{"id":"e40ef733-83ac-4850-aa8a-b859865d8dbe","arxiv_id":"2608.13146","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A quantics tensor-train time-difference formulation with direct retarded convolutions accelerates strong-coupling impurity solvers to third order in nonequilibrium DMFT and EDMFT.","lead":"This paper shows that tensor cross interpolation and quantics tensor trains can evaluate high-order strong-coupling diagrams for nonequilibrium impurity problems much faster than direct integration, including a new direct quantics convolution scheme. This makes third-order nonequilibrium DMFT and EDMFT calculations practical, enabling systematic checks of approximations used for photodoped and correlated materials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unaddressed η-dependence of the photodoped DMFT and EDMFT spectra is the load-bearing soft spot: the main results use η=0.01 without an η→0 extrapolation, so the apparent OCA-to-TOA convergence may be partly an artifact of artificial broadening.","rationale":"The reader's weakest assumption—that the pseudo-particle propagators decay within tc=327.67 or that the η=0.01 broadening is controlled—is exactly the concern I find most load-bearing. The paper introduces η in Section II A 3 and Appendix B to stabilize the Dyson equation precisely when propagators decay slowly, and Figure 20 shows that spectral peak positions and amplitudes still depend on η in the range 0.01-0.04. No η→0 extrapolation is reported for the main results, and the EDMFT section omits even a single-η comparison. This is not a disagreement with an outside consensus; it is an internal control that the paper itself identifies as necessary for convergence. The absence of that control affects the physical conclusions about order-by-order convergence, because the OCA versus TOA comparison is performed at a single finite η where Figure 20 demonstrates residual sensitivity. The method's computational contribution—TCI/QTT acceleration and direct QTT convolution—is supported by the Gaussian benchmarks and equilibrium DMFT tests, including the fluctuation-dissipation diagnostic, and I do not see a flaw in the algorithm itself that would change the verdict. However, the claimed controlled truncation-error estimate for nonequilibrium DMFT and EDMFT is conditional on the η-limit. The reader's CONDITIONAL verdict is therefore appropriate, and my read does not change it. A fourth-order check would further support the convergence claim, but the η-scan is the more direct and less expensive test of the specific weak spot already identified.","tokens_in":27635,"tokens_out":8861,"duration_ms":85811,"concrete_test":"Recompute the photodoped Bethe-lattice DMFT spectra of Figure 17 and the square-lattice EDMFT spectra of Figure 18 at η=0.02, 0.01, 0.005, 0.0025, and 0.00125, keeping all other parameters fixed, and extrapolate the OCA and TOA spectral peak positions, amplitudes, and the effective photodoping densities nex to η=0. Compare the OCA-TOA difference at η=0.01 with the extrapolation uncertainty. If the extrapolated OCA-TOA difference remains smaller than the η-extrapolation error, the concern is resolved; if the η-dependence within the scanned range is comparable to or larger than the OCA-TOA gap, the third-order convergence statement must be revised to state that order-by-order convergence is established only up to an η-dependent uncertainty.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's physical conclusions, including the claim that third-order results are 'nearly converged' and provide a 'controlled estimate of the truncation error', rest on comparisons of NCA, OCA, and TOA spectra computed at a single artificial broadening η=0.01. Section II A 3 and Appendix B introduce the damping e^{-η|t-t'|} precisely because pseudo-particle propagators may decay slowly, and they state that convergence is verified by extrapolating to η→0. However, Figure 20, which is the only η-scan in the paper, shows that both the amplitude and position of the quasiparticle peaks in the photodoped Bethe-lattice calculation still depend on η in the range 0.04 down to 0.01, with no extrapolation to η=0. The EDMFT section (Section V) reports no η-scan at all. The time cutoff tc=327.67 is large relative to the damping scale 1/η=100, so the η term is active throughout the tail of the propagators; if the intrinsic decay is slower than e^{-0.01t}, the long-time behavior and therefore the self-consistent spectra are controlled by an artificial pseudo-particle bath rather than by the physical hybridization. This is load-bearing because the central demonstration of systematic order-by-order convergence—OCA versus TOA closeness—is made at a fixed η where Figure 20 shows residual η sensitivity. If the η-induced shift is comparable to or larger than the OCA-TOA difference, the apparent third-order convergence would not survive the η→0 limit. The algorithmic acceleration claim is not invalidated