{"id":"907e7a08-e2b7-4e3b-b619-686f46ae9807","arxiv_id":"1908.00995","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The four-body SYK model is supersymmetric in all but two Altland-Zirnbauer symmetry classes, with supercharge structure fixed by k mod 8.","lead":"The paper shows that the standard four-body Sachdev-Ye-Kitaev model is supersymmetric for most Majorana mode counts, with no fine-tuning of couplings. This hidden symmetry is set by the number of modes and leaves clear signatures in time-dependent correlation functions.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Plateau prediction (Eq. 13) rests on an unproven random-vector ansatz that is already contradicted by Fig. 2 for small/large q; SUSY existence itself is sound.","rationale":"The reader's conditional verdict is appropriate. The central existence claim—that the four-body SYK model is supersymmetric for all k mod 8 other than 0 and 4—is supported by a clean, parameter-free argument from parity degeneracy; I do not see an internal inconsistency there. The only significant weakness is the long-time plateau prediction, Eq. (13), which the reader correctly identifies as relying on an unproven random-vector assumption. I agree with that assessment and add that the assumption is not merely unproven: it is in tension with the operator sum rule relating the projected coefficients to the fixed expansion of Γ_j, and it is already falsified in the paper's own numerics for q≪k. This does not undermine the existence of SUSY, but it does weaken the advertised quantitative consequences in q-body correlation functions. Therefore the verdict should remain conditional pending a derivation or more complete numerical verification of the plateau values.","tokens_in":14671,"tokens_out":25694,"duration_ms":271773,"concrete_test":"Exact-diagonalize a moderate-size realization in class D (e.g., k=12, q=1 and q=5) and compute the left side of Eq. (12) directly from eigenstates, without the random-vector approximation, for the projected operators P_μΓ_j. Compare the ensemble average to Eq. (13) and to the empirical distribution of v_{μj,a}; in particular test whether Σ_μ v_{μj,a}=v_{j,a} is satisfied and whether the variance of v_{μj,a} is q-independent as the uniform-random ansatz predicts. If systematic q-dependent deviations appear (as Fig. 2 already suggests for q≪k), Eq. (13) needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The existence proof for SUSY is sound: particle-hole symmetry T− forces parity-degenerate spectra, and the supercharges constructed from eigenstates, Eq. (4), satisfy {Qa,Qb}=2Hδab. The locality counting in Table II follows from commutation with Z and is internally consistent. The weak point is the quantitative plateau claim, Eq. (13). The step from Eq. (12) to Eq. (13) replaces the exact coefficients v_{μj,a} in P_μΓ_j = Σ_a v_{μj,a}Υ_a by a random-vector ansatz: for each μ, the vector is uniformly random on the sphere, subject only to normalization and T± selection rules. This ansatz is not derived, and it is constrained by the operator identity Σ_μ P_μΓ_j = Γ_j, so the v_{μj,a} for different μ cannot be independent; e.g., in BDI/CI Γ_1=Z has a sparse fixed expansion, imposing sum rules. The paper's own Fig. 2 shows Eq. (13) fails when (k choose q)/(k choose ⌊k/2⌋) is not near 1, so the ansatz is empirically false in those regimes. Because the abstract and conclusions present the plateau values as a consequence of SUSY, the advertised quantitative connection is not secured, even though the existence of local supercharges is unaffected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the four-body SYK model of k Majorana modes and claims that, for all k mod 8 except 0 and 4, the model is supersymmetric in the sense of supersymmetric quantum mechanics: there exist Hermitian supercharges satisfying {Q_a,Q_b}=2H δ_ab and [H,Q_a]=0. The proof is based on the antiunitary particle-hole symmetry T_-, which forces parity-degenerate spectra; the supercharges are explicitly built from pairs of degenerate eigenstates. The authors count the number of local supercharges Nloc using commutation with the chiral operator Z, classify their Majorana-operator content by k mod 8, and connect Nloc to the Dyson index and ramp shapes of q-body correlation