-time correlation functions decay on correlation-length scales, supporting the locality claim for chi_n,2 and chi_n,3.","marker":"[27]"},{"why":"Backs the statistical-mechanics statement that equilibrium correlation functions are local, the premise behind attributing the nonlocal response to chi_otoc.","marker":"[28]"},{"why":"Provides the Floquet master-equation method used to derive the steady-state density matrix and current.","marker":"[30]"}],"fun_headline_variants":["Shift current is intrinsically out-of-time-ordered","Bulk photovoltaic effect defies equilibrium response","Nonlocal shift current demands out-of-time-ordered correlators","Nonlinear optics? It's a nonequilibrium effect","OTOCs are mandatory for shift current under phonons"],"cache_read_input_tokens":21888,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Shift current is intrinsically out-of-time-ordered","Bulk photovoltaic effect defies equilibrium response","Nonlocal shift current demands out-of-time-ordered correlators","Nonlinear optics? It's a nonequilibrium effect","OTOCs are mandatory for shift current under phonons"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3457,"prompt_tokens":971,"completion_tokens":2486,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":2410}},"tokens_in":587,"tokens_out":2486,"duration_ms":21485,"temperature":1.0,"reasoning_tokens":2410,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:05:17.959040+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the quantized circular photogalvanic result whose shift-current form the semiclassical result is matched against."},{"cited_title":"Sipe and A.I","cited_arxiv_id":null,"evidence_quote":"Established the intrinsic nonlinear-response origin of the bulk photovoltaic effect, extended here by the master-equation analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Berry-curvature expression for shift current used to connect the response to resonant optical transitions."},{"cited_title":"Ogawa, M","cited_arxiv_id":null,"evidence_quote":"Introduced Keldysh and out-of-time-ordered techniques that the paper invokes for correlators beyond equilibrium diagrams."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Rice-Mele tight-binding model used for the numerical crossover study."},{"cited_title":"private communication","cited_arxiv_id":null,"evidence_quote":"Supplies the analytic-continuation argument that imaginary-time correlation functions decay on correlation-length scales, supporting the locality claim for chi_n,2 and chi_n,3."},{"cited_title":"Quantum Phase Transitions","cited_arxiv_id":null,"evidence_quote":"Backs the statistical-mechanics statement that equilibrium correlation functions are local, the premise behind attributing the nonlocal response to chi_otoc."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Floquet master-equation method used to derive the steady-state density matrix and current."}],"review_version":1}