Uncovering conformal symmetry in the $3D$ Ising transition: State-operator correspondence from a fuzzy sphere regularization

The $3D$ Ising transition, the most celebrated and unsolved critical phenomenon in nature, has long been conjectured to have emergent conformal symmetry, similar to the case of the $2D$ Ising transition. Yet, the emergence of conformal invariance in the $3D$ Ising transition has rarely been explored directly, mainly due to unavoidable mathematical or conceptual obstructions. Here, we design an innovative way to study the quantum version of the $3D$ Ising phase transition on spherical geometry, using the"fuzzy (non-commutative) sphere"regularization. We accurately calculate and analyze the energy spectra at the transition, and explicitly demonstrate the state-operator correspondence (i.e. radial quantization), a fingerprint of conformal field theory. In particular, we have identified 13 parity-even primary operators within a high accuracy and 2 parity-odd operators that were not known before. Our result directly elucidates the emergent conformal symmetry of the $3D$ Ising transition, a conjecture made by Polyakov half a century ago. More importantly, our approach opens a new avenue for studying $3D$ CFTs by making use of the state-operator correspondence and spherical geometry.


I. INTRODUCTION
Symmetry is one of the most important organizing principles in physics.As is well known, symmetries present microscopically (e.g.condensed matter systems, ultraviolet (UV) Lagrangians) can be spontaneously broken at low energies, giving rise to various distinct phases of matter such as crystals and magnets.Conversely and rather unexpectedly, symmetries absent microscopically can emerge at low energies, and such a phenomenon is called emergent symmetry.One prominent example is the order-disorder phase transition of 2D Ising model, for which Polyakov discovered emergent conformal symmetry in 1970 [1], 26 years after Onsager's exact solution [2].
Polyakov's remarkable discovery of emergent conformal symmetry in the 2D Ising transition gave birth to conformal field theory (CFT) [3], a class of quantum field theories with profound applications in various fields of physics including statistical mechanics, quantum condensed matter, string theory and quantum gravity.In statistical physics, it is a common belief that many universality classes of (classical and quantum) phase transitions are captured by CFTs, however this has not been proven for 3D transitions. 1 The emergence of conformal symmetry at phase transitions is not only aesthetically beautiful, but also useful in understanding the properties of these transitions, such as computing experimentally measurable critical exponents.In 2D the (local) conformal symmetry has an infinite-dimensional algebra, and it makes many 2D CFTs exactly solvable [3,6].In d > 2 dimensions, there is only a finite-dimensional (global) conformal symmetry, i.e.SO(d + 1, 1), with which one is not able to analytically solve CFTs as in 2D.Therefore, CFTs beyond 2D are rather poorly understood, with their solutions remaining outstanding for decades despite their broad appeal to physics and mathematics.
Historically, the study of lattice models for 2D classical phase transitions and their quantum cousins (1 + 1D quantum phase transitions) played a key role in the discovery and understanding of 2D CFTs [1,2,7].Similar progress in the study of conformal symmetry for d ≥ 3 dimensional theories, however, has stalled due to the natural limitation of the lattice formulation.There are a plenty of papers studying 3D phase transitions on the lattice, e.g.computing critical exponents.However, the perspective of conformal symmetry has rarely been explored [8][9][10][11][12][13].The conformal symmetry of a d−dimensional CFT is most transparent in geometries such as R d , S d as well as S d−1 ×R.In particular, CFTs on S d−1 ×R obey a property called state-operator correspondence (i.e.radial quantization), which is a direct consequence of conformal symmetry [7].Specifically, for a quantum Hamiltonian defined on sphere S d−1 , its eigenstates are in one-to-one correspondence with the scaling operators (including primary and descendant operators) of the infrared (IR) CFT.Moreover, the energy gaps of these eigenstates are proportional to the scaling dimensions of their corresponding scaling operators [14].This nice feature can be used to explore various properties of CFTs, including scaling dimensions of operators, operator product expansion coefficients, and even operator algebras [7].For 2D CFTs, S 1 × R is very natural as one just needs to study a 1 + 1D quantum lattice model defined on a 1D periodic chain (i.e. S 1 ) [15][16][17][18].However, simulating lattice models of d ≥ 3 dimensional CFTs on arXiv:2210.13482v3[cond-mat.stat-mech]30 Oct 2023 S d−1 × R will be problematic, because a regular lattice cannot be put on a sphere S d−1≥2 due to its nontrivial curvature. 2While efforts have nevertheless been made to discretize the sphere, no signature of state-operator correspondence has been found so far [19,20].