functions. They further derive a formula (Eq. 13) for the long-time plateau C_{q,∞} under a random-vector assumption on the expansion coefficients, and compare it to exact-diagonalization data in Figs. 2 and 4.","tokens_in":14944,"tokens_out":16011,"duration_ms":160191,"significance":"If the claims hold, the paper identifies a universal, parameter-free supersymmetric structure in the standard SYK model and ties it to the Altland-Zirnbauer classification. The explicit construction of local supercharges and the connection between their number and the Dyson index/ramp shapes are valuable for the SYK and random-matrix communities. The manuscript is careful in its symmetry analysis, and the numerical data support the qualitative ramp classification and the existence of long-time plateaus in the supersymmetric classes. The mathematical core establishing parity-degenerate eigenstates and hence the superalgebra is straightforward and sound; the paper also gives an explicit, reproducible numerical study for a range of system sizes.","major_comments":[{"comment":"The derivation of the plateau formula (13) from Eq. (12) rests on the unproven assertion that the expansion coefficients v_{μj,a} of P_μ Γ_j in the Majorana basis behave as random vectors subject only to normalization and T± constraints. This assertion is not a consequence of supersymmetry, and it is not exact: the identity Σ_μ P_μ Γ_j = Γ_j imposes sum rules that correlate the v_{μj,a} across μ, and in classes BDI/CI the operator Γ_1=Z has a sparse expansion. The numerical data in Fig. 2 show that Eq. (13) fails when (k choose q)/(k choose ⌊k/2⌋) is not close to one, so the assumption is empirically false in those regimes. Because the abstract and conclusions present the plateau values as a consequence of SUSY, the quantitative connection is not secured; please either supply a controlled derivation or clearly mark Eq. (13) as a heuristic valid only in the large-(k choose q) regime.","section":"Eqs. (12)-(13), Fig. 2"},{"comment":"The abstract's statement that SUSY has consequences away from the ground state, including in q-body dynamical correlation functions, and the concluding claim that the plateau value is due to the imprint of how Γ_j transforms are stronger than what is established. The long-time plateau formula requires the additional random-vector hypothesis discussed above; the exact consequences of SUSY are the parity degeneracy, the algebraic structure of the supercharges, and the qualitative ramp shapes. Please distinguish these exact statements from the heuristic plateau estimate in the abstract and conclusions.","section":"Abstract and Conclusions"}],"minor_comments":[{"comment":"Please state explicitly in the abstract that the SUSY construction is equivalent to the parity degeneracy induced by T_-, so that the reader does not mistake the result for a new independent symmetry beyond particle-hole symmetry.","section":"Supersymmetry from particle-hole symmetry"},{"comment":"The first line of Eq. (13) is typographically ambiguous and appears to read N/(β Nloc), which is inconsistent with the numerical plateau values quoted in Appendix C (e.g., CII q=4n+3 plateau ≈3); the intended expression is presumably N Nloc/β. Please correct the typesetting.","section":"Eq. (13)"},{"comment":"The quantity c in panel (d) of Fig. 2 is used before it is defined; please define c as the random-matrix expectation from Eq. (13).","section":"Fig. 2 caption"},{"comment":"Please specify in the caption of Fig. 4 whether the plotted plateau is C_{q,∞} or C_{q,∞}M/4, so that the quoted values (e.g., \"plateau at C_{q,∞} ≈ 2\") match the main-text notation.","section":"Appendix C b and Fig. 4"},{"comment":"The phrase that the structure of the supercharges is entirely set by the number of interacting Majorana modes should be qualified: the explicit supercharges in Eq. (4) depend on the disorder realization through the eigenstates, while what is fixed by k mod 8 is their symmetry type and Majorana-parity content.