{"id":"a109e6b0-3736-46c3-9032-bd3623069275","arxiv_id":"2507.03342","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"High-field transport in high-quality 5-SL MnBi2Te4 reveals quantized topological states governed by a parity-anomaly-based index and an anomalous Landau level that produces gate-tunable edge transport.","lead":"High-quality five-layer MnBi2Te4 samples, measured in magnetic fields up to 45 Tesla, show a rich set of quantized Hall states that do not follow the standard Landau-level ladder. The authors interpret this with a generalized topological index based on the parity anomaly of 2D Dirac fermions, and identify an anomalous Landau level that drives gate-tunable edge transport.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The experiment-to-theory linchpin is the assignment of L0, L1, L2 and the 'anomalous' LL to specific sub-bands in an underspecified slab model; if that assignment is wrong, the parity-anomaly interpretation and helical-edge claim lose their anchor.","rationale":"The paper reports impressive experimental advances: high-quality MnBi2Te4 crystals, a 30 K QAH gap, and a rich high-field fan diagram with quantized states up to nu = -8. The generalized index N = (N+ - N-)/2 is a compact way to encode the observed sequence, and the appeal to the parity anomaly is a plausible theoretical framework with prior support (Refs. 18,19). I therefore do not regard the theoretical construction as incorrect per se. The load-bearing step is the mapping between the measured fan diagram and the calculated LL spectrum. The text asserts, on the basis of Fig. 3c-3e, that L0, L1, L2 are the first LLs of three specific sub-bands and that the first LL of the surface hole band E^h_{s,k} is the anomalous level giving nu = -1. These assignments are not uniquely determined by the transport data; the DOS simulation that matches the fan diagram contains unspecified broadening and impurity-state parameters, making it possible that the fit is flexible enough to accommodate an alternative level ordering. A wrong ordering would leave the experimental plateaus intact but sever the link to the parity anomaly and to the predicted helical edge transport. The reader's weakest_assumption identified this same point, and I agree. The appropriate remedy is to require the authors to release the full model and perform a robustness check, or to provide a direct measurement of the edge transport (e.g., nonlocal resistance) in the nu = 0 phase. Since the central claim is promising but not yet anchored independently of the model, the conditional verdict stands.","tokens_in":14304,"tokens_out":9991,"duration_ms":119026,"concrete_test":"Make the full model fully public (tight-binding parameters, slab geometry, LL coupling, broadening, impurity configuration) and independently recompute Fig. 3e at 45 T. Verify that the first LL of E^h_{s,k} is the only level that (i) is the closest to the CNP on the hole side, (ii) disperses upward as B approaches 0, and (iii) crosses the Fermi level to produce nu = -1; also check that the ordering L0 = E^h_{1,k}, L1 = E^e_{1,k}, L2 = E^e_{2,k} is robust to +/-10% variations in surface potential and MnBi defect density. If the recomputation changes the ordering, the experimental assignment fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the quantized fan diagram is governed by a parity-anomaly index rests on the identification of the three lowest experimental Landau levels (L0, L1, L2) with the first LLs of E^h_{1,k}, E^e_{1,k}, E^e_{2,k}, and on the designation of the first LL of the surface hole band E^h_{s,k} as the 'anomalous' level responsible for the nu = -1 QAH state (Fig. 3a-3e, main text p.6). This identification is not derived from the experimental data alone; it comes from a model calculation of a hypothetical ferromagnetic 5-SL slab whose Hamiltonian parameters, level broadening, and impurity-state inputs are only partially described (details deferred to Supplementary Fig. S5). The DOS simulation in Fig. 3d is a fit with those inputs, so agreement with the fan diagram does not independently confirm the sub-band assignment. If, for example, surface-potential shifts or Mn_Bi antisite disorder reorder the levels, the 'anomalous' hole-derived electron-like LL could be a conventional electron LL, and the nu = -1 state would be an ordinary Landau level; then the parity-anomaly narrative and the predicted helical edge transport in the nu = 0 phase would lose their experimental anchor. The absence of zero-field QAH in the Type S devices used for the high-field data further weakens the adiabatic link to C0 = -1, though it does not by itself disprove the index.