{"id":"1f3692d2-6bd0-4746-bdf0-b460afe458ac","arxiv_id":"2608.08205","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact two-point correlation functions after quenches in an inhomogeneous XX chain, with explicit hydrodynamic-limit formulas governed by a single transmission coefficient.","lead":"This paper derives exact formulas for how fermions and correlations spread in a one-dimensional quantum chain that has different magnetic fields on its two halves. The results turn a previously conjectured quasiparticle description of entanglement growth in such inhomogeneous systems into a derived statement.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hydrodynamic formulas (66)-(73) omit the B(s,s')=π pole that Section 4.1 says is needed for distant-cell correlators, so the unqualified abstract claim is stronger than what is proved.","rationale":"The paper's central deliverable is an ab initio derivation of hydrodynamic-limit correlators in terms of the single-particle transmission coefficient. The weakest link is the evaluation of the double integral (62) by keeping only the pole at s=s′: the authors themselves concede in Section 4.1 that a second pole at B(s,s′)=π appears for N´eel-type quenches and is needed for correlations between distant mesoscopic cells, yet this pole is not included in (66)-(73). If that pole contributes at the same order for some fixed x/t and y/t, the unqualified abstract claim fails. The numerical benchmarks in Section 5 only test local correlators (x and x+j with small fixed j) and the leading across-interface decay, which are exactly the regimes the authors say the B=0 pole captures; they do not test the regime where the B=π pole matters. A concrete check is to evaluate the full even-k double integral for |x−y|∼t and compare to the B=0-only formula; this would settle whether the omitted pole affects the leading hydrodynamic behavior. This concern does not invalidate the local results, which are numerically supported, so conditional acceptance with a mandatory clarification remains appropriate. The reader's weakest_assumption identified the same pole-dominance issue; I agree. A secondary but concrete obstruction is the identically zero factor in Eqs. (60) and (62), which makes the derivation of (66) irreproducible as printed and strengthens the case for a conditional decision rather than acceptance.","tokens_in":19417,"tokens_out":15695,"duration_ms":146931,"concrete_test":"For the quench from the inhomogeneous N´eel state (25) in the homogeneous limit h_L=h_R, where exact results are available, numerically evaluate the full double integral (62) with the correct even-k denominator 1−e^{2iΔA} in a regime with x,y<0 and |x−y|∼t, e.g., x=−0.8t, y=−0.2t for large t, using stationary-phase/saddle-point methods or direct quadrature. Compare the result to the analytic B=0-only prediction (66) multiplied by the even-k factor. If the difference does not vanish at the expected power-law order as t→∞, the omitted B=π pole contributes at leading order and the hydrodynamic formulas are incomplete for such separations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The derivation of the hydrodynamic-limit formulas (66)-(73) in Section 4 starts from the double integral (62), changes variables to Q=s−s′ and K=(s+s′)/2, Taylor expands around Q=0, extends the Q-integral to ±∞, and keeps only the pole at Q=0, producing the theta-function support in (66). However, the manuscript's own Section 4.1 states that for the N´eel-type quenches (even-k sums), the sum over k gives a denominator 1−e^{2iB(s,s′)} that has a second pole at B(s,s′)=π, and that this pole is required for correlators between distant cells on the same side of the interface. That pole is not included in any of the explicit formulas (66)-(73). Therefore, in kinematic regimes where the stationary-phase point satisfies B(s,s′)=π, the omitted contribution can be of the same order as the retained B=0 contribution, so the formulas do not provide the complete hydrodynamic limit for arbitrary separations x−y. The abstract and introduction present the hydrodynamic result without this qualification, making the central claim stronger than what is actually established. In addition, Eqs. (60) and (62) as printed contain the factor (¯λ′_+(s′)−¯λ′_+(s′)), which is identically zero, so the intermediate integrand vanishes and the derivation of (66) cannot be reproduced as written without a correction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the out-of-equilibrium dynamics of the XX spin chain with a step-like magnetic field, mapping it to an inhomogeneous tight-binding fermionic model. The authors develop an exact solution for the two-point fermionic correlation function after quenches from product states by combining Laplace and Fourier transforms with a Riemann-Hilbert