An Introduction to the Fuzzy Sphere

Introduction

Conformal Field Theory

Conformal field theory (CFT) is one of the central topics of modern physics. It refers to a field theory that is invariant under conformal transformations that preserve the angles between vectors. In spacetime dimension $d>2$, the global conformal symmetry transformations form a group $\mathrm{SO}(d+1,1)$, generated by translation, $\mathrm{SO}(d)$ rotation (Here we work in Euclidean signature. In Lorentzian signature it is the Lorentz transformation $\mathrm{SO}(1,d-1)$), dilatation (scale transformation), and special conformal transformation (SCT) Each CFT operator must transform under irreducible representations of rotation and dilatation. The representations are labelled by the $\mathrm{SO}(d)$ spin $l$ and scaling dimension $\Delta$, respectively. A special kind of operators that are invariant under SCT, called 'primaries,' deserve particular attention. By acting spatial derivatives on the primaries, their 'descendants' are obtained. Each operator in CFT is a linear combination of primaries and descendants. The conformal symmetry is the maximal spacetime symmetry (except supersymmetry) that a field theory can have. It gives powerful constraints on the property of the field theory. In particular, conformal symmetry uniquely determines the form of two-point (2-pt) and three-point (3-pt) correlation functions. The 3-pt correlator of three primary operators $\Phi_i,\Phi_j,\Phi_k$ contains a universal coefficient called the OPE coefficient $f_{\Phi_i\Phi_j\Phi_k}$. The collection of scaling dimensions and the OPE coefficients of primaries $\{\Delta_{\Phi_i},f_{\Phi_i\Phi_j\Phi_k}\}$ is called the conformal data. Theoretically, with full knowledge of the CFT data, an arbitrary correlation function of a CFT can be obtained. (The full knowledge is not often possible in practice, as the number of primaries is often infinite.)

CFT has provided important insights into various aspects of theoretical physics. In condensed matter physics, it has produced useful predictions about the critical phenomena. Many classical and quantum phase transitions are conjectured to have emergent conformal symmetry in the infrared (IR), i. e. at long wavelength or low energy. The universal critical exponents are directly determined by the scaling dimensions of the primary operators. E. g., in the 3D Ising transition that spontaneously breaks $\mathbb{Z}_2$ symmetry, most critical exponents are given by the scaling dimensions of the lowest $\mathbb{Z}_2$-odd operator $\sigma$ and $\mathbb{Z}_2$-even operator $\epsilon$, such as

\[ \eta=2\Delta_\sigma-1,\qquad\nu=\frac{1}{3-\Delta_\epsilon}.\]

CFT is also closely related to string theory and quantum gravity in high-energy physics. In the string theory, CFT describes the 2D worldsheet ; in quantum gravity, there is a conjectured duality between the gravity theory in $(d+1)$-dimensional anti-de Sitter (AdS) space in the bulk and a $d$-dimensional CFT on the boundary. Moreover, CFT plays an important role in our understanding of quantum field theories. It describes many fixed points in the RG flow, and many QFTs can be seen as CFTs with perturbations. It also helps us understand how physics changes under a change of scale and reveals some fundamental structures of the RG flow.

In 2D CFTs, besides the global conformal symmetry $\mathrm{SO}(3,1)$, there also exists an infinite-dimensional local conformal symmetry. Altogether, they form the Virasoro algebra. The infinite-dimensional conformal algebra has made many theories exactly solvable, especially the rational theories such as the minimal models and, more generally, the Wess-Zumino-Witten (WZW) theories. However, going to the higher dimensions, the CFTs are much less well-studied due to a much smaller conformal group. The existing methods include numerical conformal bootstrap and Monte Carlo lattice simulations. Numerical bootstrap bounds the conformal data by making use of consistency conditions such as reflection positivity and crossing symmetry, together with some information of the CFT such as the global symmetry and a certain amount of assumptions. It has achieved great success in 3D Ising, $\mathrm{O}(N)$ Wilson-Fisher, Gross-Neveu-Yukawa CFTs, etc. On the other hand, one can study a CFT by constructing a lattice model that goes through a phase transition in the corresponding universality class, and study the phase transition by Monte Carlo simulation. This method has achieved success in many phase transitions assuming conformal symmetry, e. g. the 3D Ising model. However, the extraction of universal data usually involves complicated and expensive finite-size scaling, and only the lowest few CFT operators can be accessed.

Among these higher dimensional CFTs, we mainly focus on $d=3$, as many Lagrangians in $d\ge 4$ flow to free theories.

Fuzzy Sphere

In addition to these existing approaches, the \emph{`fuzzy sphere regularisation'} has recently emerged as a new powerful method to study 3D CFTs. It involves studying interacting electrons moving on a sphere under the influence of a magnetic monopole at its centre.

The idea begins with putting an interacting quantum Hamiltonian on a 2-sphere $S^2$. This geometry preserves the full rotation symmetry (on the contrary, lattice models often only preserve a discrete subgroup). Moreover, when the system is tuned to a critical point or critical phase, combined with the time evolution direction, the system is described by a quantum field theory living on a generalised cylinder $S^2\times\mathbb{R}$, a manifold that is conformally equivalent to flat spacetime through the Weyl transformation

\[ (\mathbf{r},\tau)\in S^2\times\mathbb{R}\ \longmapsto\ e^{\tau/R}\hat{\mathbf{n}}\in\mathbb{R}^3,\]

where $R$ is the radius of the sphere and $\hat{\mathbf{n}}$ is the unit vector corresponding to $\mathbf{r}$. This conformal transformation maps each time slice of the cylinder to a concentric sphere in the flat spacetime.

Thanks to the conformal flatness that is not owned by other manifolds (e. g., a lattice model with periodic boundary condition lives on the torus $T^2$, which is not conformally flat), we can make use of some nice properties of conformal field theories. The most important one is the state-operator correspondance. Specifically, there is a one-to-one correspondence between the eigenstates of the critical Hamiltonian on the sphere and the CFT operators. One can colloquially understand the state $|\Phi\rangle$ as the insertion of the corresponding operator $\Phi(0)$ at the origin point into the vacuum $|0\rangle$ : $|\Phi\rangle=\hat{\Phi}(0)|0\rangle$. The state and its corresponding operator have the same $\mathrm{SO}(3)$ spin and representation under global symmetry. More importantly, as the Weyl transformation maps the Hamiltonian $H$ that generates the time translation on the cylinder to the dilatation $D$ on the flat spacetime, the excitation energy of a state $|\Phi\rangle$ is proportional to the scaling dimension of the corresponding operator $\Delta_\Phi$

\[ E_\Phi-E_0=\frac{v}{R}\Delta_\Phi,\]

where $E_0$ is the ground state energy, $R$ is the radius of the sphere, and $v$ is the speed of light that is dependent on the microscopic model and is the same for every state. With this property, one can calculate the scaling dimensions simply by solving the energy spectrum of the quantum Hamiltonian without doing complicated finite-size scalings, and one can obtain the OPE coefficients simply by taking the inner product of a local operator.

Although the quantum Hamiltonians on a sphere enjoy the full rotation symmetry and the property of state-operator correspondence, it is difficult to put a lattice on the sphere due to the curvature (in particular the non-zero Euler characteristic), especially to recover an $\mathrm{SO}(3)$-symmetric thermodynamic limit. An alternative way is to fuzzify the sphere. We consider charged free particles moving on a sphere with a magnetic monopole with a flux $4\pi s$ ($s\in\mathbb{Z}/2$) placed at its centre. The monopole exerts a uniform magnetic field on the sphere, which modifies the single-particle Hamiltonian and the single-particle eigenstates. Now, the single particle eigenstates form highly degenerate spherical Landau levels. The lowest Landau level has a degeneracy $(2s+1)$. By setting the single-particle gap to be the leading energy scale, and projecting onto the lowest Landau level, we obtain a finite Hilbert space. For the purpose of numerical simulation, the system is analogous to a length-$(2s+1)$ chain with long-range interaction, where different Landau level orbitals behave like the lattice sites. The difference is that the $(2s+1)$ orbital forms a spin-$s$ representation of the $\mathrm{SO}(3)$ rotation group, and in this way, the continuous rotation symmetry is preserved. By putting multiple flavours on an orbital and adding interactions, various 3D CFTs can be realised. The interaction Hamiltonians are designed through matching the global symmetry and phase diagram. The word 'fuzzy' means the non-commutativity, in our case, due to the presence of magnetic field. The non-commutativity provides a natural length scale which serves as a UV regulator of the quantum field theory. The radius of the sphere scales as $R\sim\sqrt s$. The thermodynamic limit can be taken as $s\to\infty$, and we then recover a regular sphere without non-commutativity.

The power of this approach has been first 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 then been extended to accessing various conformal data such as the OPE coefficients, correlation functions, entropic $F$-function, conformal generators and the cross-cap coefficients, developing techniques to improve the numerical precision, such as quantum Monte Carlo, conformal perturbation and the finite-size scaling of the ground-state energy, realising various 3D CFTs such as the $\mathrm{SO}(5)$ and $\mathrm{O}(4)$ deconfined criticality, Wilson-Fisher CFTs, $\mathrm{Sp}(N)$-symmetric CFTs, exploring fractional quantum Hall transitions, such as the Ising CFT on the FQHE states, the confinement transition of the $\nu=1/2$ bosonic Laughlin state, the free Marjorana fermion theory, and the transition between bosonic Pfaffian and the Halperin 220 state, studying conformal defects and boundaries such as the magnetic line defect, various conformal boundaries in the 3D Ising CFT, and lower-dimensional CFTs like 2D Ising on the fuzzy circle. In the following sections, we shall review the existing works, technical details and numerical methods.

Review of Existing Works

In this section, we review the existing works related to the fuzzy sphere.

The pioneering work

[Zhu 2022] Uncovering conformal symmetry in the 3d Ising transition : state-operator correspondence from a quantum fuzzy sphere regularisation, Wei Zhu, Chao Han, Emilie Huffman, Johannes S. Hofmann, and Yin-Chen He, arXiv:2210.13482, Phys. Rev. X 13, 021009 (2023).

This work first proposes the idea of the fuzzy sphere and applies it to a pedagogical example of the 3D Ising CFT. Zhu et al. construct a model with two flavours of fermions that resemble the spin-up and spin-down in the lattice transverse-field Ising model. At half-filling, one can colloquially think that a spin degree of freedom lives on each orbital. The Hamiltonian contains a density-density interaction (here the density operator refers to a local fermion bilinear) that resembles the Ising ferromagnetic interaction and a polarising term that resembles the transverse field. By tuning the ratio between the two terms, a transition between quantum Hall ferromagnet (a two-fold degenerate state where either of the two flavours is completely occupied) and paramagnet (a one-fold degenerate state where the superposition of the two flavours at each orbital is occupied) occurs. This transition spontaneously breaks a $\mathbb{Z}_2$ symmetry and falls into the Ising criticality. They then make use of a unique feature of spherical models described by CFT — state-operator correspondence — at the critical point to extract the scaling dimensions of the scaling local operators. They find evidence for the conformal symmetry, including that (1) there exists a conserved stress tensor with $\Delta=3$ (which is used as the calibrator), and (2) all the levels can be classified into conformal multiplet where the spacings between operators' scaling dimensions are very close to integer. This is one of the first numerical evidence that the 3D Ising transition has emergent conformal symmetry. More remarkably, the scaling dimensions of primaries such as $\sigma,\epsilon,\epsilon'$ are already very close to the most accurate known value by numerical bootstrap with an error within $1.2\%$ at a small system size with the number of orbitals $N_m=16$, for which the computational cost is comparable to a $4\times4$ lattice system. The structure of the Ising CFT operator spectrum already starts to show up at an even smaller system size $N_m=4$. All these clues point towards a curious observation that the fuzzy sphere suffers from a remarkably small finite-size effect. The detail for the construction of models is presented in the Sections 'Density-Density Interaction' and 'Interaction in Terms of Pseudopotentials', and the detail for the analysis of the spectrum is presented in Section 'Operator Spectrum and Search for Conformal Point'.

