An introduction to fuzzy sphere

Introduction

Conformal field theory (CFT) is one of the central topics of modern physics. It has provided invaluable insights into critical phenomena in condensed matter physics, string theory and AdS/CFT correspondence in quantum gravity, and enhanced our understanding of the renormalisation group and other fundamental structures and dynamics of quantum field theory (QFT). In 2d, many CFTs are well understood thanks to their integratibility. Going to higher dimensions, CFTs are much less well-studied due to a much smaller conformal group. Existing approaches, including Monte Carlo lattice simulation and numerical conformal bootstrap, despite having achieved many successes, can only handle a limited number of CFTs and obtain a limited number of conformal data.

Recently, fuzzy sphere regularisation has emerged as a new powerful method to study 3D CFTs. By studying interacting quantum systems on the fuzzy (non-commutative) sphere, the method realises $(2+1)$D quantum phase transitions on the geometry $S^2\times\mathbb{R}$. Compared with conventional methods that involve simulating lattice models, this approach offers distinct advantages including exact preservation of rotation symmetry, direct observation of emergent conformal symmetry, and the efficient extraction of conformal data. In the fuzzy sphere method, the state-operator correspondence plays an essential role. Specifically, there is a one-to-one correspondence between the eigenstates of the critical Hamiltonian on the sphere and the CFT operators, where the energy gaps are proportional to the scaling dimensions. The power of this approach has been demonstrated in the context of the 3D Ising transition, where the presence of emergent conformal symmetry has been convincingly established and a wealth of conformal data has been accurately computed. The study has been extended to the magnetic line defect, various conformal boundaries in 3D Ising CFT, conformal generators, Wilson-Fisher theory, and deconfined criticality.

Review of existing work

The study of 3d CFTs on fuzzy sphere can mainly be devided into four catagories :

Accessing various conformal data,
Realising various 3d CFTs,
Studying conformal defects and boundaries, and
Exploring applicable numerical techniques.

Accessing various conformal data

The first direction is to develop methods to calculate as many data and quantities of 3d CFTs on fuzzy sphere. Typically, these methods are tested on the simplest example of 3d Ising CFT. For many of those CFT data, fuzzy sphere is the first non-perturbative method to access them ; for the others, fuzzy sphere has achieved great consistency with previous methods such as quantum Monte Carlo and conformal bootstrap. So far, the accessible CFT data include operator spectrum, OPE coefficients, correlation functions, entropic $F$-function and conformal generators.

Operator spectrum [Zhu 2022] This seminal paper opens a new avenue for studying 3d conformal field theories. It calculates and analyses the energy spectra at the 3d Ising transition, and explicitly demonstrate the state-operator correspondence as a fingerprint of conformal field theory, thus directly elucidates the emergent conformal symmetry of the 3d Ising transition.
OPE coefficients [Hu 2023Mar] This paper computes 17 OPE coefficients of low-lying CFT primary fields with high accuracy, including 4 that has not being reported before.
Correlation functions [Han 2023Jun] This paper computes the four-point correlators and verify the crossing symmetry.
Entropic $F$-function [Hu 2024] This paper have performed the first non-perturbative computation of the $F$-function for the paradigmatic 3d Ising conformal field theory through entanglement entropy.
Conformal generators [Fardelli 2024, Fan 2024] These papers investigate the conformal generators of translations and special conformal transformations which are emergent in the infrared and construct these generators using the energy momentum tensor.

Realising various 3d CFTs

The second direction is study various other CFTs beyond 3d Ising. Fuzzy sphere has revealed many new information about these theories ; the previously known data are also consistent with fuzzy sphere results. So far, the accessible CFTs include $\mathrm{SO}(5)$ deconfined criticality, $\mathrm{O}(3)$ Wilson-Fisher and a series of new theories with $\mathrm{Sp}(N)$ symmetry.

