{"id":"8027f3cd-147e-4593-b5f3-00626ebf487b","arxiv_id":"2506.19913","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a strongly disordered Su-Schrieffer-Heeger chain, the phase is controlled by the ratio of disorder strengths sigma_v/sigma_w, and the transition at sigma_v=sigma_w is an infinite-randomness fixed point with effective central charge ln 2.","lead":"This paper studies a one-dimensional chain of electrons with random hopping strengths and finds that, under very strong randomness, the chain's topological character is set by the widths of the two random distributions, not by their average values. The transition between trivial and topological behavior occurs at an infinite-randomness critical point, which may point to a new family of disorder-driven topological transitions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The SDRG phase determination rests on a nonstandard list-doubling protocol and on arithmetic means of broad distributions; neither is shown to reproduce the true IRFP flow, so the weff/veff limit may be an artifact.","rationale":"The reader's weakest assumption correctly identifies the SDRG list-doubling step as a nonstandard operation with no justification. In my own reading of the manuscript, this is indeed the most load-bearing concern: although exact-diagonalization polarization and LDOS provide independent support for the phase diagram, the paper explicitly advertises SDRG as one of the principal methods, and the physical picture that the phases are atomic insulators with only intra- or only inter-cell hoppings rests on the weff/veff limit. The extra technical detail that the observable is the arithmetic mean of absolute values strengthens the reader's concern: under broad, log-normal-like IRFP distributions, the mean is dominated by rare large couplings, so the ratio of means can diverge even when typical couplings do not. The proposed concrete test, comparing the standard SDRG (without doubling) to the paper's list-doubling protocol on the same ensembles, would settle whether the phase boundary and the weff/veff flow are physical or artifacts. I therefore agree with the CONDITIONAL verdict: the central claim is plausible but not fully established until this protocol is validated or replaced. I do not find a reason to move to REJECT or UNVERDICTED because the polarization and entanglement-entropy results are credible independent evidence, and the IRFP value c_eff=ln2 is the known random-singlet result.","tokens_in":13037,"tokens_out":14221,"duration_ms":162184,"concrete_test":"Implement the standard SDRG without list doubling on the same disorder ensembles: start with L v-bonds and L w-bonds, repeatedly decimate the largest remaining coupling using Eqs. (2)-(3), and after L-1 decimations record the last surviving v- and w-bond values; alternatively, stop when only a few bonds remain and compare the empirical distribution of ln|w_eff/v_eff|. Determine the phase by the median of this distribution and map the lambda=1 boundary at sigma_w=6.0 for w/v=0.5, 1.0, and 2.0. If the standard SDRG gives a phase boundary different from the paper's list-doubling result (e.g., not at lambda=1 for all w/v), then the doubling protocol is the source of the claimed behavior; if it agrees, the concern is mitigated. Also report both mean and median ratios to check the mean-vs-typical issue.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is the SDRG protocol itself (main text, 'SDRG methods', steps 4 and 5). After L/2 decimations, the remaining v- and w-lists are doubled, so that each coupling appears twice, then the two lists are independently shuffled and the process is iterated N times. This doubling-and-shuffling is not a legitimate renormalization-group step: it discards the spatial ordering of the renormalized chain, which is precisely what distinguishes inter-cell from intra-cell bonds in the SSH model, and it replaces the true renormalized Hamiltonian by independent resampling from the empirical marginal distributions of v and w. No argument is given that the fixed point of this map coincides with the infinite-randomness fixed point of the physical model. The observable is then the ratio of arithmetic means of the absolute values of the two lists (SM Fig. S1). At an IRFP the single-bond distributions are extremely broad, so arithmetic means are controlled by rare large couplings and do not represent the typical scale; the '|weff/veff| -> 0 or infinity' dichotomy could therefore be an artifact of the mean diagnostic rather than a property of the RG flow. Because this SDRG result is one of the three pillars (with polarization and p_top) used to infer the lambda=1 phase boundary, the central claim is not fully settled until this protocol is validated or replaced.