{"id":"6338491a-9dc4-42c2-99d6-462a721ddee2","arxiv_id":"1909.01402","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"DQPTs in multiband quenches occur on a d-1 dimensional Fisher-zero set, a three-band Chern insulator can avoid them entirely, and supercell numerics indicate they persist under disorder.","lead":"This paper studies when dynamical quantum phase transitions, sharp cusps in the time evolution of a suddenly changed quantum system, appear in multiband topological materials and survive in disordered wires. It contributes analytical criteria, a counterexample, and numerical evidence relevant to cold-atom and photonic experiments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Disorder-stability conclusion rests on the unproven l→∞ supercell extrapolation; direct aperiodic-chain numerics are needed to confirm the cusps survive the loss of translation invariance.","rationale":"The paper has two independent thrusts. The multiband part is on solid ground where it matters: condition (5) is an exact inequality, the counterexample in Section II D follows directly from it, and the Hofstadter simulations in Section III are consistent with the dimensional counting in Section II C. Even if one worries that the ergodicity/codimension argument is heuristic, it is not the hinge of the paper's title claim. The disorder part is the hinge: it asserts stability of DQPTs in disordered topological insulators, and the presented evidence is supercell numerics plus a conjecture about independence of disorder contributions. The authors explicitly mark the missing analytical bridge. Since the central claim depends on the l→∞ extrapolation, and since no data or code are released for independent reproduction, the appropriate verdict remains conditional rather than accept. This is exactly the assumption the reader flagged, and no new concern changes that verdict.","tokens_in":13694,"tokens_out":15722,"duration_ms":165949,"concrete_test":"Diagonalize a genuinely aperiodic disordered Kitaev chain (independent random on-site potentials, no supercell repetition) for system sizes N=200, 400, 800 at Δµ_max=1/2, with the same parameters and Slater-determinant initial state as in Section IV B. Compute g_N(t)=-(1/N) log|<Ψ_0|e^{-iHt}|Ψ_0>|² and its time derivative. If a cusp at a time approaching the supercell critical time t_c sharpens as N grows, the l→∞ extrapolation is supported; if the nonanalyticity weakens or its location drifts without bound, the disorder-stability conclusion does not follow from the presented numerics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central disorder claim — that DQPT rate-function cusps persist up to significant disorder strength — is established only for periodic supercells of finite period l. Section IV A introduces the periodic approximation with the heuristic statement 'For sufficiently large l, the system resembles a disordered system'; Section V concedes that 'a non-trivial order of limits problem renders an analytical proof for the truly disordered case of an infinite spatial period of the random potential elusive.' The numerics in Fig. 4 go to l=100, but as l→∞ the supercell Brillouin zone shrinks to zero, so each fixed-l calculation is a different periodic system (one random l-site pattern repeated infinitely) rather than a converging sequence to a single disordered chain. The variance argument in Section IV B additionally relies on Eq. (16), the zero-disorder factorization, which is only asserted to hold approximately at finite disorder. The headline stability conclusion is therefore a numerical extrapolation across an unproven interchange of limits, not a demonstrated property of the disordered thermodynamic limit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies dynamical quantum phase transitions (DQPTs) in multiband and disordered free-fermion systems. It derives necessary and sufficient conditions for Fisher zeros of the Loschmidt amplitude, uses a dimensional/ergodicity argument to claim that for more than two bands the Fisher zeros generically form a (d-1)-dimensional submanifold in momentum-time space, and constructs a three-band Chern-insulator scenario in which no DQPTs occur despite nontrivial band Chern numbers. The predictions are illustrated by Hofstadter-model simulations. For disordered systems, the paper introduces a supercell approach in which disorder is made periodic with period l, and presents numerical evidence that rate-function cusps persist up to significant disorder strength as l is