To overcome this geometric obstacle, in this paper we are pursuing a different direction, namely we fuzzify a sphere [21].Specifically, we study a 2 + 1D quantum Ising transition defined on a fuzzy (non-commutative) sphere in light of Landau level regularization [22].As a result of this innovative discretization, we have observed almost perfect state-operator correspondence in surprisingly small system sizes.We use exact diagonalization to calculate properties of the 2+1D Ising transition for up to 16 effective spins, and we have found its low lying eigenstates (up to 70 lowest states) split into representations of the 3D conformal symmetry (i.e.conformal multiplets), hence directly demonstrating the emergence of conformal symmetry.Among these low energy states, we have found 15 conformal primary states, most of which have not been discovered in any previous model studies of the 3D Ising transition.Specifically, we have found 13 parityeven primaries, whose scaling dimensions agree well with state-of-the-art conformal bootstrap results [23,24] with discrepencies smaller than 1.6%.We have also identified two parity-odd primaries which were unknown before.
Our observations directly verify conformal symmetry for the 3D Ising transition, which was conjectured by Polyakov 50 years ago [1].Before our results, the most compelling evidence for the 3D Ising transition being conformal was from numerical conformal bootstrap [23][24][25][26][27], which assumes conformal symmetry and found critical exponents close to the values obtained by various methods such as Monte Carlo simulation [28,29] and measured by experiments [30].In addition, there was an effort [13] to justify the conformal invariance of the 3D Ising by showing that the virial current operator does not exist. 3Our obtained operator spectrum from the state-operator correspondence indeed convincingly shows that the 3D Ising transition does not have the virial current, which is a structural explanation of the 3D Ising being conformal [32].A major surprise of our results is that an incredibly small system size (8 ∼ 16 total spins) is already enough to yield accurate conformal data of the 3D Ising CFT.So we expect this approach to open a new avenue for studying higher dimensional phase transitions and CFTs.Firstly, there is a zoo of universalities that can be studied using our approach, which is amenable to various numerical techniques such as exact diagnolization (ED), density-matrix renormalization group (DMRG) and determinantal Monte Carlo.
This offers an opportunity to tackle many open questions regarding phase transitions, critical phases and CFTs.Secondly, a number of new universal quantities can be computed once the 3D CFT is simulated on a sphere, such as operator product expansion coefficients, F (of Ftheorem) [33][34][35][36], and the spherical binder ratio [37], just to name a few.
The paper is organized as follows.In Sec.II A we will review background knowledge including the radial quantization of CFTs and the state-operator correspondence.The spherical Landau level quantization and related fuzzy sphere are discussed in Sec.II B. Readers familiar with these topics can skip some of these subsections.In Sec.III, we formulate spherical Landau levels to regularize the 3D Ising transition on a fuzzy sphere.A global quantum phase diagram is presented.In Sec.IV, we present the low-lying energy spectra at the phase transition point, and analyze their one-to-one correspondence with the scaling operators as predicted by the Ising CFT.This is the main result of this paper.At last, we present a discussion and outlook in Sec.V.

A.
Radial quantization of CFTs: state-operator correspondence In this subsection we review some basics of radial quantization, and for an elaborated discussion we refer the readers to CFT lecture notes such as those in [6,38].
The conformal group in From the operator point of view, a CFT can be thought of as a theory whose operators form an infinite-dimensional representation of the conformal group.Specifically, one can write CFT operators { Ôα } as eigen-operators (i.e.irreducible repre- Weyl transformation τ = log r FIG. 1. Through a Weyl transformation, Euclidean flat spacetime R d is mapped to the manifold of cylinder S d−1 × R. As a result, a CFT on R d quantized on equal radius slices can be described equivalently in terms of a CFT on S d−1 × R quantized on equal time slices.The states defined on the S d−1 × R have well-defined quantum numbers of SO(d) Lorentz rotation and dilatation, and thus they are in one-to-one correspondence with operators of the CFT, dubbed as state-operator correspondence.
sentations) of the dilatation and Lorentz rotation SO(d).