by this concern, but the physical validation that motivates the method is conditional on a controlled η limit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops tensor-cross-interpolation (TCI) and quantics tensor-train (QTT) parametrizations for evaluating high-order strong-coupling (hybridization expansion) diagrams for nonequilibrium steady-state impurity solvers. Four parametrizations are compared: absolute-time TCI, quantics absolute-time, time-difference TCI, and quantics time-difference, the last using a newly derived direct carry-bit retarded convolution in fused QTT form. Gaussian benchmarks are used to compare accuracy, bond dimensions, and runtime; the methods are then applied to self-consistent equilibrium DMFT with a fluctuation-dissipation diagnostic, to photodoped nonequilibrium Bethe-lattice DMFT up to third order (TOA), and to square-lattice EDMFT with retarded density-density interactions. The central claim is that the quantics time-difference approach makes OCA and TOA self-consistent iterations practical, enabling controlled order-by-order convergence checks in regimes previously limited to NCA/OCA.","tokens_in":27941,"tokens_out":4683,"duration_ms":45251,"significance":"If the efficiency and accuracy claims hold, this is a substantial methodological advance. The carry-based direct QTT retarded convolution is a reusable algorithmic ingredient, and the demonstration of self-consistent TOA-level nonequilibrium DMFT and EDMFT would open a regime previously inaccessible to systematic strong-coupling expansions. Credit is due for the careful benchmark design: TCI errors are measured against direct quadrature on the same grid, convergence is scanned independently in bond dimension and grid size, and the equilibrium fluctuation-dissipation relation is used as a nontrivial self-consistency check. The main weakness is that the central physical convergence claims are made at a single artificial broadening parameter in the pseudo-particle Dyson equation, with only partial evidence that the results are controlled in that parameter.","major_comments":[{"comment":"The central physical conclusion that the TOA results are 'nearly converged' and provide a 'controlled estimate of the truncation error' is established at a single value of the artificial broadening, eta=0.01. Figure 20 shows that for the same photodoped Bethe-lattice parameters the quasiparticle-peak amplitude and position still vary visibly as eta is reduced from 0.04 to 0.01, for NCA, OCA, and TOA alike, and no extrapolation to eta->0 is presented. The text in App. B states that annealing to smaller eta is possible but omits it because the focus is the accelerated solver. This leaves open the possibility that the apparent OCA-TOA closeness in Fig. 17 is partly a broadening artifact rather than genuine order-by-order convergence. Please provide an eta scan for the order-by-order comparison in Fig. 17, or otherwise quantify the eta-induced uncertainty on the claimed truncation error.","section":"Sec. IV C, Fig. 17, App. B, Fig. 20"},{"comment":"The EDMFT results are presented without stating the broadening parameter used or providing an eta-scan, although the bosonic spectra show low-frequency sensitivity and the NCA/OCA/TOA differences are used to claim convergence. The statement that the TOA results are 'already nearly converged' in EDMFT therefore rests on the same unverified broadening assumption as the DMFT results. Please state the value of eta (and the time cutoff) used in Sec. V and provide a convergence check with respect to eta, or explicitly characterize the regime in which the presented results are independent of this parameter.","section":"Sec. V, Fig. 18"}],"minor_comments":[{"comment":"The main text says 'eta=0.01 below' in Sec. IV B, but Fig. 17's caption does not state the broadening value; please state the eta used in the caption or in the text immediately preceding the figure.","section":"Fig. 17 caption and Sec. IV C"},{"comment":"The notation Theta_alpha(omega) for the smooth distribution function conflicts with the time-ordering mask Theta_{s1,...,sD} introduced in Eq. (37); please use a different symbol, such as H_alpha(omega) or F_alpha(omega).","section":"Eq. (47)"},{"comment":"The statement that the pseudo-particle propagators are 'sufficiently decayed' at t_c=327.67 is asserted for NCA but not quantified. Since eta=0.01 gives a damping scale 1/eta=100 that is active throughout the tail, please report the actual decay of G(t_c) for OCA and TOA, or otherwise provide evidence that the finite-window error is negligible.","section":"Sec. IV B and App. B"},{"comment":"The self-convergence test uses the largest available bond dimension chi=160 as the reference, which is reasonable, but the TOA slope is noticeably slower; please state explicitly that this is a self-convergence test and not an absolute error estimate.","section":"App. C"},{"comment":"The conclusion states that the quantics