This seminal work opens a new avenue for studying 3D conformal field theories. After that, most of the research on the fuzzy sphere can roughly be categorised into the following directions :

1. Accessing various conformal data,
2. Developing techniques to improve the numerical precision,
3. Realising various 3D CFTs,
4. Exploring fractional quantum Hall transitions and
5. Studying conformal defects and boundaries.

Accessing Various Conformal Data

The first direction is to develop methods to calculate various universal data of 3D CFTs on the fuzzy sphere. Typically, these methods are tested with the simplest example of the 3D Ising CFT. For many of those CFT data, the fuzzy sphere is the first non-perturbative method to access them ; for the others, the 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, conformal generators, the cross-cap coefficients.

OPE coefficients

[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).

Apart from the operator spectrum, a wealth of CFT data can be obtained from the local operators. This work studies the local observables on the fuzzy sphere, including the density operators and certain four-fermion operators. These observables can be expressed as the linear combination of local scaling operators in the CFT. After a finite-size scaling with the data from different system sizes, the subleading contribution can be subtracted, and only the leading contribution is left. In this way, the lowest primaries in Ising CFT in each symmetry sector, viz. the $\mathbb{Z}_2$-odd $\sigma$ and the $\mathbb{Z}_2$-even $\epsilon$, can be realised. The OPE coefficients are then evaluated by taking the inner product of a fuzzy sphere local observable with two CFT states $\langle\Phi_1|\Phi_2(\mathbf{r})|\Phi_3\rangle$. Hu et al. compute 17 OPE coefficients of low-lying CFT primary fields with high accuracy, including four that have not been reported before. The rest are consistent with numerical bootstrap results. It is also worth noting that this work starts to apply DMRG to the fuzzy sphere. The maximal system size is increased from $N_m=18$ by ED to $N_m=48$ by DMRG. The detail for calculating the OPE coefficient is presented in Section 'Local Observables'.

Correlation functions

[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).

In addition to the OPE coefficients, the local observables can also be used to calculate correlation functions. By taking the inner product of two local observables (density operators) at a time displacement $\langle\Phi_1|\Phi_2(\mathbf{r}_0)\Phi_3(\mathbf{r},\tau)|\Phi_4\rangle$ with two CFT states, a general 4-pt function can be calculated. In practice, this piece of CFT data cannot be derived from the scaling dimensions and the OPEs due to the existence of infinitely many primaries. Han et al. calculate the 4-pt functions in the 3D Ising CFT with DMRG. A non-trivial check of conformality, the crossing symmetry, is verified for the correlator $\langle\sigma\sigma\sigma\sigma\rangle$. The special case — 2-pt functions by taking $\Phi_1=\Phi_4=\mathbb{I}$ — is also studied and compared with the expected results by conformal symmetry. The detail for calculating the correlation functions is presented in the Section 'Local Observables'.

Entropic $F$-function

[Hu 2024] Entropic $F$-function of 3d Ising conformal field theory via the fuzzy sphere regularisation, Liangdong Hu, Wei Zhu, and Yin-Chen He, arXiv:2401.17362, Phys. Rev. B 111, 155151 (2025).

Beyond the correlators of local operators, a wealth of information can be learnt from the entanglement entropy and entanglement spectrum. A remarkable quantity is called the $F$-function, which is defined through the scaling behaviour of the entanglement entropy. Specifically, consider a quantum system that lives on $\mathbb{R}^2$. A circle with radius $R_d$ divides the system into inner part $A$ and outer part $B$. The entanglement entropy is defined and expected to scale with $R_d$ as

\[ S_A(R_d)=-\operatorname{tr}_A\rho\log\rho=\alpha R_d/\delta-F,\]

where $\delta$ is a UV-regulator. The constant piece is known as the $F$-function of a 3D CFT. The $F$-function is proved to be RG-monotonic, i. e., along a renormalisation group flow from UV to IR, the value of $F$-function is non-increasing, analogous to the central charge in 2D CFTs. Despite its importance, it has never been calculated before through non-perturbative approaches in interacting 3D CFTs. This work has performed the first non-perturbative computation of $F$ function for the 3D Ising CFT on the fuzzy sphere. The sphere is cut in the real space into two crowns along a latitude circle $\theta$, and the entanglement entropy $S_A(\theta)$ as a function of $\theta$ is calculated. The $F$-function is extracted from the $S_A(\theta)$ in the vicinity of the equator, and the result yields $F=0.0612(5)$ after a finite-size scaling.

Conformal generators

[Fardelli 2024] Constructing the infrared conformal generators on the fuzzy sphere, Giulia Fardelli, A. Liam Fitzpatrick, and Emanuel Katz, arXiv:2409.02998, SciPost Phys. 18, 086 (2025).

[Fan 2024] Note on explicit construction of conformal generators on the fuzzy sphere, Ruihua Fan, arXiv:2409.08257.

Within the generators of conformal symmetry, the $\mathrm{SO}(3)$ rotation and the dilatation are manifest and act as rotation and time translation on the fuzzy sphere. The rest two, viz. translation $P^\mu$ and special conformal transformation (SCT) $K^\mu$, need to be emergent in the IR at the conformal point. It is worthwhile to construct these IR generators by the UV operators on the fuzzy sphere. These works investigate such construction with the help of stress tensor $T^{\mu\nu}$. The time component $T^{\tau\tau}$ of stress tensor equals the Hamiltonian density $\mathscr{H}$ and it integrates into the generator $\Lambda^\mu=P^\mu+K^\mu=\int\mathrm{d}^2\mathbf{r}\,2n^\mu\mathscr{H}$. The action of this generator sends a scaling operator to other operators in the same multiplet, with the number of spatial derivatives increased or decreased by one. Fardelli et al. and Fan calculate the matrix elements of the generators $\Lambda^\mu$ and find good agreement with the theoretical values in the CFT, which is another non-trivial verification of conformal symmetry. Furthermore, the separate generators $P^\mu$ and $K^\mu$ can be obtained by considering the commutator $[H,\Lambda^\mu]$, which is useful in determining the primaries. The detail for constructing the conformal generators is presented in Section 'Conformal Generators'.

The crosscap coefficients

[Dong 2025] Numerical extraction of crosscap coefficients in microscopic models for $(2+1)$D conformal field theory, Jia-Ming Dong, Yueshui Zhang, Kai-Wen Huang, Hong-Hao Tu, and Ying-Hai Wu, arXiv:2507.20005.

This work focuses on the 3D CFTs on the space-time manifold of the the real projective space $\mathbb{R}\mathrm{P}^3$ obtained by identifying antipodal points on $S^2$. On $\mathbb{R}\mathrm{P}^3$, one-point functions of scalar primary fields are generally non-vanishing and encodes the 'cross-cap coefficients.' Dong et al. extracts the cross-cap coefficients of the 3D Ising CFT through simulating microscopic models on the lattice models on polyhedrons and continuum models in Landau levels and entangle the degrees of freedom at anti-podal points in Bell-type states.

Developing Techniques to Improve the Numerical Precision

There has also been techniques to improve the precision through accessing larger system size and detailed analysis of finite-size data, such as quantum Monte Carlo simulation, conformal perturbation, and finite-size scaling the ground-state energy.

Quantum Monte Carlo on the fuzzy sphere

[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).

Until this work, the numerical methods that have been applied to the fuzzy sphere include exact diagonalisation (ED) and density matrix renormalisation group (DMRG). Hofmann et al. further present the numerical studies of the fuzzy sphere with quantum Monte Carlo (QMC) simulation, which is known for its potential for studying criticalities in $(2+1)$ dimensions at larger system size. Specifically, they make use of the determinant quantum Monte Carlo (DQMC) method that converts the simulation of fermions into the simulation of bosonic auxiliary fields. To overcome the sign problem, they consider two copies of the original model and construct the Ising CFT on a 4-flavour model. They determine the lowest energy spectra within each symmetry sector by calculating the time-displaced correlation functions. They also calculate the equal-time correlation functions and compare them with the 2-pt functions of CFT.

Conformal perturbation

[Läuchli 2025] Exact diagonalization, matrix product states and conformal perturbation theory study of a 3d Ising fuzzy sphere model, Andreas M. Läuchli, Loïc Herviou, Patrick H. Wilhelm, and Slava Rychkov, SciPost Phys. 19, 076 (2025), arXiv:2504.00842.

The energy spectrum calculated numerically at a finite size does not coincide with that of the CFT. Part of the finite-size correction comes from the higher irrelevant operators that are not precisely tuned to zero (e. g., in the Ising CFT, the irrelevant operators include $\epsilon', C_{\mu\nu\rho\sigma}, T'_{\mu\nu}$, etc., and the lowest singlets $\epsilon$ and $\epsilon'$ are tuned away through the two parameters). These irrelevant operators exert perturbations on the states and their energies. This work captures this kind of correction using the conformal perturbation theory. By making use of the fact that the corrections from an irrelevant operator on the energy of the primary and its descendants are not independent, the coefficients of the irrelevant operators can be fitted and their corrections can be removed.

The conformal perturbation theory is first studied on the 2D CFTs and the icosahedron and then applied to the fuzzy sphere. This work opens up a new route to improving the precision of scaling dimensions on the fuzzy sphere by making better use of the existing data. The method to partly remove the finite-size correction through conformal perturbation theory is widely used by the following works.

Finite-size scaling of the ground-state energy

[Wiese 2025] Locating the Ising CFT via the ground-state energy on the fuzzy sphere, Kay Joerg Wiese, arXiv:2510.09482.

This work proposes a new approach to locate the phase-transition line from a finite-size scaling analysis of its ground-state energy with the example of the 3D Ising CFT. Wiese performs a finite-size scaling with the ansatz $E_{\text{GS}}=E_0 R^2+E_1+E_{3/2}R^{-1}$ and identify the minima of $E_{3/2}/E_0$ as the critical curve and the 'sweet spot.'

Realising Various 3D CFTs

The third direction is to study various other CFTs beyond 3D Ising. The fuzzy sphere has revealed lots of new information about these theories~; the previously known results are also consistent with the fuzzy sphere. So far, three classes of CFTs are widely studied on the fuzzy sphere :

1. the free-scalar and Wilson-Fisher CFTs realised as Heisenberg bilinear and truncated quantum rotor model,
2. CFTs with $\mathrm{Sp}(N)$ global symmetry related to the non-linear $\sigma$ models (NLSM) with a Wess-Zumino-Witten topological term, including $\mathrm{SO}(5)$ deconfined criticality and $\mathrm{O}(4)$ deconfined criticality through a symmetry-breaking perturbation, and a series of new theories with $\mathrm{Sp}(N)$ symmetry and
3. fractional quantum Hall transitions, which will be discussed in more detail in the next section.

Other accessible CFTs include the 3-state Potts model and the Yang-Lee non-unitary CFT.

Free-Scalar and Wilson-Fisher CFTs

The free real scalar theory

[He 2025Jun] Free real scalar CFT on fuzzy sphere : Spectrum, algebra and wavefunction ansatz, arXiv:2506.14904.

[Taylor 2025] Conformal scalar field theory from Ising tricriticality on the fuzzy sphere, Joseph Taylor, Cristian Voinea, Zlatko Papić, Ruihua Fan, arXiv:2506.22539.