In particular, the eigenvalue ∆ of dilatation is called scaling dimension of the operator, and it corresponds to the exponent in the power law correlation function of the operator, e.g.⟨O(x)O(0)⟩ ∼ 1/|x| 2∆ .One can further categorize operators into primary operators and descendant operators: 1) primary operators are operators that are annihilated by the special conformal transformation K µ ; 2) descendant operators are not annihilated by K µ , and all of them can be obtained by applying translations P µ (multiple times) to the primary operators.Therefore, one can organize CFT operators as primary operators and their descendants, and each primary and its descendants form a set of operators called a conformal multiplet. 4A CFT has an infinite number of primary operators, which makes it hard to tackle theoretically.A major task of solving a CFT is thus to obtain its low lying (if not full) spectrum of primary operators.
To facilitate later analysis of our numerical results, we will elaborate a bit more about the operator contents of a 3D CFT.In 3D the Lorentz rotation group is the familiar SO(3) group, all the irreducible representations of which are rank-ℓ symmetric traceless representations, i.e., spin-ℓ representations.So all (primary and descendant) operators have two quantum numbers (∆, ℓ).A primary operator O with quantum number ℓ = 0 is called a scalar operator, and any of its descendants can be written as with quantum number (∆ + 2n + j, j).We note □ = ∂ 2 .
Here and hereafter all the free indices shall be symmetrized with the trace subtracted.The descendants of a spin-ℓ primary operator O µ1•••µ ℓ are a bit more complicated as there are two different types.The first type can be written as, with quantum number (∆ + 2n + j + i, ℓ + j − i) for ℓ ≥ i ≥ 0, n, j ≥ 0.Here and hereafter the repeated indices shall be contracted.The other type will involve the ε tensor of SO(3), and can be written as, with quantum number (∆ + 2n + j We note that the ε tensor alters spacetime parity symmetry of We also remark that conserved operators (i.e.global symmetry current J µ and energy momentum tensor T µν ) should be treated a bit differently, because they satisfy the conservation equations ∂ µ J µ = 0 and ∂ µ T µν = 0. Therefore, their descendants in Eq. ( 2) and (3) should have i = 0. 5Now we turn to the state perspective of CFTs.To define states of a CFT, we first need to quantize it, or in other words find a Hilbert space construction of it.A quantum phase transition, namely a quantum Hamiltonian realization of a d-dimensional CFT in d − 1 space dimensions, can be viewed as a way to quantize the CFT.The states of the CFT are nothing but the quantum Hamiltonian's eigenstates.Formally, the quantization of CFTs can be more general than quantum phase transitions.Specifically, one can foliate d-dimensional spacetime into d − 1-dimensional surfaces, and each leaf of the foliation is endowed with its own Hilbert space.One convenient quantization is radial quantization, which has the d-dimensional Euclidean space R d foliated to S d−1 × R, as shown in the left hand side of Fig. 1.In the radial quantization, the SO(d) Lorentz rotation acts on the S d−1 sphere, while the dilatation acts as the scaling of sphere radius.Therefore, the states defined on the foliation S d−1 have well-defined quantum numbers of SO(d) rotation and dilatation, and they are indeed in one-toone correspondence with operators of the CFT, dubbed as state-operator correspondence.
For a quantum Hamiltonian realization, the radial quantization described above is not natural, and instead one may want a quantization scheme that has an identical Hilbert space on each leaf of foliation.A quantum Hamiltonian is usually defined on the M d−1 ×R manifold: R is the time direction, while M d−1 is a d−1-dimensional space manifold (e.g.sphere, torus, etc.), the leaf of foliation, on which the Hilbert space (and the quantum state) lives.In order to discuss state-operator correspondence in such a quantization scheme, one needs to map R d to the cylinder S d−1 ×R using a Weyl transformation [7,14], as shown in Fig. 1.Under the Weyl transformation the dilatation r → e λ r of R d becomes the translation along the time direction τ → τ + λ of S d−1 × R. If the theory has conformal symmetry, we can simply relate correlators and states on R d to those on S d−1 ×R.Moreover, we still have the state-operator correspondence on the cylinder S d−1 × R. In particular, the state-operator correspondence on the cylinder has a nice physical interpretation, namely the eigenstates |ψ n ⟩ of the CFT quantum Hamiltonian on S d−1 are in one-to-one correspondence with the CFT operators, and the energy gaps δE n of these states are proportional to the scaling dimensions ∆ n of CFT operators [7,14], where R is the radius of sphere S d−1 and v is the velocity of light that is model dependent.Also the SO(d) rotation symmetry of S d−1 is identified with the SO(d) Lorentz rotation of the conformal group, so the SO(d) quantum numbers of |ψ n ⟩ are identical to those of CFT operators.We emphasize that in contrast to radial quantization on R d , conformal symmetry is indispensable for the state-operator correspondence of radial quantization on the cylinder S d−1 × R. Therefore, observing the stateoperator correspondence on the cylinder S d−1 ×R will be direct evidence for the conformal symmetry of the theory or phase transition.For d = 2, the cylinder S 1 × R corresponds to nothing but a quantum Hamiltonian defined on a periodic chain, and there are very nice results studying the resulting state-operator correspondence [15][16][17][18].In higher dimensions, one needs to study a quantum Hamiltonian defined on S d−1 , however, it is highly nontrivial for a discrete lattice model as S d−1≥2 has a curvature.