The $\mathrm{SO}(5)$ deconfined criticality [Zhou 2023] This paper provides clear evidence that the DQCP exhibits approximate conformal symmetry, and demonstrate that the DQCP is more likely pseudo-critical.
The $\mathrm{O}(3)$ Wilson-Fisher [Han 2023Dec] This paper design a microscopic model of Heisenberg magnet bilayer and study the underlying Wilson-Fisher $\mathrm{O}(3)$ transition through the lens of fuzzy sphere regularization.
A series of new $\mathrm{Sp}(N)$-symmetric CFTs [Zhou 2024Oct] This paper discovers a series of new CFTs with global symmetry $\mathrm{Sp}(N)$ in the fuzzy sphere models that are closely related to the SO(5) deconfined phase transition, and are related to non-linear sigma model with a Wess-Zumino-Witten term and Chern-Simons-matter theories. The emergent conformal symmetry is numerically verified by observing the integer-spaced conformal multiplets and the quality of conformal generators.

Studying conformal defects and boundaries

Apart from the bulk CFTs, fuzzy sphere can also be used to study their conformal defects and boundaries. So far, the accessible defects/boundaries include the magnetic line defect of 3d Ising CFT, including its defect operator spectrum, correlators, $g$-function, defect changing operators, and its cusp, and the conformal boundaries of 3d Ising CFT.

Conformal magnetic line defect [Hu 2023Aug] This paper studies the magnetic line defect of 3D Ising CFT and clearly demonstrates that it flows to a conformal defect fixed point. The authors have identified 6 low-lying defect primary operators and extract their scaling dimensions, as well as computing one-point bulk correlators and two-point bulk-defect correlators.
The $g$-function and defect changing operators [Zhou 2024Jan] This paper have performed the non-perturbative computations of the scaling dimensions of defect-changing, creation operators and the $g$-function for the pinning defect in 3d Ising model.
Cusp [Cuomo 2024] This paper study the general properties of the cusp anomalous dimension and in particular calculates the pinning field defects in the 3d Ising model on fuzzy sphere.
Conformal boundaries of 3d Ising CFT [Zhou 2024Jul, Dedushenko 2024] These papers demonstrates that conformal field theory (CFT) with a boundary, known as surface CFT in three dimensions, can be studied with the setup of fuzzy sphere, and in particular in the example of surface criticality, proposes two schemes by cutting a boundary in the orbital space or the real space to realise the ordinary and the normal surface CFTs on the fuzzy sphere.

Exploring applicable numerical techniques

So far, the numerical methods that has been applied to fuzzy sphere to include exact diagonalisation (ED), density matrix renormalisation group (DMRG) and determinant quantum Monte Carlo (DQMC). The former two has been reviewed in previous sections.

Quantum Monte Carlo on fuzzy sphere [Hofmann 2023] This paper presents a numerical quantum Monte Carlo (QMC) method for simulating the 3D phase transition on the recently proposed fuzzy sphere.

Model construction

Projection onto the lowest Landau level

To build the setup of fuzzy sphere, we consider a sphere with radius $R$ and put a $4\pi s$-monopole at its centre. Consider free electrons moving on the sphere. The monopole will modify the single particle Hamilltonian.

\[ H_0=\frac{1}{2MR^2}(\partial^\mu+iA^\mu)^2\]

where $\mu=\theta,\phi$ and we take

\[ A_\theta=0, A_\phi=-\frac{s}{R}\operatorname{ctg}\theta\]

The eigenstates of the Hamiltonian are the monopole spherical harmonics

\[ Y_{lm}^{(s)}(\hat{\mathbf{r}}),\quad l=s,s+1,\dots,\quad m=-s,\dots,s-1,s\]

with the eigenenergies

\[ E_l=\frac{1}{2MR^2}(l(l+1)-s^2)\]

Each level, known as a Landau level, has a degeneracy of $(2l+1)$. Specifically, the wavefunctions on the lowest Landau level (LLL) $l=s$ is easy to write out :

\[ Y_{sm}^{(s)}(\hat{\mathbf{r}})=C_me^{im\phi}\cos^{s+m}\frac{\theta}{2}\sin^{s-m}\frac{\theta}{2},\quad C_m=\frac{1}{{\sqrt{4\pi\Beta(s+m+1,s-m+1)}}}\]

where $C_m$ is the normalising factor. The LLL has a degeneracy $N_m=2s+1$

We now consider $N_f$ flavours of fermions moving on the sphere, characterised by the second-quantised fermion operator $\hat{\psi}_f(\hat{\mathbf{r}})$, with a flavour index $f=1,\dots,N_f$. We partially fill the lowest Landau level and set the single energy gap to be much larger than the scale of interaction $H_0\gg H_\mathrm{int}$, so that the quantum fluctuation can be constrained on the lowest Landau level. In practice, we often fill integer number of flavours $N_e=kN_m$ ($k\in\mathbb{Z}$) so that a quantum Hall ferromagnet (i.e., the state where integer number of LLLs are filled) can exist on the phase diagram.

