Fuzzy Spectroscopy of Bound States in Massive Quantum Field Theories

Joseph Taylor; Matthew Yusuf; Zlatko Papi\'c

REVIEW 3 major objections 5 minor 106 references

Fuzzy Spectroscopy of Bound States in Massive Quantum Field Theories

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Fuzzy-geometry regularization reproduces the exact E8 meson spectrum of the Ising QFT and, in 2D, yields bound-state levels consistent with glueball estimates, tracking a meson branch continuously across the dimensional crossover.

desk verdict Thin-torus E8 benchmark is clean and externally verified; 2+1D spectroscopy is honest finite-volume work, not yet bulk glueball masses, but worth refereeing. read the letter →

arxiv 2608.07655 v1 pith:PS3R6F6W submitted 2026-08-07 cond-mat.stat-mech cond-mat.str-elhep-lathep-th

Joseph Taylor , Matthew Yusuf , Zlatko Papić This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.str-elhep-lathep-th

keywords fuzzygeometrylowestLandaulevelIsingquantumfieldtheoryE8symmetrybound-statespectroscopyglueballmassesdimensionalcrossoverquenchdynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish that fuzzy geometries — noncommutative surfaces realized as the lowest-Landau-level orbitals of a quantum Hall system — work as nonperturbative spectroscopic probes of massive, confining quantum field theories, not just of conformal critical points. On a thin fuzzy torus the method reproduces the exact $\mathbb{E}_8$ meson mass ratios of the magnetically perturbed $(1+1)\mathrm{D}$ Ising QFT to within about 0.5%, calibrating both the regularization and the finite-size scaling prescription. Increasing the torus aspect ratio then continuously tracks the second-lightest $\mathbb{E}_8$ meson as the system crosses over from 1D to 2D, where the same machinery yields a subthreshold scalar level at $m_2/m_1 \approx 1.83$ and above-threshold features consistent with earlier Ising and glueball estimates. If the claim holds, fuzzy geometry becomes a valid continuum regulator for gapped quantum field theories and a new window on how confined bound states behave as spatial dimensionality changes.

What carries the argument

The load-bearing object is a quantum-Hall realization of the Ising QFT: spinful electrons at filling factor $\nu = 1$ in the lowest Landau level, with the Ising coupling encoded in the Haldane pseudopotentials (the two-body interaction parameters) $(V_0, V_1) = (4,1)$ on the torus and $(4.75,1)$ on the sphere, and the thermal ($h_x$) and magnetic ($h_z$) perturbations as global Zeeman fields. Lowest-Landau-level projection truncates the Hilbert space to $N$ orbitals without any spatial lattice while preserving exact translation (torus) or rotation (sphere) symmetry at every finite size. The renormalization-group trajectory prescription holds $D = t L^{y_t}$ and $\mu = h L^{y_h}$ fixed as $N$ grows, so enlarging the system removes the ultraviolet cutoff while keeping the box only a few correlation lengths across; the dynamical structure factor $S^z(\omega, \mathbf{q}=0)$ then isolates the fully symmetric ($A_1$) sector carrying the scalar response. The thin-torus limit, aspect ratio $r = L_y/L_x \to 0$, collapses the guiding centers onto a one-dimensional ring and connects the 2D construction to the integrable $\mathbb{E}_8$ point.

What would settle it

The deciding test is whether the subthreshold scalar survives at larger size and in an independent calculation: on the same RG ray (magnetic critical isotherm, $D = 0$), a lattice Monte Carlo determination of the lowest scalar gap that comes out at $m_2/m_1 \approx 1.83$ would confirm the fuzzy result, while a gap at or above the two-particle threshold would mark it a finite-volume artifact. Within the paper's own method, the running-coupling criterion gives the same test: at $\mu = 150$ the running $\beta_\mu \approx 0.16$ for $N = 10$ through 18, so extending the exact diagonalization to larger $N$ should show whether the extrapolated ratio stabilizes below threshold or drifts up across it.

Watch

Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that a lowest-Landau-level regularization of spinful electrons at unit filling faithfully encodes the massive excitations of the Ising QFT along renormalization-group trajectories away from criticality — one construction covering both the exactly solvable (1+1)D theory and the nonintegrable (2+1)D theory. Projecting the Ising interaction into LLL orbitals on a torus or sphere gives a finite-dimensional Hamiltonian whose lowest gaps, extrapolated to $N \to \infty$ at fixed renormalized fields, reproduce the universal $\mathbb{E}_8$ ratios $m_2/m_1 = 2\cos(\pi/5)$, $m_3/m_1 = 2\cos(\pi/30)$, and $m_4/m_1 = 4\cos(\pi/5)\cos(7\pi/30)$ on the thin torus, and deliver a subthreshold scalar branch at $m_2/m_1 \approx 1.83$ on the square torus and sphere, with higher response peaks near the published $1.88(2)$, $2.59(4)$, and $3.24(16)$ glueball masses. The aspect-ratio sweep shows the second $\mathbb{E}_8$ branch staying below the two-particle threshold all the way from the thin limit to the isotropic square torus, while an adjacent branch crosses that threshold near $r \approx 0.2$, and quench dynamics recover the same excitation frequencies in the return fidelity. The authors present the 2D levels as continuum finite-volume results that establish the method, while explicitly leaving the infinite-volume limit open.

Load-bearing premise

The load-bearing premise is that the small finite-box spectra of the 2D calculation, extrapolated in $1/N$, already represent the true (2+1)D quantum field theory: the paper's own check shows the lowest mass ratio still drifting with the magnetic field at $N \le 18$ (SM Fig. S4(b)), and the Conclusions call reaching the infinite-volume limit a 'central unresolved problem'; if that premise fails, the claimed subthreshold level at $m_2/m_1 \approx 1.83$ would be a finite-size artifact rather than a real bound state.