B. Spherical Landau levels, fuzzy two-sphere and lowest Landau level projection
As originally shown by Landau, electrons moving in 2D space under a magnetic field will form completely flat bands called Landau levels, which is the key to the quantum Hall effect.Landau level quantization can be considered on any orientable manifold, and Haldane [39] first introduced Landau levels on spherical geometry to study the fractional quantum Hall physics.
For electrons moving on the surface of a radius-r sphere with a 4πs monopole (2s ∈ Z) placed at the origin (Fig. 2), the Hamiltonian is where M e is the electron's mass and Λ µ = ∂ µ + iA µ is the covariant angular momentum, A µ is the gauge field of the monopole.As usual we take ℏ = e = c = 1.The eigenstates will be quantized into spherical Landau levels, whose energies are Here (θ, φ) is the spherical coordinate.
These LLL Landau orbitals indeed form a SO(3) spins irreducible representation.This can be understood by constructing the SO(3) angular momentum operator [41], which satisfies the SO(3) algebra [L µ , L ν ] = iε µνρ L ρ .Projecting the system into the LLL, the kinetic energy of the covariant angular momentum will be quenched, so effectively we have L µ ∼ sx µ /r.(x µ denotes the coordinates in the projected LLL.)As a result, the coordinates xµ of electrons will not actually commute, instead we have This defines a fuzzy two-sphere [21].Moreover, Landau orbitals (6) are in one-to-one correspondence with states on the fuzzy two-sphere.Formally, a system defined on the LLL can be equivalently viewed as a system defined on a fuzzy two-sphere.We will not delve into details along that direction, and refer the reader to [42] for more discussions.
As is usually done in the literature, we will consider the limit where the interaction strength is much smaller than the Landau level gap, so we can project the system into the LLL.Technically, this can be done by rewriting the annihilation operator ψ(θ, φ) on the LLL as ĉm stands for the annihilation operator of Landau orbital m, and is independent of coordinates (θ, φ).The density operator n(θ, φ) = ψ † ψ can be written as, Any interaction can be straightforwardly (though perhaps tediously) written in the second quantized form using Landau orbital operators c † m , c m .For example, the density-density interaction H I = d 2 r a d 2 r b U (r a − r b )n(r a )n(r b ) can be written as, where V m1,m2,m3,m4 can be further expanded using the so-called Haldane pseudopotential V l [39], corresponding to the two-fermion scattering in the spin-2s − l channel (see Appendix Sec.A).
In summary, the model we are working with is a fermonic Hamiltonian enclosing 2s+1-Landau orbitals with long-range SO(3) invariant interactions.Interestingly, all the orbitals form an SO(3) spin-s irrep.Furthermore, the length scale of the system is √ 2s + 1 instead of 2s + 1 since the spatial dimension is d = 2, and the thermodynamic limit corresponds to taking s to infinity.

A. Hamiltonian
Here we explicitly define the model, which is spinful electrons in the LLL. 6In spatial space, the Hamiltonian takes the form where n α (θ, φ) is a local density operator given by with σ x,y,z being Pauli matrices, σ 0 = I 2×2 , and U (Ω ab ) the local density-density interactions (defined below).The first term behaves like an Ising ferromagnetic interaction, while the second term is the transverse field.By projecting the Hamiltonian into the LLL, we obtain where is the fermion creation operator on the m th Landau orbital.The parameter V m1,m2,m3,m4 is connected to the Haldane pseudopotential V l by where j 1 j 2 j 3 m 1 m 2 m 3 is the Wigner 3j-symbol.In this paper we will only consider ultra-local density-density interactions in real space, i.e.U (Ω ab ) = g 0 1 , and the associated Haldane pseudopotentials involve V 0 , V 1 (see Appendix Sec.A).Next we will set V 1 = 1 as energy unit and vary V 0 , h to study the phase diagram.
