{"id":"daeae9d4-103c-4fb1-9c2c-0e5707045545","arxiv_id":"2412.01538","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For free fermion quenches in d≥2, the entanglement Hamiltonian at the ballistic scale is given by a quasiparticle-picture kernel built from the mode occupation function, extending the 1D result to higher dimensions.","lead":"This paper derives analytic formulas for the time-dependent entanglement Hamiltonian after quantum quenches in free fermion systems in two or more spatial dimensions, using the quasiparticle picture. It extends a previously one-dimensional result to strips and arbitrary regions, and checks the formulas against exact numerics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (3.25)/(4.25)/(6.13) left-mover kernel has the wrong light-cone sign: for v(k)<0 it is nonzero throughout A instead of only near the right boundary, so the central K_A formula as printed is internally inconsistent.","rationale":"The reader's weakest_assumption is the semiclassical propagation ansatz and the GGE relaxation condition. I agree those are fundamental approximations, but they are plausibly supported by the existing 1D results and by the 2D strip numerics. The more immediate, falsifiable problem is that the printed central formula does not even encode the pair-sharing condition derived in §3.2. If Eq (3.25) were taken literally, K_L would have support across the whole subsystem, contradicting the 'light cone structure' described in the text and the figures. This is not a matter of outside consensus; it is an internal consistency failure in the equation quoted in the strongest_claim. The error propagates to the general-geometry kernel (6.13), which the reader already judged untested. Because the numerical data in §5 can only be consistent with the corrected sign, I suspect a sign typo rather than a conceptual failure; but the manuscript as written must be corrected and re-verified. This leaves the overall verdict conditional: the strip result is likely correct after the sign fix, and the general-geometry claim remains heuristic pending direct numerical corroboration. Hence the reader's CONDITIONAL verdict is unchanged, with the added explicit condition that Eqs (3.25), (4.25), and (6.13) be corrected and the numerical comparison rerun. My agreement is partial because the concrete sign issue is different from the reader's weakest_assumption, though both would be resolved by a clean independent re-derivation and test of the kernel.","tokens_in":32350,"tokens_out":9747,"duration_ms":90018,"concrete_test":"For the 1D dimer state (n(k)=(1−cos k)/2), take ℓ=100 and t/ℓ=0.1. Compute the exact correlation matrix (Eq B.6) and invert (2.11) to get K_A,exact. Compare it to Eq (3.23) using (i) the printed left-mover kernel Θ(max(ℓ+2v t,0)+x) and (ii) the corrected kernel Θ(x−max(ℓ+2v t,0)); quantify the difference using the operator norm of K_A,QP−K_A,exact on the central half of the strip, where the two candidates differ most. Repeat the same comparison for Eq (6.13) with v<0. If (ii) matches to numerical precision and (i) does not, the sign error is settled; if neither matches, the quasiparticle propagation ansatz or the pure/mixed factorization is at fault.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing problem is not the semiclassical ansatz itself but a concrete inconsistency in the printed formula that the strip and general-geometry claims reduce to. In Eq (3.25) the left-mover kernel is written K_L = ∫_{k<0} dk/(2π) η(k) Θ(max(ℓ+2v(k)t,0)+x) e^{-ikz}. For k<0, v(k)<0, so the Heaviside argument is ℓ−2|v|t+x when ℓ−2|v|t>0 and just x when ℓ−2|v|t≤0. In either case it is positive for every x∈[0,ℓ], so Θ is identically 1 on the subsystem: the printed K_L has no light-cone structure at all. Shared-pair counting from §3.2 gives the opposite condition: a left mover at position x is paired with a right mover at x+2|v|t, and the pair is shared only if x+2|v|t>ℓ, i.e. x>max(ℓ−2|v|t,0). The kernel should therefore be Θ(x−max(ℓ+2v(k)t,0)). The same erroneous argument appears in Eq (6.13), so the general-geometry formula repeats the error. Since the numerical figures in §5 appear to show boundary-anchored light cones, the intended formula is presumably the corrected one; nevertheless the central equation as printed does not follow from the model and cannot reproduce the stated checks. This is a required correction, not merely a stylistic issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a quasiparticle-picture (QPP) description of the time-dependent entanglement Hamiltonian for free-fermion