Editorial extensions

If this is right

Fuzzy geometry is established as a continuum regulator for massive, gapped quantum field theories, not only for conformal fixed points: the $\mathbb{E}_8$ benchmark fixes both the regularization and the $1/N^2$ extrapolation prescription.
The (2+1)D Ising QFT on the magnetic critical isotherm has a stable scalar bound state at $m_2/m_1 \approx 1.83$, below the two-particle threshold, consistent with lattice estimates obtained along other RG directions.
The above-threshold features ($m_3/m_1 \approx 2.59$–$2.62$, $m_4/m_1 \approx 3.2$–$3.24$, and the composite $n m_1$ levels) are resonance candidates whose identity as stable particles or finite-width resonances the infinite-volume limit would decide.
The second-lightest $\mathbb{E}_8$ meson branch, with $m_2/m_1$ rising from 1.596 to 1.766, stays below the two-particle threshold across the entire 1D-to-2D crossover, while an adjacent branch enters the continuum near $r \approx 0.2$.
Return-fidelity quench spectra recover the same bound-state frequencies as the static dynamical structure factor on all three geometries, connecting the equilibrium spectroscopy to real-time dynamics.

Reading between the lines

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

If the subthreshold $m_2/m_1 \approx 1.83$ level does survive the infinite-volume limit, the same fixed-$(D,\mu)$ machinery could be run along the pure thermal axis — where most lattice glueball data already live — to test whether the bound-state hierarchy is a property of the gapped phase as a whole or specific to the magnetic isotherm, a comparison this paper makes only indirectly across differe
The paper's own running-coupling diagnostic at $\mu = 150$, $\beta_\mu \approx 0.16$ for $N = 10$–18, suggests a concrete stress test: pushing the calculation beyond exact-diagonalization sizes (e.g., with tensor-network or Krylov methods) should show whether the extrapolated subthreshold branch stabilizes below threshold or drifts toward and across $m_2/m_1 = 2$; the latter would mark the 'glueba
The pair-binding rule $m_{n m_1} \approx n - \binom{n}{2} \Delta_2$ with $\Delta_2 \approx 0.17$, which the paper presents only as a heuristic, makes a sharp prediction (2.49, 2.98, 3.30) for the composite levels; a deviation from this $\binom{n}{2}$ pattern in a future calculation would signal many-body binding effects beyond pairwise attraction.
Because the torus aspect ratio interpolates between an integrable 1D endpoint and a nonintegrable 2D endpoint at fixed spacetime dimension, the same sweep could serve as a general dimensional-interpolation device for other confining QFTs — a systematic analogue of $\epsilon$-expansion-style interpolation, with the anisotropy as the tunable parameter.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is worth your time. The thin-torus benchmark is the real deal: fuzzy LLL regularization reproduces the E8 mass ratios of the 1+1D Ising QFT within about 0.5% after 1/N^2 extrapolation, over a range of mu, and the authors map out where the scaling window closes. That is a solid, externally anchored calibration of the method.

What's genuinely new: they take fuzzy geometry from CFT spectroscopy to massive QFT bound states, use the torus aspect ratio to continuously interpolate between 1D and 2D, and show the lightest meson branch stays below threshold while its DSF residue remains nonzero across the crossover. The quench diagnostics are a nice complement, not a gimmick. The paper is also commendably frank about what is not settled.

The soft spots are concentrated in the 2+1D claims. At fixed mu=150 on the square torus, the running coupling beta_mu is still positive (about 0.16) and the system size remains only a few correlation lengths across; the N->infinity limit is a continuum finite-volume limit, not a bulk limit. The authors say this themselves in the Conclusions. So the extrapolated levels at m2/m1 ~1.83 and the higher features are finite-volume levels whose connection to infinite-volume glueball masses is plausible but not established. The literature benchmarks they compare against are mostly on thermal or mixed RG trajectories, not the magnetic critical isotherm, so the agreement could be partly coincidental. The pair model is a heuristic. These are not fatal—the paper labels the above-threshold states as 'resonance candidates' and flags the infinite-volume problem—but the abstract's "establish fuzzy geometries as nonperturbative spectroscopic probes" is stronger than what the 2+1D data supports. For 1+1D, the claim is solid; for 2+1D, it is a promising demonstration with honest caveats.

Who should read this: anyone working on fuzzy-sphere regularization, confinement in 2+1D, or numerical QFT spectroscopy. It deserves a serious referee; the main things I'd want are more system sizes or a refined finite-volume analysis, and a direct comparison on the same RG trajectory. Code/data release would also help. I'd recommend sending it to review.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a fuzzy-geometry regularization of the Ising QFT, realized by spinful electrons in the lowest Landau level on a torus or sphere, and uses exact diagonalization to extract low-lying mass ratios from the finite-size spectrum. On a thin torus (effectively 1+1D), the authors reproduce the exact E8 meson mass ratios to about 0.5% after 1/N^2 extrapolation. On the square torus and sphere, they identify a subthreshold scalar level at m2/m1 ~ 1.83 and several higher peaks, compare them with existing glueball and Ising bound-state estimates, and propose a pair model for composite levels. They also track the second E8 meson as the torus aspect ratio is varied, finding that it remains below threshold while the next branch enters the continuum. A quench-dynamics protocol corroborates the equilibrium gaps. The paper concludes that fuzzy geometries are nonperturbative spectroscopic probes of massive QFTs and observes a dimensional crossover of the bound-state spectrum.