We then project the system onto the LLL. Technically, this can be done by write the fermion operators in terms of the annihilation operators of the LLL orbitals

\[ \hat{\psi}_f(\hat{\mathbf{r}})=\sum_{m=-s}^s Y^{(s)}_{sm}(\hat{\mathbf{r}})\hat{c}_{mf}\]

where $\hat{c}^{(\dagger)}_{mf}$ annihilates/creates an electron with $L^z$-quantum number $m$ at the $f$-th flavour of the lowest Landau level. In the following sections, we will omit the hats on the operators.

Density operator

The simplest building block of an interacting many-body Hamiltonian is density operators, which are local fermion bilinears.

\[ n_M(\hat{\mathbf{r}})=\psi_{f'}^\dagger(\hat{\mathbf{r}})M_{f'f}\psi_f(\hat{\mathbf{r}})\]

Here the matrix insertion $M$ put the density operators at a certain representation of the flavour symmetry. Like the fermion operator, the density operator can also be expressed in the orbital space.

\[ n_M(\hat{\mathbf{r}})=\sum_{lm}Y_{lm}(\hat{\mathbf{r}})n_{lm}\]

Conversely,

\[ \begin{aligned} n_{lm}&=\int\mathrm{d}^2\hat{\mathbf{r}}\,\bar{Y}_{lm}n_M(\hat{\mathbf{r}})\\ &=\int\mathrm{d}^2\hat{\mathbf{r}}\,\bar{Y}_{lm}\left(\sum_{m_1}\bar{Y}^{(s)}_{sm_1}c^\dagger_{m_1f_1}\right)M_{f_1f_2}\left(\sum_{m_2}Y^{(s)}_{sm_2}c_{m_1f_2}\right)\\ &=\sum_{m_1m_2}c^\dagger_{m_1f_1}M_{f_1f_2}c_{m_1f_2}\int\mathrm{d}^2\hat{\mathbf{r}}\,\bar{Y}_{lm}\bar{Y}^{(s)}_{sm_1}Y^{(s)}_{sm_2}\\ &=\sum_{m_1}c^\dagger_{m_1f_1}M_{f_1f_2}c_{m+m_1,f_2}(-1)^{s+m+m_1}(2s+1)\sqrt{\frac{2l+1}{4\pi}}\begin{pmatrix}s&l&s\\m_1&m&-m_1-m\end{pmatrix}\begin{pmatrix}s&l&s\\m_1&m&-m_1-m\end{pmatrix} \end{aligned}\]

where various properties of the monopole spherical harmonics are used. In this way, we have fully expressed the density operator in terms of the operators in the orbital space $c^{(\dagger)}_{mf}$.

Density-density interaction

The most straightforward way to construct an interaction term is to add a density-density interaction potential $U(r)$. We note that this is not the simplest construction and we will present the simpler construction in terms of pseudopotentials in the next section.

\[ H_\mathrm{int}=\int\mathrm{d}^2\hat{\mathbf{r}}_1\,\mathrm{d}^2\hat{\mathbf{r}}_2\,U(|\hat{\mathbf{r}}_1-\hat{\mathbf{r}}_2|)n_M(\hat{\mathbf{r}}_1)n_M(\hat{\mathbf{r}}_2)\]

The interacting potentials can be expanded in terms of the Legendre polynomials

\[ U(\theta_{12})=\sum_lU_lP_l(\cos\theta_{12})=\sum_{lm}\frac{4\pi}{2l+1}\bar{Y}_{lm}(\hat{\mathbf{r}}_1)Y_{lm}(\hat{\mathbf{r}}_2)\]

Conversely

\[ U_l=\int \sin\theta_{12}\mathrm{d}\theta_{12}\,\frac{2l+1}{2}U(\theta_{12})P_l(\cos\theta_{12})\]

