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Four-qubit states reproduce known CFTs, plus 25 new ones

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A symmetry-free entropy search on four-site states recovers known CFTs and yields unclassified candidate CFTs with 1<c<2.

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load-bearing objection Instructive method paper that recovers known CFTs from four-site states, but the ~25 'Mystery CFT' candidates are not yet supported as true CFTs. the 4 major comments →

arxiv 2509.04596 v1 pith:OGE5DQJB submitted 2025-09-04 hep-th cond-mat.stat-mechcond-mat.str-elquant-ph

A systematic search for conformal field theories in very small spaces

Xiang Li , Ting-Chun Lin , John McGreevy This is my paper
classification hep-th cond-mat.stat-mechcond-mat.str-elquant-ph
keywords conformal field theoryvector fixed point equationentanglement Hamiltonianentanglement bootstrapquantum criticalitycentral chargereconstructed Hamiltonianc-d conjecture
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the space of 1+1-dimensional conformal field theories (CFTs) can be explored without writing down any Hamiltonian: a single local entropic condition, the vector fixed point equation, picks out states that behave like CFT groundstates even when the state has only four sites. Scanning that condition across all translation-invariant, time-reversal-invariant states on four qubits and four qutrits, the authors find a small set of critical points. Reconstructing a nearest-neighbor Hamiltonian from each state, they identify the Ising, tricritical Ising and three-state Potts models, many c=1 theories, and roughly 25 unclassified candidate CFTs with central charge between 1 and 2. This matters because CFTs with c>=1 are largely uncharted and no explicit irrational CFT is known with certainty; this route needs no symmetry input and could in principle produce new ones.

Core claim

The central claim is that a four-site state is an approximate CFT ground state exactly when it is a critical point of the entropy function c_delta = ⟨ψ|ĉ_delta|ψ⟩, where ĉ_delta is proportional to the reconstructed Hamiltonian H_rec = cot(π/L) Σ_i (K_{i,i+1}−K_i). For such states H_rec is local and non-negative, and its low-lying spectrum equals 2π/L times the CFT scaling dimensions, so the state itself encodes the theory. The qubit search exhaustively finds eight critical points, three of which are CFTs (Ising, XX, Heisenberg); the qutrit search finds the Ising, tricritical Ising and Potts models, many c=1 points, and about 25 states with 1<c<2 whose spectra look conformal but match no know

What carries the argument

The vector fixed point equation (VFPE), K_Delta |ψ⟩ ∝ |ψ⟩ for the operator K_Delta = η Δ(A,B,C) + (1−η) I(A:C|B) built from entanglement Hamiltonians of three adjacent intervals. It acts as a local, finite-size-suppressed criterion that singles out RG fixed points. The search algorithm minimizes the variance of ĉ_delta (equivalently the error of the VFPE) by gradient descent followed by Newton's method, then uses the reconstructed Hamiltonian H_rec to convert each state into a spectrum of scaling dimensions and a central-charge estimate c_delta.

Load-bearing premise

The paper's enumeration relies on the assumption that a single state on only four sites, satisfying the fixed-point equation at one cross-ratio (η=1/2), is a faithful enough representative of a continuum CFT that the Hamiltonian reconstructed from it describes the infinite-size limit; the paper itself notes that proving this requires L to infinity and all intervals.