increased, while explicitly acknowledging an order-of-limits problem for the truly disordered limit.","tokens_in":13821,"tokens_out":8318,"duration_ms":81299,"significance":"If its claims hold, the paper extends the topology-DQPT connection beyond two-band translation-invariant models and provides a dimension-dependent classification of Fisher-zero sets. The exact inequalities (5) and (6) and the explicit n=2 analysis are clean and useful, and the Hofstadter numerics support the dimensional picture. The disorder-stability result, however, is supported only by finite-period supercell numerics, and the paper itself concedes the lack of an analytical proof for the truly disordered case; as presented, the disorder claim is a numerical extrapolation rather than a demonstrated property. The multiband counterexample is conditional on an unproven existence assumption. These gaps are significant but addressable within the manuscript's scope.","major_comments":[{"comment":"The central claim that DQPTs survive up to significant disorder strength in the truly disordered thermodynamic limit is not established by the presented numerics. All calculations in Figs. 3-5 are for periodic supercells of finite period l, and the paper itself states in Sec. V that a non-trivial order of limits problem renders an analytical proof for the truly disordered case elusive. Each fixed l corresponds to a different translationally invariant system (one random l-site pattern repeated infinitely), and the rate function g_l(t) in Eq. (21) is defined through a supercell Brillouin-zone integral that has no direct counterpart in the disordered thermodynamic limit. The statement in Sec. IV A that 'for sufficiently large l, the system resembles a disordered system' is a heuristic, not a controlled approximation. To support the headline conclusion, the authors should provide direct real-space simulations of a disordered Kitaev chain without supercell periodicity, with finite-size scaling, or else substantially soften the claim in the title and abstract.","section":"Sec. IV B and Sec. V"},{"comment":"The proposed counterexample is conditional on the existence of an initial Hamiltonian whose lowest band has overlap >1/sqrt(2) with the central band of the post-quench Hamiltonian at all momenta, but no such Hamiltonian is constructed or even argued to exist. Since the text says 'we construct a basic counter-example', the authors should provide an explicit initial Hamiltonian H_i(k) realizing such a Bloch state as its occupied band, or at least explain why such a state is guaranteed to be representable as the lowest band of a gapped trivial Hamiltonian. Without that, the no-DQPT conclusion is a statement about an assumed state rather than a demonstrated counterexample to the general claim.","section":"Sec. II D"},{"comment":"The claim that Fisher zeros generically form a (d-1)-dimensional manifold for n>2 rests on a codimension/ergodicity heuristic rather than a proof. The argument that the real and imaginary parts of G_k(t) can be independently tuned to zero assumes transversality of the zero set and rational independence of the band energies on the admissible region; neither condition is stated or verified, even for the Hofstadter example. The authors should formulate this as a precise conjecture with explicit non-resonance conditions, and clearly separate the rigorous n=2 result from the generic-n expectation.","section":"Sec. II C"}],"minor_comments":[{"comment":"There is a typo: 'DQOTs' in the sentence 'no Fisher zeros or DQOTs occur at any time' should read 'DQPTs'.","section":"Sec. II D"},{"comment":"The definition of the rate function in Eq. (21) uses log|G^l_k(t)|, whereas the rate function defined in the introduction and used in Eq. (8) is based on log|G|^2. This introduces a factor of 2 inconsistency; please make the definitions uniform.","section":"Eq. (21)"},{"comment":"The claimed ~1/l scaling of the variance is justified by invoking Eq. (16), which is exact only at zero disorder, and the text acknowledges this as an approximation. The figure caption should label the scaling as conjectural, since the independence assumption is not proven.","section":"Sec. IV B, Fig. 5"},{"comment":"The statement 'For sufficiently large l, the system resembles a disordered system' is not quantified. Please specify in what sense (e.g., local observables, finite-order correlation functions) the resemblance is expected to hold, or replace it with a more precise statement.","section":"Sec. IV