{"id":"d9b3b408-6bda-4a12-9390-c0d860d520ab","arxiv_id":"2412.19205","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Phonon cooling stabilizes a long-lived prethermal eta-paired superconducting-like state in photodoped Mott insulators, and steady-state DMFT reproduces its spectral properties.","lead":"Using computer simulations, this paper shows that coupling a laser-excited Mott insulator to a cold phonon bath can keep a transient superconducting-like order alive for thousands of hopping times. It also shows that a cheaper steady-state simulation method reproduces this long-lived state, which could speed up searches for hidden light-induced orders.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The long-time eta-SC plateau is not verified against the memory-truncation cutoff; the paper reports no tc value or convergence study, so the central quasi-steady-state claim rests on an unchecked approximation.","rationale":"In good faith, the paper's central claim is physically plausible: phonon cooling suppresses heating, and large U slows doublon recombination, so a long-lived prethermal eta-SC state is a reasonable expectation. The U-dependence and the contrast between runs with and without phonon coupling show coherent qualitative trends. However, the specific assertion of a quasi-steady state sustained beyond 1000 inverse hoppings depends on the ability of the memory-truncated DMFT solver to propagate accurately to those times. The paper does not state the cutoff tc, does not show a tc-convergence study, and uses the same NCA impurity solver in both the real-time and NESS calculations. The NESS comparison is fitted, so it is not an independent validation. These are missing checks rather than demonstrated errors, which is exactly why the appropriate verdict remains conditional rather than acceptance. Supplying a tc-convergence test and one higher-order impurity-solver cross-check would substantially strengthen the long-time conclusion.","tokens_in":12707,"tokens_out":5498,"duration_ms":60111,"concrete_test":"Report the tc used for Fig. 5(b) and rerun the U=10, d~0.4 phonon-coupled simulation with tc doubled and quadrupled, monitoring eta_x(t), d(t), and A(omega) at t=2000. If eta_x at t=1000-2000 shifts by more than ~0.02, or if the plateau slope changes sign, the quasi-steady state is a truncation artifact; if the results are invariant, the memory-truncation concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central long-time result is produced with memory-truncated Kadanoff-Baym equations. Sec. II states 'the self energy is assumed to decay within a cutoff window tc on the real time axis,' but no tc value or convergence test appears anywhere in the paper. This matters because the phonon self-energy (Eq. 7) contains the undamped free-boson propagator D(t,t'), which oscillates rather than decays; the electron-phonon memory kernel can only decay through G(t,t'). In the eta-SC state, the anomalous component of G is precisely the long-lived, symmetry-broken quantity that controls the plateau in Fig. 5(b). If tc is shorter than the decay time of that component, the truncation can artificially force the system into a steady state. The NESS comparison in Sec. III D cannot rescue this: the fermion-bath parameters are adjusted to reproduce the real-time effective temperature and doublon density, so the agreement is a consistency check, not an independent confirmation. Reproducibility is also compromised by an internal inconsistency in the phonon frequency (omega0=0.4 in Sec. II and III D, omega0=0.2 in Sec. III A).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The authors study the long-time dynamics of photo-doped Mott insulators using nonequilibrium DMFT with memory-truncated Kadanoff-Baym equations and the NCA impurity solver, after preparing photodoped states via entropy cooling with chirped pulses. They show that without phonon coupling the η-pairing order decays due to heating, while coupling to a cold phonon bath can cool the carriers and, for U≳10 with doping d∼0.4, stabilize ηx at a plateau that persists beyond t=2000 (in units of inverse hopping). They further argue that the resulting quasi-steady state is well reproduced by a steady-state (NESS) DMFT calculation with suitably adjusted fermion-bath parameters.","tokens_in":12897,"tokens_out":5765,"duration_ms":54120,"significance":"If the central claim holds, the paper establishes a concrete mechanism — dissipative phonon cooling — by which