{"id":"ed142faa-be5c-4140-a6e8-b758dd7e9671","arxiv_id":"2505.04138","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Interaction-driven band folding on a triangular-lattice Chern band produces integer and fractional quantum anomalous Hall crystals whose Hall conductivity equals the filling of the folded mini-band, in both fermionic and bosonic models.","lead":"This paper computes, with exact diagonalization and tensor-network methods, that repulsive interactions on a triangular-lattice Chern band create a charge-density-wave order that folds the band into a topological mini-band, and doping that mini-band produces integer and fractional quantum anomalous Hall crystals. The result gives a concrete microscopic route to Hall crystals in moiré and cold-atom settings, including a bosonic example.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"ED 'series' of FQAHC states is computed with only 1–3 particles in the C=-1 mini-band; the ν*=1/5 diagnostics match a single-particle Chern-band effect, not a Laughlin state.","rationale":"I read the paper's mechanism in good faith: a V1-driven CDW at ν=2/3 folds the BZ and leaves a C=-1 mini-band; doping it should give (F)QAHC with σH=-ν*. The robust iDMRG evidence at ν=7/9 (charge pumping, ES counting, XTRG incompressibility) is substantial and multi-method, so I do not see grounds to reject the central mechanism. However, the 'series' of FQAHC states is partly built on ED states where the mini-band contains only one to three particles. For N=1, the observed 5-fold 'topological' degeneracy, the -1/5 Chern number, and the 10π spectral flow are exactly the single-particle properties of a C=-1 band with five states; no interactions among mini-band particles are needed. The paper even notes there are only 5 momenta on the 30-site torus but does not draw this conclusion. This makes the ν*=1/5 ED state an unreliable witness for a Laughlin-like FQAHC. The same concern applies in milder form to the N=2,3 ED states. The ν*=1/3 iDMRG state is better supported because the infinite cylinder has finite density and the ES shows Laughlin counting, but a larger-circumference check (Ny=9 or 12) would be valuable. Overall, the reader's CONDITIONAL verdict is appropriate; I would add to the conditions: rule out the single-particle interpretation of the ED states (e.g., by the spectral-flow comparison above) and provide a larger-Ny ES for ν*=1/3. No change to the verdict itself.","tokens_in":14056,"tokens_out":29715,"duration_ms":312178,"concrete_test":"Compute the noninteracting single-particle spectral flow of the C=-1 HF mini-band from SI Fig. 6 on a 5-state torus with the same folded CDW potential and the same twist angles, and compare it with the ED spectral flow at ν=11/15 in Fig. 2(c). If the 15-state manifold, the -1/5 many-body Chern number of each ground state, and the 10π return are all reproduced by the N=1 Slater determinant in the mini-band, then the ED result is a single-particle band effect, not a Laughlin FQAHC; this would require reducing the 'series' claim and re-examining the σH=-ν* rule on larger systems, e.g., iDMRG with Ny≥10 at ν*=1/5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The ED evidence for the 'series' of FQAHC states at ν*=1/5, 2/5, 3/5 is obtained on 30-site tori, where the folded C=-1 mini-band has only 5 momentum states (the paper notes this on p.3). The corresponding particle numbers in the mini-band are N=1, 2, 3. For the headline ν*=1/5 case (ν=11/15, Fig. 2(c)), N=1. A single particle in a flat C=-1 band with N_s=5 has a 5-fold ground-state degeneracy (the five Bloch states), a many-body Chern number C/N_s = -1/5, and a spectral flow that returns after N_s·2π = 10π; the CDW sector count gives 15 states. All three diagnostics in Fig. 2(c) are therefore reproduced by a trivial N=1 Slater determinant in the C=-1 band, with no fractional statistics or Laughlin physics. The paper's statement that the 10π return 'suggests a Laughlin state' is thus unsupported at this filling. The robust iDMRG/ES evidence is currently limited to ν*=1/3 (and SI pumping for 2/5, 2/3); the 'series' and the claimed σH=-ν* rule for fractional ν* would partially rest on few-particle ED points that do not distinguish a single-particle band effect from a FQAHC.