A"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its limitations, but the title and abstract overstate the disorder-stability result relative to the evidence presented. The multiband counterexample also needs an explicit construction to be a true counterexample. If the authors can provide direct disordered-chain numerics for the Kitaev chain and make the heuristic status of the dimensional counting explicit, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the multiband half of this paper is the stronger half, and the disorder half is an honest numerical extrapolation rather than a proof. The authors make a real contribution by clarifying when topology-induced DQPTs actually occur: they give a dimensional-counting argument that in n>2 bands Fisher zeros generically form a (d-1)-dimensional set in momentum-time space, and they build a three-band Chern insulator counterexample where the post-quench Hamiltonian has Chern numbers (1,0,-1) but no Fisher zeros appear because the initial state is concentrated in the trivial middle band. The Hofstadter simulations illustrate the criteria and the vortex dynamics cleanly. That part is solid.\n\nThe disorder part is more fragile. The idea of approaching disorder by enlarging a supercell with random onsite potentials is reasonable, and the numerics for the Kitaev chain show cusps persisting up to significant disorder strength for l up to 100. But the stress-test concern is legitimate: as l→∞, each l system is a different periodic chain, not a converging sequence to a single disordered chain. The authors themselves say the order-of-limits problem makes an analytical proof 'elusive,' and the variance argument uses the zero-disorder factorization (16) only approximately. So the headline 'stability against disorder' is supported by numerics but not proven. They are transparent about this, which earns credit, but it would be better if they released code or data so the figures could be independently checked.\n\nMinor points: the dimensional counting is heuristic, not a rigorous theorem, though it is plausible and backed by the two-band exact result. The counterexample is conditional on a >1/√2 overlap assumption at all momenta; no explicit initial Hamiltonian is constructed, so it is more an argument that such quenches can avoid DQPTs than a demonstration with a concrete lattice model.\n\nWho is this for? People working on DQPTs in topological or disordered systems, especially experimental groups in cold atoms or photonics who want to know when quenches will show cusps. The multiband criteria and counterexample are directly useful. I'd send it to peer review—it deserves a serious referee despite the soft disorder extrapolation, because the multiband analysis is new and the methods are transparent.","headline":"Solid multiband analysis and a useful counterexample; the disorder-stability claim is an honest but unproven extrapolation from finite-supercell numerics.","tokens_in":14411,"tokens_out":2106,"would_cite":true,"duration_ms":20798,"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":"This paper establishes that in quenched multiband topological insulators the Fisher zeros underlying dynamical quantum phase transitions generically form a (d-1)-dimensional set in momentum-time space, and that a quench into a three-band…","keywords":["dynamical quantum phase transitions","Fisher zeros","Loschmidt amplitude","quench dynamics","topological insulators","disordered Kitaev chain","Hofstadter model","Pancharatnam geometric phase"],"falsifier":"Compute the rate function and its time derivative for a genuinely aperiodic disordered Kitaev chain (no supercell repetition) at disorder strength $\\Delta\\mu_{\\max}=2$ for system sizes $L=100$, $400$, $1600$: if the cusp near the critical time rounds off and disappears as $L$ grows, the supercell extrapolation would be falsified. Alternatively, test the three-band $(1,0,-1)$ counterexample with initial-state overlap exactly $1/\\sqrt{2}+\\varepsilon$ at all momenta; any Fisher zero at finite time would falsify the no-DQPT claim.","tokens_in":13417,"feed_emoji":"⚛️","tokens_out":8556,"duration_ms":71642,"temperature":0.7,"pith_summary":"This paper asks when dynamical quantum phase transitions (DQPTs) — the cusps in the return probability of a quantum system after a quench — occur in multi-band topological insulators and in disordered wires. It shows that in systems with more than two bands the zeros of the Loschmidt amplitude (Fisher zeros) that produce those cusps generically form