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reports transport measurements on high-quality five-septuple-layer (5-SL) MnBi2Te4 devices. The authors observe zero-field quantum anomalous Hall (QAH) effect in Type A devices with quantization precision better than 1 part in 10^4 and a QAH gap up to 29 K. In Type S devices under magnetic fields up to 45 T, they observe quantized Hall plateaus at filling factors ν = +1, -1, -2, -3, -4, -6, -8 and an intervening fractional state. The central claim is that the Landau fan diagram is governed by a generalized topological index N = (N+ - N-)/2 rooted in the parity anomaly of Dirac fermions in 2+1 dimensions, and that an anomalous Landau level—identified as the first Landau level of the surface hole sub-band E^h_{s,k}—explains the ν = -1 QAH state and gives rise to gate-tunable helical edge transport in the ν = 0 phase. The experimental data are extensive and the Landau fan is mapped carefully, but the theoretical interpretation rests on a slab-model calculation whose parameters and level assignments are not fully specified in the main text.","tokens_in":14634,"tokens_out":3125,"duration_ms":36860,"significance":"If the interpretation holds, the observation of a generalized topological index rooted in the parity anomaly and an anomalous Landau level in an intrinsic magnetic topological insulator would be a significant advance, connecting high-field Landau quantization with zero-field topology. The paper provides strong evidence for high-quality samples, reproducible quantized plateaus, and a careful mapping of the Landau fan. The prediction of helical edge transport in the ν = 0 phase is specific and falsifiable, and the observation of three distinct regimes within that phase is a notable experimental finding. However, the load-bearing identification of the experimental Landau levels L0, L1, L2 and the anomalous level with specific sub-bands of a hypothetical ferromagnetic MnBi2Te4 slab is not independently established, and the generalized index is presented more as a reformulation than as a derivation. The experimental contributions themselves are solid, but the theoretical narrative needs stronger support.","major_comments":[{"comment":"The identification of L0, L1, L2 as the first Landau levels of E^h_{1,k}, E^e_{1,k}, E^e_{2,k}, and the designation of the first Landau level of E^h_{s,k} as the anomalous level responsible for ν = -1, rest entirely on the slab-model calculation of hypothetical ferromagnetic MnBi2Te4. The Hamiltonian parameters, surface potential, level broadening, and impurity-state inputs are not provided in the main text; they are deferred to Supplementary Fig. S5. Because the DOS simulation in Fig. 3d is a fit that uses these inputs, agreement with the experimental fan diagram does not independently confirm the sub-band assignment. If, for example, surface-potential shifts or disorder reorder the low-lying levels, the anomaly-based interpretation and the helical-edge picture lose their experimental anchor. Please provide the full set of model parameters, a sensitivity analysis of the level ordering to those parameters, and an explicit comparison of the calculated LL spectra (not just the DOS) with the experimentally extracted fan diagram.","section":"Fig. 3c–3e and main text p.6"},{"comment":"The reformulation of the topological number N as (N+ - N-)/2 and the universal relation N_- = N_+ - 2C_0 are stated without a derivation in the main text. The connection to the parity anomaly via the spectral asymmetry η is mentioned but not made explicit; the reader is left to infer how the spectral asymmetry is computed for the multiple sub-bands in the 5-SL slab and why each Landau level carries Chern number -1. Please provide a self-contained derivation or a precise citation showing how the LL-resolved index follows from the parity anomaly in a system with several occupied and unoccupied sub-bands, and clarify the role of the regularization scheme in a finite slab. As written, the index appears to be assumed rather than demonstrated.","section":"Page 6, generalized topological index N = (N+ - N-)/2"},{"comment":"The argument that the high-field ν = -1 state is a QAH state adiabatically connected to the zero-field QAH state relies on continuity, but Type S devices do not show zero-field QAH quantization (as stated on page 5). The authors argue that the high mobility of surface electrons in the FM state enables the observation, yet the absence of zero-field quantization means that the C0 = -1 assignment for the high-field state is inferred, not directly measured. Please discuss whether the high-field ν = -1 state in Type S devices is indeed the same topological state as the zero-field QAH state in Type A devices, and whether the adiabatic connection can be justified given that zero-field quantization is absent.","section":"Page 5, Type S devices and adiabatic continuity"}],"minor_comments":[{"comment":"The abstract states that the anomaly 'gives rise to gate-tunable helical edge transport,' but the report of helical edge transport in the ν = 0 phase is based on a comparison of transport data with a theoretical model, not a direct measurement of edge-state spin