3d Ising Field Theory with Magnetic Deformation: Fuzzy Sphere Meets TCSA

A. Liam Fitzpatrick; Emanuel Katz; Giulia Fardelli; Yuan Xin

REVIEW

3d Ising Field Theory with Magnetic Deformation: Fuzzy Sphere Meets TCSA

Giulia Fardelli, A. Liam Fitzpatrick, Emanuel Katz, Yuan Xin

High Energy Physics - Theory · Submitted 9 Sep 2026 · 29 references

Pith's verdict: The paper gives solid nonperturbative numbers for the 3d Ising field theory mass gap and vacuum energy, but the bound-state claim rests on limited extrapolation data and should be treated as suggestive rather than settled.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv ID2609.10670 (version 1)

Pith IDWAATH4LX, Pith's permanent record of this paper and its review

CitationA. Liam Fitzpatrick, Emanuel Katz, Giulia Fardelli, Yuan Xin. "3d Ising Field Theory with Magnetic Deformation: Fuzzy Sphere Meets TCSA." arXiv:2609.10670 (2026). https://pith.science/paper/WAATH4LX

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Claim paper

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Editor's note

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Let me give you a direct take. The paper has two clear strengths. First, it produces the first nonperturbative infinite-volume numbers for the 3d Ising Field Theory with magnetic deformation: the vacuum energy density and the mass gap. The combination of Fuzzy Sphere and TCSA, with careful 1/N and curvature fits, is sensible. The cross-checks across angular momentum sectors add confidence. The result m_gap ≈ 1.65 μ and the relation ⟨σ⟩ = −1.28 m_gap^{Δ_σ} are usable benchmarks.

The weaker part is the bound state claim m_BS/m_gap ≈ 1.96. The evidence comes entirely from Exact Diagonalization at N ≤ 16, with a 1/N extrapolation from four points (N=10,12,14,16) and a curvature fit using a single 1/μ^2 term. DMRG, which is applied to the vacuum and gap up to N=24, is not used here because accessing higher scalar states is harder. The ratio is close to the two-particle threshold 2, so a small systematic bias could push it above threshold and eliminate the bound state interpretation. The authors acknowledge this, but they do not quantify the uncertainty from the limited N range or test the stability of the extrapolation to a higher-order polynomial. The result is plausible, but not at the same level of confidence as the vacuum and gap.

Who is this for? People working on numerical methods for QFT, the 3d Ising model, or the fuzzy sphere will get value. The vacuum and gap numbers are likely correct and worth citing. The bound state is interesting but needs further confirmation.

Final recommendation: this paper deserves a serious referee. The central claims about the vacuum and gap are sound; the bound state needs a more honest error budget or a caveat-heavy discussion. I would accept a peer-review invitation, and I would cite the gap and vacuum numbers. For a reading group, I'd put it on the maybe pile — the method discussion is good but the bound state uncertainty limits the punchline.

Referee report

Major objections (3)

[§5.1, Fig. 9 and surrounding text] The claim of a bound state with m_BS/m_gap ≈ 1.96 rests entirely on Exact Diagonalization data up to N=16. The infinite-N extrapolation uses a quadratic fit in 1/N to only four points (N=10,12,14,16). No error or stability estimate is given for this extrapolation, and the ratio is dangerously close to the two-particle threshold at 2. Since DMRG is used successfully for the vacuum and mass gap up to N=24, the authors should either provide a similar analysis for the bound state (or explain convincingly why this is computationally prohibitive) and quantify the systematic error from the limited N range—e.g., by showing the sensitivity to the fit ansatz or by including results from alternative extrapolation forms.
[§4, Eq. (4.5) and Eq. (4.7)] The curvature corrections are modelled only to leading order in 1/μ². Given that the extraction of infinite-volume quantities (especially the bound-state mass) uses this leading term for data points that extend only to moderate μR values (see Fig. 7–9), it is important to assess whether subleading 1/μ⁴ corrections could bias the extrapolated constants A0, A1 and the bound-state mass ratio. The authors should either test the stability of the fit to adding a 1/μ⁴ term, or provide an estimate of the truncation error using the EFT scaling arguments sketched in §4.
[§6, Fig. 19–21 and Appendix A.3] The TCSA comparison relies on CFT data (dimensions and OPE coefficients) obtained partly from the same Fuzzy Sphere setup in prior work [7]. Although the OPE coefficients are also checked against the conformal bootstrap, the fact that the same FS data are used both to calibrate the TCSA Hamiltonian and as a benchmark against TCSA results introduces some circularity. The authors should clarify which OPE coefficients come from independent bootstrap data and which from FS, and discuss to what extent the agreement in Figs. 16–18 is a genuine cross-check versus a self-consistency test.