problem on the unit circle. They then derive hydrodynamic-limit formulas (x,y,t→∞ with fixed ratios) in which correlation functions are expressed through the single-particle transmission coefficient across the interface, and they benchmark these formulas against exact numerics for local densities, short-range correlators, and across-interface correlators. A final consistency check reproduces the quasiparticle entropy formula of Ref. [35] from the hydrodynamic correlator.","tokens_in":19700,"tokens_out":12669,"duration_ms":126902,"significance":"The paper addresses a paradigmatic exactly solvable inhomogeneous free-fermion quench and provides an ab initio route from the exact solution to the hydrodynamic/quasiparticle picture. The main strength is that the hydrodynamic formulas are derived, not postulated, and they contain no free parameters; the numerical benchmarks for the domain-wall and Néel-type quenches support the central formulas, and the reproduction of the entropy result of Ref. [35] is a valuable consistency check. If the issues below are fixed, the paper would be a substantial contribution to the exact treatment of defects and inhomogeneous fields in integrable systems. The current version, however, contains several load-bearing typos and an explicit incompleteness of the same-side hydrodynamic formulas that must be resolved before the abstract's unqualified claims are justified.","major_comments":[{"comment":"The integrand contains the factor (λ'_+(s)-ρ'_+(s)) times (\\bar λ'_+(s')-\\bar λ'_+(s')), whose second factor is identically zero. As printed, the double integral in Eq. (60) and Eq. (62) vanishes, so the derivation of Eq. (66) cannot be reproduced. Please correct the typo, presumably to (\\bar λ'_+(s')-\\bar ρ'_+(s')), and re-check the subsequent algebra leading to the transmission-coefficient formula.","section":"Section 4, Eqs. (60) and (62)"},{"comment":"The authors explicitly state that Eqs. (66)-(73) keep only the pole at B(s,s')=0 and that the second pole at B(s,s')=π is needed for correlators between distant cells on the same side. Since the hydrodynamic limit allows separations x-y of order t, formulas (66)-(69) are therefore not the complete hydrodynamic limit for arbitrary same-side separations. The abstract and introduction present the hydrodynamic result without this qualification; please either include the B(s,s')=π contribution or explicitly restrict the claims to local correlators and to the leading across-interface behavior.","section":"Section 4.1, paragraph after Eq. (80)"},{"comment":"The result for k<-1 is stated without derivation (\"We do not show the derivation of the results\"), although it is needed for the exact solution and is used to obtain the hydrodynamic formulas (68), (69), and (73) through the negative-k sector. Please provide a derivation or at least a detailed proof sketch of Eq. (54), including the exchange h_L↔h_R and the case separation x≤-1 versus x>-1.","section":"Section 3.3, Eq. (54)"},{"comment":"The conditions h_R<h_L and h_L-h_R<1 are imposed without explanation, while Section 3.3 discusses the regime h_L-h_R≤2. The case 1≤h_L-h_R<2 is neither treated nor excluded from the abstract's claims. Please explain the origin of the h_L-h_R<1 restriction and either extend the derivation to the full transmission window or state this restriction prominently in the abstract and introduction.","section":"Section 4, immediately after Eq. (62)"},{"comment":"As written, the identity has an incorrect sign for the + branch: ∫ dQ/(2πi) e^{iQx}/(Q+i0+) equals -Θ(-x), not Θ(-x); only the - branch gives Θ(x). Because this identity controls the theta-function supports in Eqs. (66)-(73), please correct Eq. (65) and specify unambiguously which regulator branch is used in the derivation for each term.","section":"Section 4, Eq. (65)"},{"comment":"The symbol T denotes both the transmission amplitude in Eq. (13) and the real transmission probability in Eq. (15). Eq. (66) refers to \"T(k) is the transmission coefficient (15)\" but then writes Re[T(k)], which is redundant if T is the real probability and meaningful only if T is the amplitude. Since the hydrodynamic formulas and the entropy formula (84) depend on which object is used, please introduce distinct symbols (for example T_amp and T_prob) and state explicitly which one enters each final formula.","section":"Section 2, Eqs. (13) and (15); Section 4, Eqs. (66)-(73) and (84)"}],"minor_comments":[{"comment":"The coefficients A^{(j)}_\\pm are said to be defined in Table 4, but the table containing these coefficients is Table 3; please renumber the cross-reference.","section":"Section 3.3, below Eq. (56)"},{"comment":"The notation G^{[σ1σ2σ3]}_{xy} is used without an explicit definition of the three labels; please define them (for example, signs of x