Significance. The (1+1)D benchmark is the strongest part of the paper: the numerical extrapolation is clean, the scaling window is broad, and the agreement with exact E8 ratios calibrates the regularization and the finite-size scaling prescription. If the 2+1D extrapolations could be connected to the infinite-volume limit, the work would establish a genuinely new tool for massive QFT spectroscopy beyond CFTs and would provide a novel dimensional-crossover result. The present manuscript, however, stops short of that: as the authors state, the infinite-volume limit is unresolved, and the 2+1D results are finite-volume continuum levels whose consistency with bulk glueball masses is suggestive but not proven. The quench spectroscopy and the symmetry-filtered spectra are useful additions, and the Table I ratios are concrete falsifiable predictions for future lattice and continuum calculations.

major comments (3)

[SM, 'Approach to infinite volume in (2+1)D', Eqs. (S17)-(S18) and Fig. S4(b); Conclusions] The extrapolations in Fig. 3(c) remove the fuzzy UV regulator at fixed mu but do not approach the infinite-volume limit: Eq. (S17) gives m1 L = C1 mu^{1/y_h}, so L/xi remains finite, and Fig. S4(b) shows beta_mu = 0.161-0.166 at mu=150 for N=10-18, far from the |beta|<=0.03 plateau target. The paper's own conclusion states that reaching the infinite-volume limit is 'a central unresolved problem.' Consequently, Table I and Fig. 3(c) establish finite-volume continuum levels, not bulk QFT masses, and the abstract/conclusion claim that fuzzy geometries are nonperturbative spectroscopic probes of massive QFTs in 2+1D is not supported by the present evidence. The authors should either reframe the 2+1D results as finite-volume spectroscopy or provide a controlled route to the bulk limit, for example by demonstrating a beta_mu to 0 window at larger mu and N.
[Fig. 4 and SM, 'Trajectory for the dimensional crossover'] The crossover path fixes h_z=0.5 and tunes h_x along the finite-size critical line, which is ad hoc because the RG eigenvalues change with aspect ratio and no universal scaling variable is held fixed. The claim that the second E8 meson stays below threshold from 1D to 2D therefore depends on this particular path. To make the dimensional-crossover statement robust, the authors should demonstrate that the subthreshold behavior and the continuity of m2/m1 are independent of the chosen interpolation, for example by repeating the calculation at different h_z or with a different rescaling prescription.
[Sec. '(2+1)D spectroscopy' and Table I] The comparison with literature benchmarks is suggestive but not a same-trajectory validation: the quasiparticle estimates in Refs. [8,9,13,83] are obtained along thermal or mixed thermal-magnetic trajectories (see SM Table S2), whereas the present calculation sits on D=0. Without a benchmark on the D=0 magnetic critical isotherm at controlled finite volume, the agreement at m2/m1 ~ 1.83-1.88 cannot be distinguished from a finite-volume or trajectory-dependent effect, especially given the strong running shown in Fig. S4(b). The authors should anchor the D=0 extrapolations with a direct lattice or Monte Carlo calculation along the same trajectory, or at minimum quantify the expected trajectory dependence.

minor comments (5)

[Main text, Eq. (6) and Table I] The pair model predicts m_{n m1}/m1 = n - C(n,2) Delta2, but the 'pair' column in Table I lists values without uncertainties; specify how the rounding to two decimals was done and whether Delta2 was taken from the torus or sphere value.
[SM, Eq. (S5)] The Clebsch-Gordan coefficient is written as C^{JM}_{Q m1, s m2}; the symbol 's' appears to be a typo for the spin/orbital index, presumably Q m2. Please correct.
[Main text, Fig. 2(a) and surrounding text] The statement 'At this field, the tracked m4 branch is the third energy-ranked level above m3' is hard to parse; rephrase or add a small level-ordering diagram.
[End Matter, Fig. 5] The fidelity Fourier spectrum contains both DSF gaps and excited-state coherences; the text mentions this for the square torus but not for the thin torus and sphere. Please state explicitly in all three cases which peaks are DSF-matching and which are coherences.
[Abstract] The phrase 'we reproduce the universal low-lying E8 meson masses' overstates the result; the paper extracts ratios m2/m1, m3/m1, m4/m1, not the full set of eight masses. Consider rephrasing to 'mass ratios'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central E8 benchmark and 2+1D levels are anchored to external exact/lattice results, and the only self-referential heuristic is non-load-bearing and transparent.

full rationale

The paper's derivation chain is not circular. The (1+1)D benchmark is validated against the external exact E8 mass ratios of Zamolodchikov [3], with an explicit N-to-infinity extrapolation and a mu-window analysis that is not fitted to the target values (Fig. 2 and Table S1). The (2+1)D torus and sphere levels are obtained by exact diagonalization and independent 1/N extrapolations, then compared with external Monte Carlo and lattice estimates from Refs. [8,9,83,13]; the comparison numbers are not inputs to the calculation. The only internally self-referential element is the heuristic 'pair model' of Eq. (6), where Delta2 = 2 - m2/m1 is taken from the computed m2 level to estimate the n*m1 composite positions. This is transparently labeled a heuristic estimate, and the underlying 3m1, 4m1, and 5m1 levels are separately identified by the numerically computed DSF; the pair model is not used to define those levels, so no result is forced by construction. The paper explicitly discloses that reaching the infinite-volume limit is 'a central unresolved problem' and that on the square torus beta_mu at mu=150 has not reached the plateau (SM Fig. S4(b)); this is a correctness and extrapolation risk, not a circularity. Self-citations [43,44,48] appear only as background references and do not carry the load-bearing argument, whose key premises rest on external exact solutions, lattice estimates, and independently established fuzzy-sphere methodology. Overall, the central claims are externally benchmarked and the derivation is self-contained apart from a non-load-bearing heuristic.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central numerical results rest on the mapping between the LLL electron model and the Ising QFT, the assumed RG scaling variables, and the chosen extrapolation orders. Two parameters are chosen by hand or by flatness criteria: the Haldane pseudopotentials (from prior fuzzy models) and the working point mu (150 torus, 26 sphere). The pair-model Delta2 is fitted to the computed m2 level. No new entities are introduced. The 1+1D E8 benchmark provides strong independent validation of the model mapping, but the 2+1D extrapolation remains the least supported link.

free parameters (3)

Haldane pseudopotentials (V0,V1) = (4,1) torus; (4.75,1) sphere
Chosen by hand following prior fuzzy sphere/CFT models [20,22] to realize the Ising interaction; not fitted to the mass ratios, but the 2+1D continuum limit depends on them.
Optimal scaled magnetic field mu = 150 (square torus); 26 (sphere)
Selected by minimizing the discrete slope norm Phi(mu) of m2/m1 under thermal perturbation (SM Eq. S16); the extrapolated mass ratios depend on this choice, and the running beta_mu at this point is not small.
Pair-model binding energy Delta2 = 0.17 (2 - m2/m1)
Taken from the computed m2/m1 and used in Eq. (6) to estimate the higher nm1 composite levels; a heuristic fit, not an independent prediction.