Minor objections (5)

[§1, page 3] Equation (1.3): the definition μ ≡ h_z^{1/(3-Δ_σ)} appears earlier in the text but is not repeated near (1.3); including it there would improve readability.
[§5.1, Fig. 7 caption] The caption mentions 'large N extrapolation vs fit to curvature corrections' but does not specify the fitting range in h_z used for the solid line. The range (1 < h_z < 4) is given in the text but should also appear in the caption for clarity.
[§5.3, page 22, first paragraph] The text refers to 'the CFT operator ϵ′′ and descendants of ϵ and ϵ′' but ϵ′′ is not listed in Table 1; only ϵ, ϵ′, ϵ′′, ϵ′′′ appear. Please check the notation consistency.
[§6, Fig. 16] The TCSA data in Fig. 16 are shown only for h_z up to 8, but the text mentions fitting in the range 0.7 < h_z < 2.7. The reason for the upper cutoff (2.7) should be stated briefly (e.g., truncation effects become significant beyond this point) to avoid the impression that the fit range is arbitrary.
[Appendix A.3, Table 2] The OPE coefficients list 'bold' for bootstrap data, but the legend in the table footnote uses 'shown in bold' while the table itself appears to use normal font throughout. Please verify formatting so that the source of each coefficient is clear.

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The paper

The paper claims

The 3d Ising field theory has a light bound state and its mass gap is now known numerically.

The reading

This paper extracts universal, infinite-volume properties of the (2+1)-dimensional Ising Field Theory (IFT) with a magnetic deformation, using two complementary numerical methods on a spatial sphere. The authors compute the mass gap relating it to the coupling by $m_{\text{gap}} = 1.65\,h_z^{1/(3-\Delta_\sigma)}$ and find a bound state with mass $m_{\text{BS}} / m_{\text{gap}} \approx 1.96$, just below the two-particle threshold. They also determine the vacuum energy density and demonstrate consistency between the Fuzzy Sphere regulator and the Truncated Conformal Space Approach (TCSA), with curvature corrections modelled by a local effective field theory. The work establishes a reliable nonperturbative benchmark for the simplest gapped deformation of the 3d Ising CFT.

What carries the argument

The central machinery is the combination of the Fuzzy Sphere (a UV completion of non-relativistic fermions on a sphere that preserves rotational symmetry) and the Truncated Conformal Space Approach (TCSA, which builds the IFT Hamiltonian directly from Ising CFT data and truncates the Hilbert space by conformal dimension). Both are used on a spatial sphere of radius $R$, and the finite-volume effects are removed by modelling them with a local effective field theory that includes a cosmological constant term and a $1/\mu^2$ curvature correction for the vacuum energy and mass gap, following Eqs. (4.5) and (4.7). The CFT data (OPE coefficients and scaling dimensions) come from conformal bootstrap [23–25] and from earlier Fuzzy Sphere extractions [7].

Extended reading notes

Core claim

The central discovery is that the 3d Ising Field Theory with only a magnetic $\sigma$ deformation contains a stable bound state with a small binding energy and that the infinite-volume mass gap, vacuum energy, and bound-state mass are universal numbers that can be extracted from finite-sphere data by fitting curvature corrections $\propto 1/\mu^2$. Specifically, after canonically normalizing $\sigma$, the vacuum energy is $E_{\text{vac}} \approx -1.37\,h_z^{3/(3-\Delta_\sigma)}$ (density $\approx -0.109\,\mu^3$), the mass gap is $m_{\text{gap}} \approx 1.65\,\mu$, and the bound-state mass ratio is $m_{\text{BS}}/m_{\text{gap}} \approx 1.96$, where $\mu = h_z^{1/(3-\Delta_\sigma)}$. These results are obtained by modelling the sphere's curvature as a small local perturbation and extrapolating the Fuzzy Sphere data to infinite $N$ and infinite volume, with consistency checks from spinning sectors and TCSA.

Load-bearing premise

The infinite-volume extraction assumes that curvature corrections to energy levels at large $\mu R$ are dominated by a single local term proportional to $1/\mu^2$; if higher-order curvature terms (e.g., $1/\mu^4$) are significant, the extrapolated values of $A_0$, $A_1$, and the bound-state mass could be biased.