light-induced η-paired superconducting order can be made long-lived in large-gap Mott insulators, and it demonstrates that the cheaper NESS formalism can capture the relevant quasi-steady state. The paper's strengths are its systematic parameter scans in U and photodoping, the two-time spectral-function analysis including the anomalous component, and the use of a standard, documented numerical framework. The main numerical evidence is nevertheless missing a set of convergence and validation checks that are needed before the long-time plateau can be regarded as established.","major_comments":[{"comment":"The long-time plateau in Fig. 5(b) is computed within the memory-truncation scheme described in Sec. II, but no value of the cutoff tc is reported anywhere and no convergence test with respect to tc is shown. This is not a merely numerical detail: in Eq. (7) the phonon contribution contains the free-boson propagator D(t,t′), which oscillates without decaying, so the decay of the memory kernel relies entirely on the decay of G(t,t′); in the η-SC state the anomalous component of G is precisely the long-lived quantity whose plateau is the central result. If tc is shorter than the decay time of that anomalous component, the truncation can artificially force the appearance of a steady state. Please report the value of tc used in Figs. 5(b) and 7 and provide a convergence study of ηx(t) and the spectral functions as tc is increased.","section":"§II and §III B"},{"comment":"The manuscript states in Sec. II that 'In all our calculations we take ω0 = 0.4', but Sec. III A and Fig. 2 describe the phonon coupling as having frequency ω0 = 0.2; Sec. III D again uses ω0 = 0.4. Because g^2/ω0 changes from 0.1 to 0.2 between these values and the phonon cooling rate enters the central stabilization argument, this inconsistency must be resolved by specifying the value used in each figure or correcting the typo.","section":"§II, §III A, and §III D"},{"comment":"The comparison in Fig. 7 is presented as evidence that the NESS approach 'well describes' the real-time quasi-steady state, but the fermion-bath coupling Γ and temperature Tb are adjusted so that the NESS state matches the effective temperature and doublon density extracted from the real-time simulation. The agreement is therefore a consistency check rather than an independent confirmation. To make the NESS claim meaningful, the authors should report the adjusted bath parameters, test the sensitivity of the spectra to these parameters, and ideally predict at least one observable (e.g. the anomalous spectral function or the order parameter) without fitting.","section":"§III D"},{"comment":"The order parameter is measured with a seed field Pseed = 0.001, and Sec. II asserts that in a symmetry-broken phase the order parameter becomes independent of the seed. No such independence is demonstrated. A seed-field dependence study (e.g. Pseed = 0.0001 and 0.01) is required to rule out that the plateau in Fig. 5(b) is a driven response rather than spontaneous η-SC order.","section":"§II, Eq. (5)"},{"comment":"The noncrossing approximation (NCA) is used for both real-time and NESS calculations, but no accuracy assessment is provided for the parameter regime U ≈ 10, d ≈ 0.4, g = 0.2. NCA is an uncontrolled approximation, and the quantitative values of ηx and of the decay rates in Figs. 3 and 5 may depend on it. Please add a comparison with an alternative solver (e.g. IPT or a numerically exact method for one representative case) or at least a discussion of the expected NCA accuracy in this regime.","section":"§II (impurity solver)"}],"minor_comments":[{"comment":"The section title 'Effect fo η-SC order' should read 'Effect of η-SC order'.","section":"§III C"},{"comment":"The text 'coordination numberz' should read 'coordination number z'.","section":"§II"},{"comment":"The text says 'Ωfin > Ωin' but the variable introduced is Ωini; please fix the typo.","section":"§III A"},{"comment":"The next-nearest-neighbor hopping vNN_h is introduced through the self-consistency equation, but no corresponding term is added to the Hamiltonian in Eq. (1); please clarify whether vNN_h is part of the model Hamiltonian or only an effective hopping introduced in the DMFT self-consistency.","section":"§II, Eq. (3)"},{"comment":"The arrow indicating the fully thermalized kinetic energy should be defined in the caption (e.g. the value from a thermal equilibrium calculation at β = 30).","section":"Fig. 2(d)"},{"comment":"The