representation scales as O(log N_t) for sampling; this is logarithmic in N_t only at fixed diagram order and fixed bond dimension, as the complexity is O(2^D R chi^3). Please phrase this more precisely.","section":"Sec. VI"}],"recommendation":"major_revision","confidential_remarks":"The algorithmic core of the paper—the direct QTT convolution and the comparative benchmarks—appears sound and valuable. The main risk is that the physical convergence claims are presented at a single broadening value without a controlled eta->0 analysis; if the authors can supply that analysis or appropriately soften the claims, the paper would be suitable for publication. I do not see grounds for rejection, as the issue is addressable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this paper delivers a real algorithmic step—the direct carry-based QTT convolution for retarded convolutions—and validates it with controlled benchmarks. The method section is the strongest part. The physical conclusions, especially the claim of a controlled truncation error at third order, rest on spectra computed at a single artificial broadening eta=0.01, with no eta-to-0 extrapolation in the main results.\n\nWhat is new: the rank-2 carry MPO for the retarded convolution in fused QTT, with trapezoidal endpoint corrections, is clean and practically useful. The systematic comparison of four TCI/QTT parametrizations on Gaussian integrals is done carefully: TCI errors are measured against direct quadrature on the same grid, bond-dimension and grid-size scans are separated, and the equilibrium fluctuation-dissipation relation is a sensible self-consistency check. The extension to third-order EDMFT with retarded density-density interactions, including the diagram counting and symmetry reduction, is a legitimate new application. The performance numbers (seconds for OCA, minutes for TOA per DMFT iteration) are believable given the scaling analysis.\n\nSoft spots, in order of weight:\n1. The eta issue is real. The text says convergence is verified by extrapolating to eta->0 (Section II A 3 and App. B), but App. B only shows eta from 0.04 down to 0.01, and Figure 20 shows non-negligible shifts of quasiparticle peak position and amplitude over that range, with no extrapolation. The EDMFT section does not report an eta scan at all. The OCA-vs-TOA closeness—the basis for the 'controlled estimate of the truncation error'—is demonstrated at a single eta value where the artificial broadening is still visibly active. This does not invalidate the algorithmic contribution, but it makes the physical convergence statement conditional.\n2. No code or data is released, so exact reproduction is harder, though the benchmarks are described carefully enough that a determined group could reimplement them.\n3. Minor: the EDMFT 'nearly converged' claim rests on OCA/TOA agreement without an independent error estimate, such as a fourth-order check in a reduced setting. That is a standard limitation at third order, not a flaw in the method itself.\n\nWho this is for: anyone working on nonequilibrium DMFT, EDMFT, or QTT-based impurity solvers. The direct QTT convolution is likely to be reused beyond this application.\n\nRecommendation: send it out. The referee should ask for an eta-to-0 extrapolation, or at minimum an eta scan at OCA and TOA for one photodoped case, before the 'controlled truncation error' claim is accepted at face value. That is addressable; the core method is sound.","headline":"Useful QTT acceleration of strong-coupling NESS impurity solvers, with a real algorithmic contribution; the physical convergence claims are weakened by not extrapolating the eta=0.01 broadening.","tokens_in":28531,"tokens_out":2910,"would_cite":true,"duration_ms":25055,"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":"The paper claims that a fused quantics time-difference parametrization with tensor cross interpolation and a direct carry-based QTT retarded convolution reduces the cost of high-order strong-coupling diagrams enough to make…","keywords":["tensor cross interpolation","quantics tensor train","strong-coupling expansion","nonequilibrium steady state","dynamical mean-field theory","extended DMFT","Keldysh contour","retarded convolution"],"falsifier":"Run the photodoped DMFT and EDMFT calculations at $\\eta = 0.01$, $0.02$, $0.04$ and with an annealed $\\eta \\to 0$ protocol; if the quasiparticle-peak amplitude and Mott-gap edges drift as strongly as the $\\eta$-scan in Figure 20, then the TOA-vs-OCA differences could be an artifact of the broadening and the third-order convergence claim is not controlled. A complementary check is to compare the fused-QTT TOA self-energy against a direct quadrature evaluation on a coarser grid where direct evaluation is still feasible.","tokens_in":27385,"feed_emoji":"⚛️","tokens_out":8549,"duration_ms":72619,"temperature":0.7,"pith_summary":"This paper attacks the bottleneck of the self-consistent