Editorial extensions

If this is right

The universal numbers $m_{\text{gap}}/\mu \approx 1.65$ and $m_{\text{BS}}/m_{\text{gap}} \approx 1.96$ provide concrete nonperturbative predictions that can be tested by other methods, such as lattice Monte Carlo or the $\epsilon$-expansion after resummation.
The same curvature-modelling strategy can be applied to extract infinite-volume quantities for other gapped deformations of 3d CFTs, including those with both $\epsilon$ and $\sigma$ turned on, or deformations of gauge-theory CFTs.
The agreement between Fuzzy Sphere and TCSA validates both approaches for studying non-integrable QFTs defined as CFTs plus relevant deformations, establishing TCSA as a viable method in $d>2$.
The existence of a weakly-bound state suggests that the 3d IFT, like its 2d cousin, may host a rich spectrum of resonances and that some effective description (e.g., a cubic scalar theory) might capture the low-energy physics.
The spinning-sector analysis provides an independent determination of $m_{\text{gap}}$ and $E_{\text{vac}}$ that agrees with the scalar-sector extraction, confirming Lorentz invariance restoration at large volume.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the bound state is as weakly bound as measured ($\approx 0.04 m_{\text{gap}}$ binding energy), the effective range of the two-particle interaction is large, which may make the binding energy accessible to a Schr\"odinger-equation analysis with a known potential derived from the $\sigma$ four-point function.
The TCSA truncation appears to converge in steps controlled by the inclusion of new scalar primaries; this suggests that systematic TCSA extrapolations in $d=3$ may require a complete set of OPE coefficients for all primaries up to a given dimension, motivating further bootstrap or Fuzzy Sphere determinations of missing operators (e.g., a potential $\Delta \approx 9$ primary).
The fact that the curvature correction to the mass gap is negative ($-0.25/\mu^2$) and relatively large at accessible volumes implies that finite-volume studies of other theories using the same EFT ansatz must carefully check the stability of the $1/\mu^2$ fit against higher-order terms, especially when the gap is not large compared to $1/R$.
The bound-state ratio of approximately 1.96 is tantalizingly close to the golden ratio $\phi \approx 1.618$ or $2\cos(\pi/5)$, which appears in the 2d E8 spectrum; while no such exact relation is claimed here, the coincidence invites a search for hidden integrability or an approximate algebraic structure in the 3d theory.

Original abstract

We study the magnetic deformation of the $(2+1)$d Ising CFT, also known as Ising Field Theory (IFT), on a spatial sphere. We use the Fuzzy Sphere (FS) Ising setup and compare it to the Truncated Conformal Space Approach (TCSA). Universal, infinite volume, IFT quantities are extracted by modeling the effects of curvature, as expected from a local effective theory on the sphere. We find evidence that IFT contains a bound-state with a small binding energy. Locality also allows us to extract the same IFT quantities from different angular momentum sectors, providing additional consistency checks.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,198 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.35 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the magnetic deformation of the $(2+1)$d Ising CFT, also known as Ising Field Theory (IFT), on a spatial sphere. We use the Fuzzy Sphere (FS) Ising setup and compare it to the Truncated Conformal Space Approach (TCSA). Universal, infinite volume, IFT quantities are extracted by modeling the effects of curvature, as expected from a local effective theory on the sphere. We find evidence that IFT contains a bound-state with a small binding energy. Locality also allows us to extract the same IFT quantities from different angular momentum sectors, providing additional consistency checks.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the magnetic deformation of the $(2+1)$d Ising CFT, also known as Ising Field Theory (IFT), on a spatial sphere.	contribution 0.95	0.91	0.04	asserted 0.95
s1	We use the Fuzzy Sphere (FS) Ising setup and compare it to the Truncated Conformal Space Approach (TCSA).	method 0.98	0.93	0.04	asserted 0.98
s2	Universal, infinite volume, IFT quantities are extracted by modeling the effects of curvature, as expected from a local effective theory on the sphere.	method 0.60	0.79	0.18	asserted 0.97
s3	We find evidence that IFT contains a bound-state with a small binding energy.	result 1.00	0.95	0.46	hedged possibility 0.51
s4	Locality also allows us to extract the same IFT quantities from different angular momentum sectors, providing additional consistency checks.	method 0.32	0.89	0.09	asserted 0.94

What kind of paper the classifier reports this is: simulation study 0.97 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (29)

[7]