We consider the half-filling case with the LLL filled by N = 2s + 1 electrons.When h = 0 and V 0 , V 1 > 0, the ground state is an Ising ferromagnet that spontaneously breaks Z 2 symmetry.In quantum Hall literature this phase is called quantum Hall ferromagnetism [43,44].The two-fold degenerate ground states are

Particle-hole symmetry:
Electric charges of fermions are gapped in the entire phase diagram (see Appendix Sec.C 1), while the Ising spins of fermions are the degrees of freedom that go through the phase transitions.Therefore, all the gapless degrees of freedom at the phase transition are chargeneutral.In particular, the order parameter of the transition is a particle-hole excitation of fermions, We emphasize an important point for the Landau level regularization of the Ising transition: the electrons are sitting on a fuzzy sphere due to the monopole, but the Ising spins are sitting on a normal sphere (for any finite N = 2s+1) since they are charge neutral.This is the key difference between our Landau level regularization and the non-commutative field theory [45], namely the latter always has quantum fields defined on a fuzzy manifold as long as the physical volume is finite.
To further analyze the Ising transition in our system, we will relate the UV symmetries of our Landau level model to the IR symmetries of the 3D Ising CFT.It is obvious we can identify the Ising Z 2 and SO(3) sphere rotation between UV and IR.A slightly non-trivial symmetry is the particle-hole symmetry, which turns out to be the spacetime parity symmetry of 3D Ising CFT.To understand this relation, we can write an SO(3) vector, and find it transforms as under particle-hole transformation.The particle-hole acts as an improper Z 2 of O(3), so it can be identified as the spacetime parity of the 3D Ising CFT.

C. Finite size scaling
The phase diagram in Fig. 2(b) is obtained by the conventional finite size scaling of the Z 2 order parameter M in Eq. ( 16).We have simulated N = 2s + 1 = 8, 10, • • • 24 using ED for smaller sizes (N ≤ 16) and DMRG for larger sizes N > 16 (the length scale in this 2 + 1D system is L x = √ N ).At the phase transition point, the Z 2 order parameter should scale as where ∆ ≈ 0.5181489 is the scaling dimension of Ising order parameter [23,24].Fig. 3 (a) depicts ⟨M 2 ⟩/N 2−∆ with respect with the transverse field strength h of different N for V 0 = 4.75.All the curves nicely cross at h c ≈ 3.16, which we identify as the transition point.Similarly for other V 0 we have identified the critical h c and obtained the phase diagram as shown in Fig. 2(b).
We have also computed the binder cumulant U 4 is a RG-invariant quantity, and U 4 = 1, 0 at the thermodynamic limit corresponds to the ordered phase and disordered phase, respectively.At the phase transition U 4 will be a universal quantity related to the four point correlator of the order parameter field σ of CFT [37].Fig. 3 (b) shows U 4 with respect to the transverse field strength h for different N for V 0 = 4.75.Clearly, at small h the model is in the Ising ferromagnetic phase, while at large h the model is in the disordered phase.To estimate the value of binder ratio at the critical point U c 4 , we perform a detailed crossing-point analysis (Appendix Sec.B).With the data on hand, the best estimate we can give is 0.28 ≤ U c 4 ≤ 0.40.It will be interesting to evaluate U 4 from conformal bootstrap and compare with our estimate. 7 In practice, for small N (as we simulated numerically), finite-size effects are inevitable.One common source is from the couplings of irrelevant operators, which are typically present in microscopic models.Tuning along the critical line in the 2-dimensional parameter space (V 0 , h) shown in Fig. 2(b) generically modifies the coupling strength of irrelevant operators and therefore the magnitude of finite-size effects (while the relevant operators flow to the same fixed point).In the following section, we will present the data of the state-operator correspondence at a particular point V 0 = 4.75, h c = 3.16, where we find the finite size effects are smallest (Appendix Sec.D).