strong-coupling (hybridization) expansion for quantum impurity models in nonequilibrium steady states: the high-dimensional, time-ordered integrals that enter each self-energy diagram. It claims that a fused quantics tensor-train representation of the integrand, built by tensor cross interpolation, compresses these integrals so strongly that third-order diagrams become routine, and that retarded convolutions can be performed directly inside the quantics format using a carry-bit matrix-product operator. If true, the method turns NCA/OCA-only nonequilibrium DMFT into a systematically improvable hierarchy, and opens impurity models with retarded interactions in EDMFT and GW+EDMFT to controlled truncation checks. The paper demonstrates this with Gaussian benchmarks, equilibrium and photodoped DMFT up to TOA, and square-lattice EDMFT with retarded density-density interactions.","feed_headline":"Quantics tensor trains make third-order DMFT practical","feed_subtitle":"A direct quantics convolution cuts high-order diagrams to seconds per DMFT step, enabling systematic convergence checks up to third order.","key_machinery":"The central object is the fused quantics time-difference tensor train of the diagrammatic integrand, together with a direct retarded convolution in the quantics format. A time-difference index is split into binary bits and cores at the same bit level are fused, so the integrand becomes a product of small bit-indexed cores. The time-ordering Heaviside mask has an exact fused QTT representation with bond dimension two. The retarded convolution is implemented as a rank-2 MPO that carries the binary addition $u_r+v_r+\\gamma_r = w_r + 2\\gamma_{r+1}$ through an auxiliary carry bit $\\gamma_r$, which increases the bond dimension by a factor of two, or up to four with trapezoidal endpoint corrections, instead of the roughly eleven-fold growth of a quantics Fourier transform. This object is what converts the nested time-ordered sum into a sequence of small matrix contractions over the bit cores, and it is the component that makes the third-order self-consistent loops fast enough for routine use.","core_discovery":"The paper's central claim is that the practical limit of nonequilibrium strong-coupling solvers is lifted: by writing each nth-order self-energy diagram in the Keldysh time-difference parametrization, expanding every time-difference index into binary bits, and grouping bits by scale into a fused quantics tensor train via tensor cross interpolation, the expensive time-ordered integral becomes a product of small matrix contractions. The retarded convolutions that enforce time ordering are applied directly within the quantics format through a rank-2 carry-bit transfer operator, avoiding Fourier transforms. The paper argues, and demonstrates on Gaussian benchmarks, that the resulting bond dimensions are independent of grid size and only grow modestly with diagram order, so that self-consistent DMFT iterations at NCA, OCA, and TOA can be run routinely, and the same machinery handles impurity models with retarded density-density interactions inside nonequilibrium EDMFT. Its headline numerical results are that OCA Bethe DMFT iterations take seconds on a laptop and TOA iterations minutes on a 96-core node.","pith_inferences":["Because the carry-based convolution is exact at the bit level, the same operation should accelerate the pseudo-particle Dyson-equation step itself; the paper only sketches this as a future possibility.","The method's crossover depends on the time cutoff: for short simulation windows, the non-quantics time-difference parametrization with higher-order Gregory quadrature can remain faster, so an adaptive choice based on the required $t_c$ and decay rate would be a natural extension.","The $\\eta$-dependence shown in the paper's Figure 20 implies that spectral features sharper than roughly $\\eta$ may be systematically suppressed; an annealed $\\eta \\to 0$ procedure, which the paper does not present, is the cleanest way to test whether TOA corrections to the quasiparticle peak remain significant.","The moderate bond-dimension growth from OCA to TOA in the Gaussian benchmark suggests the same machinery can reach still higher orders before rank growth becomes prohibitive, rather than encountering a topological ceiling."],"forward_implications":["Nonequilibrium DMFT on the Bethe lattice can be iterated self-consistently at NCA, OCA, and TOA, giving controlled order-by-order convergence checks instead of stopping at OCA.","In the Mott-regime the strong-coupling series converges quickly, while in the correlated metal higher orders are needed; TOA availability directly improves quasiparticle-peak predictions.","Impurity models with retarded density-density interactions, the core of EDMFT and GW+EDMFT, can now be solved with an error estimate from comparing NCA, OCA, and TOA; the results indicate NCA overestimates Mott-gap stability while OCA and TOA are close.","With