G. Fardelli, A. L. Fitzpatrick and E. Katz,Improving 3d Ising OPE Coefficients with Fuzzy Sphere Conformal Generators,2602.04958

View on Pith arXiv
[1]

S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi,Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents,J. Stat. Phys.157(2014) 869, [1403.4545]

View on Pith Pith review arXiv 2014
[2]

A. B. Zamolodchikov,Integrable field theory from conformal field theory,Adv. Stud. Pure Math. 19(1989) 641–674

View on Pith 1989
[3]

V. P. Yurov and A. B. Zamolodchikov,TRUNCATED CONFORMAL SPACE APPROACH TO SCALING LEE-YANG MODEL,Int. J. Mod. Phys. A5(1990) 3221–3246

View on Pith 1990
[4]

M. Hogervorst, S. Rychkov and B. C. van Rees,Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?,Phys. Rev. D91(2015) 025005, [1409.1581]

View on Pith Pith review arXiv 2015
[5]

Y.-C. He and W. Zhu,A Fuzzy Sphere Journey in Critical Phenomena,Ann. Rev. Condensed Matter Phys.17(2026) 1–25, [2607.01310]

View on Pith Pith review arXiv 2026
[6]

W. Zhu, C. Han, E. Huffman, J. S. Hofmann and Y.-C. He,Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization,Phys. Rev. X13(2023) 021009, [2210.13482]

View on Pith Pith review arXiv 2023
[8]

A. M. Läuchli, L. Herviou, P. H. Wilhelm and S. Rychkov,Exact diagonalization, matrix product states and conformal perturbation theory study of a 3D Ising fuzzy sphere model, SciPost Phys.19(2025) 076, [2504.00842]

View on Pith arXiv 2025
[9]

M. Caselle, M. Hasenbusch and P. Provero,Nonperturbative states in the 3-D phi**4 theory, Nucl. Phys. B556(1999) 575–600, [hep-lat/9903011]

View on Pith Pith review arXiv 1999
[10]

M. Caselle, M. Hasenbusch, P. Provero and K. Zarembo,Bound states and glueballs in three-dimensional Ising systems,Nucl. Phys. B623(2002) 474–492, [hep-th/0103130]

View on Pith Pith review arXiv 2002
[11]

J. Taylor, M. Yusuf and Z. Papić,Fuzzy Spectroscopy of Bound States in Massive Quantum Field Theories,2608.07655

View on Pith Pith review arXiv
[12]

T. T. Wu,Theory of toeplitz determinants and the spin correlations of the two-dimensional ising model. i,Physical Review149(1966) 380

View on Pith 1966
[13]

V. Fateev,The exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories,Physics Letters B324(1994) 45–51

View on Pith 1994
[14]

V. P. Yurov and A. B. Zamolodchikov,Truncated fermionic space approach to the critical 2-D Ising model with magnetic field,Int. J. Mod. Phys. A6(1991) 4557–4578

View on Pith 1991
[15]

P. Fonseca and A. Zamolodchikov,Ising spectroscopy. I. Mesons at T<T(c),hep-th/0612304

View on Pith Pith review arXiv
[16]

A. Zamolodchikov,Ising Spectroscopy II: Particles and poles atT >Tc,1310.4821

View on Pith Pith review arXiv
[17]

B. Gabai and X. Yin,On the S-matrix of Ising field theory in two dimensions,JHEP10(2022) 168, [1905.00710]

View on Pith Pith review arXiv 2022
[18]

E. Brezin and D. J. Wallace,CRITICAL BEHAVIOR OF A CLASSICAL HEISENBERG FERROMAGNET WITH MANY DEGREES OF FREEDOM,Phys. Rev. B7(1973) 1967. – 37 –

View on Pith 1973
[19]

G. Fardelli, A. L. Fitzpatrick and E. Katz,Constructing the Infrared Conformal Generators on the Fuzzy Sphere,2409.02998

View on Pith Pith review arXiv
[20]

L. Hu, Y.-C. He and W. Zhu,Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Spheres,Phys. Rev. Lett.131(2023) 031601, [2303.08844]

View on Pith Pith review arXiv 2023
[21]

Z. Zhou,FuzzifiED : Julia Package for Numerics on the Fuzzy Sphere,2503.00100

View on Pith arXiv
[22]

M. S. Costa, J. Penedones, D. Poland and S. Rychkov,Spinning Conformal Correlators,JHEP 11(2011) 071, [1107.3554]

View on Pith Pith review arXiv 2011
[23]