This work introduces a simple model to realise the free real scalar CFT on the fuzzy sphere that is structurally similar to the 3D Ising CFT. A weakly-broken $\mathrm{U}(1)$ symmetry in the fuzzy sphere model realises the $\mathbb{R}$-shift symmetry of the free scalar in the thermodynamic limit, so that the fix point can be accessed with only a single tuning parameter. He numerically demonstrates that our model correctly reproduces the operator spectrum, correlation functions, and, crucially, the harmonic oscillator algebra of the real scalar CFT. He generalises the Girvin-MacDonald-Platzman algebra to the fuzzy sphere algebra of the density operators, which is potentially useful for defining quantum field theories on non-commutative geometries. He proposes a wavefunction ansatz for the ground states which exhibit remarkable agreement with the CFT ground state wavefunctions of the fuzzy sphere model.

The bilayer Heisenberg transition

[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).

The $\mathrm{O}(N)$ Wilson-Fisher (WF) theories are probably one of the most studied theories for 3D criticalities with a wide range of applications. Specifically, this work studies the $\mathrm{O}(3)$ WF CFT on the set-up of a bilayer Heisenberg model. The construction involves two copies of $\mathrm{SU}(2)$ ferromagnet with altogether four flavours. Briefly speaking, the model contains two competing terms : (1) a $\mathrm{SU}(2)$ ferromagnetic interaction which favours a Heisenberg ferromagnetic phase where each of the two copies is half-filled and the symmetry-breaking order parameter lives on a $S^2$ manifold, (2) a polarising term which favours one of the two copies being completely filled, corresponding to a Heisenberg paramagnet. The transition between these two phases falls into the $\mathrm{O}(3)$ Wilson-Fisher universality. Through the energy spectrum at the transition, Han et al. provide evidence that $\mathrm{O}(3)$ Wilson-Fisher fixed point exhibits conformal symmetry, as well as revealing a wealth of information about the CFT, e. g. the instability to cubic anisotropy. They also calculate several OPE coefficients.

The $\mathrm{O}(N)$ Wilson-Fisher CFT

[Dey 2025] Conformal data for the $\mathrm{O}(3)$ Wilson-Fisher CFT from fuzzy sphere realization of quantum rotor model, Arjun Dey, Loic Herviou, Christopher Mudry, Andreas Martin Läuchli, arXiv:2510.09755.

Although an instance — the $\mathrm{O}(3)$ WF — has been realised in the Heisenberg bilayer, seeking a general realisation of $\mathrm{O}(N)$ WF theories on the fuzzy sphere is helpful as a foundation to study the interacting scalar theories as well as conformal defects and boundaries. This work proposes such a construction analogous to a truncated rotor model. The set-up contains altogether $(N+1)$ flavours with total filling $\nu=1$, where one flavour $c_0$ is $\mathrm{O}(N)$ singlet and the rest $N$ flavours $c_i$ transform as $\mathrm{O}(N)$ vector. The scalar field is realised as the bilinear $\phi_i=c_i^\dagger c_0+c_0^\dagger c_i$, and the Hamiltonian contains (1) a particle density interaction, (2) a $\phi$ density interaction, and (3) a relative chemical potential of the singlet flavour. Tuning the chemical potential realises a phase transition between a paramagnetic phase with $c_0$ fully filled and an $\mathrm{O}(N)$ symmetry-breaking phase where electrons are in a superposed state between a vector flavour and the singlet flavour.

Dey et al. study the $N=3$ instance using ED and DMRG. They locate the critical point using the conformal perturbation. They obtain scaling dimensions from finite-size spectra and operator product expansion coefficients through conformal perturbation. The results are benchmarked against conformal bootstrap and perturbative calculations.

NLSM-WZW with Symplectic Symmetry

The $\mathrm{SO}(5)$ deconfined criticality

[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).

The first theory besides Ising CFT to which fuzzy sphere is applied is the $\mathrm{SO}(5)$ deconfined quantum critical point (DQCP). Deconfined quantum critical point (DQCP) is one of the pioneering examples of phase transitions beyond Landau paradigm. It has led to numerous theoretical surprises, including the emergent $\mathrm{SO}(5)$ symmetry and the duality between interacting theories. Despite extensive studies over the past two decades, its nature remains controversial. Numerical simulations have shown no signal of discontinuity, but abnormal scaling behaviours have been observed. A plausible proposal to reconcile the tension is that DQCP is pseudocritical, i. e. a weakly first-order phase transition that has approximate critical behaviour, and is controlled by a pair of complex fixed points very close to the pseudocritical region.

The DQCP can be conveniently studied on the fuzzy sphere by constructing a non-linear sigma model (NLSM) on target space $S^4$ with a level-1 topological Wess-Zumino-Witten (WZW) term, which serves as a dual description of the DQCP with an exact $\mathrm{SO}(5)$ symmetry. The idea is to construct a 4-flavour model with symmetry $\mathrm{Sp}(2)/\mathbb{Z}_2=\mathrm{SO}(5)$ ($\mathbb{Z}_2$ means to gauge the pseudoreal representations). At half-filling, it can be described by a NLSM on the Grassmannian $\tfrac{\mathrm{Sp}(2)}{\mathrm{Sp}(1)\times\mathrm{Sp}(1)}\cong S^4$ and the WZW level can be matched. This work provides evidence that the DQCP exhibits approximate conformal symmetry. Zhou et al. have identified 19 conformal primaries and their 82 descendants. Furthermore, by examining the renormalisation group flow of the lowest symmetry singlet, they demonstrate that the DQCP is more likely pseudocritical, with the approximate conformal symmetry plausibly emerging from nearby complex fixed points. Several works appear later to follow up.

The $\mathrm{O}(4)$ deconfined criticality

[Yang 2025Jul] Conformal Operator Flows of the Deconfined Quantum Criticality from $\mathrm{SO}(5)$ to $\mathrm{O}(4)$, Shuai Yang, Liang-Dong Hu, Chao Han, Wei Zhu, Yan Chen, arXiv:2507.01322.

Apart from the $\mathrm{SO}(5)$ deconfined criticality, one can reach the $\mathrm{O}(4)$, or easy-plane DQCP by adding a perturbation in the $\mathrm{SO}(5)$ symmetric tensor representation that breaks the global symmetry to $\mathrm{O}(4)$. The $\mathrm{O}(4)$ DQCP appears in several lattice models and has several gauge theory descriptions, e. g. QED with two flavours of Dirac fermions. This work studies the renormalisation group flow from the $\mathrm{SO}(5)$ to the $\mathrm{O}(4)$ DQCP and traces the $\mathrm{O}(4)$ operators back to the $\mathrm{SO}(5)$ operators.

A series of new $\mathrm{Sp}(N)$-symmetric CFTs

[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, Phys. Rev. Lett. 135, 026504 (2025).

The quest to discover new 3D CFTs has been intriguing for physicists. A virgin land on this quest is the parity-breaking CFTs. In 3D, the Chern-Simons-matter theories stand out as the most well-known and possibly the only known type of parity-breaking CFTs. The fuzzy sphere is a promising platform for studying these theories. This work makes a concrete construction by generalising the DQCP to the WZW-NLSM on the target space of a general symplectic Grassmannian

\[ \frac{\mathrm{Sp}(N)}{\mathrm{Sp}(M)\times\mathrm{Sp}(N-M)}.\]

Several candidate Chern-Simons-matter theories are known to exist on its phase diagram with $N$ flavour of gapless bosons or fermions coupled to a non-Abelian (viz. $\mathrm{Sp}(1)$, $\mathrm{Sp}(2)$, etc.) Chern-Simons gauge field. On the fuzzy sphere, this WZW-NLSM can be realised by a $2N$ layer model with $\mathrm{Sp}(N)$ flavour symmetry, and $2M$ out of the $2N$ layers are filled. Zhou et al. numerically verify the emergent conformal symmetry by observing the integer-spaced conformal multiplets and studying the finite-size scaling of the conformality.

Other CFTs on the Fuzzy Sphere

The 3-state Potts model

[Yang 2025Jan] Microscopic study of 3d Potts phase transition via fuzzy sphere regularisation, Shuai Yang, Yan-Guang Yue, Yin Tang, Chao Han, Wei Zhu, and Yan Chen, arXiv:2501.14320

The Potts models describe transitions that spontaneously break $S_Q$ symmetries where $Q\in\mathbb{Z}$ is known as the number of states. In 2D, the transitions with $Q\leq Q_c=4$ are continuous and captured by CFTs, while $Q>Q_c$ are first order. Specifically, 2D 5-state Potts transition is pseudocritical and described by a pair of complex CFTs in its vicinity in a similar manner with the conjectured $\mathrm{SO}(5)$ DQCP. In 3D, the 3-state Potts model is found to be first-order. This work constructs a 3-flavour model on the fuzzy sphere with $S_3$ permutation symmetry among flavours. The interacting Hamiltonian resembles the Ising model. Interestingly, Yang et al. find out that the transition point of the 3D 3-state Potts model, despite being probably first-order, exhibits approximate conformal symmetry, indicating that there might be an underlying CFT describing it. However, it is difficult to determine the nature of the transition from the operator spectrum (specifically, from the relevance of the second singlet $\epsilon'$) due to the complicated finite-size effect.

Yang-Lee non-unitary CFT

[Fan 2025] Simulating the non-unitary Yang-Lee conformal field theory on the fuzzy sphere, Ruihua Fan, Junkai Dong, and Ashvin Vishwanath arXiv:2505.06342.

[Arguello Cruz 2025] Yang-Lee quantum criticality in various dimensions, Erick Arguello Cruz, Igor R. Klebanov, Grigory Tarnopolsky, and Yuan Xin, arXiv:2505.06369.

[Elias Miro 2025] Flowing from the Ising model on the fuzzy sphere to the 3d Lee-Yang CFT, Joan Elias Miro, Olivier Delouche, JHEP 10 (2025) 037, arXiv:2505.07655.

The Yang-Lee singularity is one of the simplest non-unitary CFTs. It is a natural extension of the Ising CFT triggered by the $i\sigma$ deformation, and plays an important role in understanding order-disorder transformation. While the 2D Yang-Lee theory is described by $M(2,5)$ minimal model and 3D Yang-Lee CFT can be solved perturbatively up to five-loops, Fan et~al., Arguello Cruz et~al., and Elias Miro et~al. presents the non-perturbative results of the Yang-Lee CFT on the fuzzy sphere. They have studied the operator spectrum, the OPE coefficients, and the RG flow from the Ising CFT to the Yang-Lee CFT. They have also shown how the finite-size effects can be controlled by finite-size scaling and conformal perturbation.

Exploring Fractional Quantum Hall Transitions

The lowest Landau level at fractional filling realises various topological orders (TO) with long-range entanglement and anyon excitations. E. g., the Laughlin states with filling $\nu=1/k$ realise the Abelian topological orders captured by Chern-Simons theories $\mathrm{U}(1)_{-k}$, the Jain sequence with $\nu=p/(mp+1)$ realises the Abelian topological orders with composite fermion descriptions, and the Read-Rezayi sequence with $\nu=p/(mp+2)$, as a natural extension of the Moore-Read states, realises in general a series of non-Abelian topological orders.

The fuzzy sphere provides a suitable platforms to study the transitions between or out of these fractional quantum Hall (FQH) states. Theoretically, these transitions are often described by Chern-Simons theories coupled to matter, many of which are conjectured to be conformal with enhanced symmetries and field theory dualities. Experimentally, the Moir\'e materials provide an exciting opportunity to study these transitions. The critical points can be conveniently reached by tuning many knobs like potential amplitude.

The exploration starts at realising the Ising CFT on the background of a FQHE state, for which the topological order stays the same across the transition. The transition between $\nu=1/2$ bosonic Laughlin state and a $\nu=2$ fermionic Laughlin state, realising the confinement transition of $\nu=1/2$ bosonic Laughlin state, is the first fractional quantum Hall transition on the fuzzy sphere. Using similar strategy, the phase diagram between fermionic integer quantum Hall and bosonic Laughlin state realising two transitions of free Majorana fermion and gauged Ising CFT, and the transition between bosonic Pfaffian and Halperin 220 state, are also studied.