the fused quantics time-difference solver, OCA Bethe DMFT iterations take seconds on a laptop and TOA iterations minutes on a 96-core node.","The direct QTT convolution algorithm is reusable for other Volterra or convolution-type integral equations, and may eventually allow the entire DMFT self-consistency cycle to run inside the QTT representation."],"supporting_citations":[{"why":"Introduces the Keldysh time-difference parametrization and the FFT-based retarded convolution that this paper generalizes to a direct QTT formulation.","marker":"[55]"},{"why":"Supplies the tensor cross interpolation algorithm and the fused/interleaved quantics layouts used to construct all tensor trains.","marker":"[43]"},{"why":"Establishes the quantics representation that gives logarithmic compression of smooth functions, motivating the QTT decomposition.","marker":"[42]"},{"why":"Defines the self-consistent strong-coupling expansion and pseudo-particle Dyson equation that the solver evaluates.","marker":"[32]"},{"why":"Provides the steady-state pseudo-particle Dyson equation and the frequency-domain broadening scheme used to stabilize slow decay.","marker":"[52]"},{"why":"Earlier frequency-domain QTT strong-coupling solver whose computational strategy this work compares against and extends to real-time direct convolution.","marker":"[56]"},{"why":"Formulates EDMFT with retarded density-density interactions through a Luttinger-Ward double expansion, the framework used in Section V.","marker":"[58]"},{"why":"Supplies the photodoped steady-state distribution and the nonthermal protocol used for the nonequilibrium DMFT benchmarks.","marker":"[64]"}],"fun_headline_variants":["Quantics tensor trains bring third-order DMFT to the laptop","Third-order nonequilibrium DMFT now practical via quantics tensor trains","Tensor cross interpolation yields fast high-order impurity solver","Quantics tensor trains cut DMFT high-order diagram cost to seconds","Direct quantics convolutions enable OCA and TOA DMFT iterations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the pseudo-particle propagators decay within the numerical cutoff $t_c=327.67$, or equivalently that the artificial broadening $\\eta=0.01$ used in the main DMFT runs is a controlled approximation to the $\\eta \\to 0$ limit, so that neither the finite window nor the damping distorts the spectral peaks enough to change the apparent order-by-order convergence.","fun_headline_variants_meta":{"raw":{"variants":["Quantics tensor trains bring third-order DMFT to the laptop","Third-order nonequilibrium DMFT now practical via quantics tensor trains","Tensor cross interpolation yields fast high-order impurity solver","Quantics tensor trains cut DMFT high-order diagram cost to seconds","Direct quantics convolutions enable OCA and TOA DMFT iterations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000322,"raw_usage":{"total_tokens":1813,"prompt_tokens":952,"completion_tokens":861,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":773}},"tokens_in":568,"tokens_out":861,"duration_ms":6982,"temperature":1.0,"reasoning_tokens":773,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:46:34.373990+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the photodoped DMFT and EDMFT calculations at $\\eta = 0.01$, $0.02$, $0.04$ and with an annealed $\\eta \\to 0$ protocol; if the quasiparticle-peak amplitude and Mott-gap edges drift as strongly as the $\\eta$-scan in Figure 20, then the TOA-vs-OCA differences could be an artifact of the broadening and the third-order convergence claim is not controlled. A complementary check is to compare the fused-QTT TOA self-energy against a direct quadrature evaluation on a coarser grid where direct evaluation is still feasible.","supporting_citations":[{"cited_title":"Núñez Fernández, M","cited_arxiv_id":null,"evidence_quote":"Supplies the tensor cross interpolation algorithm and the fused/interleaved quantics layouts used to construct all tensor trains."},{"cited_title":"Oseledets and E","cited_arxiv_id":null,"evidence_quote":"Establishes the quantics representation that gives logarithmic compression of smooth functions, motivating the QTT decomposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the self-consistent strong-coupling expansion and pseudo-particle Dyson equation that the solver evaluates."},{"cited_title":"Erpenbeck, W.-T","cited_arxiv_id":null,"evidence_quote":"Provides the steady-state pseudo-particle Dyson equation and the frequency-domain broadening scheme used to stabilize slow decay."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier frequency-domain QTT strong-coupling solver whose computational strategy this work compares against and extends to real-time direct convolution."},{"cited_title":"Paprotzki and M","cited_arxiv_id":null,"evidence_quote":"Supplies the photodoped steady-state distribution and the nonthermal protocol used for the nonequilibrium DMFT benchmarks."}],"review_version":1}