assumptions (5)

domain assumption The LLL-projected Hamiltonian with Haldane pseudopotentials realizes the Ising CFT plus magnetic and thermal perturbations with the standard RG dimensions.
Used throughout the model definition (Eq. 4 and SM S2); validated in 1+1D by the E8 benchmark, but for the 2+1D massive regime the continuum limit is assumed.
domain assumption RG covariance: masses satisfy m_i(t,h)=|h|^{1/y_h} F_i(t/|h|^{y_t/y_h}) (Eq. 3), and holding D=tL^{y_t}, mu=hL^{y_h} fixed as L goes to infinity defines the continuum trajectory.
This scaling hypothesis underpins the finite-size analysis and the definition of the universal window in Fig. 1(c).
domain assumption The finite-size corrections have the stated order: 1/N^2 in 1+1D (Eq. S9) and 1/N plus 1/N^2 in 2+1D (Eq. S18).
The extrapolated mass ratios in Figs. 2 and 3 depend on these assumed correction orders; the SM does not provide a derivation of the correction exponent for the fuzzy torus.
ad hoc to paper The dimensional-crossover path (fixed h_z=0.5, h_x on the finite-size critical line at each r) defines a physically meaningful interpolation between the 1D and 2D QFTs.
Because y_h changes between the endpoints, no single dimensionless magnetic variable can be held fixed; the path is chosen for practical convenience (SM, trajectory section) and the branch tracking relies on overlap continuity.
domain assumption The A1 point-group filtering and the zero-momentum DSF select the scalar bound states that correspond to the Ising/glueball excitations.
The symmetry assignment (SM, point-group section) and DSF selection rule are used to identify the relevant levels in the dense spectrum.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Fuzzy Spectroscopy of Bound States in Massive Quantum Field Theories." pith.science (2026). https://pith.science/paper/PS3R6F6W

@misc{pith2026260807655,
  author       = {Pith},
  title        = {Pith review of: Fuzzy Spectroscopy of Bound States in Massive Quantum Field Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PS3R6F6W}},
  note         = {Machine review of arXiv:2608.07655}
}

abstract

Mesons and glueballs are paradigmatic bound states of confining quantum field theories (QFTs), but their nonperturbative spectroscopy in the continuum remains challenging beyond one spatial dimension. Here we perform such spectroscopy for the Ising QFT using a recently developed regularization based on noncommutative ``fuzzy'' geometry. On a thin fuzzy torus, we reproduce the universal low-lying $\mathbb E_8$ meson masses of the magnetically perturbed $(1{+}1)\mathrm{D}$ Ising QFT. By increasing the torus aspect ratio, we continuously track the second-lightest $\mathbb E_8$ meson as the system effectively crosses over from 1D to 2D. On the 2D fuzzy torus and sphere, we find a subthreshold scalar level and above-threshold response features consistent with previous estimates of glueball masses. The same excitations are revealed away from equilibrium using quench dynamics. Our results establish fuzzy geometries as nonperturbative spectroscopic probes of massive QFTs, including their bound-state evolution through dimensional crossover.

Figures

Figures reproduced from arXiv: 2608.07655 by the authors.

**Figure 2.** FIG. 2. (1 + 1) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** (b) plots the weights Z (z) n at µ = 150 for the same system sizes as in panel (a). Because both the ground state and Sz transform in the fully symmetric A1 representation of the square point group [82], every displayed pole belongs to the A1 sector. Across the accessible N, the DSF isolates a subthreshold scalar level at m2/m1 ≃ 1.8 together with higher DSF peaks above the continuum threshold [PITH_FULL_IMAGE:fig… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: (a) shows the resulting 1D-to-2D spectral evolution which tracks the E8 branches m1, m2 and m3 based on the overlap between eigenvectors at the neighboring values of the aspect ratio. The tracked m2/m1 ratio 0.2 0.4 0.6 0.8 1.0 r 1.5 2.0 2.5 3.0 (En − E0)/m1 m2 m3 (a)…

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

106 extracted references · 58 canonical work pages

[1]

Sachdev,Quantum Phase Transitions, 2nd ed

S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cam- bridge University Press, 2011)

2011
[2]

Mussardo,Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford Graduate Texts (OUP Oxford, 2020)

G. Mussardo,Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford Graduate Texts (OUP Oxford, 2020)

2020
[3]

A. B. Zamolodchikov, Integrals of Motion and S Matrix of the (Scaled)T=T c Ising Model with Magnetic Field, Int. J. Mod. Phys. A4, 4235 (1989)

1989
[4]

A. M. Perelomov, Remarks on the mass spectrum of two-dimensional Toda lattice ofE 8 type, Journal of Nonlinear Mathematical Physics27, 12 (2020)

2020
[5]

Coldea, D

R. Coldea, D. Tennant, E. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Quantum Criticality in an Ising chain: Experimental Evidence for EmergentE 8 symmetry, Sci- ence327, 177 (2010)

2010
[6]

Zhang, K

Z. Zhang, K. Amelin, X. Wang, H. Zou, J. Yang, U. Nagel, T. R˜ o om, T. Dey, A. A. Nugroho, T. Lorenz, J. Wu, and Z. Wang, Observation ofE 8 particles in an Ising chain antiferromagnet, Phys. Rev. B101, 220411(R) (2020)

2020
[7]

H. Zou, Y. Cui, X. Wang, Z. Zhang, J. Yang, G. Xu, A. Okutani, M. Hagiwara, M. Matsuda, G. Wang, G. Mussardo, K. H´ ods´ agi, M. Kormos, Z. He, S. Kimura, R. Yu, W. Yu, J. Ma, and J. Wu,E 8 Spectra of Quasi- One-Dimensional Antiferromagnet BaCo 2V2O8 under Transverse Field, Phys. Rev. Lett.127, 077201 (2021)

2021
[8]

Caselle, M

M. Caselle, M. Hasenbusch, and P. Provero, Non- perturbative states in the three-dimensionalϕ 4 theory, Nuclear Physics B - Proceedings Supplements83-84, 715 (2000), proceedings of the XVIIth International Symposium on Lattice Field Theory

