{"id":"71ca2c54-4e2d-46a1-9a90-844251bc1123","arxiv_id":"2502.02686","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Parton wave functions for the magnetoroton and neutral fermion modes of the anti-Pfaffian state are evaluated for large systems, showing their long-wavelength gaps are close for second Landau level Coulomb, signaling weak supersymmetry.","lead":"Physicists constructed new trial wave functions for neutral excitations of paired fractional quantum Hall states and computed their energies on large model systems. The results suggest that two distinct excitations at filling 5/2, a spin-2 graviton and a spin-3/2 gravitino, have nearly matching energies, a hint of emergent supersymmetry, but the match is not exact.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Trial parton modes are validated only against PH-conjugated Jack states at q=0; without exact SLL-Coulomb eigenstate overlaps at intermediate wave numbers, the all-q mode identification underlying the SUSY conclusion is under-supported.","rationale":"The reader identifies the weakest assumption as the faithfulness of Eqs. (3) and (4) to the true low-lying SLL Coulomb modes across wave numbers. I agree, and I would elevate this above the other stated concerns. The odd-N ground-state interpolation and missing error bars affect the accuracy of the quoted gaps, but the mode-identification issue is upstream: if the trial states are not the physical magnetoroton and neutral fermion, then no amount of interpolation refinement or error-bar tightening can make the SUSY inference valid for the actual SLL Coulomb system. The present evidence consists of two overlaps with PH-conjugated Jack model states at the smallest angular momenta; these are encouraging but do not establish the all-wave-number validity claimed in the abstract and Discussion. The proposed exact-diagonalization overlap check is feasible at the system sizes already used in the paper and would directly settle whether the parton trial states are the dominant physical modes. The conditional verdict is therefore appropriate, with the requested evidence being a necessary condition for full acceptance.","tokens_in":31012,"tokens_out":12512,"duration_ms":123269,"concrete_test":"Exact-diagonalize the SLL Coulomb Hamiltonian (equivalently, the effective interaction of Eq. (7)) on the Haldane sphere for N=12 and N=14 at the fluxes 2Q=2N+1, and compute the squared overlap of the parton states of Eqs. (3) and (4) with the exact eigenstates in each accessible L sector, from L=2 and L=3/2 up to intermediate L. If the trial state is the dominant component of the lowest exact state in its L sector at q=0 (overlap approximately 0.8 or higher) and remains the dominant component at intermediate L, the all-q identification holds; if the overlap drops toward 0.5 or the trial state spreads over several exact states, the parton dispersion and the SUSY inference do not describe the SLL Coulomb modes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central SUSY conclusion in Sec. II C rests on the claim that the parton wave functions of Eqs. (3) and (4) are the physical magnetoroton and neutral fermion modes of the SLL Coulomb system, and moreover that they remain faithful at all wave numbers. The only quantitative support offered for this identification is the absolute overlap of the L=2 parton exciton with the PH-conjugate of the Pf GMP/Jack L=2 state (0.87 at N=12) and of the L=3/2 neutral fermion with the PH-conjugate Jack state (0.91 at N=13). These are overlaps with model Jack states at the smallest accessible angular momenta, not with the low-lying eigenstates of the SLL Coulomb Hamiltonian at intermediate L. A variational trial state can have an energy above the exact mode; two such upper bounds can be close in the q->0 limit even if the exact gaps are not. Since the abstract and Discussion promote the parton construction as valid across all wave numbers, the absence of any exact-diagonalization comparison at intermediate wave numbers leaves the central identification under-supported. If Eqs. (3) and (4) mix with other excitations at moderate q, the dispersion curves in Figs. 2-7 and the inference that a SUSY interaction lies near SLL Coulomb do not transfer to the actual physical modes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs parton wave functions for the magnetoroton and neutral fermion modes of the anti-Pfaffian (aPf) universality class at filling 5/2, evaluates their dispersions for large systems using Monte Carlo, and finds that their long-wavelength gaps approach each other for the second Landau level Coulomb interaction, suggesting an emergent supersymmetry near that point. The construction is then generalized to a variety of Abelian and non-Abelian parton states, with dispersions, clustering properties, and chiralities of the collective modes discussed, and a parton-exciton ansatz is proposed for the gapped neutral mode of the composite fermion Fermi liquid at 1/4.","tokens_in":31308,"tokens_out":8491,"duration_ms":76005,"significance":"If the mode identification is correct, this work provides a numerically tractable tool for studying neutral collective modes of paired and other partonic fractional quantum Hall states at system sizes far beyond previous approaches, and it offers a concrete test of the SUSY conjecture for the anti-Pfaffian. The paper's strengths include large-N Monte Carlo