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The core existence result is sound and the paper is well suited to the journal, but the plateau formula is over-sold relative to the evidence. I would be willing to accept after a major revision in which the random-vector assumption is either justified or explicitly reframed as a regime-dependent heuristic, and after the typesetting of Eq. (13) is corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result holds up. The paper shows that the standard four-body SYK model, with no fine-tuning, is supersymmetric for all symmetry classes except AI and AII. The mechanism is really that simple: particle-hole symmetry forces parity-degenerate spectra, and the standard SUSY QM construction turns each degenerate pair into supercharges satisfying {Qa,Qb}=2Hδab. That part is rigorous and clean. What is genuinely new is the locality analysis: counting which supercharges are local to the SYK model by commuting with Z, giving the Nloc pattern in Table II and the link to the Altland-Zirnbauer class. The topological-phase interpretation of the supercharges as emergent Majoranas is also nice. The numerics for the ramp shapes (Fig. 1) convincingly follow the Dyson index, which supports the SUSY classification.\n\nThe soft spot is Eq. (13), the plateau prediction. The step from Eq. (12) to Eq. (13) assumes the expansion coefficients v_{μj,a} are random vectors subject only to normalization and symmetry constraints. That is an unproven modeling assumption, and it is not even asymptotically exact: the operator identity Σ_μ P_μ Γ_j = Γ_j imposes sum rules on those vectors, so they cannot be independent across μ. The paper's own Fig. 2 shows the prediction fails when (k choose q)/(k choose floor(k/2)) is not close to one. The existence of SUSY is not affected, but the advertised quantitative consequence for correlation-function plateaus is not secured. I would like the abstract and conclusions to say clearly that Eq. (13) is a random-matrix-inspired estimate, not a derived result. The circularity of constructing supercharges from eigenstates is a minor issue; the nontrivial content is in the locality and symmetry classification, which do not reduce to the input degeneracy.\n\nThis paper is for the SYK and random-matrix communities, and for anyone interested in hidden SUSY in disordered many-body systems. The central existence claim is important; the plateau prediction is a secondary, weaker part. I would send this to a serious referee, but with a clear request to revise the status of Eq. (13) and soften the quantitative conclusions. The paper deserves a chance; just not as-is.","headline":"The central claim—that the four-body SYK model has hidden supersymmetry for generic couplings—is sound and worth refereeing; the quantitative plateau prediction is a heuristic that the paper overstates.","tokens_in":15439,"tokens_out":1446,"would_cite":true,"duration_ms":17632,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q60","82B44","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"The plain four-body SYK model is supersymmetric for six of its eight symmetry classes.","keywords":["Sachdev-Ye-Kitaev model","supersymmetry","Majorana fermions","Altland-Zirnbauer classification","random matrix theory","many-body zero modes","fermion parity","correlation functions"],"falsifier":"Diagonalize $H$ for a specific supersymmetric instance, such as $k=6$ or $k=10$, and directly test $\\{Q_a,Q_b\\}=2H\\delta_{ab}$ for supercharges built from parity-degenerate eigenstates; then compare the exact long-time plateau $C_{q,\\infty}$ for $q=1,3,5$ against Eq. (13). A violation of the anticommutator would refute the existence of supersymmetry, while a plateau pattern that does not alternate with $q$ even at large $k$ would falsify the random-vector modeling assumption.","tokens_in":14467,"feed_emoji":"⚛️","tokens_out":5113,"duration_ms":49562,"temperature":0.7,"pith_summary":"This paper argues that the simplest Sachdev-Ye-Kitaev model, consisting of $k$ Majorana fermions with random four-body couplings, is exactly supersymmetric for all but two values of $k \\bmod 8$, with no fine-tuning of the couplings. The supercharges square to the Hamiltonian, and their number and structure are fixed by $k \\bmod 8$ and the model's Altland-Zirnbauer symmetry class. The discovery matters because supersymmetry in many-body physics is usually imposed by hand; here it emerges automatically from the model's particle-hole symmetry, with observable consequences for time-dependent correlation functions away from the