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a two-band triangular-lattice Chern model (Eq. 1) with repulsive interactions V1, V2, V3. At ν=2/3 filling of the lower Chern band, strong V1 drives a commensurate CDW that triples the unit cell; a Hartree-Fock analysis fed by iDMRG mean fields indicates that the mini-band just above the CDW gap carries Chern number C=-1. The authors argue that doping this mini-band to (fractional) integer fillings ν* produces integer and fractional quantum anomalous Hall crystals with Hall conductivity σH=-ν*, where ν*=3ν-2, coexisting with the CDW order. Evidence is presented from ED on 30-site tori, iDMRG on cylinders, and XTRG for finite temperatures. The most robust example is a σH=-1/3 FQAHC at ν=7/9 with V1=10, V2=V3=2, supported by charge pumping after 6π flux, entanglement-spectrum shift by one charge sector, and edge-mode counting {1,1,2,3,5,...}. The paper also reports a possible V1-only FQAHC at the same filling, a compressible CDW phase at intermediate temperature, and a bosonic σH=-1/2 FQAHC at ν=5/6.","tokens_in":14400,"tokens_out":15294,"duration_ms":153587,"significance":"If the central claim holds, the paper provides a concrete and unbiased microscopic realization of a generic band-folding mechanism for fractional Hall crystals, going beyond the previously studied integer QAHC cases and offering a natural reinterpretation of the earlier topological pinball liquid states. The robust iDMRG/ES evidence for the ν=7/9 state is a genuine strength, as are the bosonic extension and the finite-temperature phase sequence. The significance is currently tempered by the fact that part of the claimed 'series' of FQAHC states rests on few-particle ED data that do not distinguish fractional Hall physics from single-particle band effects; this needs to be addressed before the general claim is fully established.","major_comments":[{"comment":"The ED example at ν=11/15 (ν*=1/5) on the 30-site torus cannot by itself support a fractional Hall crystal. With 15 unit cells folded into 5 mini-band momenta, the C=-1 mini-band holds exactly one particle. A single particle in a C=-1 band with N_s=5 reproduces all the reported diagnostics: a 5-fold topological degeneracy from the five Bloch states, a many-body Chern number -1/5 per state, a spectral-flow period 5×2π=10π, and, after multiplying by the three CDW sectors, a 15-fold ground-state manifold. These are single-particle band properties, not signatures of a Laughlin state or fractional statistics. The statement that the 10π return 'further suggests FQAHC state is a Laughlin state' is therefore unsupported. Please either remove this example from the FQAHC evidence or provide an interaction-sensitive diagnostic (e.g., many-body gap behavior and ES counting with more than one particle in the mini-band).","section":"Robust FQAHC ground states, Fig. 2(c)"},{"comment":"The claimed 'series' of FQAHC states at ν*=2/5 and 3/5 needs to be separated into large-scale iDMRG/ES evidence and few-particle ED points. On the same 30-site torus, ν*=2/5 (ν=4/5) and ν*=3/5 (ν=13/15) correspond to two and three particles in the five-state mini-band; noninteracting Slater determinants in a C=-1 band already give Hall conductance C×N/N_s = -2/5 and -3/5 and the same flux-pumping periods. The many-body Chern number and spectral flow therefore do not, by themselves, distinguish an FQAH state from a trivial few-particle band-filling effect. The iDMRG charge pumping and ES for ν*=2/5 in SI D are the right kind of evidence; the main text should explicitly state which members of the series have such large-scale support and should avoid presenting the ED few-particle diagnostics as establishing FQAHC.","section":"SI D, Fig. 11; main-text 'Robust FQAHC ground states'"},{"comment":"The V1-only FQAHC claim at ν=7/9 (V1=10, V2=V3=0) rests on a single iDMRG charge-pumping curve that the authors themselves describe as 'less straight' than the V2=V3=2 case. No entanglement-spectrum counting, ground-state degeneracy, or gap estimate is presented for this parameter set. Given that the integer QAHC at ν=1 with only V1 has a vanishingly small gap (0.017–0.053, SI C), the V1-only FQAHC should be presented as preliminary evidence, and the abstract's wording that 'some FQAHC state might even exist in less ideal conditions' should be correspondingly tempered unless additional diagnostics are provided.","section":"Robust FQAHC ground states, V1-only paragraph"}],"minor_comments":[{"comment":"The caption states 'σH =ν∗ =−1/5' for the ν=11/15 state, but the text and the Hall-conductivity rule require σH = −ν∗ = −1/5; please correct the sign inconsistency.","section":"Fig. 2 caption"},{"comment":"The SI contains two sections labelled 'C' ('THE QAHC∗ AT ν=1' and 'ROBUST CDW ORDER IN THE (F)QAHC STATES'); please renumber them sequentially.","section":"Supplementary Information, section numbering"},{"comment":"The sentence 'we find a series of new FQAHC states at fractional fillings of this C = −1 mini-band and (more details in the SI [47])' is grammatically incomplete; please