evaluation with stated chain statistics, thermodynamic extrapolations, explicit overlap checks with Jack states, and a broad set of falsifiable predictions for dispersions of collective modes. The main uncertainties are the limited validation of the trial modes against exact eigenstates at intermediate wave numbers and the unquantified interpolation used for the odd-electron ground-state energy in the neutral fermion gap.","major_comments":[{"comment":"The identification of the wave functions in Eqs. (3) and (4) as the physical magnetoroton and neutral fermion modes of the SLL Coulomb system at all wave numbers rests on overlaps of 0.87 (L=2, N=12) and 0.91 (L=3/2, N=13) with particle-hole-conjugated Jack states. These are only at the smallest accessible angular momenta and are not overlaps with exact SLL Coulomb eigenstates. The statement in Sec. II C that 'the agreement between the above trial wave functions and the exact states (not shown here) is similar to ...' is not quantified. Since the SUSY conclusion compares two variational upper bounds in the long-wavelength limit, the absence of exact-diagonalization comparisons at intermediate L leaves open the possibility that the true gaps do not approach each other. Please provide overlaps with exact SLL Coulomb eigenstates for accessible system sizes at several L values, or otherwise bound the variational error.","section":"Sec. II B and II C"},{"comment":"The neutral fermion gap at 5/2 requires the ground-state energy for an odd number of particles, which is obtained by interpolating the ground-state energies of even-particle systems. The systematic error from this interpolation is neither quantified nor shown. Because the thermodynamic extrapolation in Fig. 3 is central to the SUSY conclusion, this missing error estimate is a load-bearing gap. Please display the odd-N data, the interpolation curve, and an estimate of the interpolation uncertainty (for example, by comparing against exact odd-N ground-state energies at small N where they are available).","section":"Sec. II C, paragraph 2"},{"comment":"The paper repeatedly claims that the parton wave functions 'can provide a good description of the modes at all wave numbers' and that they are 'valid across all wave numbers'. The numerical evidence shown is limited to the long-wavelength limit. This overclaim is not essential for the long-wavelength SUSY test, but it is a central selling point of the construction. Either provide supporting evidence (e.g., overlaps with exact eigenstates or spectral functions at finite q) or temper the claim to avoid overstating the validated range of validity.","section":"Sec. II B and Sec. VII (Discussion)"}],"minor_comments":[{"comment":"The overlap comparisons involve particle-hole conjugation on the sphere, which maps a state with N particles at flux 2Q to a state with 2Q+1-N particles; this convention should be stated explicitly for clarity.","section":"Sec. II B"},{"comment":"The fit lines and error bars are described, but the fit parameters and goodness-of-fit (e.g., chi-squared) are not given; please include them to support the thermodynamic extrapolation.","section":"Fig. 3"},{"comment":"The coefficients B1, B3, B5, and C0-C6 are taken from Ref. [80] but are not listed; providing them or a pointer to a table would make the effective interaction self-contained.","section":"Eq. (7)"},{"comment":"The discussion of the possible counterexample to the inversion-symmetry claim of Ref. [125] is interesting but appears tangential to the main results; consider moving it to a footnote or an appendix.","section":"Sec. IV C"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of cond-mat.str-el and addresses a timely question about neutral modes and supersymmetry in fractional quantum Hall systems. The main results are plausible and the parton framework is promising, but the missing validation against exact eigenstates at intermediate wave numbers and the unquantified interpolation for odd-N ground-state energies are significant enough that the manuscript needs revision before the central claims can be fully endorsed. The authors' reliance on their own prior work is extensive but appropriate given the continuity of the parton program."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Buddy,\n\nThe headline: this paper does something new. It builds explicit parton wave functions for the magnetoroton and neutral fermion of the anti-Pfaffian class that can be evaluated for N up to ~30, and it uses them to test the SUSY conjecture at 5/2. The test is honest: the gaps are close but not equal for SLL Coulomb, so they conclude SUSY is not exact but might be nearby. That is a solid, well-scoped result.\n\nWhat the paper does well: the wave functions in Eqs. (3)-(4) are explicit, the Monte Carlo is large-scale with stated chain statistics, and the overlaps 0.87 and 0.91 with PH-conjugated Jack states at L=2 and L=3/2 are respectable. The generalization to other parton states (2213, 2215, the 3/8 and 2/5 non-Abelian candidates, 4/11, etc.) gives a useful toolbox for neutral-mode dispersions. The parton-exciton ansatz for the 1/4 CFFL is a nice addition.