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies a one-dimensional Su-Schrieffer-Heeger (SSH) chain with intra-cell and inter-cell hoppings drawn from normal distributions with means \\bar v, \\bar w and standard deviations \\sigma_v, \\sigma_w. For large \\sigma_w, the authors report that the phase is controlled only by \\lambda = \\sigma_v/\\sigma_w: topological for \\lambda<1 and trivial for \\lambda>1, independent of the mean ratio \\bar w/\\bar v. They support this with three methods: (i) a numerical strong-disorder RG (SDRG) procedure that yields |w_eff/v_eff| tending to infinity or zero depending on \\lambda; (ii) exact-diagonalization polarization P, which jumps from approximately 1 to approximately 0 at \\lambda=1; and (iii) an analytic single-cell probability p_top. In both phases the bulk gap is reported to be gapless with Griffiths scaling \\epsilon~L^{-z}; at \\lambda=1 the authors find activated scaling \\epsilon~exp(-a\\sqrt L) and entanglement entropy S_l \\approx (\\ln 2)/3 \\, \\ln l, which they interpret as an infinite-randomness fixed point with central charge \\ln 2.","tokens_in":13283,"tokens_out":8818,"duration_ms":87427,"significance":"The claimed phenomenon—a topological transition driven purely by the relative fluctuation scales of the two hoppings, with the quantum critical point at an infinite-randomness fixed point—is novel and would significantly extend the SDRG/IRFP paradigm to fermionic topological systems. The manuscript's strongest evidence is the direct polarization calculation, which is a standard, parameter-free numerical probe and shows a clear P=1 to P=0 transition at \\lambda=1 for several \\bar w/\\bar v values. The activated-scaling collapse and the logarithmic entanglement growth at \\lambda=1 are also convincing qualitative signatures. The p_top calculation is analytic but relies on the clean local criterion; the SDRG protocol contains a nonstandard step that requires justification. Overall, if the identified issues are addressed, the result would be of broad interest to the disordered-topological-phases community.","major_comments":[{"comment":"The list-doubling and random reshuffling step is not a legitimate renormalization-group transformation. After L/2 decimations, the remaining coupling lists are doubled and independently shuffled, which destroys the spatial adjacency pattern that distinguishes intra-cell from inter-cell bonds in the SSH model; the renormalized chain is thus replaced by independent resampling from the empirical marginal distributions of v and w. No argument or numerical test is provided that the fixed point of this map coincides with the true infinite-randomness fixed point of the model. Since Fig. 2(c,d) and the |w_eff/v_eff| limit are used to identify the phases, this is a load-bearing issue; please validate the protocol (e.g., by comparing against exact RG decimation without doubling or against an alternative scheme) or appropriately weaken the SDRG claims.","section":"SDRG methods, steps 4 and 5"},{"comment":"The calculation of p_top builds the clean limit criterion |w|>|v| into the definition of a topological unit cell. Consequently, p_top=1/2 at \\lambda=1 in the large-\\sigma_w limit is a consequence of the assumed local criterion together with equal variances, not an independent derivation of the transition. This is a consistency check rather than a confirmation. Please either derive p_top from a criterion that emerges from the disordered system or explicitly present it as a heuristic consistency check.","section":"Determination of QCP from probability distributions, Eq. (7)"},{"comment":"The claim c_eff \\approx \\ln(2) is made without any uncertainty estimate, and the dynamical exponent z=2.1 in Fig. 4(b)/SM Fig. S2(a) is also quoted without error bars. Because the \"irrational central charge\" and the divergence of z at \\lambda\\to1 are central claims, please provide fit errors (e.g., bootstrap over realizations) and show the sensitivity of the extracted slopes to the chosen fit range.","section":"Fig. 5(b) and SM Sec. III"}],"minor_comments":[{"comment":"Specify explicitly which \\lambda values are shown and confirm that both curves are for the same \\bar w/\\bar v; the text says \"both \\lambda<1 and \\lambda>1\" but the caption does not list the curves.","section":"Fig. 4(a)"},{"comment":"The phrase \"for large \\sigma_w\" is used throughout but never quantified; please state the range of \\sigma_w for which the \\lambda=1 boundary and the p_top=1/2 condition hold.","section":"General text"},{"comment":"The universality claim \"regardless of the value of \\bar w/\\bar v\" is demonstrated for \\bar w/\\bar v=0.5, 1.0, and 2.0; please state this explicitly and, if