D. Simmons-Duffin,The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT,JHEP03 (2017) 086, [1612.08471]

View on Pith Pith review arXiv 2017
[24]

C.-H. Chang, V. Dommes, R. S. Erramilli, A. Homrich, P. Kravchuk, A. Liu et al., Bootstrapping the 3d Ising stress tensor,JHEP03(2025) 136, [2411.15300]

View on Pith Pith review arXiv 2025
[25]

M. Reehorst,Rigorous bounds on irrelevant operators in the 3d Ising model CFT,JHEP09 (2022) 177, [2111.12093]

View on Pith Pith review arXiv 2022
[26]

J. Elias-Miro, S. Rychkov and L. G. Vitale,High-Precision Calculations in Strongly Coupled Quantum Field Theory with Next-to-Leading-Order Renormalized Hamiltonian Truncation, JHEP10(2017) 213, [1706.06121]

View on Pith Pith review arXiv 2017
[27]

J. Elias-Miro, S. Rychkov and L. G. Vitale,NLO Renormalization in the Hamiltonian Truncation,Phys. Rev. D96(2017) 065024, [1706.09929]

View on Pith Pith review arXiv 2017
[28]

J. Henriksson, S. R. Kousvos and J. Roosmale Nepveu,EFT meets CFT: multiloop renormalization of higher-dimensional operators in generalϕ4 theories,JHEP05(2026) 005, [2511.16740]

View on Pith arXiv 2026
[29]

E. Huffman, Z. Zhou, Y.-C. He and J. S. Hofmann,Generalizing deconfined criticality to 3D N-flavor SU(2) quantum chromodynamics on the fuzzy sphere,Phys. Rev. D114(2026) 034503, [2602.11255]. – 38 –

View on Pith Pith review arXiv 2026

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Effective Field Theory Origin of the Fuzzy Sphere Method
hep-th 2026-10 conditional novelty 8.0 of 10

Fuzzy sphere Hamiltonians descend from local mother theories on S^2 x R, and a new easy-plane descendant has an exactly solvable one-magnon sector and a z=2 critical point.
Matrix-vector models for the 3D Ising CFT
hep-th 2026-11 conditional novelty 6.0 of 10

A class of matrix-vector and matrix models derived from the fuzzy-sphere regularization of the 3D Ising CFT is shown to reproduce the CFT spectrum for a range of a Gaussian deformation parameter.

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [7]

G. Fardelli, A. L. Fitzpatrick and E. Katz,Improving 3d Ising OPE Coefficients with Fuzzy Sphere Conformal Generators,2602.04958

View on Pith arXiv

[2] [1]

S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi,Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents,J. Stat. Phys.157(2014) 869, [1403.4545]

View on Pith Pith review arXiv 2014

[3] [2]

A. B. Zamolodchikov,Integrable field theory from conformal field theory,Adv. Stud. Pure Math. 19(1989) 641–674

View on Pith 1989

[4] [3]

V. P. Yurov and A. B. Zamolodchikov,TRUNCATED CONFORMAL SPACE APPROACH TO SCALING LEE-YANG MODEL,Int. J. Mod. Phys. A5(1990) 3221–3246

View on Pith 1990

[5] [4]

M. Hogervorst, S. Rychkov and B. C. van Rees,Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?,Phys. Rev. D91(2015) 025005, [1409.1581]

View on Pith Pith review arXiv 2015

[6] [5]

Y.-C. He and W. Zhu,A Fuzzy Sphere Journey in Critical Phenomena,Ann. Rev. Condensed Matter Phys.17(2026) 1–25, [2607.01310]

View on Pith Pith review arXiv 2026

[7] [6]

W. Zhu, C. Han, E. Huffman, J. S. Hofmann and Y.-C. He,Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization,Phys. Rev. X13(2023) 021009, [2210.13482]

View on Pith Pith review arXiv 2023

[8] [8]

A. M. Läuchli, L. Herviou, P. H. Wilhelm and S. Rychkov,Exact diagonalization, matrix product states and conformal perturbation theory study of a 3D Ising fuzzy sphere model, SciPost Phys.19(2025) 076, [2504.00842]

View on Pith arXiv 2025

[9] [9]

M. Caselle, M. Hasenbusch and P. Provero,Nonperturbative states in the 3-D phi**4 theory, Nucl. Phys. B556(1999) 575–600, [hep-lat/9903011]

View on Pith Pith review arXiv 1999

[10] [10]