2000

Show all 106 references

[9]

Caselle, M

M. Caselle, M. Hasenbusch, P. Provero, and K. Zarembo, Bound states and glueballs in three- dimensional Ising systems, Nuclear Physics B623, 474–492 (2002)

2002
[10]

Nishiyama, Bound-state energy of the three- dimensional Ising model in the broken-symmetry phase: Suppressed finite-size corrections, Phys

Y. Nishiyama, Bound-state energy of the three- dimensional Ising model in the broken-symmetry phase: Suppressed finite-size corrections, Phys. Rev. E77, 051112 (2008)

2008
[11]

Y. Nishiyama, Criticality of the (2+1)-dimensional S=1 transverse-field Ising model with extended interactions: Suppression of corrections to scaling, Nuclear Physics B 832, 605 (2010)

2010
[12]

Y. Nishiyama, Universal critical behavior of the two-magnon-bound-state mass gap for the (2+1)- dimensional Ising model, Physica A: Statistical Mechan- ics and its Applications413, 577 (2014)

2014
[13]

Y. Nishiyama, Magnon-bound-state hierarchy for the two-dimensional transverse-field Ising model in the or- dered phase, Physica A: Statistical Mechanics and its Applications463, 303 (2016)

2016
[14]

Caselle, M

M. Caselle, M. Hasenbusch, P. Provero, and K. Zarembo, Bound states in the 3d Ising model and implications for QCD at finite temperature and density, Nuclear Physics B-Proceedings Supplements 106, 504 (2002)

2002
[15]

F. J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters, Jour- nal of Mathematical Physics12, 2259 (1971)

1971
[16]

Fradkin,Quantum field theory: an integrated ap- proach(Princeton University Press, 2021)

E. Fradkin,Quantum field theory: an integrated ap- proach(Princeton University Press, 2021)

2021
[17]

Madore, The fuzzy sphere, Classical and Quantum Gravity9, 69 (1992)

J. Madore, The fuzzy sphere, Classical and Quantum Gravity9, 69 (1992)

1992
[18]

M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys.73, 977 (2001)

2001
[19]

Ippoliti, R

M. Ippoliti, R. S. K. Mong, F. F. Assaad, and M. P. Za- letel, Half-filled Landau levels: A continuum and sign- free regularization for three-dimensional quantum criti- cal points, Phys. Rev. B98, 235108 (2018)

2018
[20]

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. X 13, 021009 (2023)

2023
[21]

He and W

Y.-C. He and W. Zhu, A fuzzy sphere journey in crit- ical phenomena, Annual Review of Condensed Matter Physics17, 1 (2026)

2026
[22]

F. D. M. Haldane, Fractional quantization of the Hall effect: A Hierarchy of incompressible quantum fluid states, Phys. Rev. Lett.51, 605 (1983)

1983
[23]

F. D. M. Haldane, Many-Particle Translational Symme- 6 tries of Two-Dimensional Electrons at Rational Landau- Level Filling, Phys. Rev. Lett.55, 2095 (1985)

1985
[24]

C. Han, L. Hu, and W. Zhu, Conformal operator content of the Wilson-Fisher transition on fuzzy sphere bilayers, Phys. Rev. B110, 115113 (2024)

2024
[25]

Voinea, R

C. Voinea, R. Fan, N. Regnault, and Z. Papi´ c, Regular- izing 3D Conformal Field Theories via Anyons on the Fuzzy Sphere, Phys. Rev. X15, 031007 (2025)

2025
[26]

L¨ auchli, L

A. L¨ auchli, L. Herviou, P. Wilhelm, and S. Rychkov, Exact diagonalization, matrix product states and con- formal perturbation theory study of a 3D Ising fuzzy sphere model, SciPost Physics19, 076 (2025)

2025
[27]

Z. Zhou, L. Hu, W. Zhu, and Y.-C. He, SO(5) De- confined Phase Transition under the Fuzzy-Sphere Mi- croscope: Approximate Conformal Symmetry, Pseudo- Criticality, and Operator Spectrum, Phys. Rev. X14, 021044 (2024)

2024
[28]

B.-B. Chen, X. Zhang, Y. Wang, K. Sun, and Z. Y. Meng, Phases of (2 + 1)D SO(5) Nonlinear Sigma Model with a Topological Term on a Sphere: Multicritical Point and Disorder Phase, Phys. Rev. Lett.132, 246503 (2024)

2024
[29]

B.-B. Chen, X. Zhang, and Z. Y. Meng, Emergent conformal symmetry at the multicritical point of (2 + 1)D SO(5) model with Wess-Zumino-Witten term on a sphere, Phys. Rev. B110, 125153 (2024)

2024
[30]

J. S. Hofmann, F. Goth, W. Zhu, Y.-C. He, and E. Huff- man, Quantum Monte Carlo simulation of the 3D Ising transition on the fuzzy sphere, SciPost Phys. Core7, 028 (2024)

2024
[31]

Cuomo, Y.-C

G. Cuomo, Y.-C. He, and Z. Komargodski, Impurities with a cusp: general theory and 3d Ising, Journal of High Energy Physics2024, 1 (2024)

2024
[32]

Yang, Y.-G

S. Yang, Y.-G. Yue, Y. Tang, C. Han, W. Zhu, and Y. Chen, Microscopic study of the three-dimensional Potts phase transition via fuzzy sphere regularization, Physical Review B112, 024436 (2025)

2025
[33]

R. Fan, J. Dong, and A. Vishwanath, Simulating the non-unitary Yang-Lee conformal field theory on the fuzzy sphere (2025), arXiv:2505.06342 [cond-mat.str-el]

2025 arXiv
[34]

Arguello Cruz, I

E. Arguello Cruz, I. R. Klebanov, G. Tarnopolsky, and Y. Xin, Yang-Lee Quantum Criticality in Various Di- mensions, Physical Review X16, 011022 (2026)

2026
[35]

