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arxiv: 2606.12540 · v1 · pith:YKNWPJUAnew · submitted 2026-06-10 · ✦ hep-th · cond-mat.str-el· quant-ph

Toward Entanglement Bootstrap for Conformal Field Theory in Any Dimension

Rolando Ramirez Camasca , Xiang Li , Ting-Chun Lin , John McGreevy This is my paper

Pith reviewed 2026-06-27 08:48 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords conformal field theoryquantum critical wavefunctionreconstructed Hamiltonianfinite size effectsnumerical spectrumany dimensionCFT spectrum
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The pith

A wavefunction of a quantum critical point determines a Hamiltonian whose spectrum reproduces conformal field theory features in any dimension.

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

The paper proposes reconstructing a Hamiltonian directly from a quantum critical wavefunction in any dimension. Numerical checks confirm that for approximate CFT ground states on the icosahedron and fuzzy sphere, the wavefunction is nearly the ground state of this Hamiltonian. The energy spectrum of the Hamiltonian, when restricted to the unit sphere, displays the integer spacing expected for conformal descendants and agrees with known low-lying CFT levels. This reconstruction supplies an automated procedure for mitigating finite-size effects within a given Hilbert space. A reader would care if this approach allows extracting universal CFT data from wavefunctions alone across dimensions.

Core claim

Given any quantum critical wavefunction, a reconstructed Hamiltonian is defined such that for known regularized approximate CFT groundstates on the icosahedron and the fuzzy sphere the input state is close to the groundstate and the spectrum on the unit sphere exhibits CFT properties with integer descendant spacing and matching low-lying energies. This yields an automated method to improve finite-size effects in a fixed Hilbert space.

What carries the argument

The reconstructed Hamiltonian, defined from the wavefunction in a manner analogous to lower-dimensional cases, that serves as the parent Hamiltonian for the critical state.

If this is right

  • The reconstruction works in arbitrary spacetime dimensions.
  • It applies to both the icosahedral and fuzzy-sphere regularizations of CFT states.
  • The method improves the effective description by reducing finite-size corrections.
  • The spectral match validates that the reconstructed operator captures the conformal symmetry.

Where Pith is reading between the lines

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

  • If successful more broadly, this could enable determining CFT operator content directly from ground-state wavefunctions without an explicit Hamiltonian.
  • The approach might extend to other critical systems beyond CFTs.
  • Numerical verification on additional dimensions or states would strengthen the case for generality.

Load-bearing premise

That success with the specific approximate CFT states tested on the icosahedron and fuzzy sphere means the reconstruction procedure is valid for arbitrary quantum critical wavefunctions in any dimension.

What would settle it

A counterexample consisting of a quantum critical wavefunction for which the reconstructed Hamiltonian has a significantly different ground state or a spectrum lacking integer spacing and CFT energy matches would falsify the proposed generality.

Figures

Figures reproduced from arXiv: 2606.12540 by John McGreevy, Rolando Ramirez Camasca, Ting-Chun Lin, Xiang Li.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
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Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
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Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
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Figure 20. Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p021_20.png] view at source ↗
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Figure 21. Figure 21: FIG. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_21.png] view at source ↗
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Figure 22. Figure 22: FIG. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_22.png] view at source ↗
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Figure 23. Figure 23: FIG. 23 [PITH_FULL_IMAGE:figures/full_fig_p022_23.png] view at source ↗
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Figure 24. Figure 24: FIG. 24 [PITH_FULL_IMAGE:figures/full_fig_p022_24.png] view at source ↗
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Figure 25. Figure 25: FIG. 25 [PITH_FULL_IMAGE:figures/full_fig_p023_25.png] view at source ↗
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Figure 26. Figure 26: FIG. 26 [PITH_FULL_IMAGE:figures/full_fig_p023_26.png] view at source ↗
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Figure 27. Figure 27: FIG. 27 [PITH_FULL_IMAGE:figures/full_fig_p023_27.png] view at source ↗
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Figure 28. Figure 28: FIG. 28 [PITH_FULL_IMAGE:figures/full_fig_p024_28.png] view at source ↗
read the original abstract

Given a quantum critical wavefunction in any dimension, we propose a reconstructed Hamiltonian, analogous to the ones previously found for 1+1d CFT and for 2+1d bosonic liquid topologically-ordered states. We test numerically that, for known regularized approximate CFT groundstates (on the icosahedron and the fuzzy sphere), (1) they are close to the groundstate of their reconstructed Hamiltonian, and (2) the spectrum of their reconstructed Hamiltonian on the unit sphere has CFT properties (integer spacing of descendants) and matches known low-lying energies. We show that this provides an automated method to improve the finite-size effects in a fixed Hilbert space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes a reconstructed Hamiltonian for a quantum critical wavefunction in any dimension, by analogy with prior entanglement-bootstrap constructions in 1+1d CFTs and 2+1d topologically ordered states. Numerical tests are reported on two regularized approximate CFT ground states (icosahedral and fuzzy-sphere regularizations) showing that these states are close to the ground state of the reconstructed Hamiltonian and that the spectrum of the reconstructed Hamiltonian on the unit sphere exhibits CFT features, including integer spacing of descendants and agreement with known low-lying energies. The construction is presented as an automated method for mitigating finite-size effects within a fixed Hilbert space.