What would settle it

For any specific 'Mystery CFT' state, take the reconstructed Hamiltonian H_rec to L=24, 32, 48 and check whether the low-lying eigenvalues keep converging to fixed values of L·E_i and whether S(l) still fits (c/3) log sin(πl/L) to the largest available l; if instead the levels drift linearly with L (as the paper shows for gapped or relevant-perturbed states), the four-site state is not a CFT groundstate.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Enumerating CFTs is reduced to finding critical points of an entropy function on a finite-dimensional state space; no Hamiltonian, symmetry, or integrability input is required.
  • Each solution yields a fully normalized local Hamiltonian, so the speed of light and the central charge are obtained without manual rescaling.
  • The answer to the long-open question of whether irrational CFTs exist could come from the unclassified 1<c<2 candidates if they survive more careful continuum-limit checks.
  • Scanning the VFPE error over a parameter family detects continuous phase transitions, as demonstrated on the antiferromagnetic Ising chain with transverse and longitudinal fields.
  • The observed empty regions ('voids') in error-versus-c_delta plots hint that allowed central charges form a discrete constrained set even where no bootstrap bound is known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural next check is to run the same search on four d=4 sites (or with the blocking step the paper suggests); if the c=1 orbifold line and the Ginsparg archipelago appear, the small-state enumeration is mapping the true CFT moduli space rather than a finite-size artefact.
  • The paper's observation that the number of negative Hessian directions of c_delta equals the number of relevant operators suggests c_delta is a genuine landscape function on state space; if established, it would give a variational principle for RG flow.
  • The mystery CFTs, if confirmed, would be prime candidates for irrational CFTs; comparing their OPE coefficients (computable from eigenstates via a wavefunction-overlap method) against known rational data would settle the match.
  • The c-d conjecture could be tested directly: pushing the search to larger local dimension d and finding any CFT state with c > d−1 would disprove it; the paper reports consistency so far.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a systematic search for 1+1d CFT groundstates by finding critical points of an entropic function c_Δ (proportional to a reconstructed Hamiltonian H_rec) on the space of translation-invariant real states of four qubits and four qutrits. For each critical state, the authors reconstruct a nearest-neighbor Hamiltonian H_rec and study its low-energy spectrum and entanglement entropy for system sizes L=4,6,8,10,12,14 using periodic MPS. They report recovering several known CFTs (Ising, tricritical Ising, 3-state Potts, the c=1 free-boson line) and, in the qutrit search, about 25 previously unidentified 'Mystery CFT' candidates with 1<c<2, alongside gapped fixed points and states with relevant perturbations. The central claim is that small-system VFPE solutions encode enough universal data to enumerate CFTs without input from a Hamiltonian or symmetry.

Significance. If the central claim were established, this would be a very significant result: a Hamiltonian-free, entropy-based enumeration of 1+1d CFTs that could in principle reach irrational or otherwise inaccessible theories in 1<c<2. The paper's concrete successes are real and should be credited: the reconstructed Hamiltonians for the qubit XX and Heisenberg states are exactly the known XX and Heisenberg Hamiltonians with the correct prefactor (Section 3), and the identification of Ising, TCI, Potts and several c=1 points from four-qutrit states is convincing and provides strong external validation of the method. The authors also honestly list many states that are gapped or have relevant perturbations, and they make the raw states available as supplementary material. However, the headline claim of discovering previously unknown CFTs is not yet supported: the finite-size diagnostics used to classify the 'Mystery CFTs' cannot distinguish a true CFT from a gapped/near-critical Hamiltonian whose correlation length exceeds the largest system size studied, and the paper itself contains a concrete false-positive example (the 'Ising in Fib' state). The work is therefore best viewed as a promising m