possible, add one more value or a scan.","section":"General claims"},{"comment":"The constant a is introduced as \"a>0\" but not defined; define it as a nonuniversal constant.","section":"Eq. (5)"},{"comment":"Clarify the definition of \\tilde p (density versus cumulative probability) and the range of \\epsilon used for the linear fits from which z is extracted.","section":"SM Eq. (S4)"}],"recommendation":"major_revision","confidential_remarks":"For the editor: The exact-diagonalization results (polarization, gap scaling, entanglement entropy) are the most convincing part of the paper and, in my view, already support the central phase-boundary claim. The main risk is the unvalidated SDRG protocol; if the authors can validate it or explicitly downgrade it to a heuristic, the paper would be publishable. The analytic p_top section should be reframed as a consistency check. I recommend major revision rather than rejection because the core claim is defensible and the issues are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central claim is that a disordered SSH chain with bond variances sigma_v and sigma_w has a phase boundary at lambda = sigma_v/sigma_w = 1 when disorder is strong, with an infinite-randomness fixed point governing the transition. The idea is new, as far as I can tell, and the numerical evidence is more than a one-off: polarization, SDRG flow, and an analytic p_top heuristic all point to the same boundary, and the gap distributions show Griffiths scaling off criticality and activated scaling at lambda = 1 with c_eff = ln2.\n\nWhat the paper does well: it checks the transition for various w/v, for normal and uniform distributions, uses large samples (10^6 realizations for gap distributions), and gives collapse plots. The polarization calculation is an independent ground-state probe and is the most convincing piece.\n\nWhere it gets shaky: the SDRG protocol in the main text is nonstandard. After L/2 decimations, the remaining coupling lists are doubled, shuffled, and iterated. That is not a standard RG step; it discards spatial structure and replaces the renormalized Hamiltonian by independent resampling from empirical marginals. No justification is given that the fixed point of this map is the physical IRFP. The observable is then the ratio of arithmetic means of the two lists, which at an IRFP is dominated by rare large couplings rather than typical ones. So the '|weff/veff| -> 0 or infinity' result may be an artifact of the procedure. This matters because the SDRG result is one of the three pillars, though not the only one.\n\nAlso, the analytic p_top is a local heuristic: it assumes the clean |w|>|v| criterion cell by cell. It is not a topological invariant. And the abstract's 'solely governed by the fluctuation scales' overstates things; the SM shows that at finite sigma_w the means matter, e.g. sigma_w-driven transitions occur at finite sigma_w. The headline numbers (z, c_eff, phase boundary) come without error bars.\n\nBottom line: I think the central scenario is likely correct, but it is not fully settled until the SDRG protocols are validated or replaced, and the claims are tightened. This is worth a serious referee: the question is good, the numerical evidence is substantial, and the flaws are addressable. I would send it to review, with a request to either justify the doubling step or drop it in favor of a more standard real-space RG.","headline":"A plausible new axis for tuning disorder-driven topology in 1D, but the SDRG protocol behind one pillar of evidence is nonstandard and needs validation.","tokens_in":13839,"tokens_out":4155,"would_cite":true,"duration_ms":42752,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A disordered Su–Schrieffer–Heeger chain changes between topological and trivial phases by tuning the ratio of hopping disorder widths, with the critical point at ratio one.","keywords":["Su–Schrieffer–Heeger chain","strong disorder","infinite randomness fixed point","topological phase transition","strong-disorder renormalization group","Griffiths phase","activated dynamical scaling","polarization"],"falsifier":"Compute the disorder-averaged polarization and bulk-gap distribution for the same Hamiltonian by exact diagonalization at sizes up to $L=400$, without the list-doubling protocol: if the polarization crossing moves away from $\\lambda=1$ as $\\bar w/\\bar v$ is varied, or if the gap distribution at $\\lambda=1$ no longer collapses under $\\ln(\\epsilon)/\\sqrt{L}$, the central claim is refuted.","tokens_in":12767,"feed_emoji":"⚛️","tokens_out":10380,"duration_ms":94083,"temperature":0.7,"pith_summary":"This paper studies a Su–Schrieffer–Heeger chain whose intra-cell and inter-cell hoppings are independently random, drawn