Ising CFT from FQHE state

[Voinea 2024] Regularising 3d conformal field theories via anyons on the fuzzy sphere, arXiv:2411.15299, Phys. Rev. X 15, 031007 (2025).

Until this work, all the constructions of CFTs on the fuzzy sphere were based on the quantum Hall ferromagnet. Specifically, in the absence of the interaction, an integer number of the lowest Landau levels are fully occupied. This state has a finite charge gap that guarantees that the gapless spin degree of freedom does not strongly couple with the charge degree of freedom when one adds the interactions.

Voinea et al. further explore the possibility of constructing CFTs on other states with charge gap — in particular, the Haldane-Laughlin states that capture the fractional quantum Hall effect (FQHE). Specifically, they study the fermionic LLL at fillings of $\nu=1/3$ and $1/5$. The model Hamiltonian contains (1) a dominant projection term that puts the ground state on the Haldane-Laughlin state, and (2) an interaction term as a perturbation that drives the Ising-type phase transition. They show that the energy spectra at the critical point exhibit conformal symmetry. Notably, they also make the construction with respect to the bosonic LLL at a filling of $\nu=1/2$.

The confinement transition of $\nu=1/2$ bosonic Laughlin state

[Zhou 2025Jul] Chern-Simons-matter conformal field theory on fuzzy sphere: Confinement transition of Kalmeyer-Laughlin chiral spin liquid, Zheng Zhou, Chong Wang, Yin-Chen He, arXiv:2507.19580.

This work studies one of the simplest instances of Chern-Simons-matter theories : one complex critical scalar coupled to a $\mathrm{U}(1)_2$ Chern-Simons gauge field and its three other dual Lagrangian descriptions. They describe the transition between a $\nu=1/2$ bosonic Laughlin state and a trivially gapped phase. This transition appears in various contexts in condensed~matter physics, viz. Kalmeyer-Laughlin chiral spin liquid, anyon superconductivity, Feshbach resonance, etc.

Zhou et al. realise this theory on a set-up with fermion-boson mixture containing two flavours of fermion and one flavour of bosons carrying electric charge $Q_b=2Q_f$. Tuning the relative chemical potential induces a transition between a $\nu_f=2$ fermionic integer quantum Hall state and a $\nu_b=1/2$ bosonic fractional quantum Hall state. They show that the transition is continuous and has emergent conformal symmetry. The operator spectrum has exactly one relevant singlet with scaling dimension $\Delta_S=1.52(18)$, signature of a critical point. This work opens up the possiblity of using the fuzzy sphere to study transitions between distinct topological order.

The free Majorana fermion theory

[Zhou 2025Sep] Free Majorana fermion meets gauged Ising conformal field theory on the fuzzy sphere, Zheng Zhou, Davide Gaiotto, Yin-Chen He, arXiv:2509.08038.

The CFTs realised on the fuzzy sphere until this work contains only bosonic CFT operators. The microscopic fermions on the fuzzy sphere have strong non-commutativity, transform under projective representations of the sphere rotation symmetry, and thus cannot flow to CFT operators. This work overcomes this challenge by constructing a boson-fermion mixed setup with both microscopic bosons $b$ and fermions $f$ with the same electric charge and let their angular momenta differ by $1/2$. Their bilinears $b^\dagger f$ or $f^\dagger b$ can be used to realise fermionic local operators in a CFTs.

Zhou et al. allow conversion between two bosons and two fermions and set the total filling $\nu=1$. On the phase diagram, there are three gapped phases, viz. a fermionic integer quantum Hall phase, an $f$-wave chiral topological superconductor, and a bosonic Pfaffian phase. They are separated by two continuous transitions described respectively by a massless free Majorana fermion and a gauged Ising CFT.

The transition between bosonic Pfaffian and Halperin 220

[Voinea 2025] Critical Majorana fermion at a topological quantum Hall bilayer transition, Cristian Voinea, Wei Zhu, Nicolas Regnault, Zlatko Papić, arXiv:2509.08036.

The fuzzy sphere can be used to answer long-standing questions on fractional quantum Hall transitions. An example is the transition between the bosonic Moore-Read Pfaffian and the Halperin 220 state on a bilayer bosonic system, which has been predicted to be described by a massless free Majorana fermion with the fermion parity $\mathbb{Z}_2$ symmetry gauged, but lacks simulations. This work identifies the low-energy spectrum with the gauged Majorana fermion.

Studying Conformal Defects and Boundaries

Apart from the bulk CFTs, the fuzzy sphere can also be used to study their conformal defects and boundaries. Deforming a CFT with interactions living on a sub-dimensional defect may trigger a RG flow towards a non-trivial interacting IR fixed point. A defect IR theory with a smaller conformal symmetry is called a defect CFT. The dCFTs have rich physical structures, such as defect operators and bulk-to-defect correlation functions. Moreover, a bulk CFT can flow to several different dCFTs. Similarly, deformation on the boundary may trigger a flow towards a boundary CFT (bCFT). So far, the accessible defects/boundaries include the magnetic line defect of the 3D Ising CFT, in particular, its defect operator spectrum, correlators, $g$-function, defect-changing operators, its cusp, and the conformal boundaries of the 3D Ising CFT. Besides the defects and boundaries of 3D CFTs, one can also studying lower dimensional bulk CFTs on a $(2+1)$D set-up, like the on a fuzzy thin torus.

Conformal magnetic line defect

[Hu 2023Aug] Solving conformal defects in 3d conformal field theory using fuzzy sphere regularisation, Liangdong Hu, Yin-Chen He, and Wei Zhu, arXiv:2308.01903, Nat. Commun. 15, 3659 (2024).

This is the first work that studies conformal defects with the fuzzy sphere. The simplest example of conformal defect is the magnetic line defect of the 3D Ising CFT, where the defect line is completely polarised and the $\mathbb{Z}_2$ symmetry is explicitly broken. A defect line along $z$-direction that passes the origin point, after the radial quantisation, corresponds to the north and south poles of the sphere being polarised. Hence, to realise the magnetic line defect on the fuzzy sphere, one only needs to apply a pinning magnetic field to the north and south poles (Since only the $m=+s$ orbital has non-zero amplitude at the north pole and $m=-s$ at the south pole due to the locality, one only need to pin the $m=\pm s$ orbitals).

This work demonstrates that the defect IR fixed point has emergent conformal symmetry $\mathrm{SO}(2,1)\times\mathrm{O}(2)$ : in the operator spectrum, there exists a displacement operator as the non-conservation of stress tensor at exactly $\Delta_\mathrm{d}=2$, and the defect primaries and descendants have integer spacing ; the bulk-to-defect 1-pt and 2-pt correlation functions follow a power law. Hu et al. have identified six low-lying defect primary operators, extracted their scaling dimensions, and computed the 1-pt function of bulk primaries and 2-pt bulk-to-defect correlators.

The $g$-function and defect changing operators

[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).

This work studies the $g$-function of conformal defects and the defect-creation and changing operators. Similar to the central charge and the $F$-function in bulk CFTs, there exists a RG-monotonic quantity called the $g$-function for the line defects that is non-increasing along the flow. It is defined as the ratio between the partition functions of the defect CFT and the bulk CFT. On a different note, consider two semi-infinite magnetic line defects pinned towards opposite directions joint at one point, a defect-changing operator lives at the joining point. Similarly, a defect-creation operator lives at the endpoint of a semi-infinite line defect. The relevance of the defect-changing operator is related to the stability of spontaneous symmetry breaking (SSB) on the line defect.

Zhou et al. realise the defect-creation and changing operators for the Ising magnetic line defect by acting a pinning field at the north pole, and opposite pinning fields at the north and south poles, respectively. The scaling dimensions are calculated through state-operator correspondence $\Delta_{\textrm{creation}}=0.108(5),\Delta_{\textrm{changing}}=0.84(5)$, indicating the instability of SSB on the Ising magnetic line. Moreover, they show that the $g$-function and many other CFT data can be calculated by taking the overlaps between the eigenstates of different defect configurations. Notably, this work has given the first non-perturbative result for the $g$-function $g=0.602(2)$.

Cusp

[Cuomo 2024] Impurities with a cusp : general theory and 3d Ising, Gabriel Cuomo, Yin-Chen He, Zohar Komargodski, arXiv:2406.10186, JHEP 11 (2024) 061.

A cusp is two semi-infinite defect lines joined at one point at an angle. This can be realised on the fuzzy sphere through pinning fields at two points at an angle. Cuomo et al. study the cusps through various theoretical and numerical approaches. In particular, on the fuzzy sphere, they calculate the cusp anomalous dimension as a function of the angle for the Ising magnetic line defect, and verify its relation with the Zamolodchikov norm of the displacement operator.

Conformal boundaries of 3D Ising CFT

[Zhou 2024Jul] Studying the 3d Ising surface CFTs on the fuzzy sphere, Zheng Zhou, and Yijian Zou, arXiv:2407.15914, SciPost Phys. 18, 031 (2025).

[Dedushenko 2024] Ising BCFTs from the fuzzy hemisphere, Mykola Dedushenko, arXiv:2407.15948.

Apart from line defects, boundaries are also important extended objects in CFT. For the Ising CFT, there exist several conformal boundaries : normal bCFT with explicitly broken $\mathbb{Z}_2$ symmetry, ordinary bCFT that is stable and has preserved $\mathbb{Z}_2$ symmetry, extraordinary bCFT with spontaneously broken $\mathbb{Z}_2$ symmetry, and special bCFT as the transition between ordinary and extraordinary bCFTs. These works focus on the normal and ordinary bCFTs and show that they can be realised by acting a polarising field on a hemisphere. By noting that the LLL orbitals are localised along latitude circles, the bCFTs can equivalently be realised by pinning the orbitals with $m<0$. By studying the operator spectrum, Zhou et al. and Dedushenko et al. show numerical evidence for conformal symmetry and estimate the scaling dimensions of the conformal primaries. They also calculate the bulk-to-boundary 1-pt and 2-pt functions and extract the corresponding OPE coefficients. Interestingly, Zhou et al. notice a certain correspondence between the boundary energy spectrum and bulk entanglement spectrum through the orbital cut.

Fuzzy circle

[Han 2025] Quantum phase transitions on the noncommutative circle, Chao Han, and Wei Zhu, Phys. Rev. B 111, 085113 (2025)

Besides the fuzzy sphere $S^2$, the regularisation with the lowest Landau level can also be used for other manifolds or dimensions. This work studies the 2D CFT on a 'fuzzy circle.' Although Landau levels can only be defined on even space dimensions, one can reach odd space dimensions by compactifying one of the even dimensions. Specifically, Han et al. construct the LLL on a thin torus $T^2$ and sends one of the lengths to infinity while fixing the other. In this way, a circle $S^1$ is recovered in the thermodynamic limit. They construct the 2D Ising and 3-state Potts CFTs on the fuzzy circle and compare the operator spectrum and OPE coefficients with the Virasoro multiplet structure and the known results in the exactly solvable minimal models.

Model Construction

In this section, we review the process of constructing a model on the fuzzy sphere and extracting the conformal data. We aim to provide the necessary technical information for those who want to get started with the research on the fuzzy sphere, especially the aspects rarely covered by other literature.