abbreviation 'KB contour' is used without definition; spell out 'Kadanoff-Baym contour' at first use.","section":"Fig. 7 caption"},{"comment":"The word 'consitent' should be 'consistent'.","section":"§III A"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The paper fits the scope of the journal and addresses a timely question. The main technical reservations are the missing memory-cutoff convergence study and the NCA accuracy assessment; both are fixable within the scope of a revision. I would not recommend rejection on the current evidence, but the manuscript should not be accepted until the convergence tests are provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of arXiv:2412.19205. The paper claims photodoped Mott insulators, when coupled to a cold phonon bath, relax to a long-lived prethermal state with eta-paired superconducting order, and that a nonequilibrium steady-state DMFT calculation reproduces that state. That's the new result, and if it holds it's a useful step: it suggests a practical route to stabilize light-induced eta pairing, and it validates the cheaper NESS approach for ordered phases.\n\nWhat the paper does well: the real-time DMFT data with entropy cooling are clearly presented, the U-dependence is physically coherent, and the spectral function fits (same effective temperature for normal and anomalous Green's functions) are a nice consistency check. The comparison between real-time and NESS spectra (Fig. 7) is reassuring even for d=0.4.\n\nThe soft spots are real, and one is load-bearing. The long-time plateau in eta_x (Fig. 5b) is computed with memory-truncated Kadanoff-Baym equations, but the cutoff tc is never stated and no convergence study is shown. The stress-test note is right: the phonon self-energy contains an oscillating free-boson propagator, and the memory kernel decay relies on G(t,t'), whose anomalous part is precisely the long-lived order parameter. If tc is shorter than that decay, the truncation could artificially force the plateau. The NESS comparison cannot rescue this because the bath parameters are adjusted to match the real-time effective temperature and doublon density; it's a consistency check, not an independent confirmation. So the central claim is not yet established to the standard I'd want.\n\nAlso, there's a sloppy internal inconsistency: Sec. II and III D use omega0=0.4, but Sec. III A text says omega0=0.2. That's easily fixed but matters for reproducibility. No code or data is provided, and no test of the NCA impurity solver accuracy or seed-field sensitivity is shown.\n\nOverall: the paper is a serious, plausible contribution from a strong group. It extends a known program in a natural direction and the results are likely correct. But the decisive long-time result needs a tc convergence study, a stated tc value, and ideally an NCA check before I'd fully trust it.\n\nWho is this for? People working on light-induced order in Mott systems and on nonequilibrium DMFT methods. It deserves a serious referee; with the convergence evidence and a corrected phonon frequency, I'd be comfortable with publication. Without them, the main claim remains unverified.\n\nMy recommendation: send it to review, but ask for the missing checks.","headline":"Plausible and well-motivated, but the central long-time eta-SC plateau rests on an unchecked memory-truncation cutoff; needs convergence evidence before I'd trust it.","tokens_in":13477,"tokens_out":2922,"would_cite":false,"duration_ms":27498,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A laser-pumped Mott insulator coupled to a cold phonon bath settles into a long-lived prethermalized state with persistent eta-pairing superconducting order, and the nonequilibrium steady-state formalism reproduces its spectral functions.","keywords":["photodoped Mott insulator","eta pairing","phonon cooling","nonequilibrium DMFT","memory truncation","hidden order","prethermal state","steady state"],"falsifier":"Recompute the $U=10$, $d\\approx0.4$ case with progressively larger truncation windows $t_c$ (for example, doubling it twice) and check that the $\\eta_x$ plateau and the extracted $\\beta\\approx16$ do not shift, and repeat the run with an exact impurity solver to confirm the spectral functions still match the NESS result.","tokens_in":12432,"feed_emoji":"⚡","tokens_out":10769,"duration_ms":92526,"temperature":0.7,"pith_summary":"This paper asks whether light-induced superconducting order in a Mott insulator can