lattice systems in d≥2 after a quantum quench. For strip geometries, dimensional reduction yields an explicit formula K_A(t)=∫dx∫dz [K_R+K_L]c†_x c_{x−z}, with kernels determined by the mode occupation function η(k) and Heaviside counting functions (Secs. 4.2–4.3). For general geometries, a formula with a geometric counting function B(A,x,v(k),t) is proposed (Sec. 6). The strip results are benchmarked against exact numerical diagonalization for collinear, staggered, and diagonal dimer initial states (Sec. 5), and analytic checks against known QPP results for Rényi entropies and full counting statistics are given. A crossed-dimer state that does not relax to a GGE is treated separately by modifying the quasiparticle basis (Sec. 5.4).","tokens_in":32600,"tokens_out":6616,"duration_ms":58547,"significance":"If correct, the paper would extend the quasiparticle-picture description of entanglement Hamiltonians from one dimension to higher dimensions, providing operator-level predictions that go beyond entanglement entropies and full counting statistics. The strip-geometry results are supported by a careful numerical comparison and by exact closed-form correlation matrices in Appendix B, and the analytic checks in Secs. 3.4 and 4.3 are useful consistency tests. However, the printed left-mover kernels contain a sign error that removes the light-cone structure from the central formula, so the main result as written is internally inconsistent. Because the intended formula is apparent from the shared-pair counting and from the numerics, the issue is likely repairable, but it is load-bearing and must be corrected before the claims can be accepted.","major_comments":[{"comment":"The left-mover kernel is printed as Θ(max(ℓ+2v(k)t,0)+x) with v(k)<0. For negative v(k) this argument is nonnegative for every x∈[0,ℓ], so the Heaviside function is identically one and the kernel has no light-cone structure; in the early-time regime it incorrectly weights the entire subsystem rather than only the region x>ℓ−2|v(k)|t. The shared-pair counting of Sec. 3.2 requires Θ(x−max(ℓ+2v(k)t,0)). The same incorrect +x argument appears in the strip formula (4.25), in the 1D consistency check (6.13), and in Appendix C, so the central formula as printed cannot reproduce the numerical checks of Sec. 5 or reduce to the 1D result of Ref. [53]. This is a required correction, not a stylistic point.","section":"Eqs. (3.25), (4.25), (6.13), (C.5)"},{"comment":"In the left-moving contribution the integration is written ∫_ℓ^{max(ℓ−2|v_x(q_x)|t,0)} dx \\hat n_{x,q_x;yσ}. Since the upper limit is never larger than ℓ, the integral has the wrong orientation; the domain of shared left movers is x∈[max(ℓ−2|v_x|t,0),ℓ], so the limits should be reversed. This appears to be the same boundary-counting error as in comment 1, propagated into the crossed-dimer construction.","section":"Eq. (5.26)"},{"comment":"The general-geometry formula is stated to follow by a 'straightforward' generalization, but the only concrete demonstrations are the circle case and the 1D consistency check (6.13), which itself contains the sign error noted above. Since the general-geometry formula is part of the paper's central claim, the derivation should be supplied or, failing that, the formula should be tested numerically for at least one non-strip geometry (e.g., a disk or square) before publication.","section":"Sec. 6.2, Eqs. (6.11)–(6.12)"}],"minor_comments":[{"comment":"The subsystem length is typeset inconsistently as both 'l' and 'ℓ'; please use a single symbol throughout.","section":"Eqs. (3.23)–(3.25) and (4.25)"},{"comment":"The sum over y runs from y=1 to Ly/2−1, whereas the crossed-dimer state in Eq. (5.21) is defined with y=0,...,Ly/2−1; please check whether the y=0 contribution is intentionally excluded and justify this if so.","section":"Eq. (5.26)"},{"comment":"The legends list several values of t/ℓ but do not identify which line corresponds to which value; please add this information, because the light-cone structure of the kernels is the main feature being compared.","section":"Figures 3–5"},{"comment":"In the second term, the notation ∫_{k>0} dk_x is ambiguous; please state explicitly that the integral is over k_x>0 and clarify how the factor L_y arises from the summation over k_y.","section":"Eq. (4.32)"}],"recommendation":"major_revision","confidential_remarks":"The discrepancy between the printed Eq. (4.25) and the numerical curves in Sec. 5 strongly suggests that the published formulas were not typeset from the expressions actually used in the numerics; the authors