major comments (4)
  1. [Section 4, Appendix B, Tables 6–10] The classification of the ~25 'Mystery CFTs' relies on spectral and entanglement diagnostics for L = 4,6,8,10,12,14. The paper's own Discussion admits that 'most of the states we find produce H_rec with a finite correlation length (of order 15−30 lattice spacing)'. If the correlation length ξ is 15–30, then all system sizes studied are within ξ, so the observed 'no visible drift' of L·E_i and the apparent CFT-like S(ℓ) are equally consistent with a gapped Hamiltonian whose correlation length is larger than 14. The analyzer's criteria (Fig. 9, Appendix B) are manual and qualitative ('Does the spectrum converge as L increases?', 'Does S(ℓ) follow the CFT prediction?'), with no quantitative threshold or extrapolation. This is a load-bearing gap: Tables 6–10 are presented as CFT candidates, but the finite-size data shown cannot establish that they are CFTs. Please either extend the analysis
  2. [Appendix B, Table 2 (cΔ = 0.63776)] The 'Ising in Fib' state is a controlled counterexample to the paper's classification criteria. It satisfies the VFPE with small error, its reconstructed spectrum on L = 4,...,14 is the low-energy Ising spectrum (below a gap that grows with L), and its entanglement entropy appears to follow the CFT scaling for small L, yet no unitary CFT exists at c ≈ 0.63. The authors themselves write 'Why c∆ on 4 sites for this state is so far off we do not know.' This demonstrates that a VFPE solution with small error and CFT-looking small-size spectra need not correspond to a CFT at all. It is therefore essential to explain how many of the 'Mystery CFTs' in Tables 6–10 might be analogous artifacts, and what additional evidence (e.g., vanishing gap extrapolated to L→∞, or matching to a known or bootstrap-consistent spectrum) distinguishes them from this false positive.
  3. [Section 1, Footnote 2] The paper explicitly relies on L=4 and a single cross-ratio η=1/2, while acknowledging that proving a VFPE solution is the groundstate of H_rec requires L→∞ and all intervals. For the known CFTs, external identification (XX, Heisenberg, Ising, TCI, Potts) provides the missing validation. For the Mystery CFTs there is no such external anchor: the only evidence is the reconstructed spectrum of H_rec on small sizes, whose connection to the continuum CFT spectrum is precisely what footnote 2 leaves unproven. Please either provide an independent validation of at least one Mystery CFT (e.g., by bootstrap or by demonstrating that the reconstructed spectrum stabilizes for L well beyond the correlation length), or soften the abstract's claim that 'others are CFTs we have not yet identified' to 'candidates requiring further evidence.'
  4. [Appendix B, Fig. 9] The analyzer's final decision is a 'manual fuzzy decision' with two qualitative checks. This is not a reproducible classification criterion. To make the central claim testable, specify quantitative thresholds: e.g., the maximum allowed drift of L·E_i from L=4 to L=14, the goodness-of-fit range for S(ℓ) vs log sin(πℓ/L), and the maximum allowed scatter of c_21 across sizes. Without such thresholds, the reader cannot assess the false-positive rate or reproduce the assignment of states to Tables 2–10 versus Tables 11–14. This is especially important because the 'Ising in Fib' state would pass the qualitative criteria as described.
minor comments (5)
  1. [Introduction] Typo: 'relativstic' should be 'relativistic'.
  2. [Eq. (2.1)] The normalization factor is printed as '3 2 log 2'; please clarify whether it is 3/(2 log 2) or 3·2·log 2. The text later says c_Δ is exactly the central charge, so the intended factor should be stated unambiguously.
  3. [Table 1] The Ising coefficients are listed as '[a1, a2, a5, a6] = [0.11036, 0.80152, 0, 0, −0.51103, 0.29023]' — six numbers for four coefficients, apparently including a3,a4. Please correct the notation.
  4. [Appendix B] The text defines c_21 in Eq. (B.1) and says it is 'UV-insensitive', but the subsequent tables report both c_fit and c_21 for each state; a brief explanation of how c_fit is computed (least-squares slope over which ℓ range?) would help the reader interpret the tables.
  5. [References] Several key background results are deferred to works 'to appear' ([20], [35]). Since the present paper's method depends heavily on [17], [20]–[22], please make the dependence explicit in the main text and, if possible, include a short summary of the relevant results from [20] in an appendix so the current paper is more self-contained.

Circularity Check

1 steps flagged

VFPE sufficiency is load-bearing and deferred to a self-citation; known-CFT matches provide independent anchors, so partial circularity only.

specific steps
  1. self citation load bearing [Section 1, after Eq. (1.6), footnote 2]
    "Elsewhere [20] we will show that a state |ψ⟩ satisfying the fixed point equation is the groundstate of its reconstructed Hamiltonian H ψ rec. ... proving the groundstate result requires taking the limit of many sites, L → ∞, and demands that the fixed-point equation hold for all possible three intervals. Whereas in this paper, we rely only on the case L = 4."