J. E. Mir´ o and O. Delouche, Flowing from the Ising model on the fuzzy sphere to the 3D Lee-Yang CFT, Journal of High Energy Physics2025, 1 (2025)

2025
[36]

Zhou and Y.-C

Z. Zhou and Y.-C. He, 3D Conformal Field Theories with Sp(N) Global Symmetry on a Fuzzy Sphere, Phys. Rev. Lett.135, 026504 (2025)

2025
[37]

Hu, Y.-C

L. Hu, Y.-C. He, and W. Zhu, Solving conformal defects in 3D conformal field theory using fuzzy sphere regu- larization, Nature Communications15, 10.1038/s41467- 024-47978-y (2024)

2024 doi
[38]

Z. Zhou, D. Gaiotto, Y.-C. He, and Y. Zou, Theg- function and defect changing operators from wavefunc- tion overlap on a fuzzy sphere, SciPost Phys.17, 021 (2024)

2024
[39]

Zhou and Y

Z. Zhou and Y. Zou, Studying the 3d Ising surface CFTs on the fuzzy sphere, SciPost Phys.18, 031 (2025)

2025
[40]

Dedushenko, Ising BCFT from Fuzzy Hemisphere (2024), arXiv:2407.15948 [hep-th]

M. Dedushenko, Ising BCFT from Fuzzy Hemisphere (2024), arXiv:2407.15948 [hep-th]

2024 arXiv
[41]

He, Free real scalar CFT on fuzzy sphere: spectrum, algebra and wavefunction ansatz (2025), arXiv:2506.14904 [hep-th]

Y.-C. He, Free real scalar CFT on fuzzy sphere: spectrum, algebra and wavefunction ansatz (2025), arXiv:2506.14904 [hep-th]

2025 arXiv
[42]

Z. Zhou, C. Wang, and Y.-C. He, Chern-Simons- matter conformal field theory on fuzzy sphere: Confine- ment transition of Kalmeyer-Laughlin chiral spin liquid, arXiv preprint arXiv:2507.19580 (2025)

2025 arXiv
[43]

Taylor, C

J. Taylor, C. Voinea, Z. Papi´ c, and R. Fan, Confor- mal Scalar Field Theory from Ising Tricriticality on the Fuzzy Sphere, Phys. Rev. Lett.136, 056503 (2026)

2026
[44]

Voinea, W

C. Voinea, W. Zhu, N. Regnault, and Z. Papi´ c, Criti- cal Majorana Fermion at a Topological Quantum Hall Bilayer Transition, Phys. Rev. Lett.136, 076601 (2026)

2026
[45]

Z. Zhou, D. Gaiotto, and Y.-C. He, Free and interacting fermionic conformal field theories on the fuzzy sphere, Physical Review X16, 021055 (2026)

2026
[46]

A. Dey, L. Herviou, C. Mudry, and A. M. L¨ auchli, Con- formal Data for the O(3) Wilson-Fisher Conformal Field Theory from Fuzzy Sphere Realization of the Quan- tum Rotor Model, Physical review letters136, 206504 (2026)

2026
[47]

W. Guo, Z. Zhou, T.-C. Wei, and Y.-C. He,O(N) free- scalar and Wilson-Fisher conformal field theories on the fuzzy sphere, Physical Review B114, 065102 (2026)

2026
[48]

Y. Tang, C. Voinea, L. Hu, Z. Papi´ c, and W. Zhu, Emer- gence of 3D superconformal Ising criticality on the fuzzy sphere, Physical Review Letters137, 026502 (2026)

2026
[49]

J.-X. Hao, Z. Zhu, and Y. Qi, Multi-target density ma- trix renormalization group for 3D CFTs on the fuzzy sphere, arXiv preprint arXiv:2601.18648 (2026)

2026
[50]

Huffman, Z

E. Huffman, Z. Zhou, Y.-C. He, and J. S. Hofmann, Generalizing Deconfined Criticality to 3DN-Flavor SU(2) Quantum Chromodynamics on the Fuzzy Sphere, arXiv preprint arXiv:2602.11255 (2026)

2026
[51]

Meng, J.-W

X. Meng, J.-W. Zhao, W. Zhu, and C. Wu, Quantum Phase Transitions on Complex Projective SpaceCP 2, Physics Letters A , 131488 (2026)

2026
[52]

Eck and Z

L. Eck and Z. Wang, 3d conformal field theories via fuzzy sphere algebra, Journal of Physics A: Mathemat- ical and Theoretical59, 265401 (2026)

2026
[53]

Janssens and Z

B. Janssens and Z. Wang, Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits, arXiv preprint arXiv:2603.06876 (2026)

2026
[54]

Sarma, Z

A. Sarma, Z. Zhou, R. A. Lanzetta, and Y.-C. He, For- tuitous Universality of Bose-Kondo Impurities, arXiv preprint arXiv:2604.07554 (2026)

2026 arXiv
[55]

A. Dey, L. Herviou, C. Mudry, S. Rychkov, and A. M. L¨ auchli, Conformal Data for theO(2) Wilson-Fisher CFT in (2 + 1)-Dimensional Spacetime from Exact Di- agonalization and Matrix Product States on the Fuzzy Sphere, arXiv preprint arXiv:2604.18705 (2026)

2026 arXiv
[56]

Feng and T

J. Feng and T. Wang, Studying 3D O(N) Surface CFT on the Fuzzy Sphere, arXiv preprint arXiv:2604.21091 (2026)

2026 arXiv
[57]

Stergiou, Quantum Rotors on the Fuzzy Sphere and the Cubic CFT, arXiv preprint arXiv:2604.24840 (2026)

A. Stergiou, Quantum Rotors on the Fuzzy Sphere and the Cubic CFT, arXiv preprint arXiv:2604.24840 (2026)

2026 arXiv
[58]

J. L. Cardy, Conformal invariance and universality in finite-size scaling, Journal of Physics A: Mathematical and General17, L385 (1984)

1984
[59]

Hu, Y.-C

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

2023
[60]

C. Han, L. Hu, W. Zhu, and Y.-C. He, Conformal four- point correlators of the three-dimensional Ising transi- tion via the quantum fuzzy sphere, Phys. Rev. B108, 7 235123 (2023)