should verify every occurrence of the left-mover kernel against their code. The paper is well within the journal's scope, and with the sign corrections and an additional non-strip numerical check the central claim would be credible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this paper is worth engaging, but do not cite Eq (3.25) (or its twins (4.25) and (6.13)) as printed. The left-mover kernel has the wrong Heaviside argument. For k<0, v(k)<0, so max(ℓ+2v(k)t,0)+x is always positive for x∈[0,ℓ]; the Θ is identically 1 and the light cone vanishes. The shared-pair condition gives x > max(ℓ−2|v|t,0), so the argument should be x−max(ℓ+2v(k)t,0). As written, the central K_A formula cannot reproduce the entropy or full-counting-statistics results derived later in the same paper. The numerical figures show the expected boundary-anchored light cones, so the authors almost certainly used the corrected formula; still, this is a required correction, not a stylistic blemish.\n\nWhat is genuinely new: explicit operator-level QPP formulas for K_A(t) in strip geometries, Eq (4.24)–(4.26); the full-counting-statistics expression (4.32); and the general-geometry ansatz (6.11)–(6.12). The strip results are a dimensional reduction of the earlier 1D result [53], and the entropies were already known from [56], but the operator-level form and the FCS are new. The numerics for the collinear, staggered, and diagonal dimers agree well and support the (presumably corrected) strip formula. The authors are honest about limitations: they flag the GGE relaxation condition (4.27), the crossed-dimer counterexample, and the absence of numerical tests for general geometries.\n\nSoft spots, in proportion. The sign error is the biggest: it affects the central equations and the claimed reduction to 1D in (6.13). Second, Section 6 is heuristic and untested; the circle-geometry entropy check is a self-consistency test within the same ansatz. Third, the spectral-cutoff comparison (ε = 10^{-4} or 10^{-6} depending on state) gets no sensitivity analysis, and the cutoff choice could mask systematic discrepancies. Fourth, the analytic checks are consistency checks rather than independent verifications; the numerics carry the load, which is fine but should be stated more sharply.\n\nWho is this for? Researchers working on quench dynamics in free-fermion systems and on entanglement Hamiltonians. It is a solid, clearly written extension of an established program, not a conceptual breakthrough. It deserves peer review, but the referee should demand corrected formulas and, ideally, a numerical test in a non-strip geometry before publication.\n\nBest.","headline":"Solid d>=2 extension of the quasiparticle-picture entanglement Hamiltonian, but check Eq (3.25): the left-mover Heaviside argument as printed destroys the light-cone structure and should be corrected before citation.","tokens_in":33219,"tokens_out":3283,"would_cite":false,"duration_ms":28874,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40","82C10","82B20"],"pacs":["03.65.Ud","05.30.-d"],"model":"deepseek-v4-flash","headline":"After a quantum quench in a free-fermion lattice in two or more dimensions, the entanglement Hamiltonian takes an explicit quasiparticle form governed by the mode occupation function and light-cone counting.","keywords":["entanglement Hamiltonian","quasiparticle picture","quantum quench","free fermions","dimensional reduction","ballistic scaling limit","full counting statistics","entanglement entropy"],"falsifier":"Compute the exact entanglement Hamiltonian, via the correlation matrix and the Peschel formula, for a two-dimensional free-fermion quench from a Gaussian initial state whose correlation length is comparable to the fluid-cell size, and compare it element by element with (4.24) for a strip: if the entries deviate by more than the chosen numerical cutoff, the rigid-propagation ansatz and the pure/mixed factorization fail. Alternatively, for a region with a curved boundary, check whether the kernel (6.12) reproduces the numerical $K_A$ for all pairs of sites.","tokens_in":32086,"feed_emoji":"⚛️","tokens_out":8637,"duration_ms":70533,"temperature":0.7,"pith_summary":"The paper claims that the quasiparticle picture, long used to predict entanglement entropies after quantum quenches, also determines the full entanglement Hamiltonian in free-fermion systems in two and higher spatial dimensions. In the ballistic scaling limit, it derives explicit closed forms for $K_A(t)$ for strip geometries by dimensional reduction and for arbitrary geometries via a