{"id":"d365484e-7994-45a7-af41-69b39795e7b2","arxiv_id":"2508.11521","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In a 2D Kitaev lattice, Loschmidt-spectrum in-gap bands appear only for quenches into topological phases and account for the boundary contribution to the return rate.","lead":"Researchers simulated quantum quenches in a 2D topological superconductor and found that special in-gap bands in a non-Hermitian matrix appear only when the post-quench Hamiltonian is topological. These bands quantitatively explain an added boundary contribution to the dynamical free energy, suggesting a 2D analogue of the bulk-boundary correspondence.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Causal link between in-gap bands and boundary return rate rests on a fitted linear-dispersion ansatz; direct eigenvalue-sum test needed.","rationale":"The paper makes a credible and well-documented observation: in-gap bands appear in the Loschmidt spectrum for quenches into the topological phase and are absent for quenches into the trivial phase, across geometries, quenches, and disorder. The finite-size scaling collapse in Fig. 7 supports the scaling forms. The load-bearing weakness is precisely the quantitative match claimed in Fig. 2: it relies on fitting the in-gap bands to a linear dispersion (Eq. (19)) using parameters extracted from the same spectra whose boundary contribution they explain. This creates a risk of circularity—a flexible fit could reproduce the scaling-extracted l_B(t) even if the in-gap modes are not truly causative. A parameter-free check, directly summing the logarithmic eigenvalues of the numerically obtained in-gap modes, would settle whether the linear ansatz is essential or merely a convenient representation. The reader's weakest assumption identified the same vulnerability, and the recommended conditional verdict remains appropriate but should be explicitly tied to passing this direct test. No independent derivation or second model is currently provided, so the causal claim should not be upgraded without further evidence.","tokens_in":17908,"tokens_out":8840,"duration_ms":102140,"concrete_test":"Recompute l_B(t) directly from the raw in-gap eigenvalues of the Loschmidt matrix for the ribbon: for each time t, form l_B^raw(t) = - (δ/N_y) ∑_{k ∈ in-gap} ln|λ_k(t)| (per edge, using the same data as Fig. 3), and compare to the l_B(t) obtained from the finite-size scaling with Eqs. (13)-(14). If the two agree within the numerical uncertainty of the scaling extrapolation, the linear-band ansatz in Eq. (19) is a convenience, not a load-bearing assumption; if they disagree, the causal attribution fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (19) is the sole quantitative bridge between the in-gap Loschmidt eigenvalues and the boundary return rate l_B(t). It assumes a strictly linear in-gap dispersion |λ_k| = v|k| over a width 2Δk with edge value λ̃ = vΔk, and the parameters δ, Δk, λ̃ are fitted from the same finite-size spectra whose boundary contribution they are meant to explain. The paper reports no goodness-of-fit for the linear form, and no test of whether the scaling forms (13)-(14) are contaminated by O(1/N^2) bulk corrections that could masquerade as a boundary term. The ribbon-versus-flake comparison in Sec. IV A only isolates the exponentially small zero-mode contribution α_i; it does not validate the linear-band formula. If the true dispersion is curved, e.g. |λ_k| = sqrt(v^2 k^2 + m^2), the integral in Eq. (19) changes and the agreement in Fig. 2 would not follow. Because the central causal claim—that the in-gap bands are directly responsible for the boundary return rate—is supported only by this consistency fit, the quantitative link is load-bearing and currently unproven.