Specifically, for local and super-local interactions

\[ \begin{aligned} U(|\mathbf{r}_{12}|)&=\delta(\mathbf{r}_{12})&U_l&=2l+1\\ U(|\mathbf{r}_{12}|)&=\nabla^2\delta(\mathbf{r}_{12})&U_l&=-l(l+1)(2l+1) \end{aligned}\]

By expanding the density operators into the orbital space and completing the integrals,

\[ H_\mathrm{int}=\sum_{lm}\frac{4\pi U_l}{2l+1}n^\dagger_{M,lm}n_{M,lm}\]

For example, for the 3d Ising CFT, we take two flavours of fermions and write down an interaction

\[ H_\mathrm{int}=\int\mathrm{d}^2\hat{\mathbf{r}}_1\,\mathrm{d}^2\hat{\mathbf{r}}_2\,U(|\hat{\mathbf{r}}_1-\hat{\mathbf{r}}_2|)(n_0(\hat{\mathbf{r}}_1)n_0(\hat{\mathbf{r}}_2)-n_z(\hat{\mathbf{r}}_1)n_z(\hat{\mathbf{r}}_2))-h\int\mathrm{d}^2\hat{\mathbf{r}}\,n_x(\hat{\mathbf{r}})\]

where the density operators are defined as

\[ n_i(\hat{\mathbf{r}})=\psi^\dagger_{f'}(\hat{\mathbf{r}})(\sigma_i)_{f'f}\psi_f(\hat{\mathbf{r}})\]

and the potentials are taken as a combination of local and super-local interactions.

Interaction in terms of pseudopotentials

Another way that is much more convenient to construct the interactions is through Haldane pseudopotential. To explain the idea, we take the 3d Ising model as an example. We first classify all the fermion bilinears $\lambda_{mm'ff'}c_{mf}c_{m'f'}$. To simplify the discussion, we can take a specific pseudo-spin index $\lambda_{mm'}c_{m\uparrow}c_{m'\downarrow}$. The fermion bilinears can be classified into irreducible representations of $\mathrm{SO}(3)$ rotation symmetry. Since $c_{mf}$ carries the spin-$s$ representation, the spin of its bilinear ranges from $0$ to $2s$ and takes integer values. The spin-$(2s-l)$ combination reads

\[ \Delta_{lm}=\sqrt{2s-2l+1}\sum_{m_1}\begin{pmatrix}s&s&2s-l\\m_1&m-m_1&-m\end{pmatrix}c_{m_1,\uparrow}c_{m-m_1,\downarrow}\]

where $m=-(2s-l),\dots,(2s-l)$. A four-fermion interaction term can be formed by contracting these paring operators with its conjugate.

\[ H=\sum_lU_lH_l,\quad H_l=\sum_m\Delta_{lm}^\dagger\Delta_{lm}\]

The coupling strength $U_l$ of the spin-$(2s-l)$ channel is called the Haldane pseudopotentials.

We need also to consider the constraint that the two fermions must be anti-symmetrised : for even $l$, the orbital index is symmetrised, so the spin index must be antisymmetrised, so the two fermions form a spin-singlet which is invariant under the $\mathrm{SU}(2)$ transformation ; for odd $l$, the orbital index is anti-symmetrised, so the spin index is symmetrised, breaking the flavour symmetry from $\mathrm{SU}(2)$ to $\mathbb{Z}_2$.

The fermion bilinears with other pseudo-spin configurations $\lambda_{mm',\pm}(c_{m\uparrow}c_{m'\uparrow}\pm c_{m\downarrow}c_{m'\downarrow})$ can be analysed in a similar way. After that, we have enumerated all possible four-fermion interaction terms.

For systems with more complicated continuous symmetries, classification in terms of representation of flavour symmetry must also be considered, and the indices must be overall antisymmetrised. For an example with $\mathrm{Sp}(N)$ symmetry, see Zhou 2024Oct.

We also note that each pseudopotential can correspond to a profile of interaction potential functions. Specifically, $U_0$ corresponds to a local interaction ; $U_l$ corresponds to a super-local interaction $\nabla^{2l}\delta(\mathbf{r}_{12})$ in the thermodynamic limit. For a more detailed expression at finite system size, see Fan 2024.