kernel built from the mode occupation function $\\eta(k)$ and a light-cone counting function that selects quasiparticle pairs shared between the subsystem and its complement. If correct, the late-time reduced density matrix of a quenched free-fermion system is completely characterized by this data, and R\\'enyi entropies and full counting statistics follow from the same formula. The paper verifies the prediction numerically against exact correlation-matrix calculations for several dimer-type initial states.","feed_headline":"Free-fermion quench entanglement Hamiltonian derived in d≥2","feed_subtitle":"For strips or curved regions, K_A(t) is fixed by occupation data and light-cone counting of quasiparticle pairs.","key_machinery":"The engine is the fluid-cell decomposition of the lattice into cells of size $\\Delta$ with $1 \\ll \\Delta \\ll \\ell$, together with the semiclassical propagation ansatz (3.10), $e^{iHt} b^\\dagger_{x_0,k} e^{-iHt} \\approx b^\\dagger_{x_0+v(k)t,k}$. This turns the initial Gaussian state into a product of independent quasiparticle pairs or multiplets, whose members either both lie in $A$, contributing to the pure part, or have exactly one member in $A$, contributing to the mixed part after tracing out the complement. The mixed part is $\\rho_{\\mathrm{mixed}} = Z^{-1} e^{-K_{A,\\mathrm{QP}}}$ with $K_{A,\\mathrm{QP}}$ a sum over shared modes of $\\eta(k) = \\log\\frac{1-n(k)}{n(k)}$ times occupation numbers. Fourier transforming back to real space produces the kernels (4.24)--(4.26) for strips and (6.12) for general geometries, with the counting function $B$ encoding the light-cone structure.","core_discovery":"The central claim is that out-of-equilibrium entanglement Hamiltonians in $d \\geq 2$ admit a quasiparticle-picture description. For a strip $A = [1,\\ell] \\times S^{d-1}$, $K_A(t)$ is given by (4.24) with kernels (4.25)--(4.26), and for a general connected region $A$ it is given by (6.11)--(6.12), where the counting function $B(A,x,v(k),t)$ selects pairs with one member inside and one outside $A$. The formula reproduces the known quasiparticle predictions for the entanglement entropy and full counting statistics, and exact numerics for collinear, staggered, diagonal and crossed dimer states agree with the kernels once the near-pure subspace is removed by an eigenvalue cutoff. In the crossed-dimer case, which does not relax to a GGE, the standard picture must be supplemented by a unitary rotation to quasiparticle modes that respect the extra conserved charges, and the correct $K_A$ is then obtained by evolving the reduced state under the transverse part of the Hamiltonian.","pith_inferences":["The factorized form of (4.24) suggests that the entire entanglement spectrum of a strip obeys dimensional reduction, not just entropies; one could test this by comparing the full spectrum of the exact correlation matrix with the spectrum of $K_{A,\\mathrm{QP}}$ for a two-dimensional strip.","Since the kernel (6.12) depends on the initial state only through $\\eta(k)$, any two Gaussian states with the same occupation function should produce identical entanglement Hamiltonians at the ballistic scale even when their pure parts differ, which is directly testable with two different squeezed states.","The counting-function structure for arbitrary $A$ suggests a route to the negativity Hamiltonian for two disjoint regions, which the paper identifies as an open problem; a formula along the lines of (6.12) with $B$ counting pairs shared between $A_1 \\cup A_2$ and its complement is a natural conjecture.","In interacting integrable models the same counting logic would predict a $K_A$ built from Bethe quasiparticle occupation data, but the paper notes the quasiparticle picture fails for R\\'enyi entropies there, so any such extension would have to be checked carefully against exact generalized-hydrodynamics results."],"forward_implications":["For strip geometries, the entanglement Hamiltonian in $d \\geq 2$ is a sum over transverse momenta of one-dimensional forms (4.24)--(4.26), so all quantities derived from it inherit the one-dimensional light-cone structure.","At long times the mixed part saturates and $K_{A,\\mathrm{QP}}$ becomes extensive in the subsystem volume, matching the volume-law growth of the entropy, and the light-cone support of the kernels makes the transition from boundary-localized to bulk-entangled explicit.","For a circular region, the kernel (6.7) yields the entropy (6.10) with early linear growth and late-time saturation, and the same counting-function construction extends