survive the heating that follows an ultrashort laser pulse. Using long-time simulations of the Hubbard model, the authors show that coupling the electrons to a cold phonon bath drains away the excess energy, and that in large-gap systems the staggered superconducting order (eta pairing) then settles into a plateau that persists for thousands of hopping times. They further show that this long-lived prethermal state is captured by a cheaper steady-state calculation, so the steady-state method can be used to explore such hidden orders. The practical upshot is a specific recipe: large Hubbard repulsion, enough photodoping, and phonon dissipation together stabilize metastable superconductivity.","feed_headline":"Phonon cooling keeps Mott insulators superconducting after pulse","feed_subtitle":"A cold phonon bath stops laser heating, letting eta-pairing order linger for thousands of hopping times.","key_machinery":"The machinery is memory-truncated Kadanoff-Baym nonequilibrium DMFT on the Bethe lattice, with entropy-cooling pulses to prepare a photodoped state and a Holstein phonon bath that acts as an energy sink. The central object is the eta order parameter $\\eta_x = \\frac{1}{2}(c^\\dagger_\\uparrow c^\\dagger_\\downarrow + \\mathrm{H.c.})$ with alternating sign between sublattices, measured in the Nambu Green's function; its plateau signals the prethermal superconducting state. The NESS counterpart replaces the laser and phonon history by permanent weak coupling to cold fermion baths at $\\pm U/2$ plus the same phonon bath, so that the two-time Green's function depends only on the time difference. The load-bearing connection is that both routes give the same spectral functions when the effective temperature and doublon density match.","core_discovery":"The central claim of the paper is that in the single-orbital repulsive Hubbard model on the Bethe lattice, after a chirped laser pulse creates roughly $d\\approx 0.4$ doublons, the eta-pairing order parameter $\\eta_x$ (a staggered superconducting order in which electron pairs have alternating phase between sublattices) grows during the pulse and then decays if the system is isolated; with a Holstein phonon coupling ($\\omega_0=0.4$, $g=0.2$) and $U \\gtrsim 10$, the order parameter instead rises to $\\eta_x \\approx 0.2$–$0.3$ and stays there until the end of the simulation at $t=2000$ (in units of inverse hopping). The system reaches a quasi-steady state with an almost constant spectral function and an effective inverse temperature $\\beta \\approx 16$ extracted identically from the normal and anomalous components. Nonequilibrium steady-state (NESS) DMFT with cold fermion baths, tuned to the same doping and temperature, reproduces the total and occupied spectral functions from the real-time simulation, with closer agreement at lower photodoping $d\\approx 0.14$ than at $d\\approx 0.4$. This establishes that the NESS construction is a valid description of the long-lived prethermalized eta-paired state in large-gap photodoped Mott insulators.","pith_inferences":["A natural follow-up is a systematic convergence study in the memory-truncation window $t_c$, which would confirm that the $\\eta_x$ plateau is a physical prethermal state rather than a truncation artefact.","The same steady-state-plus-phonon protocol could be applied to multi-orbital or frustrated Hubbard models to search for other photo-induced orders, such as spin-triplet or chiral pairing, by independently controlling doping and effective temperature.","An experimental test could look for a long-lived plateau in the transient optical or terahertz response of a large-gap Mott material with strong electron-phonon coupling; the predicted lifetime of thousands of inverse hoppings corresponds to picoseconds."],"forward_implications":["For $U \\gtrsim 10$ and photodoping $d \\sim 0.4$, the $\\eta_x$ order parameter saturates near $0.2$–$0.3$ and persists beyond $t=2000$ inverse hoppings, a timescale of several picoseconds in real materials.","The NESS formalism reproduces the real-time spectral functions, so steady-state calculations can be used in place of expensive real-time simulations to hunt for hidden orders.","Phonon coupling plays two roles: it cools the photo-carriers in large-gap systems, but in small-gap systems ($U \\lesssim 8$) it accelerates doublon-holon recombination and prevents $\\eta$ order from forming.","Without phonon coupling, the photodoped system slowly heats toward a negative-temperature