ground state.","feed_headline":"The generic SYK model is supersymmetric","feed_subtitle":"Four-body Majorana model needs no fine-tuned couplings; six of eight symmetry classes carry exact supercharges.","key_machinery":"The key machinery is the set of spectrally flattened supercharges $\\Gamma_j = Q_j/\\sqrt{H}$, which behave as many-body Majorana zero modes: they satisfy $\\{\\Gamma_j,\\Gamma_k\\}=2\\delta_{jk}$, commute with $H$, and anticommute with fermion parity $P$. The existence of $N$ such operators gives a $2^{N/2}$-fold degeneracy for each energy level. The count $N$ and the number of local supercharges $N_{\\text{loc}}$ are read off from the antiunitary symmetries $T_\\pm$; in particular, the presence of particle-hole symmetry $T_-$ forces parity degeneracy and hence supersymmetry. Expanding the local $\\Gamma_j$ in the basis of odd products of Majorana operators then determines which $q$-body observables couple to the long-time plateau of correlation functions.","core_discovery":"The central claim is that the four-body SYK Hamiltonian with structureless Gaussian couplings $J_{qrst}$ admits Hermitian supercharges $Q_a$ satisfying $\\{Q_a,Q_b\\}=2H\\delta_{ab}$ and $[H,Q_a]=0$ whenever $k \\bmod 8$ is not $0$ or $4$. The number of independent supercharges is $N=2$ in the symmetry classes D, C, BDI, and CI, and $N=4$ in classes DIII and CII. In odd-$k$ classes, all but one supercharge are local to the $k$ Majorana modes, and the remaining supercharge is the auxiliary Majorana $\\gamma_\\infty$. The microscopic structure of the local supercharges is dictated by $k \\bmod 8$: they expand only in products of $4n+1$ Majorana operators in classes with positive supercharge signature, and only in products of $4n+3$ operators when the signature is negative. The supersymmetry follows directly from particle-hole symmetry $T_-$, which pairs opposite fermion-parity sectors and forces an energy degeneracy that can be reorganized into supermultiplets.","pith_inferences":["The argument that particle-hole symmetry generates supersymmetry is not specific to the four-body interaction: any parity-conserving Hamiltonian in an Altland-Zirnbauer class with $T_-$ should admit analogous supercharges, so extended SYK models with other interaction orders may also be supersymmetric without fine-tuning.","The random-vector assumption used for the plateau coefficients is likely exact only in the large-Hilbert-space limit; the finite-$k$ growth visible in the numerics suggests a systematic finite-size correction to the plateau formula that could be derived from the exact trace structure.","If the identification with a one-dimensional topological phase is taken literally, the emergent supersymmetry should be stable under perturbations that preserve $T_-$, which is a concrete stability prediction that could be tested numerically."],"forward_implications":["The long-time plateau of the infinite-temperature $q$-body correlation function alternates with $q$: it is nonzero only for $q=4n+1$ in classes with positive supercharge signature, with additional contributions in classes DIII and CII where the product supercharge $\\Gamma_4$ enters for $q=4n+3$.","The ramp shape in $q$-body correlation functions is set by the Dyson index $\\beta$, which equals 1, 2, or 4 and is tied to the number of local supercharges.","The plateau formula $C_{q,\\infty}M/4 = N/(\\beta N_{\\text{loc}})$ for matching Majorana strings, with a $\\beta=4$ correction from $\\Gamma_4$, predicts the alternating plateau pattern observed numerically when $\\binom{k}{q}/\\binom{k}{\\lfloor k/2\\rfloor}$ is close to one.","The local supercharges can be viewed as emergent unpaired Majorana modes at the boundary of a one-dimensional topological phase, connecting supersymmetry to the $\\mathbb{Z}_8$ classification of the SYK model.","Because these correlation functions are measurable in digital quantum simulation, the predicted plateau and ramp signatures provide dynamical, far-from-equilibrium tests of supersymmetry."],"supporting_citations":[{"why":"Introduces the four-body SYK model of Majorana fermions that the paper shows is supersymmetric.","marker":"[2]"},{"why":"Maps the SYK model to the boundary of a one-dimensional topological phase, supplying the topological interpretation of the