M. Caselle, M. Hasenbusch, P. Provero and K. Zarembo,Bound states and glueballs in three-dimensional Ising systems,Nucl. Phys. B623(2002) 474–492, [hep-th/0103130]

View on Pith Pith review arXiv 2002

[11] [11]

J. Taylor, M. Yusuf and Z. Papić,Fuzzy Spectroscopy of Bound States in Massive Quantum Field Theories,2608.07655

View on Pith Pith review arXiv

[12] [12]

T. T. Wu,Theory of toeplitz determinants and the spin correlations of the two-dimensional ising model. i,Physical Review149(1966) 380

View on Pith 1966

[13] [13]

V. Fateev,The exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories,Physics Letters B324(1994) 45–51

View on Pith 1994

[14] [14]

V. P. Yurov and A. B. Zamolodchikov,Truncated fermionic space approach to the critical 2-D Ising model with magnetic field,Int. J. Mod. Phys. A6(1991) 4557–4578

View on Pith 1991

[15] [15]

P. Fonseca and A. Zamolodchikov,Ising spectroscopy. I. Mesons at T<T(c),hep-th/0612304

View on Pith Pith review arXiv

[16] [16]

A. Zamolodchikov,Ising Spectroscopy II: Particles and poles atT >Tc,1310.4821

View on Pith Pith review arXiv

[17] [17]

B. Gabai and X. Yin,On the S-matrix of Ising field theory in two dimensions,JHEP10(2022) 168, [1905.00710]

View on Pith Pith review arXiv 2022

[18] [18]

E. Brezin and D. J. Wallace,CRITICAL BEHAVIOR OF A CLASSICAL HEISENBERG FERROMAGNET WITH MANY DEGREES OF FREEDOM,Phys. Rev. B7(1973) 1967. – 37 –

View on Pith 1973

[19] [19]

G. Fardelli, A. L. Fitzpatrick and E. Katz,Constructing the Infrared Conformal Generators on the Fuzzy Sphere,2409.02998

View on Pith Pith review arXiv

[20] [20]

L. Hu, Y.-C. He and W. Zhu,Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Spheres,Phys. Rev. Lett.131(2023) 031601, [2303.08844]

View on Pith Pith review arXiv 2023

[21] [21]

Z. Zhou,FuzzifiED : Julia Package for Numerics on the Fuzzy Sphere,2503.00100

View on Pith arXiv

[22] [22]

M. S. Costa, J. Penedones, D. Poland and S. Rychkov,Spinning Conformal Correlators,JHEP 11(2011) 071, [1107.3554]

View on Pith Pith review arXiv 2011

[23] [23]

D. Simmons-Duffin,The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT,JHEP03 (2017) 086, [1612.08471]

View on Pith Pith review arXiv 2017

[24] [24]

C.-H. Chang, V. Dommes, R. S. Erramilli, A. Homrich, P. Kravchuk, A. Liu et al., Bootstrapping the 3d Ising stress tensor,JHEP03(2025) 136, [2411.15300]

View on Pith Pith review arXiv 2025

[25] [25]

M. Reehorst,Rigorous bounds on irrelevant operators in the 3d Ising model CFT,JHEP09 (2022) 177, [2111.12093]

View on Pith Pith review arXiv 2022

[26] [26]

J. Elias-Miro, S. Rychkov and L. G. Vitale,High-Precision Calculations in Strongly Coupled Quantum Field Theory with Next-to-Leading-Order Renormalized Hamiltonian Truncation, JHEP10(2017) 213, [1706.06121]

View on Pith Pith review arXiv 2017

[27] [27]

J. Elias-Miro, S. Rychkov and L. G. Vitale,NLO Renormalization in the Hamiltonian Truncation,Phys. Rev. D96(2017) 065024, [1706.09929]

View on Pith Pith review arXiv 2017

[28] [28]

J. Henriksson, S. R. Kousvos and J. Roosmale Nepveu,EFT meets CFT: multiloop renormalization of higher-dimensional operators in generalϕ4 theories,JHEP05(2026) 005, [2511.16740]

View on Pith arXiv 2026

[29] [29]

E. Huffman, Z. Zhou, Y.-C. He and J. S. Hofmann,Generalizing deconfined criticality to 3D N-flavor SU(2) quantum chromodynamics on the fuzzy sphere,Phys. Rev. D114(2026) 034503, [2602.11255]. – 38 –

View on Pith Pith review arXiv 2026