structure. Please temper the wording or provide direct evidence for the helical character.","section":"Abstract and Introduction"},{"comment":"The fractional state between ν = -2 and ν = -3 is presented as emerging above 40 T, but the available data appear limited to one device. Please state the reproducibility across devices and the sample-to-sample variation in the threshold field.","section":"Fig. 2c"},{"comment":"The caption of Fig. 3d mentions 'impurity states' as an input to the DOS simulation, but the main text does not define what these impurity states are or how they were modeled. Please specify their energy distribution, density, and effect on the simulated fan diagram.","section":"Fig. 3d and Supplementary Fig. S5"},{"comment":"The QAH gap ΔE is extracted from a line fit of ln Rxx versus 1/T. The temperature range and the quality of the fit for each magnetic field are not shown. Please provide representative fits or a statement about the fitting range and uncertainty.","section":"Page 5, Arrhenius gap extraction"},{"comment":"The equation N = (N+ - N-)/2 is not numbered, which makes it awkward to reference in the text. Please number equations consistently.","section":"Page 6, Eq. (1)"},{"comment":"The statement that 'a sharp tip' is used to trim excess MnBi2Te4 would benefit from a reference to the technique, since edge definition is important for interpreting edge transport.","section":"Page 3, sample preparation"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is a solid experimental paper with a rich high-field phase diagram, and a theoretical interpretation that is plausible but not yet airtight. The data are the real contribution.\n\nWhat's genuinely new: 5-SL MnBi2Te4 devices with much-improved QAH quantization (zero-field gap of 29 K, roughly three times previous reports), and a high-field fan diagram showing plateaus at ν = +1, -1, -2, -3, -4, -6, -8, plus a developing fractional state above 40 T. The crystal-growth and exfoliation improvements seem to work. The mapping of the Landau fan is careful, and the three-regime structure in the ν = 0 phase (0−, 0′, 0+) is a fresh observation.\n\nThe paper's central claim is that these quantized states obey a generalized topological index N = (N₊ − N₋)/2 rooted in the parity anomaly, and that an 'anomalous' Landau level—the first LL of the surface hole band E_h^s—is what produces the ν = −1 QAH state and the helical edge transport in the ν = 0 phase. This framework is borrowed from earlier work (Böttcher, Lapa) and applied here to 5-SL MnBi2Te4 with a specific level assignment.\n\nThe soft spots are where the theory meets the data. The assignment of L0, L1, L2 and the anomalous LL to specific sub-bands comes from a slab model of hypothetical ferromagnetic MnBi2Te4 whose Hamiltonian parameters, level broadening, and impurity inputs are not specified in the main text; the DOS simulation in Fig. 3d is a fit, so agreement with the fan diagram does not independently confirm the sub-band labels. If those labels are wrong, the parity-anomaly narrative and the helical-edge claim lose their anchor. Also, the helical edge transport in the ν = 0 phase is inferred from transport regimes—there are no nonlocal or edge-selective measurements shown. And the Type S devices used for high-field data do not exhibit zero-field QAH, so the adiabatic connection to C0 = −1 is an assumption, though a defensible one given the similar magnetic transitions.\n\nThe parity-anomaly index itself is, at this level, a counting statement that works for any 2D system where each LL carries Chern number −1. The connection to the parity anomaly is conceptual rather than derived here. That's acceptable as a framework, but it shouldn't be oversold.\n\nOverall: the experimental results are new, reproducible in principle, and significant enough to justify referee time. The referee should press for the supplementary model specification and, ideally, a more direct demonstration of edge transport. If the model details hold up, this becomes a benchmark paper for MnBi2Te4 high-field physics. Send it to peer review.","headline":"A strong experimental paper with a genuinely new high-field QAH phase diagram, but the parity-anomaly interpretation leans on a level assignment that needs full model disclosure to convince.","tokens_in":15265,"tokens_out":4443,"would_cite":true,"duration_ms":55670,"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 quantized Hall plateaus in five-layer MnBi2Te4 follow a generalized topological index rooted in the parity anomaly of (2+1)-dimensional Dirac fermions, carried by a single anomalous Landau level.","keywords":["MnBi2Te4","quantum anomalous Hall effect","parity anomaly","Landau quantization","topological insulator","helical edge transport","high magnetic field","spectral asymmetry"],"falsifier":"A direct spectroscopic probe of the dispersion of the anomalous Landau level—for example, tunneling spectroscopy at the