IV. STATE-OPERATOR CORRESPONDENCE
We now turn to the central results of our paper: the state-operator correspondence of the 3D Ising transition.As explained in Sec.II A, on S 2 ×R the eigenstates of the quantum Hamiltonian are in one-to-one correspondence with the scaling operators of its corresponding CFT.In particular, the energy gaps of each state will be proportional to the scaling dimensions of the scaling operators [14].Therefore, we explore energy spectra at the critical point by utilizing exact diagonalization and compare it with CFT predictions.
To match the Ising transition's energy spectra with the 3D Ising CFT's operator spectrum, we first need to rescale the energy spectrum with a non-universal (i.e.model-and size-dependent) numerical factor.The natural calibrator is the energy momentum tensor T µ1µ2 , a conserved operator that any local CFT possesses.For any 3D CFT, T µ1µ2 will be a global symmetry singlet, Lorentz spin ℓ = 2 operator with scaling dimension ∆ T = 3.Our model has exact SO(3) Lorentz rotation, Ising Z 2 , and spacetime parity symmetries, so every eigenstate has well-defined quantum numbers (Z 2 , P, ℓ) of these three symmetries.The energy-momentum tensor will be the lowest state in the (Z 2 = 1, P = 1, ℓ = 2) sector.We rescale the full spectrum by setting the energy momentum tensor to exactly ∆ T = 3, and then examine if the low-lying states form representations of 3D conformal symmetry up to a finite size correction.
We analyze the low-lying spectra according to the following steps, 1.For each Z 2 = ±1 sector, we find the lowest-lying energy state (regardless of ℓ and P ), and identify it as a primary state.
2. Based on the representation theory of the 3D conformal group as summarized in Eq. ( 1), ( 2), (3), 7 For models on the non-conformal manifold such as T 2 × R or T 3 , which Monte Carlo usually simulates, U 4 cannot be computed using the R 3 four-point correlator from conformal bootstrap.
we enumerate the descendant states of the identified primary state and examine if all of descendant states (up to ∆ = 7) exist in our energy spectrum.
3. We remove the identified conformal multiplet (i.e.primary and its descendants) from the energy spectrum, and for the remaining states we repeat the step 1,2.
Remarkably, we found that the lowest-lying 70 eigenstates8 form representations of the 3D conformal symmetry up to a small finite size correction, with no extra or missing state.This is a direct and unambiguous demonstration of the emergent conformal symmetry of the 3D Ising transition.
After verifying the emergent conformal symmetry, we further compare our scaling dimensions of the identified primary operators with the numerical conformal bootstrap data [23,24], and we find a good agreement for all of them.Table I lists all the primary operators we have identified with N = 16 ED data.We have found 12 parity-even primary operators besides the energymomentum tensor, and all of them have less than a 1.6% discrepancy from the bootstrap data [23,24].In Appendix Sec.D we list concrete values of each conformal multiplet, as one can see the numerical accuracy is unexpectedly high, particularly given that it is from a small system size (N = 16 total spins): around 10 operators have relative numerical error around 3% ∼ 5.5%, and the rest of them have relative numerical error smaller than 3%.Fig. 4 plots conformal multiplets of a few representative primary operators, which clearly illustrate the emergent conformal symmetry and agree well with numerical conformal bootstrap results.
A few remarks are in order.1) We verify the emergent conformal symmetry of the 3D Ising transition by showing that the low-lying spectra of our model form representations of 3D conformal symmetry.This procedure does not rely on any input of previous knowledge such as numerical bootstrap data.2) A spinning (ℓ > 0) parityeven (parity-odd) primary operator can have parity-odd (parity-even) descendant opertors as written in Eq. ( 3).This nontrivial structure from the CFT's algebra matches our ED spectrum. 93) The energy momentum tensor T µ1µ2 is a conserved operator, so it does not have any ℓ < 2 descendant.This structure is clearly shown in our data.4) All the parity-even primary operators that we found have been reported in the bootstrap study of mixed correlators ⟨σσσσ⟩, ⟨ϵϵϵϵ⟩, ⟨σσϵϵ⟩.The mixedcorrelator bootstrap study is only capable of detecting operators in the σ × σ, ϵ × ϵ and σ × ϵ OPE, so it will TABLE I. Low-lying primary operators identified via state-operator correspondence on a fuzzy sphere with N = 16 electrons.The operators in the first and second row are Z2 odd and even operators, respectively.We highlight that two new parity-odd primary operators σ P − and ϵ P − are found.The conformal bootstrap data is from Ref. [24].FIG. 4. Conformal multiplet of several low lying primary operators: scaling dimension ∆ versus Lorentz spin ℓ.We plot conformal bootstrap data with lines: lines in red are parity even, non-degenerate operators; lines in green are parity odd, non-degenerate operators; lines in black are parity even, two-fold degenerate operators.Symbols are our numerical data of parity even (red circle) and odd (green square) operators.The discrepancy is typically more significant for the larger ∆.