from normal distributions with means $\\bar{v}$, $\\bar{w}$ and widths $\\sigma_v$, $\\sigma_w$. It argues that in the strong-disorder regime the topological phase is governed entirely by the ratio $\\lambda = \\sigma_v/\\sigma_w$: for large $\\sigma_w$ the chain is topological when $\\lambda < 1$ and trivial when $\\lambda > 1$, independent of $\\bar{w}/\\bar{v}$ and of which phase the clean chain would occupy. Both phases turn out to be gapless, with localized zero modes in the bulk, rare regions, and a bulk gap scaling as $\\epsilon \\sim L^{-z}$ with $z$ diverging at $\\lambda=1$. At the critical point $\\lambda=1$ the gap obeys activated scaling $\\epsilon \\sim \\exp(-a\\sqrt{L})$ and the effective central charge is $c_{\\rm eff} = \\ln(2)$, identifying the transition as an infinite-randomness fixed point. The claim matters because it shows that competing fluctuation scales, not average parameters, can set the topology of a disordered system.","feed_headline":"Disorder widths, not averages, set the chain's topology","feed_subtitle":"A strongly disordered chain turns topological when intra-cell fluctuations are weaker than inter-cell ones.","key_machinery":"The argument is carried by the strong-disorder renormalization-group decimation rules for a one-dimensional tight-binding chain. When the largest remaining bond is an intra-cell hopping $\\Omega=v_r$, the two neighboring inter-cell hoppings combine into $\\tilde w_{r-1,r+1}= -w_{r-1,r}w_{r,r+1}/\\Omega$; when $\\Omega=w_{r,r+1}$ is inter-cell, the two adjacent intra-cell hoppings combine into $\\tilde v_{r,r+1}= -v_r v_{r+1}/\\Omega$. After $L/2$ decimations the remaining hopping lists are doubled, reshuffled, and re-decimated, and the limit of $|w_{\\rm eff}/v_{\\rm eff}|$ (infinity or zero) labels the phase. The analytical complement is the probability $p_{\\rm top} = \\int_{-\\infty}^{\\infty} dw\\, p_2(w) \\int_{-|w|}^{|w|} dv\\, p_1(v)$ that a unit cell satisfies $|w|>|v|$; it equals $1/2$ at $\\lambda=1$ for all $\\bar w/\\bar v$, and the polarization computed from the many-body position operator confirms the same boundary.","core_discovery":"The central discovery is that strong randomness replaces the clean SSH transition with an infinite-randomness fixed point whose location is set by the ratio of disorder widths, not by the mean hoppings. For large $\\sigma_w$, the SDRG flow drives $|w_{\\rm eff}/v_{\\rm eff}|$ to infinity when $\\lambda<1$ and to zero when $\\lambda>1$, for every $\\bar w/\\bar v$; the disorder-averaged polarization $P$ is correspondingly close to $1$ or $0$. The two phases are gapless: the density of states is nonzero at $E\\to 0$ and the bulk gap follows rare-region scaling $\\epsilon\\sim L^{-z}$, with $z$ diverging as $\\lambda\\to1$. At $\\lambda=1$, the gap shows activated scaling $\\epsilon\\sim\\exp(-a\\sqrt{L})$ and the entanglement entropy grows as $(c_{\\rm eff}/3)\\ln l$ with $c_{\\rm eff}\\approx\\ln(2)$, identifying the critical point as an infinite-randomness fixed point. The same critical behavior is found for transitions driven by $\\sigma_w$ at fixed $\\lambda$ and for uniform disorder distributions.","pith_inferences":["If the central claim is right, the clean-limit topology is irrelevant in the strong-disorder regime: a chain whose clean limit is deep in the trivial phase should become topological as soon as $\\sigma_v<\\sigma_w$. This is a sharp prediction one could test with exact diagonalization without relying on the SDRG doubling step.","The same ratio-of-fluctuations mechanism may organize other strongly disordered topological systems, such as disordered Chern insulators, where competing hopping dispersions would play the role of $\\sigma_v$ and $\\sigma_w$; this is a direction the paper names but does not develop.","A practical observable for cold-atom or mechanical SSH realizations is the disorder-averaged polarization as a function of $\\lambda$ at fixed mean hoppings: a sharp crossing from $P\\approx1$ to $P\\approx0$ at $\\lambda=1$ would confirm the phase diagram, and measuring the sample-to-sample variance of $P$ could expose the rare-region regime.","The power-law divergence of the dynamical exponent of the rare-region scaling, $z \\sim |\\lambda-1|^{-1}$ on one side, suggests a universal exponent that a future analytic strong-disorder RG treatment could be checked against."],"forward_implications":["In the strong-disorder regime both the topological and trivial phases are gapless, so topological order is carried by edge-localized zero modes sitting on a featureless bulk rather than by a finite gap.","The phase boundary at $\\lambda=1$ is independent of $\\bar w/\\bar v$; a chain whose clean limit is trivial becomes topological for $\\lambda<1$, and