    The enumeration identifies 4-site VFPE solutions as CFT groundstates and reads off scaling dimensions from H_rec via (1.6). The bridge from 'satisfies VFPE' to 'is the groundstate of H_rec' is not proved here; it is cited to [20], a paper by the same authors (Li, Lin, McGreevy) and 'to appear'. The footnote concedes the proof needs L→∞ and all intervals, while the search uses L=4. Thus the 'Mystery CFT' classification is supported by an unverified self-citation rather than by a derivation in this paper. The known-CFT matches are external anchors, but the un-identified states are exactly the ones that depend on this assumed converse.

full rationale

The paper's core numerical search—finding critical points of c∆ and comparing H_rec spectra to known CFT data—is not circular: the known CFT matches (Ising, TCI, Potts, XX/Heisenberg at c=1) are external anchors, and the reconstructed Hamiltonians for XX and Heisenberg coincide exactly with known lattice Hamiltonians. These are genuine predictions. However, the headline claim that ~25 'Mystery CFT' states in 1<c<2 are previously unknown CFTs depends on the converse of the VFPE (a VFPE state is a CFT groundstate and is the groundstate of its H_rec). That converse is not proved in the paper; it is attributed to refs. [17,20,21,22], all by the same group, with [20] 'to appear.' Footnote 2 explicitly admits the proof requires L→∞ and all intervals while the paper uses only L=4. The 'Ising in Fib' state (c∆=0.63776) shows that a 4-site VFPE solution can have a misleading c∆, so the 4-site input does not by itself determine the CFT data. Hence the un-identified states are not yet established as CFTs by the paper's own derivation; the load-bearing step is an unverified self-citation. This is partial circularity, not full: the identified CFTs are independently confirmed.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free physical parameters are fitted. The search rests on the VFPE conjecture, the spectral interpretation of H_rec, and the small-system sufficiency assumption, all inherited from the authors' earlier work; the external benchmark is provided by known CFT spectra. No new particles, forces, or mediators are introduced.

axioms (4)
  • domain assumption A state satisfying the VFPE (1.1) is an RG fixed point (a CFT or a zero-correlation-length gapped state).
    This is the premise that makes the enumeration meaningful. It is introduced as a conjecture from [17] with supporting evidence in [20-22], none of which is formalized or machine-checked in this paper (Section 1).
  • domain assumption The reconstructed Hamiltonian (1.4) is local, non-negative, and its normalized spectrum is the CFT scaling dimension list (1.6), with the source state as groundstate.
    Used to convert each solution into CFT data. Non-negativity follows from [24]; the spectrum and groundstate claims are inherited from [17] and deferred proof in [20] 'to appear' (footnote 2).
  • ad hoc to paper A single 4-site configuration (L=4, eta=1/2) captures the universal data of the continuum CFT.
    The paper intentionally relies on L=4 only and acknowledges that proving the groundstate result needs L to infinity and all interval choices (footnote 2, Section 2). This small-system sufficiency is the main risk to the candidate lists.
  • domain assumption Spectral convergence and linear S(l) vs log sin(pi*l/L) diagnose a CFT.
    The analyzer uses these criteria (Appendix B, Fig. 9) with manual fuzzy decisions. They are calibrated on TFIM and XXZ examples but not derived.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of A systematic search for conformal field theories in very small spaces." pith.science (2026). https://pith.science/paper/OGE5DQJB

@misc{pith2026250904596,
  author       = {Pith},
  title        = {Pith review of: A systematic search for conformal field theories in very small spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGE5DQJB}},
  note         = {Machine review of arXiv:2509.04596}
}
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read the original abstract

Groundstates of 1+1d conformal field theories (CFTs) satisfy a local entropic condition called the vector fixed point equation. This condition is surprisingly well satisfied by groundstates of quantum critical lattice models even at small system sizes. We perform a search in the space of states of very small systems (four qubits and four qutrits) and examine the states that satisfy this condition. By reconstructing a local Hamiltonian from each state, we are able to identify many of these solutions with known CFTs; others are gapped fixed points, or involve large relevant perturbations, and others are CFTs we have not yet identified. These ideas are also useful for identifying continuous quantum phase transitions in a given family of Hamiltonians, and for identifying the nature of the critical theory in small systems.

Figures

Figures reproduced from arXiv: 2509.04596 by John McGreevy, Ting-Chun Lin, Xiang Li.

Figure 1
Figure 1. Figure 1: ). The second part of the procedure generates approx￾imate CFT Hamiltonians from the ground states using (1.4). Once we have the Hamiltonian, we can obtain the CFT data, including scaling dimensions and OPEs, through known algorithms [26]. Note that the purpose of this algorithm is not to give accurate values for the CFT data. The purpose is to generate as many CFTs as pos￾sible, including previously unkno… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.