rewrite to name the fillings explicitly and refer to the SI in a complete sentence.","section":"Robust FQAHC ground states, first paragraph"},{"comment":"The complex hopping phase ϕij is described only through Fig. 1(a); for reproducibility the phase assignment (0, π, π/2 on the labeled bonds) should be stated explicitly in the main text.","section":"Model and methods"},{"comment":"The estimates of T* and T_CDW come from XTRG on a single cylinder size (3×18×2); please state the system-size and bond-dimension convergence checks for these transition temperatures, either in the main text or the SI.","section":"Thermodynamics of the FQAHC state"}],"recommendation":"major_revision","confidential_remarks":"The robust iDMRG/ES evidence for the ν=7/9 FQAHC with V2=V3=2 is convincing and should be preserved. The main obstacle is the overinterpretation of few-particle ED data, especially the ν*=1/5 example, as evidence for fractional Hall physics; the authors should revise the 'series' claim accordingly. The paper is otherwise well within the scope of the journal and addresses a timely topic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe real result here is the ν*=1/3 FQAHC at ν=7/9 with V1=10, V2=V3=2. That state is supported by three independent diagnostics: the ES counting {1,1,2,3,5}, one electron pumped after 6π flux, and the ground-state degeneracy structure. The band-folding mechanism—V1-driven CDW at 2/3 folding the BZ and leaving a C=-1 mini-band—is also a nice unifying explanation for the previously mysterious topological pinball liquid states in Ref. 43. The finite-temperature XTRG work showing a compressible CDW intermediate phase is a genuine addition, and the bosonic FQAHC at ν*=1/2, if it holds up, broadens the platform.\n\nBut the paper overreaches when it claims a \"series\" of FQAHC states at ν*=1/5, 2/5, 3/5. The stress-test note is correct: on the 30-site torus the folded mini-band has only 5 momentum states, so ν*=1/5 means N=1 particle in that band. A single particle in a flat C=-1 band with N_s=5 has exactly the degeneracy, many-body Chern number, and 10π spectral flow that Fig. 2(c) shows. There is no Laughlin physics to infer at that filling. The paper itself notes the five-momentum limitation but still says the 10π return \"suggests a Laughlin state,\" which is not supported. The iDMRG pumping for 2/5 and 2/3 (SI Fig. 11) is more convincing, but the 3/5 case appears to rest on ED alone.\n\nThe V1-only FQAHC at ν=7/9 is also weaker than the V2=V3=2 version: the pumping curve is visibly less straight, and the paper acknowledges the QAHC at V2=V3=0 has a vanishingly small gap. That does not kill the claim, but it means the \"generic\" and \"less ideal conditions\" phrasing oversells what the numerics show. The Hartree-Fock band analysis fed by iDMRG mean fields is a reasonable interpretative tool, not a circular proof, because the FQAHC diagnostics come from unbiased simulations of the full Hamiltonian.\n\nBottom line: the central FQAHC at ν*=1/3 is likely real and worth a serious referee. The paper deserves publication after the authors either remove or explicitly reframe the ν*=1/5 case as a single-particle effect, and after they provide error bars or system-size trends for the pumping and gap estimates. This is a solid but not complete paper; the overclaimed series is the main issue.\n\nRecommendation: accept peer review, with revision required. Also ask for simulation data/code, given how much of the case rests on finite-size numerics.\n\nMy take for the reading group: worth discussing once, mainly for the good ν*=1/3 result and the band-folding picture—but read the ED supplement skeptically.","headline":"Solid FQAHC at ν*=1/3, but the claimed series of fractional states includes at least one single-particle finite-size artifact.","tokens_in":14946,"tokens_out":2750,"would_cite":true,"duration_ms":30515,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["73.43.-f","71.10.Fd","71.27.+a"],"model":"deepseek-v4-flash","headline":"In a two-band triangular-lattice model, a nearest-neighbor interaction drives a trivial charge-density wave at ν=2/3 that folds the Brillouin zone and exposes a Chern number -1 mini-band; doping that band produces integer and fractional…","keywords":["quantum anomalous Hall crystal","fractional quantum anomalous Hall","charge density wave","band folding","topological mini-band","triangular lattice","infinite density matrix renormalization group","exact diagonalization"],"falsifier":"Perform larger-scale iDMRG on a cylinder at ν=7/9 with V1=10, V2=V3=2 and check whether the charge pump after inserting 6π flux remains exactly one electron while the CDW