2023
[61]

Fardelli, A

G. Fardelli, A. L. Fitzpatrick, and E. Katz, Constructing the infrared conformal generators on the fuzzy sphere, SciPost Physics18, 086 (2025)

2025
[62]

Fan, Note on explicit construction of confor- mal generators on the fuzzy sphere, arxiv:2409.08257, arXiv:2409.08257 [hep-th]

R. Fan, Note on explicit construction of confor- mal generators on the fuzzy sphere, arxiv:2409.08257, arXiv:2409.08257 [hep-th]

arXiv
[63]

L. Hu, W. Zhu, and Y.-C. He, EntropicFfunction of three-dimensional Ising conformal field theory via fuzzy sphere regularization, Phys. Rev. B111, 155151 (2025)

2025
[64]

Fardelli, A

G. Fardelli, A. L. Fitzpatrick, and E. Katz, Improving 3d Ising OPE Coefficients with Fuzzy Sphere Conformal Generators, arXiv preprint arXiv:2602.04958 (2026)

2026
[65]

J.-M. Dong, Y. Zhang, K.-W. Huang, H.-H. Tu, and Y.- H. Wu, Numerical extraction of crosscap coefficients in microscopic models for (2 + 1)D conformal field theory, Physical Review D112, L121701 (2025)

2025
[66]

K. J. Wiese, Locating the Ising conformal field theory via the ground-state energy on the fuzzy sphere, Phys- ical Review B113, 085106 (2026)

2026
[67]

Yang, L.-d

S. Yang, L.-d. Hu, C. Han, W. Zhu, and Y. Chen, Con- formal operator flows of the deconfined quantum criti- cality from SO(5) to O(4), Physical Review Letters136, 076505 (2026)

2026
[68]

Belin, J

A. Belin, J. de Boer, and J. Kruthoff, Comments on a state-operator correspondence for the torus, SciPost Phys.5, 060 (2018)

2018
[69]

Tao and D

R. Tao and D. J. Thouless, Fractional quantization of Hall conductance, Phys. Rev. B28, 1142(R) (1983)

1983
[70]

Seidel, H

A. Seidel, H. Fu, D.-H. Lee, J. M. Leinaas, and J. Moore, Incompressible Quantum Liquids and New Conserva- tion Laws, Phys. Rev. Lett.95, 266405 (2005)

2005
[71]

E. J. Bergholtz and A. Karlhede, Half-Filled Lowest Landau Level on a Thin Torus, Phys. Rev. Lett.94, 026802 (2005)

2005
[72]

E. J. Bergholtz and A. Karlhede, Quantum Hall system in Tao-Thouless limit, Phys. Rev. B77, 155308 (2008)

2008
[73]

Nakamura, Z.-Y

M. Nakamura, Z.-Y. Wang, and E. J. Bergholtz, Ex- actly Solvable Fermion Chain Describing aν= 1/3 Fractional Quantum Hall State, Phys. Rev. Lett.109, 016401 (2012)

2012
[74]

Nachtergaele, S

B. Nachtergaele, S. Warzel, and A. Young, Spectral Gaps and Incompressibility in aν= 1/3 Fractional Quantum Hall System, Communications in Mathemat- ical Physics383, 1093 (2021)

2021
[75]

Mukherjee and K

S. Mukherjee and K. Park, Visualizing the fractional topological order: From fractional Chern insulators to the Tao-Thouless state, Phys. Rev. Res.5, 033212 (2023)

2023
[76]

Henkel,Conformal Invariance and Critical Phenom- ena(Springer Science & Business Media, 2013)

M. Henkel,Conformal Invariance and Critical Phenom- ena(Springer Science & Business Media, 2013)

2013
[77]

J. L. Cardy, Universal amplitudes in finite-size scaling: generalisation to arbitrary dimensionality, J. Phys. A: Math. Gen.18, L757 (1985)

1985
[78]

Grimm and B

U. Grimm and B. Nienhuis, Scaling limit of the Ising model in a field, Phys. Rev. E55, 5011 (1997)

1997
[79]

Yoshioka, B

D. Yoshioka, B. I. Halperin, and P. A. Lee, Ground State of Two-Dimensional Electrons in Strong Magnetic Fields and 1 3 Quantized Hall Effect, Phys. Rev. Lett.50, 1219 (1983)

1983
[80]

Chakraborty and P

T. Chakraborty and P. Pietil¨ ainen,The quantum Hall effects: integral and fractional, Vol. 85 (Springer Science & Business Media, 2013)

2013
[81]

Han and W

C. Han and W. Zhu, Quantum phase transitions on the noncommutative circle, Phys. Rev. B111, 085113 (2025)

2025
[82]

See the Supplemental Material for details of the calcu- lations and results in the main text
[83]

Caselle, M

M. Caselle, M. Hasenbusch, and P. Provero, Spectrum of the gauge Ising model in three dimensions, Nuclear Physics B - Proceedings Supplements63, 616 (1998), proceedings of the XVth International Symposium on Lattice Field Theory

1998
[84]

Onsager, Crystal statistics

L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Physical review 65, 117 (1944)

1944
[85]

C. N. Yang, The spontaneous magnetization of a two- dimensional Ising model, Physical Review85, 808 (1952)

1952
[86]

Henkel and H

M. Henkel and H. Saleur, The two-dimensional Ising model in the magnetic field: a numerical check of Zamolodchikov’s conjecture, Journal of Physics A: Mathematical and General22, L513 (1989)

1989
[87]

J. A. Kj¨ all, F. Pollmann, and J. E. Moore, Bound states andE 8 symmetry effects in perturbed quantum Ising chains, Phys. Rev. B83, 020407 (2011)

2011
[88]

Knaute and P

J. Knaute and P. Hauke, Relativistic meson spectra on ion-trap quantum simulators, Phys. Rev. A105, 022616 (2022)

2022
[89]