\n\nThe soft spots are real but not fatal. The paper claims the parton wave functions describe the modes at all wave numbers, but the only quantitative validation is at the two smallest angular momenta. The sentence that agreement with exact states is \"similar to\" earlier work is not backed by a figure or table, so the intermediate-q dispersion curves and the all-q selling point are under-supported. Second, the neutral fermion gap uses odd-N ground-state energies obtained by interpolating even-N energies, and the systematic error from that interpolation is not quantified. That affects the SUSY conclusion somewhat, but the trend with N looks reasonable. Third, some dispersion figures (Figs. 4-7) lack error bars or have them only for the largest runs, so fine features of those curves should not be overinterpreted.\n\nThe central SUSY conclusion holds as a statement about the trial wave functions: for the parton state in the aPf class, the two long-wavelength gaps approach each other but are not degenerate for SLL Coulomb. Whether that transfers to the actual SLL Coulomb modes depends on the trial states being faithful, which is supported at q->0 but not at all q. That is a limitation, not a fatal flaw.\n\nThis paper deserves a serious referee. The construction is novel and the numerics are reproducible in spirit. A referee should ask for the missing exact-diagonalization comparison and an estimate of the interpolation error. I would bring it to a reading group for the parton-exciton construction alone.","headline":"Solid parton construction for aPf neutral modes with an honest SUSY test; the all-q validation claim is under-supported by the shown evidence.","tokens_in":31839,"tokens_out":2372,"would_cite":true,"duration_ms":23640,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Parton wave functions for the anti-Pfaffian state put the magnetoroton and neutral fermion nearly degenerate at long wavelengths for second Landau level Coulomb interactions.","keywords":["fractional quantum Hall effect","parton theory","anti-Pfaffian state","neutral fermion mode","magnetoroton mode","supersymmetry","composite fermions","second Landau level"],"falsifier":"Compute the exact second Landau level Coulomb low-energy spectrum at intermediate angular momenta for N=14 to 20 and compare the lowest states with the parton trial wave functions; if the overlaps drop substantially or another state appears below the trial dispersion at intermediate q, the conclusion that a supersymmetric interaction lies near the second Landau level Coulomb interaction would not be supported for the real system.","tokens_in":30782,"feed_emoji":"🧲","tokens_out":8741,"duration_ms":77392,"temperature":0.7,"pith_summary":"At filling factor 5/2, two low-lying neutral excitations of the Moore-Read Pfaffian family are conjectured to be supersymmetric partners: the bosonic magnetoroton (the density-wave mode) and the neutral fermion (a fermionic mode), with equal energy gaps in the long-wavelength limit. This paper tests that conjecture for the anti-Pfaffian, the particle-hole conjugate of the Pfaffian, using parton wave functions obtained by placing a particle-hole pair in the same or in different factors of a $\\Phi_{-2}$ parton. Unlike earlier trial states, these wave functions can be evaluated for systems up to roughly 30 to 40 electrons, so the long-wavelength gaps can be reliably extrapolated to the thermodynamic limit. For the second Landau level Coulomb interaction the extrapolated gaps of the two modes approach each other, indicating that an emergent supersymmetry point lies near the second Landau level Coulomb interaction. The same construction is then applied to a catalogue of other Abelian and non-Abelian parton states and to the composite fermion Fermi liquid at quarter filling.","feed_headline":"Anti-Pfaffian's two neutral modes nearly merge at long wavelengths","feed_subtitle":"Large-system parton wave functions put the magnetoroton and neutral fermion nearly at equal energy under SLL Coulomb.","key_machinery":"The machinery is the parton construction of trial wave functions. Parton theory splits each electron into fictitious particles filling integer Landau levels, so the ground state is a projected product of Slater determinants; at $\\nu=1/2$ the state $\\bar{2}\\bar{2}111$ lies in the anti-Pfaffian universality class. Neutral excitations are generated by creating a particle-hole pair: placing both in the same $\\Phi_{-2}$ factor gives the magnetoroton (starting at total orbital angular momentum L=2), while placing them in different $\\Phi_{-2}$ factors gives the neutral fermion (starting at L=3/2). The projected forms factorize into composite-fermion exciton wave functions, which can be evaluated by Jain-Kamilla projection for large systems. The operational supersymmetry test is whether the L=2 and L=3/2 gaps coincide as momentum $q\\to 0$.","core_discovery":"The paper's central claim is that the parton wave functions in Eqs. (3) and (4) describe the magnetoroton and neutral fermion modes of the anti-Pfaffian universality class across the full range of wave numbers, not just in the long-wavelength limit. Supporting evidence is given by overlaps of 0.87 for the L=2 magnetoroton at N=12 and 0.91 for the L=3/2 neutral fermion at N=13 with particle-hole-conjugated Jack states of the Pfaffian, and by dispersion curves computed for systems up to N=30. For the second Landau level Coulomb interaction, the thermodynamic extrapolations of the $q\\to 0$ gaps come close, so the paper concludes that a supersymmetric interaction likely lies near the second Landau level Coulomb interaction, while exact supersymmetry would still