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents numerical evidence for a dynamical bulk-boundary correspondence in a two-dimensional Kitaev lattice model. Following quenches into a topologically nontrivial phase, bands of in-gap eigenvalues appear in the spectrum of the Loschmidt matrix between successive critical times, while such bands are absent for quenches into the trivial phase. The authors propose Eq. (19), a formula for the boundary contribution to the return rate based on a linear-dispersion in-gap band, and fit parameters δ, Δk, and λ̃ to the same spectra. They report agreement between this fitted contribution and the boundary return rate extracted from finite-size scaling, concluding that the in-gap bands are directly responsible for the boundary term. Additional quenches and disorder checks are presented.","tokens_in":18211,"tokens_out":8540,"duration_ms":94962,"significance":"If established, the result would extend the dynamical bulk-boundary correspondence to two-dimensional topological matter and connect the spectrum of a non-Hermitian Loschmidt matrix to boundary contributions of the dynamical free energy. The clean qualitative observation—in-gap bands appear only for topological quenches and are robust to geometry and disorder—is valuable and likely correct. The paper includes data availability statements and multiple cross-checks. However, the quantitative causal link currently rests on a fitted linear-dispersion ansatz, so the paper's strongest claim is not yet fully supported.","major_comments":[{"comment":"The central quantitative claim is supported only by a consistency fit. Eq. (19) assumes a strictly linear in-gap dispersion |λ_k| = v|k| = λ̃|k|/Δk over a width 2Δk, and the parameters δ, Δk, and λ̃ are extracted from the same finite-size Loschmidt spectra whose boundary contribution they are meant to explain. No goodness-of-fit, error bars, or independent test of the linear form are provided. The agreement in Fig. 2 therefore does not distinguish the linear-band model from a generic curved dispersion, which would change the integral in Eq. (19). To make the causal claim load-bearing, please compute the boundary return rate directly from the sum over the in-gap eigenvalues, e.g. \\tilde l_B^{direct}(t) = -1/(2N_x) ∑_{k∈in-gap} ln|λ_k(t)| for the ribbon and the analogous four-edge sum for the flake, without fitting, and compare with l_B(t) from the scaling forms. This would also validate t","section":"§IV A, Eq. (19)"},{"comment":"The finite-size scaling forms are assumed to isolate the boundary term. In Eq. (13), with N_x fixed at 202 and N_y ∈ {200,...,700}, the 'bulk correction' A/(N_x N_y) is of the same order in 1/N_y as the boundary term itself, so the extraction of l_B(t) depends on the flake-derived A(t) being transferable to the ribbon geometry and on the absence of an O(1/N_y^2) correction. The ribbon scaling collapse in Fig. 7 is helpful, but the flake scaling and the extracted A(t) are not shown, and no sensitivity test to including a 1/N_y^2 term is reported. Please provide these, or justify the scaling forms from a controlled expansion, to rule out contamination of l_B(t) by bulk finite-size effects.","section":"§IV A, Eqs. (13)–(14), Fig. 7"},{"comment":"The quantitative identification of l_B(t) with the in-gap-band contribution is demonstrated for one quench (ν:0→1) only. For the other topological quenches (ν:1→−1 and ν:1(Δ>0)→1(Δ<0)), the text states that 'a direct comparison to l_B(t) is hindered by insufficient system sizes to perform a stable scaling analysis' (Appendix E). The qualitative in-gap bands are visible, but the claim that they 'directly account' for the boundary return rate in general is not yet supported. Either provide a stable scaling comparison for at least one additional topological quench, or soften the central claim to a conjecture for these cases.","section":"Appendix E and §IV A"}],"minor_comments":[{"comment":"The notation λ_i → (1, λ_k) is unclear. Please state explicitly that half the eigenvalues are identically 1 and only the λ_k sector is nontrivial.","section":"§II, Eq. (7)"},{"comment":"The symbols are not defined in the main text; please state clearly that they are \\tilde l_B and \\tilde l_{B+0}, and indicate the time points at which they are evaluated.","section":"Fig. 2 caption"},{"comment":"The index i in α_i is not defined; specify that it labels the exponentially small zero modes of the ribbon.","section":"Eq. (20)"},{"comment":"The y-axis label '104lB(t)/Ny' should read '10^4 l_B(t)/N_y'.","section":"Appendix C, Fig. 7"},{"comment":"The text introduces a 'velocity' v, but v never appears explicitly in Eq. (19). This is fine if v cancels in the derivation, but it should be noted or shown explicitly.","section":"§IV A"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a condensed-matter physics journal and the qualitative numerical observation is timely. The main risk is overclaiming the quantitative attribution. The authors should be encouraged to perform the direct eigenvalue-sum test; it is straightforward and would considerably strengthen the paper. I see no citation or novelty issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: the paper's central observation is real and well documented; its headline claim is somewhat oversold. The in-gap Loschmidt bands for quenches into a 2D topological phase are there, and the absence for trivial quenches is consistent. But the statement that those bands 'directly cause' the boundary return rate rests on Eq. (19), which is a fitted linear-dispersion ansatz, not a derivation.