to arbitrary connected regions.","Because the formula reproduces the known quasiparticle predictions for R\\'enyi entropies and full counting statistics, the entanglement Hamiltonian is consistent with every observable previously computed in this framework.","For states that do not relax to a GGE, such as the crossed dimer, the same framework still works after a unitary rotation to the quasiparticle basis that diagonalizes the extra conserved charges."],"supporting_citations":[{"why":"supplies the one-dimensional entanglement Hamiltonian (3.23)-(3.25) that the paper generalizes to higher dimensions.","marker":"[53]"},{"why":"gives the dimensional-reduction treatment of strip quenches in 2D whose entropy predictions the paper reproduces and whose states it uses for numerical checks.","marker":"[56]"},{"why":"provides the multiplet structure of shift-symmetric initial states and the quasiparticle framework for free-fermion lattices in $d>1$.","marker":"[55]"},{"why":"provides the exact free-fermion entanglement entropy growth that the quasiparticle formula must match in one dimension and by extension in strips.","marker":"[30]"},{"why":"establishes the Peschel formula linking the correlation matrix to the entanglement Hamiltonian, used for all exact numerical comparisons.","marker":"[61]"},{"why":"gives the exact time-dependent Bessel-function correlation matrix elements for the dimer states used in the numerical checks.","marker":"[62]"},{"why":"introduces the quasiparticle picture of entanglement growth that the paper lifts from entropy to the Hamiltonian level.","marker":"[27]"}],"fun_headline_variants":["Entanglement Hamiltonian from quasiparticles in d≥2","Quasiparticle picture for entanglement in d≥2","Higher-D entanglement Hamiltonian via quasiparticles","Free-fermion quasiparticle entanglement Hamiltonian in d≥2","Quasiparticle description of entanglement Hamiltonians in d≥2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the quasiparticle propagation ansatz (3.10), that a fluid-cell creation operator moves rigidly from cell $x_0$ to cell $x_0+v(k)t$ without dispersing, so that the reduced density matrix factorizes into pure and mixed parts at the ballistic scale.","fun_headline_variants_meta":{"raw":{"variants":["Entanglement Hamiltonian from quasiparticles in d≥2","Quasiparticle picture for entanglement in d≥2","Higher-D entanglement Hamiltonian via quasiparticles","Free-fermion quasiparticle entanglement Hamiltonian in d≥2","Quasiparticle description of entanglement Hamiltonians in d≥2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000693,"raw_usage":{"total_tokens":3109,"prompt_tokens":892,"completion_tokens":2217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":2135}},"tokens_in":508,"tokens_out":2217,"duration_ms":14707,"temperature":1.0,"reasoning_tokens":2135,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:17:04.984829+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact entanglement Hamiltonian, via the correlation matrix and the Peschel formula, for a two-dimensional free-fermion quench from a Gaussian initial state whose correlation length is comparable to the fluid-cell size, and compare it element by element with (4.24) for a strip: if the entries deviate by more than the chosen numerical cutoff, the rigid-propagation ansatz and the pure/mixed factorization fail. Alternatively, for a region with a curved boundary, check whether the kernel (6.12) reproduces the numerical $K_A$ for all pairs of sites.","supporting_citations":[{"cited_title":"Yamashika, F","cited_arxiv_id":null,"evidence_quote":"gives the dimensional-reduction treatment of strip quenches in 2D whose entropy predictions the paper reproduces and whose states it uses for numerical checks."},{"cited_title":"Gibbins, A","cited_arxiv_id":null,"evidence_quote":"provides the multiplet structure of shift-symmetric initial states and the quasiparticle framework for free-fermion lattices in $d>1$."},{"cited_title":"Peschel and V","cited_arxiv_id":null,"evidence_quote":"establishes the Peschel formula linking the correlation matrix to the entanglement Hamiltonian, used for all exact numerical comparisons."},{"cited_title":"Eisler and I","cited_arxiv_id":null,"evidence_quote":"gives the exact time-dependent Bessel-function correlation matrix elements for the dimer states used in the numerical checks."},{"cited_title":"Calabrese and J","cited_arxiv_id":null,"evidence_quote":"introduces the quasiparticle picture of entanglement growth that the paper lifts from entropy to the Hamiltonian level."}],"review_version":1}