order parameter stays nonzero; a deviation in the pumped charge or a melting of the CDW would refute the FQAHC claim. Alternatively, compute the many-body Chern number of the 15 quasi-degenerate ground states at ν=11/15 on a 36-site or 42-site torus to verify the 5×3 degeneracy and σH=-1/5.","tokens_in":13867,"feed_emoji":"🧊","tokens_out":5890,"duration_ms":52968,"temperature":0.7,"pith_summary":"The paper claims a generic mechanism for quantum anomalous Hall crystals (QAHCs)—states that combine a nonzero Hall conductivity with a charge-density wave. In a two-band triangular-lattice model, the V1 interaction at ν=2/3 stabilizes a commensurate, topologically trivial CDW that triples the unit cell. The Brillouin-zone folding that results leaves an isolated band above the CDW gap with Chern number C=-1. Doping this mini-band to integer or fractional filling produces QAHC and fractional QAHC ground states whose Hall conductivity is σH=-ν*, where ν*=3ν-2, not the original band filling. The authors demonstrate one FQAHC at ν=7/9 even without competing interactions, a bosonic FQAHC at ν=5/6, and a compressible CDW phase at intermediate temperatures that precedes the FQAHC.","feed_headline":"A CDW-folded band gives fractional Hall crystals","feed_subtitle":"In a triangular-lattice model, doping the C=-1 mini-band at ν=7/9 yields a σH=-1/3 state without any projection.","key_machinery":"The load-bearing object is the C=-1 mini-band that appears above the CDW gap in the folded Brillouin zone. The paper defines its filling as ν*=3ν-2: at ν=2/3, ν*=0; at ν=1, ν*=1; and at ν=7/9, ν*=1/3. The mini-band is obtained from Hartree-Fock bands built from mean-field parameters measured in iDMRG simulations of the ν=2/3 CDW; adding V2=V3=2 flattens the band (bandwidth 0.11 versus 1.25 with only V1) and makes the Berry curvature more uniform. The topological C=-1 character of this band, together with the CDW's 3-fold ground-state degeneracy, sets the Hall conductivity σH=-ν* and combines with the topological degeneracy to determine the total ground-state degeneracy of each (F)QAHC state.","core_discovery":"The central discovery is that interaction-driven band folding, not just external potentials or lattice geometry, creates a topological band that can host fractional Hall states. Starting from a two-band triangular-lattice model with complex hoppings, the V1-driven CDW at ν=2/3 (electron density 1/3) triples the unit cell and folds the original Brillouin zone. The resulting mini-band above the CDW gap has C=-1; at full filling (ν*=ν=1) the ground states are threefold-degenerate QAHC states, each with quantized σH=-1, which are stabilized by adding longer-range repulsion V2=V3=2. At fractional fillings of the same band, exact diagonalization and infinite DMRG show FQAHC states with σH=-ν* (e.g. -1/5 at ν=11/15, -1/3 at ν=7/9, -2/5 at ν=4/5, -2/3 at ν=8/9), with ground-state degeneracy equal to the topological degeneracy times the 3-fold CDW degeneracy. The same scheme works for hard-core bosons at ν=5/6, giving a σH=-1/2 bosonic FQAHC.","pith_inferences":["In moiré or cold-atom systems where a fractional Hall conductivity is observed at a filling that does not match the Chern band's filling, the FQAHC mechanism would be identifiable by a triple-unit-cell CDW and by the relation σH=-ν* rather than σH=ν.","The compressible CDW phase at intermediate temperatures could show anisotropic longitudinal resistivity, while the transverse resistivity would not be quantized; this is a concrete experimental signature that distinguishes the precursor phase from the low-temperature FQAHC.","The analogy the paper draws to doping a solid into a supersolid suggests that doping any trivial commensurate CDW whose folded band has nonzero Chern number could generically produce fractional Hall states, making the mechanism a promising search principle for new lattice models.","Because the band-mixing at V2=V3=0 weakens the FQAHC, improving the quantum geometry of the folded band—by engineered hoppings, magnetic fields, or other perturbations—could be a practical route to realizing these states in experiments."],"forward_implications":["At V1=10, V2=V3=2, a series of FQAHC states appears at fractional fillings of the C=-1 mini-band, with σH=-ν*, including σH=-1/5, -1/3, -2/5, and -2/3; the topological degeneracy of each is multiplied by the 3-fold CDW degeneracy.","With only V1 interaction and no competing terms, a σH=-1/3 FQAHC at ν=7/9 still survives, though with a less straight charge-pumping curve indicating less uniform Berry curvature.","The finite-temperature study of the σH=-1/3 FQAHC shows a compressible CDW phase at intermediate temperatures T*