Projection onto the Lowest Landau Level

To build the setup of the 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 modifies the single particle Hamiltonian

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

where $\mu=\theta,\phi$ and the gauge connexion is taken as

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

The eigenstates of the Hamiltonian are the monopole spherical harmonics

\[ \Phi_{nm}(\mathbf{r})=\frac{1}{R}Y_{lm}^{(s)}(\hat{\mathbf{n}}),\qquad n=0,1,\dots,\quad l=n+s,\quad m=-l,\dots,l-1,l,\]

where $\hat{\mathbf{n}}$ is the unit vector of the point on the sphere specified by angular coordinates $\theta$ and $\phi$, and the energies are

\[ E_n=\frac{1}{2MR^2}n(2s+n+1).\]

Each level, known as a Landau level, has a degeneracy $(2l+1)$. Specifically, the wavefunctions on the lowest Landau level (LLL) $n=0,l=s$ are easy to write out :

\[ \Phi_{0m}(\mathbf{r})=\frac{1}{R}Y_{sm}^{(s)}(\hat{\mathbf{n}}),\qquad Y_{sm}^{(s)}(\hat{\mathbf{n}})=C_me^{im\phi}\cos^{s+m}\frac{\theta}{2}\sin^{s-m}\frac{\theta}{2}\]

where $C_m=1/\sqrt{4\pi\Beta(s+m+1,s-m+1)}$ is the normalising factor, and $\Beta$ is the Euler's beta function. 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 $\psi_f(\mathbf{r})$, 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 $H_0\gg H_\text{int}$, 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 fully filled) is preferred in the absence of interaction, for which the charge degree of freedom is gapped and does not couple strongly to the gapless CFT degree of freedom when the interactions are introduced.

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

\[ \psi^\dagger_f(\mathbf{r})=\sum_{m=-s}^s \Phi_{0m}(\mathbf{r})c^\dagger_{mf}=\frac{1}{R}\sum_{m=-s}^s Y^{(s)}_{sm}(\hat{\mathbf{n}})c^\dagger_{mf},\]

where $c^{(\dagger)}_{mf}$ annihilates/creates an electron with $L^z$-quantum number $m$ at the $f$-th flavour of the lowest Landau level. Hereafter, we will omit the hats on the operators. In the old version, we used a different convension $\psi_f(\hat{\mathbf{n}})_\text{(old)}=\sum_{m=-s}^s Y^{(s)}_{sm}(\hat{\mathbf{n}})c_{mf}$. In the old convension, the components of density operator $n_{M,lm}$ are extensive and their action decreases the angular momentum $L^z$ by $m$ ; in the current convension, $n_{M,lm}$ is intensive and its action increases the angular momentum $L^z$ by $m$. In the code, two parameters ObsNormRadSq and ObsMomIncr controls the convension.

After the projection, we obtain a finite Hilbert space on which numerical simulations can be carried out. For this purpose, the system is analogous to a length-$(2s+1)$ spin chain with long-range interaction, where different Landau level orbitals behave like the lattice sites. The difference is that the $(2s+1)$ orbital forms a spin-$s$ representation of the $\mathrm{SO}(3)$ rotation group, and in this way the continuous rotation symmetry is preserved. The exact rotation symmetry shortens the RG flow from the UV to the IR and reduces the finite-size effect, so that the numerical results are considerably accurate even at a small system size.

The word 'fuzzy' means non-commutativity. Here, the magnetic field results in the non-commutativity of the coordinates. More concretely, we write the coordinate operators as a matrix on the lowest Landau level

\[ X^\mu_{m_1m_2}=\int\mathrm{d}^2\mathbf{r}\,x^\mu \frac{Y_{sm_1}^{(s)}(\mathbf{r})}{R}\frac{\bar{Y}_{sm_2}^{(s)}(\mathbf{r})}{R}.\]

These matrices $\mathbf{X}^\mu$ ($\mu=x,y,z$) satisfy relation

\[ \mathbf{X}_\mu\mathbf{X}^\mu=\frac{s}{s+1}R^2\mathbb{I},\qquad [\mathbf{X}^\mu,\mathbf{X}^\nu]=\frac{1}{s+1}i\epsilon^{\mu\nu\rho}R\mathbf{X}_\rho.\]

The first equation involves a renormalised radius $\tilde{R}$ of the sphere, and the second equation involves the magnetic length $l_B$ that determines the non-commutativity.

\[ \mathbf{X}_\mu\mathbf{X}^\mu=\tilde{R}^2\mathbb{I},\qquad [\mathbf{X}^\mu,\mathbf{X}^\nu]=l_B^2\,i\epsilon^{\mu\nu\rho}(\mathbf{X}_\rho/R).\]

We can take $l_B=1$ as the unit length. In this way, the radius scales with the square root of the number of orbitals

\[ \tilde{R}/l_B=[s(s+1)]^{1/4}\sim\sqrt{N_m}.\]

The thermodynamic limit can be taken as $N_m\to\infty$, where a regular sphere is recovered. Hereafter we take the radius of the sphere $R=\sqrt{N_m}$

Density Operator

Having constructed the single-particle states, we then consider the interacting many-body Hamiltonian. The simplest building block is the density operator, i. e., local fermion bilinear

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

Here, the matrix insertion $M$ puts the density operators in a certain representation of the flavour symmetry. For example, for a 2-flavour system, $M$ can be taken as the Pauli matrices $\mathbb{I},\sigma^x,\sigma^y,\sigma^z$ ; for a system with $N_f$ flavours in the fundamental representation of $\mathrm{SU}(N_f)$ flavour symmetry, one can put $n_M$ in the singlet or adjoint representation

\[\begin{aligned} n_S(\mathbf{r})&=\psi_{c}^\dagger(\mathbf{r})\psi^c(\mathbf{r})\nonumber\\ n_a{}^b(\mathbf{r})&=\psi_{a}^\dagger(\mathbf{r})\psi^b(\mathbf{r})-\tfrac{1}{N}\delta_{a}{}^b\psi_c^\dagger(\mathbf{r})\psi^c(\mathbf{r}). \end{aligned}\]

Like the fermion operator, the density operator can also be expressed in the orbital space

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

Conversely,

\[\begin{aligned} n_{M,lm}&=\frac{1}{R^2}\int\mathrm{d}^2\mathbf{r}\,\bar{Y}_{lm}(\hat{\mathbf{n}})n_M(\mathbf{r})\nonumber\\ &=\int\mathrm{d}^2\hat{\mathbf{n}}\,\bar{Y}_{lm}(\hat{\mathbf{n}})\left(\frac{1}{R}\sum_{m_1}Y^{(s)}_{sm_1}(\hat{\mathbf{n}})c^\dagger_{m_1f_1}\right)M_{f_1f_2}\left(\frac{1}{R}\sum_{m_2}\bar{Y}^{(s)}_{sm_2}(\hat{\mathbf{n}})c_{m_1f_2}\right)\nonumber\\ &=\sum_{m_1m_2}c^\dagger_{m_1f_1}M_{f_1f_2}c_{m_1f_2}\int\mathrm{d}^2\hat{\mathbf{n}}\,\bar{Y}_{lm}(\hat{\mathbf{n}})Y^{(s)}_{sm_1}(\hat{\mathbf{n}})\bar{Y}^{(s)}_{sm_2}(\hat{\mathbf{n}})\nonumber\\ &=\frac{1}{R^2}\sum_{m_1}c^\dagger_{m_1f_1}M_{f_1f_2}c_{m-m_1,f_2}\times\nonumber\\ &\qquad\qquad(-1)^{s+2m-m_1}(2s+1)\sqrt{\frac{2l+1}{4\pi}}\begin{pmatrix}l&s&s\\-m&m_1&-m_1+m\end{pmatrix}\begin{pmatrix}l&s&s\\0&-s&s\end{pmatrix}. \end{aligned}\]

Here, we have used the properties of the monopole spherical harmonics

\[\begin{aligned} \bar{Y}_{lm}^{s}&=(-1)^{s+m}Y_{l,-m}^{(-s)}\\ \int\mathrm{d}^2\hat{\mathbf{n}}\,Y_{lm}^{(s)}\bar{Y}_{lm}^{(s)}&=\delta_{ll'}\delta_{mm'}\\ \int\mathrm{d}^2\hat{\mathbf{n}}\,Y_{l_1m_1}^{(s_1)}Y_{l_2m_2}^{(s_2)}Y_{l_3m_3}^{(s_3)}&=\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}}\begin{pmatrix}l_1&l_2&l_3\\m_1&m_2&m_3\end{pmatrix}\begin{pmatrix}l_1&l_2&l_3\\-s_1&-s_2&-s_3\end{pmatrix}, \end{aligned}\]

where $\left(\begin{smallmatrix}\bullet&\bullet&\bullet\\\bullet&\bullet&\bullet\end{smallmatrix}\right)$ is the $3j$-symbol, and we denote the common spherical harmonics by $Y_{lm}^{(0)}=Y_{lm}$. 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 with a potential function (This is, however, not the simplest construction and we will present the simpler construction in terms of pseudopotentials in the next section)

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

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

\[ U(|\mathbf{r}_{12}|)=\sum_l\tilde{U}_lP_l(\cos\theta_{12})=\sum_{lm}\frac{4\pi\tilde{U}_l}{2l+1}\bar{Y}_{lm}(\hat{\mathbf{n}}_1)Y_{lm}(\hat{\mathbf{n}}_2),\]

where $\mathbf{r}_{12}=\mathbf{r}_1-\mathbf{r}_2$ and $|\mathbf{r}_{12}|=2R\sin\theta_{12}/2$. Conversely

\[ \tilde{U}_l=\int\sin\theta_{12}\mathrm{d}\theta_{12}\,\frac{2l+1}{2}U(|\mathbf{r}_{12}|)P_l(\cos\theta_{12}).\]

Specifically, for local and super-local interactions

\[\begin{aligned} U(|\mathbf{r}_{12}|)&=g_0\delta(\mathbf{r}_{12}),&\tilde{U}_l&=\frac{g_0}{R^2}(2l+1)\nonumber\\ U(|\mathbf{r}_{12}|)&=g_1\nabla^2\delta(\mathbf{r}_{12}),&\tilde{U}_l&=-\frac{g_1}{R^4}l(l+1)(2l+1). \end{aligned}\]

Here we make use of the conversion relations

\[ \delta(\mathbf{r})=\frac{1}{R^2}\delta(\hat{\mathbf{n}}),\qquad \nabla^2_\mathbf{r}=\frac{1}{R^2}\nabla^2_{\hat{\mathbf{n}}}\]

for $\mathbf{r}=R\hat{\mathbf{n}}$. By expanding the density operators into the orbital space and completing the integrals,

\[ H_\text{int}=\sum_{lm}\frac{4\pi R^4\tilde{U}_l }{2l+1}n^\dagger_{M,lm}n_{M,lm}.\]

With these ingredients, we can now consider how to construct models. This comes down to matching the symmetry and phase diagram. E. g., for the Ising model, the $\mathbb{Z}_2$ global symmetry is realised as the exchange of the two flavours $\psi_\uparrow(\mathbf{r})\leftrightarrow\psi_\downarrow(\mathbf{r})$. We need a phase diagram with a paramagnetic (PM) phase where the $\mathbb{Z}_2$ symmetry is conserved and a ferromagnetic (FM) phase where the $\mathbb{Z}_2$ symmetry is spontaneously broken. The PM phase is favoured by a polarising term that resembles a transverse field

\[ -h\int\mathrm{d}^2\mathbf{r}\,n_x(\mathbf{r})\]

and the FM phase where either of the two flavours is fully filled is favoured by a repulsion between the two flavours

\[ \int\mathrm{d}^2\mathbf{r}_1\,\mathrm{d}^2\mathbf{r}_2\,U(|\mathbf{r}_{12}|)n_\uparrow(\mathbf{r}_1)n_\downarrow(\mathbf{r}_2),\]

where the density operators are defined as

\[ n_x(\mathbf{r})=\psi^\dagger_\downarrow(\mathbf{r})\psi_\uparrow(\mathbf{r})+\psi^\dagger_\uparrow(\mathbf{r})\psi_\downarrow(\mathbf{r}),\quad n_\uparrow(\mathbf{r})=\psi^\dagger_\uparrow(\mathbf{r})\psi_\uparrow(\mathbf{r}),\quad n_\downarrow(\mathbf{r})=\psi^\dagger_\downarrow(\mathbf{r})\psi_\downarrow(\mathbf{r}),\]

and the potentials can be most conveniently taken as a combination of local and super-local interactions. Altogether, the model Hamiltonian reads

\[ H_\text{int}=\int\mathrm{d}^2\mathbf{r}_1\,\mathrm{d}^2\mathbf{r}_2\,U(|\mathbf{r}_{12}|)n_\uparrow(\mathbf{r}_1)n_\downarrow(\mathbf{r}_2)-h\int\mathrm{d}^2\mathbf{r}\,n_x(\mathbf{r}).\]

By tuning the ratio between $h$ and $U(|\mathbf{r}_{12}|)$, a phase transition described by the Ising CFT is realised.