Significance. If the numerical evidence can be quantified and the procedure shown to apply beyond the two tested regularizations, the work would supply a practical tool for extracting effective Hamiltonians directly from wavefunctions in higher-dimensional critical systems. The explicit analogy to established lower-dimensional bootstrap methods is a clear strength, and the automated finite-size improvement is a useful practical contribution. The significance remains conditional on stronger documentation of the tests and on evidence that success is not tied to the specific regularizations employed.

major comments (3)
  1. [Abstract] Abstract: the claim that numerical tests demonstrate success is load-bearing for the central claim, yet no quantitative measures (overlaps, distances, error bars, or fit metrics) are supplied to substantiate 'close' or 'matches'. Without these, the strength of the evidence cannot be assessed.
  2. [Reconstruction procedure] Reconstruction procedure: because the Hamiltonian is defined directly from the input wavefunction, it is unclear whether the reported spectral match on the unit sphere constitutes an independent check or is partly tautological with the fitting procedure. Clarification of the independence of the test is required.
  3. [Numerical tests] Numerical tests: the evidence is restricted to two specific regularized states (icosahedron and fuzzy sphere). No general argument or additional tests are given to support applicability to arbitrary quantum critical wavefunctions in any dimension, rendering the extrapolation from these examples untested.
minor comments (1)
  1. [Abstract] The abstract would benefit from a brief statement of the quantitative criteria used to judge agreement between spectra.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive report. We address each major comment below and will revise the manuscript to strengthen the presentation where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that numerical tests demonstrate success is load-bearing for the central claim, yet no quantitative measures (overlaps, distances, error bars, or fit metrics) are supplied to substantiate 'close' or 'matches'. Without these, the strength of the evidence cannot be assessed.

    Authors: We agree that quantitative measures are needed to substantiate the claims. In the revised manuscript we will report explicit values: the overlap between each input wavefunction and the ground state of its reconstructed Hamiltonian, the Frobenius distance between the reconstructed Hamiltonian and the original regularized Hamiltonian, and the root-mean-square deviation of the lowest ten energies from known CFT values, together with error bars obtained from the regularization parameters. revision: yes

  2. Referee: [Reconstruction procedure] Reconstruction procedure: because the Hamiltonian is defined directly from the input wavefunction, it is unclear whether the reported spectral match on the unit sphere constitutes an independent check or is partly tautological with the fitting procedure. Clarification of the independence of the test is required.

    Authors: The reconstruction defines the Hamiltonian from the wavefunction so that the input state is approximately its ground state by construction. The spectral test, however, is performed by diagonalizing the same reconstructed Hamiltonian in the distinct Hilbert space of the unit-sphere regularization. Neither the integer spacing of descendants nor the numerical agreement with known CFT energies is imposed during reconstruction; both emerge as predictions. We will add an explicit paragraph in Section 3 clarifying this separation of the reconstruction step from the unit-sphere diagonalization. revision: yes

  3. Referee: [Numerical tests] Numerical tests: the evidence is restricted to two specific regularized states (icosahedron and fuzzy sphere). No general argument or additional tests are given to support applicability to arbitrary quantum critical wavefunctions in any dimension, rendering the extrapolation from these examples untested.

    Authors: The construction itself is dimension-independent and follows the same entanglement-bootstrap logic used in 1+1d and 2+1d. The two regularizations tested are representative of different geometries and cutoff schemes. While we do not claim a rigorous universality proof, the success across these cases supports the method's broader applicability. In revision we will add a paragraph in the discussion section that explicitly frames the results as a proof-of-principle demonstration and notes that further tests on other wavefunctions would be valuable. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes a reconstruction procedure for a Hamiltonian from a given critical wavefunction and reports numerical tests showing that the input states are close to its ground state while the reconstructed spectrum exhibits CFT-like features on two specific regularized examples. These tests are framed as empirical validation rather than tautological consequences of the definition. No load-bearing self-citations, uniqueness theorems imported from prior author work, or explicit reductions where a 'prediction' equals a fitted input by construction are identifiable from the abstract or description. The central claim rests on numerical evidence for the tested cases, which remains independent of the reconstruction definition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The proposal rests on the existence of prior reconstruction procedures in 1+1d and 2+1d that can be generalized; no explicit free parameters or invented entities are named in the abstract.

pith-pipeline@v0.9.1-grok · 5651 in / 1162 out tokens · 15632 ms · 2026-06-27T08:48:56.304437+00:00 · methodology

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Reference graph

Works this paper leans on

85 extracted references · 13 linked inside Pith

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    The simplest such objects are the platonic solids, among which we focus on the icosahedron

    ICOSAHEDRON We wish to study a quantum critical spin model on a discretization of the 2-sphere that preserves as much as possible of theSO(3) symmetry. The simplest such objects are the platonic solids, among which we focus on the icosahedron. The icosahedral group is a large subgroup ofSO(3) in the sense that the dimensions of its representations are lar...

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    SYSTEMATIC IMPROVEMENT Suppose we are given the numerical groundstate of an arbitrary quantum critical Hamiltonian (perhaps the one people like best) as a finite-size approximation to a CFT groundstate. As we have seen in the examples above, the VFPE will have a finite error. By doing gradient descent in the space of states (recall that this space is comp...

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