a set of dimension one less than the spatial dimension in momentum-time space, and that the familiar two-band guarantee fails: a quench into a three-band Chern insulator with band Chern numbers $(1,0,-1)$ can produce no Fisher zeros at any time if the initial filled band leans on the central trivial band. To approach disorder, the paper enlarges a periodic supercell and presents numerical evidence that rate-function cusps persist up to significant disorder strength as the supercell grows. The authors state that an analytical proof for the truly aperiodic disordered case remains open.","feed_headline":"Topological quench can avoid dynamical phase transitions entirely","feed_subtitle":"In a three-band Chern insulator with a trivial middle band, Fisher zeros vanish; in disordered Kitaev chains, cusps persist.","key_machinery":"The workhorse is the factorized Loschmidt amplitude $G(t)=\\prod_k G_k(t)$ whose zeros are Fisher zeros; DQPTs are the resulting cusps of the rate function $g(t)=-\\frac{1}{N}\\log|G(t)|^2$. The argument runs on two criteria: the triangle-inequality overlap condition (5), which requires that no single band overlap exceed $1/2$ for a Fisher zero to be possible, and the dynamical exclusion condition (6), which forbids zeros when all post-quench phases lie on a minor arc of the unit circle. Dimensional counting uses ergodicity of the phases $e^{-iE_{k,\\alpha}t}$ on the $n$-torus, yielding the generic $(d-1)$-dimensional Fisher-zero manifold. The Pancharatnam geometrical phase picture ties each Fisher zero to a phase vortex in momentum-time space. For disorder, the supercell representation promotes disorder coefficients into orbital indices of a $2\\ell \\times 2\\ell$ Bloch Hamiltonian, making the Loschmidt amplitude real-analytic in the noise; this continuity is what lets the cusps survive finite disorder.","core_discovery":"The central discovery is that the existence of DQPTs is decided by where the Fisher zeros of the momentum-resolved Loschmidt amplitude $G_k(t) = \\sum_{\\alpha} |\\langle u_{k,\\alpha}|\\psi_k\\rangle|^2 e^{-iE_{k,\\alpha}t}$ can be located, which is controlled by two independent conditions: the static overlap condition $|\\langle u_{k,\\alpha}|\\psi_k\\rangle|^2 \\le 1/2$ for all $\\alpha$, and a dynamical condition on the post-quench eigenvalues. For more than two bands, the overlap condition holds in an entire $d$-dimensional momentum region, so both real and imaginary parts of $G_k(t)$ must be tuned to zero, making the Fisher-zero set generically $(d-1)$-dimensional in momentum-time space; for two bands the admissible region is $(d-1)$-dimensional but zeros are guaranteed there, so the set is again $(d-1)$-dimensional. A three-band counterexample with Chern numbers $(1,0,-1)$ shows that a topologically nontrivial final Hamiltonian does not force DQPTs: if the initial filled band has overlap $> 1/\\sqrt{2}$ with the trivial middle band everywhere, the overlap condition is violated everywhere and no Fisher zero exists at any time. For disordered systems, real-analytic dependence of $G_k(t)$ on the disorder amplitudes implies that Fisher-zero vortex-antivortex pairs move continuously, so cusps cannot vanish discontinuously for finite supercell size; extensive Kitaev-chain numerics indicate they survive the large-supercell limit up to substantial disorder.","pith_inferences":["A testable extension would be to compute the dynamical topological order parameter for the $(1,0,-1)$ counterexample; the absence of Fisher zeros suggests the order parameter may also vanish, sharpening 'topology change implies DQPT' into 'only changes in the topology of fully occupied bands matter'.","The supercell approach implicitly assumes that rare disorder configurations do not generate Fisher zeros in the thermodynamic limit; studying the extreme-value statistics of $\\min_k |G_k(t)|$ across many realizations would probe this assumption.","Because the paper assumes a single filled band throughout, the criterion (5) and the counterexample both need re-derivation for partially filled bands; the determinant form in Eq. (15) suggests the factorization structure, and with it the clean dichotomy, may break down there.","If the $1/\\ell$ variance scaling holds generally, DQPT rate functions would be self-averaging observables, making them robust diagnostics in experiments with disordered optical lattices and wires."],"forward_implications":["In $d$ spatial dimensions, generic multiband quenches with $n>2$ bands have Fisher zeros on a $(d-1)$-dimensional manifold in momentum-time space: isolated points in one dimension, curves in two dimensions.","The two-band result that a change of Chern number across the quench guarantees DQPTs does not extend to multiband systems; the $(1,0,-1)$ counterexample has a topological final Hamiltonian and yet no Fisher zeros.","For any finite supercell size $\\ell$, Fisher zeros and rate-function cusps persist as continuous deformations under increasing disorder; they cannot disappear discontinuously.","Numerical evidence on the disordered Kitaev chain indicates that the cusps survive up to disorder strength comparable to the hopping and gap parameters as $\\ell \\to \\infty$, i.e., approaching the truly disordered limit.","The variance of the rate function across disorder realizations decays as $\\sim 1/\\ell$, indicating self-averaging of the return probability in large supercells."],"supporting_citations":[{"why":"Supplies the Fisher-zero overlap criterion (5) and the earlier result that quenches into Hamiltonians with all bands topologically nontrivial imply DQPTs; the present work extends this to dimension counting and constructs a counterexample when a band is trivial.","marker":"[34]"},{"why":"Introduces the Pancharatnam geometrical phase vortex picture and the dynamical topological order parameter that links Fisher zeros to DQPTs.","marker":"[16]"},{"why":"Provides the concept of Fisher zeros of the partition function, the statistical-mechanics analogy underlying the analysis.","marker":"[25]"},{"why":"Reports the two-band d=2 numerical study whose Fisher-zero curves in momentum-time space the present dimension analysis generalizes.","marker":"[41]"},{"why":"Defines the Hofstadter Hamiltonian used as the multiband numerical case study.","marker":"[42]"},{"why":"Defines the Kitaev chain used for the disordered supercell numerical study.","marker":"[44]"}],"fun_headline_variants":["Multiband quench can erase dynamical phase transitions","Three-band Chern quench: no Fisher zeros, no DQPTs","Disorder fails to destroy DQPTs in Kitaev chains","Topology alone doesn't force dynamical phase transitions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion that DQPTs survive true randomness rests on the unproven leap that, for very large supercell period $\\ell$, a periodic disorder realization behaves like an aperiodic random chain, so that the limit of large system size does not destroy the cusps.","fun_headline_variants_meta":{"raw":{"variants":["Multiband quench can erase dynamical phase transitions","Three-band Chern quench: no Fisher zeros, no DQPTs","Disorder fails to destroy DQPTs in Kitaev chains","Topology alone doesn't force dynamical phase transitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000363,"raw_usage":{"total_tokens":2035,"prompt_tokens":1103,"completion_tokens":932,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":719,"completion_tokens_details":{"reasoning_tokens":862}},"tokens_in":719,"tokens_out":932,"duration_ms":9153,"temperature":1.0,"reasoning_tokens":862,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:19:15.186773+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the rate function and its time derivative for a genuinely aperiodic disordered Kitaev chain (no supercell repetition) at disorder strength $\\Delta\\mu_{\\max}=2$ for system sizes $L=100$, $400$, $1600$: if the cusp near the critical time rounds off and disappears as $L$ grows, the supercell extrapolation would be falsified. Alternatively, test the three-band $(1,0,-1)$ counterexample with initial-state overlap exactly $1/\\sqrt{2}+\\varepsilon$ at all momenta; any Fisher zero at finite time would falsify the no-DQPT claim.","supporting_citations":[{"cited_title":"Huang and A","cited_arxiv_id":null,"evidence_quote":"Supplies the Fisher-zero overlap criterion (5) and the earlier result that quenches into Hamiltonians with all bands topologically nontrivial imply DQPTs; the present work extends this to dimension counting and constructs a counterexample when a band is trivial."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Pancharatnam geometrical phase vortex picture and the dynamical topological order parameter that links Fisher zeros to DQPTs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the concept of Fisher zeros of the partition function, the statistical-mechanics analogy underlying the analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Hofstadter Hamiltonian used as the multiband numerical case study."}],"review_version":1}