Interaction in Terms of Pseudopotentials

Another way that is much more convenient to construct the four-fermion interaction terms is through Haldane pseudopotential. To explain the idea, we take the 3D Ising model as an example. We first classify all the fermion bilinears $\lambda_{m_1f_1m_2f_2}c_{m_1f_1}c_{m_2f_2}$. To simplify the discussion, we can take a specific flavour index $\lambda_{m_1m_2}c_{m_1\uparrow}c_{m_2\downarrow}$. The fermion bilinears can be classified into irreducible representations (irreps) 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

\[ \Delta_{lm}=\sum_{m_1}\langle sm_1,s(m-m_1)|(2s-l)m\rangle c_{m_1\uparrow}c_{m-m_1\downarrow},\]

where $m=-(2s-l),\dots,(2s-l)$, and the Clebshbar-Gordan coefficients is related to the $3j$-symbol by

\[ \langle l_1m_1,l_2m_2|lm\rangle=(-1)^{-l_1+l_2-m}\sqrt{2l+1}\begin{pmatrix}l_1&l_2&l\\m_1&m_2&-m\end{pmatrix}\]

A four-fermion interaction term is formed by contracting these paring operators with its conjugate

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

Putting these together, the interaction Hamiltonian can be expressed as

\[ H=\sum_{l,m_1m_2m_3m_4}U_lC^l_{m_1m_2m_3m_4}c^\dagger_{m_1\uparrow}c^\dagger_{m_2\downarrow}c_{m_3\downarrow}c_{m_4\uparrow}-h\sum_m(c_{m\uparrow}^\dagger c_{m\downarrow}+\text{h. c.}),\]

where the matrix elements are

\[ C^l_{m_1m_2m_3m_4}=\delta_{m_1+m_2,m_3+m_4}\langle sm_1,sm_2|(2s-l)(m_1+m_2)\rangle\langle sm_3,sm_4|(2s-l)(m_3+m_4)\rangle.\]

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

We also need to consider the constraint that the two fermions must be antisymmetrised : 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 antisymmetrised, so the spin index is symmetrised, breaking the flavour symmetry from $\mathrm{SU}(2)$ to $\mathbb{Z}_2$. Hence, an odd-$l$ pseudopotential must be added (This fact escapes the construction by density-density interaction).

The fermion bilinears with other flavour configurations $\lambda_{m_1m_2,\pm}(c_{m_1\uparrow}c_{m_2\uparrow}\pm c_{m_1\downarrow}c_{m_2\downarrow})$ can be analysed in a similar way. After that, we have enumerated all possible four-fermion interaction terms.

Each pseudopotential corresponds to a profile of interaction potential functions. The conversion between the pseudopotentials $U_l$ and the Legendre expansion coefficients of the potential function $\tilde{U}_l$

\[ U(|\mathbf{r}_{12}|)=\sum_l\tilde{U}_lP_l(\cos\theta_{12})\]

is

\[ U_l=\sum_k \tilde{U}_k(-1)^l(2s+1)^2\begin{Bmatrix}2s-l&s&s\\k&s&s\end{Bmatrix}\begin{pmatrix}s&k&s\\-s&0&s\end{pmatrix}^2,\]

where $\left\{\begin{smallmatrix}\bullet&\bullet&\bullet\\\bullet&\bullet&\bullet\end{smallmatrix}\right\}$ is the $6j$-symbol. Specifically, a local interaction $\delta(\mathbf{r}_{12})$ contains only pseudopotential $U_0$ ; a superlocal interaction of form $(\nabla^2)^l\delta(\mathbf{r}_{12})$ contains $U_0,U_1,\dots,U_l$. Here we explicitly give the expressions for the lowest pseudopotentials

\[\begin{aligned} U(|\mathbf{r}_{12}|)&=\delta(\mathbf{r}_{12}),&U_0&=\frac{(2s+1)^2}{4s+1}\nonumber\\ U(|\mathbf{r}_{12}|)&=\nabla^2\delta(\mathbf{r}_{12}),&U_0&=-\frac{s(2s+1)^2}{4s+1},&U_1&=\frac{s(2s+1)^2}{4s-1}. \end{aligned}\]

More details are given in Ref.. We also note that the construction for interaction Hamiltonian for LLL with pseudopotentials is not restricted to sphere. E. g., on a torus, the interaction can also be parametrise by pseudopotentials.

For systems with more complicated continuous flavour symmetries, classification in terms of representation of flavour symmetry must also be considered, and the indices must be overall antisymmetrised. We explain that through the example of a $2N$-flavour system with $\mathrm{Sp}(N)$ symmetry. The maximal flavour symmetry is $\mathrm{SU}(2N)$, so interactions must be added to break the symmetry from $\mathrm{SU}(2N)$ to $\mathrm{Sp}(N)$. The fermion operators $c_{ma}$ live in the $\mathrm{Sp}(N)$ fundamental representation, where $a$ is the $\mathrm{Sp}(N)$ index. We shall show that all the allowed terms are

\[\begin{aligned} H&=\sum_{\substack{l\in\mathbb{Z}\\m_1m_2m_3m_4}}U_lC^l_{m_1m_2m_3m_4}c^\dagger_{m_1a}c^\dagger_{m_2b}c_{m_3b}c_{m_4a}\\ &\qquad\qquad-\frac{1}{2}\sum_{\substack{l\in 2\mathbb{Z}\\m_1m_2m_3m_4}}V_lC^l_{m_1m_2m_3m_4}\Omega_{aa'}\Omega_{bb'}c^\dagger_{m_1a}c^\dagger_{m_2a'}c_{m_3b'}c_{m_4b}. \end{aligned}\]

where $\Omega=\begin{pmatrix}0&\mathbb{I}_N\\-\mathbb{I}_N&0\end{pmatrix}$.

To find out all the four-fermion interactions allowed by the rotation symmetry $\mathrm{SO}(3)$ and flavour symmetry $\mathrm{Sp}(N)$, we classify all the fermion bilinears $c_{m_1a}c_{m_2b}$ into irreps of $\mathrm{SO}(3)\times\mathrm{Sp}(N)$. For each irrep, by contracting the bilinear with its Hermitian conjugate, we obtain an allowed four-fermion interaction term. Each fermion carries $\mathrm{SO}(3)$ spin-$s$ and $\mathrm{Sp}(N)$ fundamental. For the rotation symmetry $\mathrm{SO}(3)$, the bilinears can carry spin-$(2s-l)(l=0,\dots,2s)$ represetation ; for even $l$, the orbital indices are symmetrised ; for odd $l$, the orbital indices are antisymmetrised. For the flavour symmetry $\mathrm{Sp}(N)$, the bilinears can carry singlet $S$, traceless antisymmetric rank-2 tensor $A$ and symmetric rank-2 tensor $T$ representation ; for $S$ and $A$, the flavour indices are antisymmetrised ; for $T$, the flavour indices are symmetrised. As the two fermions altogether should be antisymmetrised, the allowed combinations are

Case 1. $\mathrm{Sp}(N)$ singlet and $\mathrm{SO}(3)$ spin-$(2s-l)$ with even $l$, the bilinears are

\[ \Delta_{lm}=\sum_{m_1m_2}\langle sm_1,sm_2|(2s-l)m\rangle\Omega_{cc'}c_{m_1c}c_{m_2c'}\delta_{m,m_1+m_2}.\]

The corresponding interaction term $H_{S,l}=\sum_m\Delta_{lm}^\dagger\Delta_{lm}$ is the even-$l$ pseudopotential for the $V$-term.

Case 2. $\mathrm{Sp}(N)$ antisymmetric and $\mathrm{SO}(3)$ spin-$(2s-l)$ with even $l$, the bilinears are

\[ \Delta_{lm,[ab]}=\sum_{m_1m_2}\langle sm_1,sm_2|(2s-l)m\rangle(c_{m_1a}c_{m_2b}-c_{m_1b}c_{m_2a}-\tfrac{1}{N}\Omega_{ab}\Omega_{cc'}c_{m_1c'}c_{m_2c})\delta_{m,m_1+m_2}.\]

The corresponding interaction term $H_{A,l}=\sum_m\Delta_{lm,[ab]}^\dagger\Delta_{lm,[ab]}$ is the even-$l$ pseudopotential for the $U$-term.

Case 3. $\mathrm{Sp}(N)$ symmetric and $\mathrm{SO}(3)$ spin-$(2s-l)$ with odd $l$, the bilinears are

\[ \Delta_{lm,(ab)}=\sum_{m_1m_2}\langle sm_1,sm_2|(2s-l)m\rangle(c_{m_1a}c_{m_2b}+c_{m_1b}c_{m_2a})\delta_{m,m_1+m_2}.\]

The corresponding interaction term $H_{T,l}=\sum_m\Delta_{lm,(ab)}^\dagger\Delta_{lm,(ab)}$ is the odd-$l$ pseudopotential for the $U$-term.

In summary, all allowed interactions are the $U_l$ terms with both even and odd $l$, and the $V_l$ terms with only even $l$.

Operator Spectrum and Search for Conformal Point

Having introduced the construction of an interacting model on the fuzzy sphere, we now turn to the verification of the conformal symmetry and the extraction of the CFT data. The most straightforward approach is to extract the scaling dimensions from the energy spectrum through the state-operator correspondence. Specifically, there is a one-to-one correspondence between the eigenstates of the Hamiltonian and the CFT operators. The state and its corresponding operator has the same $\mathrm{SO}(3)$ spin and representation under flavour symmetry, and the excitation energy of a state $|\Phi\rangle$ is proportional to the scaling dimension of the corresponding operator $\Delta_\Phi$

\[ E_\Phi-E_0=\frac{v}{R}\Delta_\Phi,\]

where $E_0$ is the ground state energy, $R$ is the radius of the sphere (here we take $R=\sqrt{N_m}$), and $v$ is a model-dependent speed of light. The constant $v/R$ can be determined through a calibration process, i. e. comparing the spectrum to some known properties of a CFT spectrum. The criteria to determine the conformal symmetry include

1. The existence of a conserved stress tensor $T^{\mu\nu}$. The stress tensor is the symmetry current of the translation transformation. It is known to be a singlet under the flavour symmetry, have spin-2 under $\mathrm{SO}(3)$ rotation and scaling dimension exactly $\Delta_{T^{\mu\nu}}=3$.
2. The existence of a conserved flavour symmetry current $J^\mu$ if there is a continuous flavour symmetry. The symmetry current typically lives in the antisymmetric rank-2 tensor representation of the flavour symmetry. E. g., if the flavour symmetry is $\mathrm{U}(1)$, then the symmetry current has charge-0 ; if the flavour symmetry is $\mathrm{O}(3)$, then the symmetry current has spin-1 and is odd under the improper $\mathbb{Z}_2$ transformation ; if the flavour symmetry is $\mathrm{O}(n)$ ($n\ge 4$) or $\mathrm{SU}(n)$ ($n\ge 3$), then the symmetry current lives in the antisymmetric rank-$2$ tensor representation.
3. The organisation of the operator spectrum into conformal multiplets. All the levels in the spectrum of a CFT can be organised into the conformal primaries and their descendants. The descendants live in the same representation under the flavour symmetry as the primary, and the difference between the scaling dimensions of a primary and its descendant is an integer. Specifically, for a scalar primary $\Phi$, its descendants have the form (Hereafter, we will presume the subtraction of trace and omit the terms)

\[ \Box^n\partial^{\mu_1}\partial^{\mu_2}\dots\partial^{\mu_l}\Phi-\textrm{(trace)}\qquad (n,l=0,1,2,\dots)\]

with $\mathrm{SO}(3)$ spin-$l$ and scaling dimension

\[ \Delta=\Delta_\Phi+2n+l,\]

where $\Box=\partial_\mu\partial^\nu$. For a spinning primary $\Phi^{\mu_1\dots\mu_s}$, its descendants has the two forms :

\[ \Box^n\partial^{\nu_1}\dots\partial^{\nu_m}\partial_{\rho_1}\dots\partial_{\rho_k}\Phi^{\rho_1\dots\rho_{k}\mu_1\dots \mu_{s-k}}\qquad (k=0,\dots,s,\ n,m=0,1,\dots)\]

with scaling dimension and $\mathrm{SO}(3)$ spin

\[ \Delta=\Delta_\Phi+k+m+2n,\qquad l=s-k+m,\]

and

\[ \Box^n\partial^{\nu_1}\dots\partial^{\nu_m}\partial_{\rho_1}\dots\partial_{\rho_k}\epsilon^{\sigma}{}_{\tilde{\mu}\tilde{\nu}}\partial^{\tilde{\nu}}\Phi^{\rho_1\dots\rho_{k}\tilde{\mu}\mu_1\dots \mu_{s-k-1}}\qquad (k=0,\dots,s-1,\ n,m=0,1,\dots)\]

with

\[ \Delta=\Delta_\Phi+k+m+2n+1,\qquad l=s-k+m.\]

For the second form, the fully antisymmetric tensor $\epsilon$ alters the parity.