require fine-tuning the Hamiltonian. The same construction is extended to several other parton states, and a parton-exciton ansatz is proposed for the gapped neutral mode of the composite fermion Fermi liquid at $\\nu=1/4$.","pith_inferences":["If the parton trial states remain faithful at intermediate wave numbers, the closeness of the two gaps at $q\\to 0$ makes the second Landau level Coulomb interaction a natural starting point for tuning an exact supersymmetric Hamiltonian, and the required tuning might be as simple as adjusting the V1 Haldane pseudopotential.","The no-neutral-fermion result for k≥3 suggests an experimental discriminator: a Read-Rezayi candidate with k≥3 that shows two low-lying neutral modes cannot be described by the parton construction in Eq. (8).","The parton-exciton ansatz at $\\nu=1/4$ could be tested in wide quantum wells by inelastic light scattering or microwave absorption; observing a gapped dispersing mode at the predicted energy would support the parton content of the quarter-filled Fermi liquid.","The overlap evidence is thin at intermediate angular momenta, so exact diagonalization for systems near N=20 at L values between 2 and N/2 would test whether the trial states are the true second Landau level Coulomb modes."],"forward_implications":["The parton wave functions provide a numerically tractable way to compute neutral-mode dispersions in the anti-Pfaffian class at all wave numbers, a regime earlier trial states could not reach.","If the conclusion is correct, the fine-tuned supersymmetry found near the Pfaffian has an analogue on the particle-hole-conjugate side, and the second Landau level Coulomb interaction sits close to, but not exactly at, the supersymmetric point.","For Read-Rezayi k≥3 states, the construction predicts no neutral-fermion analogue on the sphere, marking the Pfaffian (k=2) as the only member of the series with two low-lying neutral modes.","The dispersions and clustering properties catalogued for states such as $\\bar{3}\\bar{2}214$, $\\bar{2}314$, $4\\bar{2}13$, 2213, and 2215 give concrete predictions for multiple graviton modes and their chiralities.","The parton-exciton ansatz at $\\nu=1/4$ gives a gapped graviton whose long-wavelength energy matches the spectral-function peak, providing a candidate interpretation for the gapped neutral mode of the quarter-filled composite fermion Fermi liquid."],"supporting_citations":[{"why":"It supplies the parton ground state in the anti-Pfaffian universality class on which the magnetoroton and neutral fermion wave functions of Eqs. (3) and (4) are built.","marker":"[36]"},{"why":"It provides the earlier large-system supersymmetry test for the Pfaffian state whose conclusions the paper extends to the anti-Pfaffian side.","marker":"[34]"},{"why":"It formulates the superspace and supersymmetry partner conjecture for the neutral modes of the Pfaffian that this paper tests for the anti-Pfaffian.","marker":"[33]"},{"why":"It supplies the Jack-polynomial wave functions whose particle-hole conjugates give the overlaps 0.87 and 0.91 used to validate the parton trial modes.","marker":"[32]"},{"why":"It introduces parton theory, the construction framework used for all the trial wave functions in the paper.","marker":"[40]"},{"why":"It provides the effective interaction whose lowest-Landau-level pseudopotentials match second Landau level Coulomb, allowing the Monte Carlo evaluation of the relevant energies.","marker":"[80]"}],"fun_headline_variants":["Near-SUSY anti-Pfaffian neutral modes from parton wave functions","Parton waves show near-SUSY anti-Pfaffian modes","Anti-Pfaffian's neutral modes nearly match parton dispersion","SUSY hint: anti-Pfaffian's neutral modes close","Parton dispersions for anti-Pfaffian and beyond"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the trial wave functions in Eqs. (3) and (4) are the true low-lying neutral modes of the second Landau level Coulomb system at all wave numbers, a premise supported only by overlaps of 0.87 at L=2 and 0.91 at L=3/2; if the exact modes mix with other excitations at intermediate momenta, the computed gap difference and the supersymmetry conclusion would not transfer to the real system.","fun_headline_variants_meta":{"raw":{"variants":["Near-SUSY anti-Pfaffian neutral modes from parton wave functions","Parton waves show near-SUSY anti-Pfaffian modes","Anti-Pfaffian's neutral modes nearly match parton dispersion","SUSY hint: anti-Pfaffian's neutral modes close","Parton dispersions for anti-Pfaffian and beyond"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00156,"raw_usage":{"total_tokens":6291,"prompt_tokens":1063,"completion_tokens":5228,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":5133}},"tokens_in":679,"tokens_out":5228,"duration_ms":36970,"temperature":1.0,"reasoning_tokens":5133,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T11:27:40.472669+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact second Landau level Coulomb low-energy spectrum at intermediate angular momenta for N=14 to 20 and compare the lowest states with the parton trial wave functions; if the overlaps drop substantially or another state appears below the trial dispersion at intermediate q, the conclusion that a supersymmetric interaction lies near the second Landau level Coulomb interaction would not be supported for the real system.","supporting_citations":[],"review_version":1}