\n\nWhat's genuinely new: the DBBC was previously established for 1D chains and 2D higher-order topological insulators, where the in-gap modes are pinned at zero. Here, for an ordinary 2D topological superconductor (the 2D Kitaev lattice), the in-gap modes form dispersing bands between successive critical regions. That's a new feature and they support it with a clean numerical study: ribbon and flake geometries, several quenches (0→1, 1→0, 1→-1, and a same-Chern-number Δ-sign quench), disorder checks, and deposited data. I trust the qualitative observation.\n\nWhere it gets soft: Eq. (19) assumes |λ_k| = v k over width 2Δk and an edge value λ̃ = v Δk. The parameters δ, Δk, λ̃ are extracted from the same finite-size Loschmidt spectra whose boundary contribution they're meant to explain. So the agreement in Fig. 2 is a consistency check between two ways of slicing the same eigenvalues—not an independent confirmation. There's no reported goodness-of-fit for the linear dispersion, and the finite-size scaling forms (13)-(14) are demonstrated at only two representative times. If the true dispersion were curved, Eq. (19) would change and the agreement might not hold. The paper is honest that no complete theory exists, but the intro's 'directly responsible' and 'establishing' language goes beyond what the numerics show.\n\nThat said, I don't think the qualitative DBBC claim is in danger. The disorder robustness and the geometry dependence (zero modes only in the ribbon, contributing the α_i terms) are solid. The fix is straightforward: either derive Eq. (19) from the model, or compute the boundary return rate directly by summing the numerical in-gap eigenvalues without the linear ansatz and compare to the scaling-extracted l_B(t).\n\nThis paper deserves a serious referee. I'd send it out, but with the expectation that the causal language be softened or the quantitative link strengthened. People working on DQPTs and topological dynamics will want to read it; just don't take the causation claim at face value.\n\nRecommendation: peer review, conditional on revisions.","headline":"Real new observation of in-gap Loschmidt bands for 2D topological quenches, but the causal attribution to the boundary return rate is a fitted consistency check, not a proof.","tokens_in":18694,"tokens_out":3438,"would_cite":true,"duration_ms":38671,"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":"The paper establishes a dynamical bulk-boundary correspondence in two-dimensional topological matter: quenches into nontrivial phases produce in-gap Loschmidt bands that drive the boundary return rate.","keywords":["dynamical quantum phase transitions","Loschmidt matrix","dynamical bulk-boundary correspondence","topological superconductors","two-dimensional Kitaev model","return rate","in-gap bands","quantum quench"],"falsifier":"For the $\\nu:0\\to1$ quench, fit the in-gap band dispersion at a fixed time to $|\\lambda_k|=v k + a k^2$; a significant quadratic coefficient $a$ invalidates Eq. (19). Separately, enlarge the flake scaling to $N\\sim 200$ and check whether $l_B(t)-\\tilde l_B(t)$ vanishes or retains a $1/N^2$ term; a retained term means the scaling forms did not isolate the boundary contribution.","tokens_in":17788,"feed_emoji":"⚛️","tokens_out":8604,"duration_ms":88663,"temperature":0.7,"pith_summary":"This paper aims to establish a dynamical bulk-boundary correspondence in two-dimensional topological matter. After a quantum quench, the spectrum of the Loschmidt matrix develops bands inside the bulk gap between successive dynamical quantum phase transitions, but only when the time-evolving Hamiltonian is topologically non-trivial; quenches into a trivial phase show no such in-gap bands in any studied case. The