and y and the sign of the k-sector) when the notation is introduced.","section":"Section 4, Eqs. (59)-(73)"},{"comment":"The Heaviside arguments for the h_L terms differ in form between Eq. (17) (Θ(-x-1) and Θ(-y-1)) and Eq. (22) (Θ(-x-1) only); please check the consistency of the convention at x=0 and state the convention Θ(0)=1 clearly.","section":"Section 3, Eq. (17) versus Eq. (22)"},{"comment":"The phrase \"local correlators, i.e., for x≈y\" is potentially misleading in the hydrodynamic limit, where x and y themselves diverge; please clarify that \"local\" means x-y fixed while x,y,t→∞.","section":"Section 4, after Eq. (80)"},{"comment":"The text states that G_{x,x+j} has a discontinuity at the origin for even j, but the mechanism behind this discontinuity is not explained; a brief comment in the main text would help.","section":"Section 5, Fig. 4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the overall strategy is sound, but the current version is not publishable as is: the zero factor in Eqs. (60) and (62), the sign issue in Eq. (65), the ambiguous transmission notation, and the acknowledged omission of the B=π pole for same-side correlators are all load-bearing for the central hydrodynamic claims. I believe these are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First things first: this is the first exact derivation of the hydrodynamic-limit two-point function for the XX chain with a step potential, and it is a genuine step beyond the conjectured quasiparticle picture in Ref. [35]. The Riemann-Hilbert machinery is heavy but the result is concrete: correlators in the scaling limit are expressed through the single-particle transmission coefficient T(k), and the numerical benchmarks for the domain-wall and N'eel quenches agree well. They also reproduce the conjectured entanglement-growth formula as a consistency check. That's solid, reproducible work.\n\nThe soft spots are real but mostly addressable. The most glaring is a typo in Eqs. (60) and (62): the factor (\\bar λ_+ - \\bar λ_+) in the integrand is identically zero, so both integrals vanish as written. The final formula (66) has the right structure, so it is clearly a transcription slip, but it blocks a reader from following the derivation. Fix it. Second, Eq. (54) for k<-1 is stated as 'verified' without a derivation; a referee should ask for the argument. Third, and more substantively, the abstract claims 'explicit formulas' for the hydrodynamic limit without qualification, but Section 4.1 admits that the derivation keeps only the B=0 pole. That pole is sufficient for local correlators (x≈y) and for the leading across-origin behavior, but correlations between distant cells on the same side would require the B=π pole, which is not included in (66)-(73). So the blanket claim is stronger than what is proved. The paper should either supply that contribution or rewrite the abstract to state the limitation clearly. The conclusions also contradict the abstract on 'elementary functions' — another pass to clean up.\n\nNone of these issues break the central result, which is a first-principles derivation of the hydrodynamic description from the exact solution. But the paper needs revision, not just proofreading.\n\nI'd send it to a serious referee: the math is substantial, the benchmarks are honest, and the overclaim is fixable. Worth reading for anyone in exact quench dynamics or GHD.","headline":"Exact hydrodynamic-limit correlators for the step-potential XX chain: real novelty and good numerics, but the abstract overclaims and a typo breaks two intermediate equations.","tokens_in":20238,"tokens_out":4757,"would_cite":true,"duration_ms":40824,"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 derives the exact two-point correlator of an inhomogeneous XX chain after a quench and shows that in the hydrodynamic limit it is determined entirely by the transmission coefficient across the interface.","keywords":["XX chain","inhomogeneous magnetic field","quantum quench","Riemann-Hilbert problem","hydrodynamic limit","two-point correlation function","transmission coefficient","quasiparticle picture"],"falsifier":"Evaluate the full double integral (62) numerically for large $x,y,t$ with $x$ and $y$ far apart on the same side, or solve the linear system exactly, and compare with formulas (66)-(69); a mismatch would reveal the missing contribution of the $B(s,s')=\\pi$ pole.","tokens_in":19163,"feed_emoji":"🧲","tokens_out":10616,"duration_ms":89740,"temperature":0.7,"pith_summary":"The paper studies the time evolution of the XX spin chain with a step-like magnetic field, which after a Jordan-Wigner transformation is a free-fermion hopping problem on two half-chains with different chemical potentials. It establishes that the two-point fermionic correlation function after a quench from