sample edge that resolves a hole-derived level curving upward through the Fermi level—would settle the identification. Alternatively, a transport experiment on the ν = 0 phase that detects the predicted counter-propagating helical edge channels (e.g., via nonlocal resistance) or, conversely, shows purely chiral edge conduction, would confirm or refute the helical-edge scenario.","tokens_in":14101,"feed_emoji":"🧲","tokens_out":6122,"duration_ms":65402,"temperature":0.7,"pith_summary":"This paper claims that the quantized topological states observed in five-septuple-layer (5-SL) MnBi2Te4 under magnetic fields up to 45 tesla are not a simple superposition of quantum Hall and quantum anomalous Hall effects, but are governed by a generalized topological index rooted in the parity anomaly of (2+1)-dimensional Dirac fermions. The anomaly manifests as an anomalous Landau level that is hole-derived yet disperses like an electron level, and this level sustains the ν = −1 quantum anomalous Hall state at high fields and produces gate-tunable helical edge transport in the ν = 0 phase. If the claim is right, the paper provides a direct solid-state realization of the parity anomaly and a counting rule—half the imbalance between unoccupied and occupied Landau levels—that should apply to any quantum anomalous Hall insulator in a quantizing magnetic field.","feed_headline":"Parity anomaly sets the quantized Hall plateaus in MnBi2Te4","feed_subtitle":"Five-layer magnetic topological insulator shows a counting rule that ties zero-field QAH to high-field plateaus via one anomalous Landau…","key_machinery":"The central object is the generalized topological index N = (N+ − N−)/2, a Landau-level-resolved expression of the parity anomaly's spectral asymmetry in (2+1) dimensions. The mechanism it captures is an 'inverted' electron-like Landau level descending from a hole sub-band—the first Landau level of the surface band E^h_{s,k}—which, unlike all other hole levels, disperses upward and crosses the Fermi level at the sample boundary, sustaining the chiral edge state behind the ν = −1 plateau and the helical edge channels in the ν = 0 regime. Supporting it is a slab-model band structure of ferromagnetic MnBi2Te4 whose sub-band ordering places three bulk bands (E^e_1, E^h_1, E^h_2) inside the surface band gap of the 5-SL quantum well, so that the lowest observable Landau levels L0, L1, L2 originate from E^h_{1,k}, E^e_{1,k}, and E^e_{2,k} respectively.","core_discovery":"In the presence of Landau quantization, the topology of 5-SL MnBi2Te4 is characterized by the index N = (N+ − N−)/2, where N+ and N− count unoccupied and occupied Landau levels, with the universal relation N− = N+ − 2C0 holding at the QAH gap (here C0 = −1). The paper identifies this index with the spectral asymmetry of the parity anomaly and pinpoints the anomalous Landau level as the first Landau level of the surface hole band E^h_{s,k}, which, unlike ordinary hole levels, disperses upward and crosses the Fermi level at the sample boundary, giving rise to the ν = −1 QAH plateau and, through its crossing with the first hole Landau level, to the helical edge channels that split the ν = 0 phase into three regimes (0−, 0′, 0+). The assignment is anchored by slab-model calculations of a hypothetical ferromagnetic MnBi2Te4 quantum well, which reproduce the observed Landau fan once level broadening and impurity states are included.","pith_inferences":["If the parity-anomaly index is correct, the same anomalous Landau level should appear as a robust feature in other intrinsic magnetic topological insulators with Chern number −1, and its energy spacing relative to ordinary Landau levels could serve as a direct measure of the spectral asymmetry.","The fractional state observed between ν = −2 and ν = −3 above 40 T may be a many-body extension of the parity-anomaly picture or an unrelated fractional Chern state; its field threshold suggests interaction effects beyond the single-particle Landau-level description.","The slab-model dependence on a hypothetical ferromagnetic 5-SL stack could be tested across thicknesses: if a different layer count shifts the sub-band ordering, the anomalous level's electron-like dispersion should move accordingly, and the ν = 0 helical regime should appear or disappear in concert."],"forward_implications":["The counting rule N = (N+ − N−)/2 applies to all quantum anomalous Hall insulators in a quantizing magnetic field, so the same imbalance of two extra occupied Landau levels should appear in any |C0| = 1 Chern insulator when the field polarizes the magnetism.","The ν = −1 plateau at high field is adiabatically connected to the zero-field QAH state, meaning the zero-field and high-field quantizations share the same topological origin.","The ν = 0 phase of 5-SL MnBi2Te4 hosts gate-tunable helical edge transport with three distinct regimes controlled by the Fermi level position, analogous to the edge