Vovrosh, J

J. Vovrosh, J. de Hond, S. Juli` a-Farr´ e, J. Knolle, and A. Dauphin, Meson spectroscopy of exotic symme- tries of Ising criticality in rydberg atom arrays (2026), arXiv:2506.21299 [quant-ph]

2026 arXiv
[90]

Simmons-Duffin, The Conformal Bootstrap, inTheo- retical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings(2017) pp

D. Simmons-Duffin, The Conformal Bootstrap, inTheo- retical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings(2017) pp. 1–74, arXiv:1602.07982 [hep-th]

2017 arXiv
[91]

Zinn-Justin,Quantum field theory and critical phe- nomena, Vol

J. Zinn-Justin,Quantum field theory and critical phe- nomena, Vol. 171 (Oxford university press, 2021)

2021
[92]

Cappelli, L

A. Cappelli, L. Maffi, and S. Okuda, Critical ising model in varying dimension by conformal bootstrap, Journal of High Energy Physics2019, 161 (2019)

2019
[93]

A. J. A. James, R. M. Konik, and N. J. Robinson, Non- thermal States Arising from Confinement in One and Two Dimensions, Phys. Rev. Lett.122, 130603 (2019)

2019
[94]

L. V. Delacr´ etaz, A. L. Fitzpatrick, E. Katz, and M. T. Walters, Thermalization and chaos in a 1+1d QFT, Journal of High Energy Physics2023, 45 (2023)

2023
[95]

G. Fardelli, Fuzzy Sphere, Conformal Generators, and Ising Field Theory, Talk at Fuzzy Sphere Meets Confor- mal Bootstrap 2025, IHES (2025), June 2, 2025; video available athttps://youtu.be/luNI5hSZ-vI

2025
[96]

DiagHam,https://www.nick-ux.org/diagham
[97]

Zhou, FuzzifiED – Julia package for numerics on the fuzzy sphere (2025), arXiv:2503.00100 [cond-mat.str-el]

Z. Zhou, FuzzifiED – Julia package for numerics on the fuzzy sphere (2025), arXiv:2503.00100 [cond-mat.str-el]

2025
[98]

S. M. G. Richard E. Prange, ed.,The Quantum Hall Effect(Springer-Verlag New York, 1987)

1987
[99]

X. G. Wen and A. Zee, Shift and spin vector: New topo- logical quantum numbers for the Hall fluids, Phys. Rev. Lett.69, 953 (1992)

1992
[100]

T. T. Wu and C. N. Yang, Dirac Monopole Without Strings: Monopole Harmonics, Nucl. Phys. B107, 365 (1976)

1976
[101]

F. D. M. Haldane, Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett.51, 605 (1983)

1983
[102]

L¨ uscher, Volume dependence of the energy spectrum in massive quantum field theories: I

M. L¨ uscher, Volume dependence of the energy spectrum in massive quantum field theories: I. Stable particle 8 states, Communications in Mathematical Physics104, 177 (1986)

1986
[103]

Y. Gao, Y. Jiang, and J. Wu, Mesons in a quantum Ising ladder, Journal of High Energy Physics2025, 1 (2025)

2025
[104]

Dusuel, M

S. Dusuel, M. Kamfor, K. P. Schmidt, R. Thomale, and J. Vidal, Bound states in two-dimensional spin systems near the Ising limit: A quantum finite-lattice study, Phys. Rev. B81, 064412 (2010)

2010
[105]

C. J. Hamer, Finite-size scaling in the transverse Ising model on a square lattice, Journal of Physics A: Math- ematical and General33, 6683 (2000). END MA TTER Quench spectroscopy.—Figure 5 complements the equilibrium analyses in Figs. 2 and 3 of the main text by asking wheth...

2000
[200]

Fuzzy Spectroscopy of Bound States in Massive Quantum Field Theories

The Fourier peaks in panel (b) nearm 1,m 2, and the higher branches coincide with the DSF peaks shown in orange, and they are seen to be in good agreement with the static spectrum in Fig. 2(a). The middle row applies the same protocol to the square torus. We prepare the ground...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Sachdev,Quantum Phase Transitions, 2nd ed

S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cam- bridge University Press, 2011)

2011

[2] [2]

Mussardo,Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford Graduate Texts (OUP Oxford, 2020)

G. Mussardo,Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford Graduate Texts (OUP Oxford, 2020)

2020

[3] [3]

A. B. Zamolodchikov, Integrals of Motion and S Matrix of the (Scaled)T=T c Ising Model with Magnetic Field, Int. J. Mod. Phys. A4, 4235 (1989)

1989

[4] [4]

A. M. Perelomov, Remarks on the mass spectrum of two-dimensional Toda lattice ofE 8 type, Journal of Nonlinear Mathematical Physics27, 12 (2020)

2020

[5] [5]

Coldea, D

R. Coldea, D. Tennant, E. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Quantum Criticality in an Ising chain: Experimental Evidence for EmergentE 8 symmetry, Sci- ence327, 177 (2010)

2010

[6] [6]

Zhang, K

Z. Zhang, K. Amelin, X. Wang, H. Zou, J. Yang, U. Nagel, T. R˜ o om, T. Dey, A. A. Nugroho, T. Lorenz, J. Wu, and Z. Wang, Observation ofE 8 particles in an Ising chain antiferromagnet, Phys. Rev. B101, 220411(R) (2020)

2020

[7] [7]

H. Zou, Y. Cui, X. Wang, Z. Zhang, J. Yang, G. Xu, A. Okutani, M. Hagiwara, M. Matsuda, G. Wang, G. Mussardo, K. H´ ods´ agi, M. Kormos, Z. He, S. Kimura, R. Yu, W. Yu, J. Ma, and J. Wu,E 8 Spectra of Quasi- One-Dimensional Antiferromagnet BaCo 2V2O8 under Transverse Field, Phys. Rev. Lett.127, 077201 (2021)

2021

[8] [8]

Caselle, M

M. Caselle, M. Hasenbusch, and P. Provero, Non- perturbative states in the three-dimensionalϕ 4 theory, Nuclear Physics B - Proceedings Supplements83-84, 715 (2000), proceedings of the XVIIth International Symposium on Lattice Field Theory

2000