The most convenient way of determining the coefficient $v/R$ is by utilising criteria 1 or 2 :

\[ \frac{v}{R}=\frac{E_{T^{\mu\nu}}-E_0}{3}\quad\textrm{or}\quad\frac{E_{J^\mu}-E_0}{2}.\]

Alternatively, one can define a cost function that depends on the tuning parameter and the speed of light and compares the scaling dimensions obtained from the fuzzy sphere and the prediction by conformal symmetry. E. g., for the Ising CFT, the tuning parameters are the pseudopotentials $\{U_i\}$ and the transverse field $h$. The criteria for conformal symmetry we use include the stress tensor $T^{\mu\nu}$ and the descendants $\partial^\mu\sigma$, $\partial^\mu\partial^\nu\sigma$, $\Box\sigma$, $\partial^\mu\epsilon$. The cost function is the root-mean-square of the deviations of these criteria from the expectation of the conformal symmetry

\[\begin{aligned} Q^2(\{U_i\},h,v;N_m)&=\frac{1}{N_s}\left[(\Delta_{T^{\mu\nu}}^\text{(FS)}-3)^2+(\Delta_{\partial^\mu\sigma}^\text{(FS)}-\Delta_\sigma^\text{(FS)}-1)^2\right.\\ &\qquad\qquad\left.+(\Delta_{\partial^\mu\partial^\nu\sigma}^\text{(FS)}-\Delta_\sigma^\text{(FS)}-1)^2+(\Delta_{\Box\sigma}^\text{(FS)}-\Delta_\sigma^\text{(FS)}-1)^2+(\Delta_{\partial^\mu\epsilon}^\text{(FS)}-\Delta_\epsilon^\text{(FS)}-1)^2\right] \end{aligned}\]

where $N_s=5$ is the number of criteria, the scaling dimension of an operator $\Phi$ on the fuzzy sphere is determined as

\[ \Delta_\Phi^\text{(FS)}(\{U_i\},h,v;N_m)=\frac{E_\Phi-E_0}{v/R}.\]

The optimal conformal point and calibrator are determined by minimising this cost function for each system size $N_m$. Note that this optimal point depends on the system size. In order to do finite-size scaling, if the CFT describes a phase transition, one could fix all but one parameters at the optimal point in the largest accessible system size and tune the last parameter to determine the critical point through a finite-size scaling.

Local Observables

We have introduced how to determine the scaling dimensions from the energy spectrum. Beyond that, evaluating other CFT quantities requires realising local CFT operators on the fuzzy sphere. Any gapless local observables $\mathscr{O}(\mathbf{r})$ on the fuzzy sphere can be written as the linear combination of CFT operators that live in the same representation of flavour symmetry and parity

\[ \mathscr{O}(\mathbf{r},\tau)=\sum_\alpha \lambda_\alpha\Phi^\text{(cyl.)}_\alpha(\mathbf{r},\tau).\]

Here special care should be taken for the CFT operator $\Phi^\text{(cyl.)}_\alpha(\mathbf{r},\tau)$ on the cylinder. A conformal transformation produces a scale factor $\Lambda(\mathbf{r})^{\Delta_\Phi}$ to a primary operator $\Phi$. The scale factor is $\Lambda(\mathbf{r})=r/R$ for the Weyl transformation from the flat spacetime to the cylinder. Hence,

\[ \Phi^\text{(cyl.)}_\alpha(\mathbf{r},\tau)=\left(\frac{e^{\tau/R}}{R}\right)^{\Delta_{\Phi_\alpha}}\Phi_\alpha^\text{(flat)}(x).\]

Here we need to clarify some of the notations : $\mathscr{O}$ represents an operator in the microscopic model, and $\Phi$ represents a CFT operator ; the arguments $\Phi(\mathbf{r})$ or $\Phi(\mathbf{r},\tau)$ by default mean the operator is defined on a cylinder, and $\Phi(x)$ by default on flat spacetime. For descendants, certain other factors may be produced, but the conversion factors still scale with the radius of the sphere as $R^{-\Delta}$ where $\Delta$ is the scaling dimension of the descendants. For simplicity, hereafter we focus on the equal-time correlators with $\tau=0$, for which $\Phi^\text{(cyl.)}_\alpha(\mathbf{r})=R^{-\Delta_{\Phi_\alpha}}\Phi_\alpha^\text{(flat)}(x)$. The operator with larger system size decays faster when increasing system size.

The simplest local observable is the density operator defined in Section 'Density Operator'. From the CFT perspective, the density operators are the superpositions of scaling operators with corresponding quantum numbers, i. e. with the same representation under flavour symmetry and parity.

Take the Ising model as an example. Consider the density operators $n^x$ and $n^z$ with matrix insertion $M=\sigma^x,\sigma^z$. In the leading order, they can be used as UV realisations of CFT operators $\sigma$ and $\epsilon$.

\[\begin{aligned} n^x(\mathbf{r})&=\lambda_0+\lambda_\epsilon\epsilon(\mathbf{r})+\lambda_{\partial^\mu\epsilon}\partial^\mu\epsilon(\mathbf{r})+\lambda_{T^{\mu\nu}}T^{\mu\nu}(\mathbf{r})+\dots&\epsilon_\textrm{FS}&=\frac{n^x-\lambda_0}{\lambda_\epsilon}+\dots\nonumber\\ n^z(\mathbf{r})&=\lambda_\sigma\sigma(\mathbf{r})+\lambda_{\partial^\mu\sigma}\partial^\mu\epsilon(\mathbf{r})+\lambda_{\partial^\mu\partial^\nu\sigma}\partial^\mu\partial^\nu\sigma(\mathbf{r})+\dots&\sigma_\textrm{FS}&=\frac{n^z}{\lambda_\sigma}+\dots \end{aligned}\]

where the coefficients $\lambda_0,\lambda_\epsilon,\lambda_\sigma,\dots$ are model-dependent and need to be determined, and all the operators on the right-hand side are defined on the cylinder.

We first consider the insertion of a single operator $\langle\Phi_1|\Phi_2(\mathbf{r})|\Phi_3\rangle$. It helps us produce the OPE coefficients. For the simplest example of three scalars,

\[ f_{\Phi_1\Phi_2\Phi_3}=\lim_{r_\infty\to\infty}r_\infty^{-2\Delta_{\Phi_1}}\langle \Phi_1(x_\infty)\Phi_2(x)\Phi_3(0)\rangle_\text{(flat)}=\langle\Phi_1|\Phi_2^\text{(flat)}(x)|\Phi_3\rangle\]

where $x_\infty$ is a point on the sphere with radius $r_\infty$, $x$ is a point on the unit sphere, the states are obtained from acting the operator at the origin point on the vacuum state

\[ |\Phi_3\rangle=\Phi_3(0)|0\rangle\]

and its Hermitian conjugate is defined as

\[ \Phi_1^\dagger(\infty)=(\Phi_1(0))^\dagger=\lim_{r_\infty\to\infty}r_\infty^{2\Delta_{\Phi_1}}\Phi_1(x_\infty),\qquad\langle\Phi_1|=\langle0|\Phi_1^\dagger(\infty).\]

After the Weyl transformation from the flat spacetime to the cylinder, we obtain the expression on the fuzzy sphere

\[ f_{\Phi_1\Phi_2\Phi_3}=R^{\Delta_{\Phi_2}}\langle\Phi_1|\Phi^{(\text{(cyl.)})}_2(\mathbf{r})|\Phi_3\rangle.\]

The UV realisation of $\Phi_2$ contains many other operators with different spins. By integrating the correlation function against different spherical harmonics, i. e. take the angular modes of the operator inserted

\[ \int\mathrm{d}^2\mathbf{r}\,\bar{Y}_{lm}(\hat{\mathbf{n}})\langle\Phi_1|\Phi_2(\mathbf{r})|\Phi_3\rangle=\langle\Phi_1|\Phi_{2,lm}|\Phi_3\rangle,\]

we can filter out the subleading contributions with different spins. For the spinning operators, this also tells us about different OPE structures. By taking $\Phi_3=\mathbb{I}$, we can recover the 2-pt functions

\[\begin{aligned} \langle\Phi_2|\Phi_{2,00}|0\rangle&=R^{-\Phi_2}\nonumber\\ \Phi_2(\mathbf{r})|0\rangle&=R^{-\Phi_2}\left[|\Phi_2\rangle+\lambda'_\mu(\mathbf{r})|\partial^\mu\Phi_2\rangle+\lambda''(\mathbf{r})|\Box\Phi_2\rangle+\lambda''_{\mu\nu}(\mathbf{r})|\partial^\mu\partial^\nu\Phi_2\rangle\right]. \end{aligned}\]

It is worth noting that acting a primary $\Phi_2(\mathbf{r})$ on the vacuum also produces various descendants in the multiplet.