One can keep only terms with the smallest $l$. For Ising model, we can keep up to $l=1$. Note that at lease one term with odd $l$ must be kept to break the $\mathrm{SU}(2)$ symmetry.

\[ H=U_0\sum_m\Delta_{0m}^\dagger\Delta_{0m}+U_1\sum_m\Delta_{1m}^\dagger\Delta_{1m}-hn_{x,00}\]

Local observables

On the fuzzy sphere, the simplest local observable is the density operator. We consider the Ising model as an example specifically

\[ n^i(\Omega)=\Psi^\dagger(\Omega)\sigma^i\Psi(\Omega)=\sum_{mm'}Y^{(s)}_{sm}(\Omega)\bar{Y}^{(s)}_{sm'}(\Omega)\mathbf{c}^\dagger_m\sigma^i\mathbf{c}_{m'}\]

where $i=x,z$. From the CFT perspective, the density operators are the superpositions of scaling operators with corresponding quantum numbers. In the leading order, they can be used as UV realisations of CFT operators $\sigma$ and $\epsilon$.

\[\begin{aligned} n^x&=\lambda_0+\lambda_\epsilon R^{-\Delta_\epsilon}\epsilon+\lambda_{\partial_\tau\epsilon}R^{-\Delta_\epsilon-1}\partial_\tau\epsilon+\lambda_{T_{\tau\tau}}R^{-3}T_{\tau\tau}+\dots&\epsilon_\mathrm{FS}&=\frac{n^x-\lambda_0}{\lambda_\epsilon R^{-\Delta_\epsilon}}\nonumber\\ n^z&=\lambda_\sigma R^{-\Delta_\sigma}\sigma+\lambda_{\partial_\tau\sigma}R^{-\Delta_\sigma-1}\partial_\tau\epsilon+\lambda_{\partial^2\sigma}R^{-\Delta_\sigma-2}\partial^2\sigma+\dots&\sigma_\mathrm{FS}&=\frac{n^z}{\lambda_\sigma R^{-\Delta_\sigma}} \end{aligned}\]

The UV-dependent coefficients $\lambda_0,\lambda_\sigma,\lambda_\epsilon$ can be computed from the 1-point function. The subleading terms to the density operators contribute to the finite size corrections

\[\begin{aligned} \epsilon_\mathrm{FS}&=\epsilon+\frac{\lambda_{\partial_\tau\epsilon}}{\lambda_\epsilon}R^{-1}\partial_\tau\epsilon+\frac{\lambda_{T_{\tau\tau}}}{\lambda_\epsilon}R^{-3+\Delta_\epsilon}T_{\tau\tau}+\dots\nonumber\\ \sigma_\mathrm{FS}&=\sigma+\frac{\lambda_{\partial_\tau\sigma}}{\lambda_\sigma}R^{-1}\partial_\tau\sigma+\frac{\lambda_{\partial^2\sigma}}{\lambda_\sigma}R^{-2}\partial^2\sigma+\dots \end{aligned}\]

where $R=N^{1/2}$ in the fuzzy sphere.

Conformal generators

The conformal generator $\Lambda^\mu=P^\mu+K^\mu$ on the states is the $l=1$ component of the Hamiltonian density. For example, for Ising model, it is the local density operator and density-density interactions with some full derivatives

\[ \mathscr{H}(\mathbf{r})=n_z\left(g_0+g_1\nabla^2\right)n_z-hn_x+g_{D,1}\nabla^2n_x+g_{D,2}\nabla^2n_z^2+\dots\]

where $g_{D,i}$ are undetermined constants that does not affect the Hamiltonian $H=\int\mathrm{d}^2\mathbf{r}\,\mathscr{H}$. We have only listed a few examples of the allowed full derivatives. The generator is expressed as

\[ \Lambda_m=P_m+K_m=\int\mathrm{d}^2\mathbf{r}\,Y_{l=1,m}(\mathbf{r})\mathscr{H}(\mathbf{r}).\]

To determine those constants, we consider another strategy by combining four fermion operators into $\mathrm{SO}(3)$ spin-1 operators. Similar to what we have done for Hamiltonian, we combine the fermion bilinears $\Delta_{lm}$.

\[ \Lambda_m=\sum_{\substack{l_1l_2m_1m_2}}\tilde{U}_{l_1l_2}\Delta^\dagger_{l_1m_1}\Delta_{l_2m_2}\begin{pmatrix} 2s-l_1&2s-l_2&1\\-m_1&m_2&m \end{pmatrix}+hn_{x,1m}\]