product initial states obeys a linear system whose exact solution can be obtained by reducing the frequency-space equation to a Riemann-Hilbert problem on the unit circle. The central result is that in the hydrodynamic limit $x,y,t\\to\\infty$ with fixed ratios $x/t,y/t$, every correlator collapses to explicit integrals over the single-particle transmission coefficient $T(k)$ across the interface, such as $G^{[+--]}_{x,y}(t)=\\int_0^\\pi \\frac{dk}{2\\pi}\\mathrm{Re}[T(k)] e^{i(k+\\pi)(x-y)}\\Theta((x+y+2)/2+t v(k))\\Theta(-x)\\Theta(-y)$. The paper verifies these formulas against exact numerics and uses them to rederive the quasiparticle formula for the von Neumann entropy, giving the missing first-principles derivation of that picture for these quenches. If correct, the result reduces a non-equilibrium many-body problem to a scattering problem, with all large-scale correlation physics encoded in the interface transmission coefficient.","feed_headline":"One transmission coefficient controls all large-scale correlations","feed_subtitle":"Exact solution of the XX chain: every large-scale correlator is fixed by the interface transmission coefficient","key_machinery":"The engine of the paper is the factorization $G_{xy}=\\sum_k S_{k,x}\\bar{S}_{k,y}$ of the correlator into single-particle mode functions, followed by a Laplace transform in time and Fourier transform in space that turns the equation of motion into a Riemann-Hilbert problem on the unit circle: find analytic functions inside and outside the circle whose jump across the circle is dictated by the magnetic-field step (Eq. (37)). The solution uses the Plemelj formula and residue calculus; the inverse transforms run along branch cuts in the $s$-plane, and the hydrodynamic limit is extracted by stationary phase and by the dominance of the pole at $s=s'$ in the double integral. This machinery converts the exact but unwieldy sums into transparent integrals involving only $\\mathrm{Re}[T(k)]$ and the velocity $v(k)=J\\sin k$.","core_discovery":"In the hydrodynamic limit $x,y,t\\to\\infty$ with fixed $x/t$ and $y/t$, the two-point fermionic correlator is completely determined by the single-particle scattering data across the origin. For both points on the left of the interface the paper obtains (Eq. (66)) $G^{[+--]}_{x,y}(t)=\\int_0^\\pi \\frac{dk}{2\\pi}\\mathrm{Re}[T(k)] e^{i(k+\\pi)(x-y)}\\Theta((x+y+2)/2+t v(k))\\Theta(-x)\\Theta(-y)$, with $v(k)=J\\sin k$ and $T(k)$ the transmission coefficient (15). Analogous formulas hold for both points on the right and for one point on each side; in the across-interface case the momentum $k$ on one side is related to the momentum $k'=\\arccos(h_L-h_R+\\cos k)$ on the other by energy conservation, and the $\\theta$ functions encode the light-cone condition set by the velocity. The paper derives these limiting expressions from the exact solution, not by postulating them, and it shows that in momentum space between mesoscopic cells the correlator becomes diagonal on each side of the interface with a transmission-reduction term. Together the formulas imply that in the scaling limit the initial state enters only through the choice of occupied sector, while the interface enters only through $\\mathrm{Re}[T(k)]$.","pith_inferences":["The same Riemann-Hilbert reduction should transfer to other quadratic chains with a step-like inhomogeneity, for example the Kitaev chain with a step in the pairing term, with the transmission coefficient again playing the central role; the paper lists this as a future direction.","Including the second pole at $B(s,s')=\\pi$ would supply the correlations between distant mesoscopic cells that the explicit formulas leave out, which are exactly the inputs needed for entanglement entropy of subsystems away from the interface.","The discontinuous behavior of $\\mathrm{Re}\\,G_{x,x+j}$ at $x/t=0$ for even $j$ is a sharp experimental signature that could be looked for in optical-lattice realizations of the step potential.","The predicted $x^{-1/2}$ decay of the across-interface correlator provides a quantitative finite-size target against which experiments or exact numerics can test the hydrodynamic limit."],"forward_implications":["In the hydrodynamic limit, every large-scale fermionic correlator after a quench from the considered product states is fixed by the single-particle transmission coefficient $T(k)$ across the interface; the initial state enters only by selecting which $k$-sectors are occupied.","The mesoscopic-cell Fourier correlator is diagonal in momentum on each side of the interface, with a transmission-reduction term that accounts for particles that were transmitted away; across the interface, the two momenta are linked by energy conservation, $k'=\\arccos(h_L-h_R+\\cos k)$.","The correlator (66) yields the von Neumann