paper shows that these in-gap bands, fitted with a linear dispersion, account for the boundary contribution to the dynamical free energy (the return rate) extracted from finite-size scaling of ribbons and flakes of a two-dimensional Kitaev lattice. If correct, the boundary return rate becomes a dynamical order parameter that can distinguish quenches into different topological phases in two dimensions, and dynamical topological phenomena can be classified through spectra of a non-Hermitian Loschmidt matrix.","feed_headline":"In-gap bands drive the boundary signal after 2D topological quenches","feed_subtitle":"Boundary return rates trace the in-gap Loschmidt bands that appear only when a 2D quench lands in a topological phase.","key_machinery":"The central object is the non-Hermitian dynamical Loschmidt matrix $\\mathcal{M}(t)=1-C+C e^{-itH_1}$, where $C$ is the ground-state correlation matrix of the initial Hamiltonian; its determinant gives the Loschmidt amplitude and its eigenvalues $\\lambda_i(t)$ the return rate. The load-bearing spectral feature is a band of in-gap eigenvalues with strictly linear dispersion $|\\lambda_k|=v k$ that appears between successive critical times; Eq. (19) turns its width $2\\Delta k$, degeneracy $\\delta$, and edge value $\\tilde\\lambda$ into a quantitative prediction for the boundary return rate. The matrix also supplies the exponentially small ribbon-only eigenvalues that distinguish ribbon and flake b","core_discovery":"Using a spinless $p_x+ip_y$ superconductor on a square lattice (the two-dimensional Kitaev model), the authors quench between phases with Chern numbers $\\nu=0$, $\\nu=1$, and $\\nu=-1$, and compute the Loschmidt amplitude from the determinant of the Loschmidt matrix $\\mathcal{M}(t)=1-C+C e^{-itH_1}$. For quenches ending in a topologically non-trivial phase, the smallest eigenvalues of $\\mathcal{M}$ form linear in-gap bands, $|\\lambda_k|=v k$, between successive critical regions; for quenches into the trivial phase these bands are absent. Fitting the band width $2\\Delta k$, the degeneracy $\\delta$, and the edge eigenvalue $\\tilde\\lambda=v\\Delta k$, they compute a boundary return rate $\\tilde l_","pith_inferences":["The paper leaves implicit that the predicted $\\tilde l_B$ depends on edge orientation through $\\delta$ and $\\Delta k$; cutting flakes along different lattice directions is a direct test.","If a time-dependent topological index counting in-gap bands can be defined, the correspondence would move from empirical to derived; the band width and degeneracy are natural candidates for such an index.","The same Loschmidt-matrix mechanism should appear in three-dimensional topological matter, where Fisher zeroes occupy volumes; the boundary contribution would then scale with surface area and in-gap sheets would replace bands.","Momentum-resolved measurements of Loschmidt spectra in synthetic quantum systems should show the in-gap bands as bright lines, making the correspondence observable without finite-size scaling."],"forward_implications":["The boundary return rate $l_B(t)$ can act as a dynamical order parameter: large periodic boundary contributions appear if and only if the time-evolving Hamiltonian is topologically non-trivial.","In-gap dispersing bands, not only zero-energy modes, generate the boundary signal, so the boundary return rate slopes downward between critical times instead of sitting at a plateau.","The difference between open ribbons and fully open flakes is quantitatively the sum of exponentially small Loschmidt eigenvalues present only in the ribbon, tracing back to Majorana zero modes of the time-evolving Hamiltonian.","The correspondence extends the one-dimensional dynamical bulk-boundary correspondence to ordinary two-dimensional topological insulators and superconductors and suggests that Loschmidt-matrix spectra can classify dynamical topological phenomena.","Disorder robustness of the in-gap bands indicates the effect is topological rather than a finite-size artifact."],"supporting_citations":[{"why":"Establishes the one-dimensional dynamical bulk-boundary correspondence that this paper extends: boundary return-rate contributions arise when the time-evolving Hamiltonian is topological.","marker":"[13]"},{"why":"Extends the correspondence to multiband topological insulators and gives the in-gap-eigenvalue