one whose clean limit is topological becomes trivial for $\\lambda>1$.","The critical point exhibits activated dynamical scaling $\\epsilon\\sim\\exp(-a\\sqrt{L})$ and effective central charge $c_{\\rm eff}=\\ln(2)$, the signatures of an infinite-randomness fixed point.","Transitions can also be driven by $\\sigma_w$ alone when the mean hoppings favor the opposite phase, and those critical points show the same infinite-randomness signatures.","The results are qualitatively unchanged when the normal disorder distributions are replaced by uniform distributions of the same width."],"supporting_citations":[{"why":"Introduces infinite-randomness fixed points and activated dynamical scaling in disordered quantum chains, the paradigm the paper's critical point is identified with.","marker":"[24]"},{"why":"Supplies the numerical method for extracting the dynamical exponent z from gap distributions and for verifying activated scaling collapses.","marker":"[27]"},{"why":"Establishes logarithmic entanglement-entropy scaling with effective central charge at random quantum critical points, used to extract $c_{\\rm eff}=\\ln(2)$.","marker":"[28]"},{"why":"Gives the original strong-disorder decimation idea on which the SDRG procedure in this paper is built.","marker":"[42]"},{"why":"Provides the SDRG decimation rules for the one-dimensional tight-binding model that yield the effective hoppings in Eqs. (2) and (3).","marker":"[45]"},{"why":"Defines the SSH Hamiltonian and its clean-limit topological and trivial phases that the disordered model generalizes.","marker":"[47]"},{"why":"Supplies the quantum position operator and polarization formula used to compute P and locate the $\\lambda=1$ boundary.","marker":"[50]"}],"fun_headline_variants":["Disorder widths, not means, decide topological phase","Infinite randomness flips chain topology by width ratio","Strong disorder gap closes, width ratio sets topology","Topological transition governed by disorder fluctuation scale","Width ratio, not average, sets topological phase in random chain"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that the SDRG list-doubling procedure — decimating $L/2$ bonds, then doubling and reshuffling the remaining hoppings and repeating — converges to the true infinite-randomness fixed point, so that the fate of $|w_{\\rm eff}/v_{\\rm eff}|$ reflects the actual phase and not an artifact of the protocol.","fun_headline_variants_meta":{"raw":{"variants":["Disorder widths, not means, decide topological phase","Infinite randomness flips chain topology by width ratio","Strong disorder gap closes, width ratio sets topology","Topological transition governed by disorder fluctuation scale","Width ratio, not average, sets topological phase in random chain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000497,"raw_usage":{"total_tokens":2426,"prompt_tokens":927,"completion_tokens":1499,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":1424}},"tokens_in":543,"tokens_out":1499,"duration_ms":9197,"temperature":1.0,"reasoning_tokens":1424,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:23:52.979453+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the disorder-averaged polarization and bulk-gap distribution for the same Hamiltonian by exact diagonalization at sizes up to $L=400$, without the list-doubling protocol: if the polarization crossing moves away from $\\lambda=1$ as $\\bar w/\\bar v$ is varied, or if the gap distribution at $\\lambda=1$ no longer collapses under $\\ln(\\epsilon)/\\sqrt{L}$, the central claim is refuted.","supporting_citations":[{"cited_title":"Cybi´ nski, M","cited_arxiv_id":null,"evidence_quote":"Introduces infinite-randomness fixed points and activated dynamical scaling in disordered quantum chains, the paradigm the paper's critical point is identified with."},{"cited_title":"Thatcher, P","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical method for extracting the dynamical exponent z from gap distributions and for verifying activated scaling collapses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes logarithmic entanglement-entropy scaling with effective central charge at random quantum critical points, used to extract $c_{\\rm eff}=\\ln(2)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the original strong-disorder decimation idea on which the SDRG procedure in this paper is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the SDRG decimation rules for the one-dimensional tight-binding model that yield the effective hoppings in Eqs. (2) and (3)."},{"cited_title":"Javan Mard, J","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum position operator and polarization formula used to compute P and locate the $\\lambda=1$ boundary."}],"review_version":2}