state; with phonon coupling, that heating is suppressed on the simulated timescales, leaving a well-defined effective temperature."],"supporting_citations":[{"why":"Established that entropy-cooled photodoped states of the Hubbard model host eta-pairing order; this is the preparation protocol used here.","marker":"[23]"},{"why":"Predicted the eta-paired superconducting hidden phase in photodoped Mott insulators, the target order whose stability this paper tests.","marker":"[15]"},{"why":"Introduced the nonequilibrium steady-state theory of photodoped Mott insulators that this paper benchmarks against real-time results.","marker":"[24]"},{"why":"Developed the memory-time truncation scheme that makes the long-time real-time DMFT simulations feasible.","marker":"[26]"},{"why":"Provided the memory-truncated Kadanoff-Baym equations used for the real-time time propagation.","marker":"[27]"},{"why":"Supplied the numerical framework for nonequilibrium DMFT used in the calculations.","marker":"[21]"},{"why":"Showed that the thermalization rate of a photo-excited Mott insulator depends strongly on U, explaining the slow heating at large gap.","marker":"[10]"},{"why":"Quantified the long lifetime of double occupancies in the large-gap regime, supporting the assumption of slow doublon-holon recombination.","marker":"[9]"}],"fun_headline_variants":["Phonon cooling locks in photoinduced eta-pairing order","Phonon coupling sustains light-induced superconductivity","Cold phonons prolong superconducting order in Mott insulators","Eta-pairing survives long-term with phonon bath coupling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The long-time results assume that all memory of the initial state dies out within the self-energy truncation window $t_c$, and that the non-crossing impurity solver is accurate; if correlations persist beyond $t_c$, the apparent superconducting plateau could be a numerical artifact.","fun_headline_variants_meta":{"raw":{"variants":["Phonon cooling locks in photoinduced eta-pairing order","Phonon coupling sustains light-induced superconductivity","Cold phonons prolong superconducting order in Mott insulators","Eta-pairing survives long-term with phonon bath coupling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000467,"raw_usage":{"total_tokens":2340,"prompt_tokens":971,"completion_tokens":1369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":1312}},"tokens_in":587,"tokens_out":1369,"duration_ms":11311,"temperature":1.0,"reasoning_tokens":1312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:50:06.757293+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the $U=10$, $d\\approx0.4$ case with progressively larger truncation windows $t_c$ (for example, doubling it twice) and check that the $\\eta_x$ plateau and the extracted $\\beta\\approx16$ do not shift, and repeat the run with an exact impurity solver to confirm the spectral functions still match the NESS result.","supporting_citations":[{"cited_title":"Werner, J","cited_arxiv_id":null,"evidence_quote":"Established that entropy-cooled photodoped states of the Hubbard model host eta-pairing order; this is the preparation protocol used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Predicted the eta-paired superconducting hidden phase in photodoped Mott insulators, the target order whose stability this paper tests."},{"cited_title":"Li and M","cited_arxiv_id":null,"evidence_quote":"Introduced the nonequilibrium steady-state theory of photodoped Mott insulators that this paper benchmarks against real-time results."},{"cited_title":"Sch ¨uler, M","cited_arxiv_id":null,"evidence_quote":"Developed the memory-time truncation scheme that makes the long-time real-time DMFT simulations feasible."},{"cited_title":"Stahl, N","cited_arxiv_id":null,"evidence_quote":"Provided the memory-truncated Kadanoff-Baym equations used for the real-time time propagation."},{"cited_title":"Sch ¨uler, D","cited_arxiv_id":null,"evidence_quote":"Supplied the numerical framework for nonequilibrium DMFT used in the calculations."},{"cited_title":"Eckstein and P","cited_arxiv_id":null,"evidence_quote":"Showed that the thermalization rate of a photo-excited Mott insulator depends strongly on U, explaining the slow heating at large gap."},{"cited_title":"Sensarma, D","cited_arxiv_id":null,"evidence_quote":"Quantified the long lifetime of double occupancies in the large-gap regime, supporting the assumption of slow doublon-holon recombination."}],"review_version":1}