For $l_1\in2\mathbb{Z}$, the spin index in the pairing operator is anti-symmetrised ; For $l_1\in2\mathbb{Z}+1$, the spin index in the pairing operator is symmetrised. Therefore, $l_1-l_2\in2\mathbb{Z}$ for non-zero results. And since $|l_1-l_2|\leq 1$, we conclude $l_1=l_2$, so

\[ \Lambda_m=\sum_{\substack{lm_1m_2}}\tilde{U}_{l}\Delta^\dagger_{lm_1}\Delta_{lm_2}\begin{pmatrix} 2s-l&2s-l&1\\-m_1&m_2&m \end{pmatrix}+hn_{x,1m}\]

Here, $\tilde{U}_l$ are tuning parameters.

[Zhu 2022] Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization, Wei Zhu, Chao Han, Emilie Huffman, Johannes S. Hofmann, and Yin-Chen He, arXiv:2210.13482, Phys. Rev. X 13, 021009 (2023).
[Hu 2023Mar] Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Sphere, Liangdong Hu, Yin-Chen He, and Wei Zhu, arXiv:2303.08844, Phys. Rev. Lett 131, 031601 (2023).
[Han 2023Jun] Conformal four-point correlators of the 3D Ising transition via the quantum fuzzy sphere, Chao Han, Liangdong Hu, Wei Zhu, and Yin-Chen He, arXiv:2306.04681, Phys. Rev. B 108, 235123 (2023).
[Zhou 2023] The $\mathrm{SO}(5)$ Deconfined Phase Transition under the Fuzzy Sphere Microscope: Approximate Conformal Symmetry, Pseudo-Criticality, and Operator Spectrum, Zheng Zhou, Liangdong Hu, Wei Zhu, and Yin-Chen He, arXiv:2306.16435, Phys. Rev. X 14, 021044 (2024).
[Hu 2023Aug] Solving Conformal Defects in 3D Conformal Field Theory using Fuzzy Sphere Regularization, Liangdong Hu, Yin-Chen He, and Wei Zhu, arXiv:2308.01903, Nat. Commun. 15, 3659 (2024).
[Hofmann 2024] Quantum Monte Carlo Simulation of the 3D Ising Transition on the Fuzzy Sphere, Johannes S. Hofmann, Florian Goth, Wei Zhu, Yin-Chen He, and Emilie Huffman, arXiv:2310.19880, SciPost Phys. Core 7, 028 (2024).
[Han 2023Dec] Conformal Operator Content of the Wilson-Fisher Transition on Fuzzy Sphere Bilayers, Chao Han, Liangdong Hu, and Wei Zhu, arXiv:2312.04047, Phys. Rev. B 110, 115113 (2024).
[Zhou 2024Jan] The $g$-function and Defect Changing Operators from Wavefunction Overlap on a Fuzzy Sphere, Zheng Zhou, Davide Gaiotto, Yin-Chen He, Yijian Zou, arXiv:2401.00039, SciPost Phys. 17, 021 (2024).
[Hu 2024] Entropic $F$-function of 3D Ising conformal field theory via the fuzzy sphere regularization, Liangdong Hu, Wei Zhu, and Yin-Chen He, arXiv:2401.17362.
[Cuomo 2024] Impurities with a cusp: general theory and 3d Ising, Gabriel Cuomo, Yin-Chen He, Zohar Komargodski, arXiv:2406.10186.
[Zhou 2024Jul] Studying the 3d Ising surface CFTs on the fuzzy sphere, Zheng Zhou, and Yijian Zou, arXiv:2407.15914.
[Dedushenko 2024] Ising BCFTs from the fuzzy hemisphere, Mykola Dedushenko, arXiv:2407.15948.
[Fardelli 2024] Constructing the Infrared Conformal Generators on the Fuzzy Sphere, Giulia Fardelli, A. Liam Fitzpatrick, and Emanuel Katz, arXiv:2409.02998.
[Fan 2024] Note on explicit construction of conformal generators on the fuzzy sphere, Ruihua Fan, arXiv:2409.08257.
[Zhou 2024Oct] A new series of 3D CFTs with $\mathrm{Sp}(N)$ global symmetry on fuzzy sphere, Zheng Zhou, and Yin-Chen He, arXiv:2410.00087.