picture for periodic boundary contributions.","marker":"[14]"},{"why":"Shows the dynamical bulk-boundary correspondence for two-dimensional higher-order topological insulators, the immediate two-dimensional precedent.","marker":"[15]"},{"why":"Studies the higher-order Benalcazar-Bernevig-Hughes model and the exceptions where Fisher zeroes form lines rather than areas.","marker":"[16]"},{"why":"Supplies the Fisher-zero formula and two-band topological classification used to identify critical times for quenches.","marker":"[84]"},{"why":"Provides the bulk DQPT analysis for the two-dimensional Kitaev model, including Fisher-zero areas and critical-time structure.","marker":"[98]"},{"why":"Gives the correlation-matrix determinant formula for the Loschmidt amplitude, the basis of the Loschmidt matrix.","marker":"[101]"},{"why":"Provides the derivation of the determinant formula used to compute the Loschmidt amplitude numerically.","marker":"[102]"},{"why":"Applies the determinant formula to Loschmidt echoes, grounding the numerical method.","marker":"[103]"},{"why":"Explains orientation-dependent Majorana zero modes on edges of two-dimensional topological superconductors, used to interpret ribbon-versus-flake differences.","marker":"[105]"}],"fun_headline_variants":["2D topological quenches leave in-gap fingerprints in Loschmidt spectrum","Boundary dynamics reveal 2D topological order after quench","In-gap bands signal topological phase in 2D quantum quench","Loschmidt bands expose hidden boundary signature in 2D matter","Quench into topological phase spawns in-gap boundary modes"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The central quantitative claim assumes the in-gap bands are strictly linear in momentum over their full width and that the finite-size scaling used to extract the boundary return rate captures only the boundary contribution, with no higher-order bulk corrections contaminating the result.","fun_headline_variants_meta":{"raw":{"variants":["2D topological quenches leave in-gap fingerprints in Loschmidt spectrum","Boundary dynamics reveal 2D topological order after quench","In-gap bands signal topological phase in 2D quantum quench","Loschmidt bands expose hidden boundary signature in 2D matter","Quench into topological phase spawns in-gap boundary modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000491,"raw_usage":{"total_tokens":2246,"prompt_tokens":732,"completion_tokens":1514,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":1421}},"tokens_in":476,"tokens_out":1514,"duration_ms":10188,"temperature":1.0,"reasoning_tokens":1421,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:51:04.783932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the $\\nu:0\\to1$ quench, fit the in-gap band dispersion at a fixed time to $|\\lambda_k|=v k + a k^2$; a significant quadratic coefficient $a$ invalidates Eq. (19). Separately, enlarge the flake scaling to $N\\sim 200$ and check whether $l_B(t)-\\tilde l_B(t)$ vanishes or retains a $1/N^2$ term; a retained term means the scaling forms did not isolate the boundary contribution.","supporting_citations":[{"cited_title":"Vajna and B","cited_arxiv_id":null,"evidence_quote":"Supplies the Fisher-zero formula and two-band topological classification used to identify critical times for quenches."},{"cited_title":"Mas lowski, H","cited_arxiv_id":null,"evidence_quote":"Provides the bulk DQPT analysis for the two-dimensional Kitaev model, including Fisher-zero areas and critical-time structure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the correlation-matrix determinant formula for the Loschmidt amplitude, the basis of the Loschmidt matrix."},{"cited_title":"Klich, An Elementary Derivation of Levitov’s For- 13 mula, in Quantum Noise in Mesoscopic Physics , NATO Advanced Science Series, Vol","cited_arxiv_id":null,"evidence_quote":"Provides the derivation of the determinant formula used to compute the Loschmidt amplitude numerically."},{"cited_title":"Rossini, T","cited_arxiv_id":null,"evidence_quote":"Applies the determinant formula to Loschmidt echoes, grounding the numerical method."},{"cited_title":"Sedlmayr, V","cited_arxiv_id":null,"evidence_quote":"Explains orientation-dependent Majorana zero modes on edges of two-dimensional topological superconductors, used to interpret ribbon-versus-flake differences."}],"review_version":1}