In the example of Ising CFT, we first use the insertion of a single operator to determine the coefficients $\lambda_0,\lambda_\epsilon,\lambda_\sigma$

\[ \lambda_0=\frac{R^{-2}}{\sqrt{4\pi}}\langle 0|n^x_{00}|0\rangle,\quad\lambda_\epsilon=\frac{R^{\Delta_\epsilon-2}}{\sqrt{4\pi}}\langle \epsilon|n^x_{00}|0\rangle,\quad\lambda_\sigma=\frac{R^{\Delta_\sigma-2}}{\sqrt{4\pi}}\langle \sigma|n^z_{00}|0\rangle.\]

Take the OPE coefficient $f_{\sigma\sigma\epsilon}$ as an example. It can be expressed either as an inner product of $\sigma$ or $\epsilon$

\[\begin{aligned} f_{\sigma\sigma\epsilon}&=R^{\Delta_\sigma}\langle\epsilon|\sigma(\mathbf{r})|\sigma\rangle=\frac{\langle\epsilon|n_{00}^z|\sigma\rangle}{\langle 0|n_{00}^z|\sigma\rangle}+\mathscr{O}(R^{-2})\nonumber\\ &=R^{\Delta_\epsilon}\langle\sigma|\epsilon(\mathbf{r})|\sigma\rangle=\frac{\langle\sigma|n^x_{00}|\sigma\rangle-\langle0|n^x_{00}|0\rangle}{\langle\epsilon|n^x_{00}|0\rangle}+\mathscr{O}(R^{-(3-\Delta_\epsilon)}). \end{aligned}\]

For the first line, the subleading contribution comes from the contribution of the descendant $\Box\sigma$ to $n_{00}^z$. As $\sigma(\mathbf{r})$ scales as $R^{-\Delta_\sigma}$ and $\Box\sigma(\mathbf{r})$ as $R^{-\Delta_\sigma-2}$,

\[\begin{aligned} \langle\epsilon|n_{00}^z|\sigma\rangle&=f_{\sigma\sigma\epsilon}\lambda_\sigma R^{-\Delta_\sigma}(1+c_1R^{-2}+\dots)\nonumber\\ \langle\epsilon|n_{00}^z|\sigma\rangle&=\lambda_\sigma R^{-\Delta_\sigma}(1+c'_1R^{-2}+\dots)\nonumber\\ \frac{\langle\epsilon|n_{00}^z|\sigma\rangle}{\langle 0|n_{00}^z|\sigma\rangle}&=f_{\sigma\sigma\epsilon}+\mathscr{O}(R^{-2}). \end{aligned}\]

Here $c_1$ and $c'_1$ are constant factors that represent the contribution of $\Box\sigma$ and do not scale with system size. Hence, the subleading contribution scales as $R^{-2}$. For the second line, the subleading contribution comes from the stress tensor $T^{\mu\nu}$. Similarly, the power of the scaling is the difference of the scaling dimension $R^{-(\Delta_{T^{\mu\nu}}-\Delta_\epsilon)}=R^{-(3-\Delta_\epsilon)}$.

We then proceed to the insertion of two operators. This can help us determine up to a 4-pt function. Through conformal transformation, any 4-pt function can be expressed in the form of

\[ \langle\Phi_1|\Phi^{\textrm{(cyl.)}}_2(\mathbf{r},\tau)\Phi^{\textrm{(cyl.)}}_3(\hat{\mathbf{z}})|\Phi_4\rangle=\frac{e^{\Delta_{\Phi_2}\tau/R}}{R^{\Delta_{\Phi_2}+\Delta_{\Phi_3}}}\langle\Phi_1^\dagger(\infty)\Phi_2(x)\Phi_3(\hat{\mathbf{z}})\Phi_4(0)\rangle,\]

where the time-displaced operator can be defined as

\[ \Phi_2(\mathbf{r},\tau)=e^{-H\tau}\Phi_2(\mathbf{r})e^{H\tau}.\]

As a sanity check, By taking $\Phi_1=\Phi_4=\mathbb{I}$, $\Phi_2=\Phi_3$ and $\tau=0$, the 2-pt function on the unit sphere is recovered

\[ \langle0|\Phi^{\textrm{(cyl.)}}_2(\mathbf{r})\Phi^{\textrm{(cyl.)}}_2(\hat{\mathbf{z}})|0\rangle=R^{-2\Delta_{\Phi_2}}\langle\Phi_2(\mathbf{r})\Phi_2(\hat{\mathbf{z}})\rangle\\=\frac{1}{R^{2\Delta_{\Phi_2}}|\mathbf{r}-\hat{\mathbf{z}}|^{2\Delta_{\Phi_2}}}=\frac{1}{R^{2\Delta_{\Phi_2}}(1-\cos\theta)^{\Delta_{\Phi_2}}}.\]

Conformal Generators

So far, in the conformal group, we know that the rotation and the dilatation are manifest on the fuzzy sphere. The rest, viz. translation and SCT, are emergent. In this section, we consider how to express the generators of these emergent symmetries in terms of the microscopic operators.

A general Noether current and corresponding generator of the infinitesimal spacetime transformation $x^\mu\mapsto x^\mu+\epsilon^\mu(x)$ can be expressed in terms of the stress tensor

\[ j_\epsilon^\mu(x)=\epsilon^\nu(x)T^\mu{}_\nu(x),\quad Q_\epsilon=\int_\Sigma\mathrm{d}^{d-1}x\,\sqrt{g}j_\epsilon^0(x),\]

where for the second equation, the integral is evaluated on a closed surface $\Sigma$. Specifically, for the generators $P^\mu,K^\mu$ of translation and SCT in the embedded sphere

\[\begin{aligned} P^\mu&=\int\mathrm{d}^2\mathbf{r}\,(r^\mu T^0{}_0+iT^{0\mu}),\nonumber\\ K^\mu&=\int\mathrm{d}^2\mathbf{r}\,(r^\mu T^0{}_0-iT^{0\mu}). \end{aligned}\]

Hence, the conformal generator $\Lambda^\mu=P^\mu+K^\mu$ is the $l=1$ component of the Hamiltonian density $\mathscr{H}=T^0{}_0$

\[ \Lambda_m=P_m+K_m=\sqrt\frac{16\pi}{3}\int\mathrm{d}^2\mathbf{r}\,\bar{Y}_{1m}(\hat{\mathbf{n}})\mathscr{H}(\mathbf{r}).\]

Here the indices $\mu$ and $m$ are two equivalent way to express the components. By acting it on the states, the number of derivatives is increased or decreased by $1$, e. g., for a primary $\Phi$

\[\begin{aligned} \Lambda^\mu|\Phi\rangle&=\textrm{const.}\times|\partial^\mu\Phi\rangle\nonumber\\ \Lambda^\mu|\partial_\mu\Phi\rangle&=\textrm{const.}\times|\Phi\rangle+\textrm{const.}\times|\partial^\mu\partial^\nu\Phi\rangle+\textrm{const.}\times|\Box\Phi\rangle. \end{aligned}\]

The derivation of the expression and the constant factors are calculated and given in Ref..

We then need to find the expression for the Hamiltonian density. For example, for the 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.

To determine those constants, we consider another strategy by considering all the possible two-fermion and four-fermion operators that are singlet under flavour symmetry and spin-1 under $\mathrm{SO}(3)$. We consider the example of Ising CFT. The two-fermion terms include the density operators

\[ n^x_{1m}\quad\textrm{and}\quad n^0_{1m}.\]

Similar to what we have done for Hamiltonian, the four-fermion operators can be obtained by combining the fermion bilinears $\Delta_{lm}$

\[ \sum_{\substack{l_1l_2m_1m_2}}\tilde{U}_{l_1l_2}\Delta^\dagger_{l_1m_1}\Delta_{l_2m_2}\langle (2s-l_1)m_1,(2s-l_2)(-m_2)|1m\rangle\]

For $l_1\in2\mathbb{Z}$, the spin index in the pairing operator is antisymmetrised ; 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

\[ \Lambda_m=\sum_{\substack{lm_1m_2}}\tilde{U}_{l}\Delta^\dagger_{lm_1}\Delta_{lm_2}\begin{pmatrix}2s-l&2s-l&1\\-m_1&m_2&m \end{pmatrix}+\tilde{h}n^x_{1m}+\tilde{\mu}n^0_{1m}\]

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

After obtaining $\Lambda^\mu=P^\mu+K^\mu$, the separate $P^\mu$ and $K^\mu$ can be obtained by considering the commutator with the dilatation generator $D$, which is proportional to the Hamiltonian. As $[D,P^\mu]=P^\mu$ and $[D,K^\mu]=-K^\mu$.

\[\begin{aligned} P^\mu&=\tfrac{1}{2}\Lambda^\mu+\tfrac{1}{2}[D,\Lambda^\mu]\nonumber\\ K^\mu&=\tfrac{1}{2}\Lambda^\mu-\tfrac{1}{2}[D,\Lambda^\mu]. \end{aligned}\]

Numerical Methods

In this section, we briefly review the numerical methods supported in FuzzifiED. The numerical methods that have been applied to the fuzzy sphere include exact diagonalisation (ED), density matrix renormalisation group (DMRG) and determinant quantum Monte Carlo (DQMC). Among these, ED and DMRG have been implemented in FuzzifiED.

Exact Diagonalisation (ED)

Exact diagonalisation (ED) might be the most straightforward method for solving a quantum many-body Hamiltonian. In ED, one constructs a many-body basis and writes down all the non-zero elements of the Hamiltonian as a sparse matrix on this basis. The eigenstates of the Hamiltonian with the lowest energy can be solved without finding the full eigensystem by Arnoldi or Lanczos algorithm.

Briefly speaking, the Arnoldi algorithm is an iterative method. Each iteration constructs an orthonormal basis of the Krylov subspace from an initial vector and finds an approximation to the eigenvector on that basis. This approximate eigenvector is then used as the initial vector for the next iteration. An example of Krylov subspace is spanned by acting the matrix $H$ repeatedly on the initial vector $|i\rangle$

\[ \mathscr{K}_r(H,|i\rangle)=\operatorname{span}\left\{|i\rangle,H|i\rangle,H^2|i\rangle,\dots,H^{r-1}|i\rangle\right\}.\]

The ED calculation can be optimised in several ways. The storage of the Hamiltonian matrix may be compressed by data structure tailored for sparse matrix such as compressed sparse column (CSC). The Hamiltonian matrix is usually block diagonal due to the symmetry of the Hamiltonian. The Hilbert space is divided into several sectors that carry different representations under the symmetry, and acting the Hamiltonian on a state in a sector results in a state in the same sector. E. g., in the ED calculation for the Ising model on the fuzzy sphere, the symmetries we can use include two $\mathrm{U}(1)$ symmetries, viz. the conservation of particle number and the angular momentum in the $z$-direction, and three $\mathbb{Z}_2$ symmetries, viz. the Ising $\mathbb{Z}_2$ flavour symmetry, the particle-hole symmetry and the $\pi$-rotation along the $y$-axis.

The ED method enjoys several advantages, including (1) the full knowledge of the eigenstate wavefunction and (2) the ability to access relatively high excited states. However, despite these optimisations, the dimension of the Hilbert space scales exponentially with the number of orbitals. This results in exponentially growing space and time complexity. E. g., for the Ising model on the fuzzy sphere, for $N_m=14$, the dimension of Hilbert space $\dim\mathscr{H}=1.8\times10^5$ and the number of elements in the Hamiltonian is $N_ \text{el}=1.1\times 10^7$ ; for $N_m=16$, the numbers have already grown to $\dim\mathscr{H}=2.2\times10^6$ and $N_\text{el}=2.1\times 10^8$, which translates to a memory demand of $3.1$ gigabytes.

In FuzzifiED, we use the Fortran library Arpack to perform the Arnoldi algorithm.

Density Matrix Renormalisation Group (DMRG)

To overcome the size limit of ED, the density matrix renormalisation group (DMRG) is a powerful method to calculate the ground state of a quasi-one-dimensional system. It was first invented by White as an improvement to the numerical renormalisation group (NRG) used in the Kondo problem. Since its proposal, it has been proven potent in various problems in condensed matter physics, such as the static and dynamic properties of one-dimensional models such as the Heisenberg, $t$–$J$ and Hubbard models. Later, Schollw\"ock has discovered a new point of view that implements the DMRG in the language of matrix product states (MPS).

Briefly speaking, in this language, DMRG is a variational method that optimises the fidelity between the exact ground state and the variational MPS. During each 'sweep,' DMRG solves a local eigenvalue problem for the active tensors to improve the approximation. To find the excited states, one needs to add projection $|0\rangle\langle 0|$ of the ground state $|0\rangle$ to the Hamiltonian by hand.

Although the fuzzy sphere deals with $(2+1)$-dimensional quantum systems, the basis of the lowest Landau level provides a natural way to express it as a quasi-1D problem. Therefore, DMRG has been a powerful numerical method for the fuzzy sphere. However, like other $(2+1)$D models, the DMRG on the fuzzy sphere also suffers from the divergence of the required maximal bond dimension with system size. One should thus be careful with the convergence of the results when doing DMRG.

In FuzzifiED, we use the ITensor library in Julia to perform the DMRG calculations.