transport of a quantum spin Hall insulator.","Improved crystal growth—slow cooling instead of quenching—suppresses Mn-Bi anti-site defects and raises the zero-field QAH gap to about 30 K, making MnBi2Te4 a practical platform for reaching the extreme quantum limit and for applications in topological electronics."],"supporting_citations":[{"why":"Introduces the Chern-insulator paradigm whose parity anomaly the paper invokes as the origin of the generalized index.","marker":"[15]"},{"why":"Establishes the axial anomaly in odd dimensions that the paper identifies as the field-theoretic source of the spectral asymmetry.","marker":"[16]"},{"why":"Provides the Hamiltonian formulation of the parity anomaly that justifies the spectral-asymmetry definition used here.","marker":"[17]"},{"why":"Sets the Landau-level counting rule N = (N+ − N−)/2 for QAH insulators in external fields, which the experiment is said to realize.","marker":"[18]"},{"why":"Predicts the survival of the QAH effect in orbital magnetic fields as a consequence of the parity anomaly, providing the notion of an anomalous Landau level.","marker":"[19]"},{"why":"Reports the zero-field QAH effect in 5-SL MnBi2Te4, the material system and baseline that this work improves and extends to high fields.","marker":"[7]"},{"why":"Supplies the theory of the ferromagnetic MnBi2Te4 slab as a Weyl-semimetal quantum well, used for the sub-band and Landau-level calculations.","marker":"[2]"},{"why":"Presents an alternative model of anomalous Landau quantization which the paper explicitly distinguishes from its parity-anomaly-based picture.","marker":"[61]"}],"fun_headline_variants":["Parity anomaly ties MnBi2Te4's quantum plateaus","Anomalous Landau level fuels MnBi2Te4's edge flow","MnBi2Te4: parity anomaly unlocks quantized states","One anomaly explains MnBi2Te4's Hall quantization","MnBi2Te4's parity anomaly dictates its plateaus"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The interpretation rests on the slab-model assignment of the observed Landau levels to the calculated sub-band Landau levels of a hypothetical ferromagnetic MnBi2Te4; if that band-structure labeling is wrong, the parity-anomaly reading and its edge-transport picture lose their experimental anchor.","fun_headline_variants_meta":{"raw":{"variants":["Parity anomaly ties MnBi2Te4's quantum plateaus","Anomalous Landau level fuels MnBi2Te4's edge flow","MnBi2Te4: parity anomaly unlocks quantized states","One anomaly explains MnBi2Te4's Hall quantization","MnBi2Te4's parity anomaly dictates its plateaus"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000567,"raw_usage":{"total_tokens":2687,"prompt_tokens":945,"completion_tokens":1742,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":1653}},"tokens_in":561,"tokens_out":1742,"duration_ms":14928,"temperature":1.0,"reasoning_tokens":1653,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:12:50.827304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct spectroscopic probe of the dispersion of the anomalous Landau level—for example, tunneling spectroscopy at the sample edge that resolves a hole-derived level curving upward through the Fermi level—would settle the identification. Alternatively, a transport experiment on the ν = 0 phase that detects the predicted counter-propagating helical edge channels (e.g., via nonlocal resistance) or, conversely, shows purely chiral edge conduction, would confirm or refute the helical-edge scenario.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Chern-insulator paradigm whose parity anomaly the paper invokes as the origin of the generalized index."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the axial anomaly in odd dimensions that the paper identifies as the field-theoretic source of the spectral asymmetry."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Hamiltonian formulation of the parity anomaly that justifies the spectral-asymmetry definition used here."},{"cited_title":"& Hankiewicz, E","cited_arxiv_id":null,"evidence_quote":"Sets the Landau-level counting rule N = (N+ − N−)/2 for QAH insulators in external fields, which the experiment is said to realize."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Predicts the survival of the QAH effect in orbital magnetic fields as a consequence of the parity anomaly, providing the notion of an anomalous Landau level."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the zero-field QAH effect in 5-SL MnBi2Te4, the material system and baseline that this work improves and extends to high fields."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theory of the ferromagnetic MnBi2Te4 slab as a Weyl-semimetal quantum well, used for the sub-band and Landau-level calculations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents an alternative model of anomalous Landau quantization which the paper explicitly distinguishes from its parity-anomaly-based picture."}],"review_version":1}