local supercharges.","marker":"[22]"},{"why":"Presents explicitly supersymmetric SYK extensions that require coupling fine-tuning, the contrast that motivates the no-fine-tuning claim.","marker":"[25]"},{"why":"Identified ramps and long-time plateaus in SYK correlation functions, the dynamical signatures that the paper links to supercharges.","marker":"[34]"},{"why":"Provides the random-matrix description of ramp shapes through the Dyson index, used to connect $N_{\\text{loc}}$ to the observable ramp.","marker":"[35]"},{"why":"Supplies the one-dimensional topological classification of interacting fermions and the Majorana zero-mode counting underlying the locality analysis.","marker":"[45]"},{"why":"Establishes the eightfold Altland-Zirnbauer symmetry classification of the SYK model and the parity degeneracy from particle-hole symmetry that generates supersymmetry.","marker":"[46]"},{"why":"Defines the Altland-Zirnbauer symmetry classes used throughout the paper's classification.","marker":"[48]"}],"fun_headline_variants":["SYK model unwittingly supersymmetric for most mode counts","No fine-tuning: SYK inherits SUSY from its symmetry class","Majorana four-body model has hidden supercharges","SYK supersymmetry emerges from Altland-Zirnbauer class","Exact SUSY in SYK without coupling constraints"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The plateau predictions assume that the coefficients expanding the projected supercharges into Majorana strings behave as random vectors subject only to normalization and symmetry constraints; if this fails, the plateau values would differ, although the existence of supersymmetry itself would not be affected.","fun_headline_variants_meta":{"raw":{"variants":["SYK model unwittingly supersymmetric for most mode counts","No fine-tuning: SYK inherits SUSY from its symmetry class","Majorana four-body model has hidden supercharges","SYK supersymmetry emerges from Altland-Zirnbauer class","Exact SUSY in SYK without coupling constraints"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000882,"raw_usage":{"total_tokens":3812,"prompt_tokens":949,"completion_tokens":2863,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":2777}},"tokens_in":565,"tokens_out":2863,"duration_ms":20454,"temperature":1.0,"reasoning_tokens":2777,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:26:28.182460+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Diagonalize $H$ for a specific supersymmetric instance, such as $k=6$ or $k=10$, and directly test $\\{Q_a,Q_b\\}=2H\\delta_{ab}$ for supercharges built from parity-degenerate eigenstates; then compare the exact long-time plateau $C_{q,\\infty}$ for $q=1,3,5$ against Eq. (13). A violation of the anticommutator would refute the existence of supersymmetry, while a plateau pattern that does not alternate with $q$ even at large $k$ would falsify the random-vector modeling assumption.","supporting_citations":[{"cited_title":"We refer to the first case as time- reversal symmetry: The level spacing statistics of each subblock are determined by the presence of T and by the signT 2 =±1","cited_arxiv_id":null,"evidence_quote":"Introduces the four-body SYK model of Majorana fermions that the paper shows is supersymmetric."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Maps the SYK model to the boundary of a one-dimensional topological phase, supplying the topological interpretation of the local supercharges."},{"cited_title":"Sannomiya, H","cited_arxiv_id":null,"evidence_quote":"Provides the random-matrix description of ramp shapes through the Dyson index, used to connect $N_{\\text{loc}}$ to the observable ramp."},{"cited_title":"Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi 44, 131 (2000)","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional topological classification of interacting fermions and the Majorana zero-mode counting underlying the locality analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the eightfold Altland-Zirnbauer symmetry classification of the SYK model and the parity degeneracy from particle-hole symmetry that generates supersymmetry."},{"cited_title":"Behrends, J","cited_arxiv_id":null,"evidence_quote":"Defines the Altland-Zirnbauer symmetry classes used throughout the paper's classification."}],"review_version":1}