entropy of a subsystem of length $\\ell$ next to the origin as $S_A(t)=\\int_0^\\pi \\frac{dk}{2\\pi} \\min(v(k)t,\\ell)\\, s(\\mathrm{Re}[T(k)]/2)$, which coincides with the previously conjectured quasiparticle formula.","Numerical benchmarks confirm the hydrodynamic formulas at finite times: for the domain-wall-type quench the agreement is good already around $t\\simeq 80$, while the N\\'eel-state quench shows larger finite-time corrections, and the across-interface correlator decays as $|G_{xy}|\\sim x^{-1/2}$ along $t=2x$."],"supporting_citations":[{"why":"Supplies the mode-function factorisation and the Laplace/Fourier transform strategy for free fermions with a localized source that the present solution extends to an inhomogeneous step.","marker":"[15]"},{"why":"Supplies the linear-system factorization and the hydrodynamic-limit method for free fermions with localized losses; the paper's derivation follows it closely.","marker":"[17]"},{"why":"Provides the exact factorised solution for inhomogeneous quenches in a free fermionic chain, the starting point for the present Riemann-Hilbert formulation.","marker":"[42]"},{"why":"Contains the conjectured quasiparticle picture for entanglement spreading that Eq. (84) here derives ab initio from the correlator.","marker":"[35]"},{"why":"Provides the boundary-value-problem solution (Plemelj formula, homogeneous solution X(z)) used to solve the Riemann-Hilbert problem.","marker":"[43]"},{"why":"Supplies the strategy of extracting the von Neumann entropy from the Fourier transform of the correlator between mesoscopic cells, used to obtain Eq. (84) and to point to the second pole needed for entanglement.","marker":"[24]"}],"fun_headline_variants":["Transmission coefficient governs all scaling correlations","Exact XX chain: hydrodynamic correlators from one coefficient","Single scattering data sets all large-scale fermion correlations","Hydrodynamic limit of XX chain: one transmission number decides all","Step-field XX chain: exact solution, one coefficient rules correlations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that in the limit $x,y,t\\to\\infty$ the double spectral integral (62) is dominated by the pole at $s=s'$ after stationary phase; the paper notes that a second pole at $B(s,s')=\\pi$ is needed for correlations between distant cells, so if that pole dominance fails the explicit formulas would be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Transmission coefficient governs all scaling correlations","Exact XX chain: hydrodynamic correlators from one coefficient","Single scattering data sets all large-scale fermion correlations","Hydrodynamic limit of XX chain: one transmission number decides all","Step-field XX chain: exact solution, one coefficient rules correlations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000615,"raw_usage":{"total_tokens":2873,"prompt_tokens":979,"completion_tokens":1894,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":1816}},"tokens_in":595,"tokens_out":1894,"duration_ms":13828,"temperature":1.0,"reasoning_tokens":1816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:16:25.061883+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the full double integral (62) numerically for large $x,y,t$ with $x$ and $y$ far apart on the same side, or solve the linear system exactly, and compare with formulas (66)-(69); a mismatch would reveal the missing contribution of the $B(s,s')=\\pi$ pole.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the mode-function factorisation and the Laplace/Fourier transform strategy for free fermions with a localized source that the present solution extends to an inhomogeneous step."},{"cited_title":"Inhomogeneous quenches in a fermionic chain: exact results","cited_arxiv_id":"1507.08132","evidence_quote":"Provides the exact factorised solution for inhomogeneous quenches in a free fermionic chain, the starting point for the present Riemann-Hilbert formulation."},{"cited_title":"Entanglement dynamics after quenches with inhomogeneous Hamiltonians","cited_arxiv_id":"2605.01595","evidence_quote":"Contains the conjectured quasiparticle picture for entanglement spreading that Eq. (84) here derives ab initio from the correlator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the boundary-value-problem solution (Plemelj formula, homogeneous solution X(z)) used to solve the Riemann-Hilbert problem."},{"cited_title":"Alba,Unbounded entanglement production via a dissipative impurity, SciPost Phys.12, 11 (2022), doi:10.21468/SciPostPhys.12.1.011","cited_arxiv_id":null,"evidence_quote":"Supplies the strategy of extracting the von Neumann entropy from the Fourier transform of the correlator between mesoscopic cells, used to obtain Eq. (84) and to point to the second pole needed for entanglement."}],"review_version":1}