miss (Z 2 = 1, P = 1, odd ℓ) primary operators (in addition to P = −1 primaries).Our approach should be able to detect operators in these quantum number sectors, including the candidate of virial current [4]10 , namely the lowest primary in the (Z 2 = 1, P = 1, ℓ = 1) sector.We have not observed any primary operators in the (Z 2 = 1, P = 1, odd ℓ) sector below ∆ = 7, and so this gives a lower bound for the virial current candidate, which is higher than the previous estimate [13].5) We have identified two previously unknown (parityodd) primary operators in the (Z 2 = 1, P = −1, ℓ = 0) and (Z 2 = −1, P = −1, ℓ = 0) sectors with ∆ ≈ 10.01 and ∆ ≈ 11.19, respectively.To access P = −1 primary operators in the bootstrap calculation, one has to bootstrap correlation functions of the spinning operator: for example, the energy momentum tensor.Such study has only been initiated in Ref. [46] but no P = −1 primary has been identified by conformal bootstrap or any other methods so far.6) In all previous lattice model studies, only several primary fields (σ, ϵ and ϵ ′ ) were found, and their scaling dimensions are related to the critical exponents η, ν and ω [28,29].

V. SUMMARY AND DISCUSSION
We have designed an innovative scheme to numerically study the 3D Ising transition on the space-time geometry S 2 × R, and in our calculation we have found almost perfect state-operator correspondence of the 3D CFT, supporting the conjecture that the 3D Ising transition has emergent conformal symmetry.In detail, we considered the 3D Ising transition realized in a fermionic model defined on a fuzzy sphere, which we achieved by projecting spinful electrons into the lowest spherical Landau level where the spin degrees of freedom go through an order-disorder transition.We are able to identify 13 parity-even primary operators and 2 parity-odd primary operators, and around 60 descendant operators, in agreement with the predictions of underlying CFT within a high accuracy.
Our results have now offered a novel solution to the long-standing quest of simulating 3D CFTs on the sphere (more generally on the curved space), and even more remarkably, the finite size effect of our model is much smaller than the conventional approach (i.e.3D classical Ising model) used to study 3D CFTs.Therefore, our results open a new avenue for studying 3D CFTs in a microscopic way.Thanks to the state-operator correspondence on S 2 ×R, many universal quantities such as operator product expansion coefficients, four-point correlators, and thermal correlators of CFTs are ready to compute directly.These will lead to many insights of CFTs that are important for several purposes.For example, the thermal correlators will not only be useful for predicting experiments of 2 + 1 dimensional quantum phase transitions at the finite temperature, but also help to understand the properties of quantum black holes using the AdS/CFT duality. 11Another interesting quantity is the RG monotonic quantity F of the F -theorem [33][34][35][36], which can be extracted from the quantum entanglement [47,48].
We also expect our approach can be used to tackle many open problems of the 3D CFTs.Specifically, our approach can be applied to many universalities such as O(N ) Wilson-Fisher transitions (i.e.XY universality, etc.) and critical gauge theories (e.g.see Ref. [49]). 12  11 A CFT at finite temperature on boundary is dual to a black hole of quantum gravity in the bulk. 12A guiding principle for the model design will be kinematical properties such as symmetries and anomalies.Here we start with spinful fermions, whose maximal spin symmetry is SU (2).We further add interactions that break SU (2) down to the anomalyfree Ising Z 2 symmetry.Similarly, for other transitions one can With a straightforward examination of the emergent conformal symmetry and a precise determination of the scaling dimensions of various primary operators, one may eventually solve the question of the conformal window of 3D critical gauge theories, a problem puzzled the high energy physics and condensed matter community for decades.
In our paper the fuzzy sphere regularization is formulated using the language of lowest Landau level projection.It will be interesting to translate our formulation into the formal language of non-commutative geometry.Such perspective of the fuzzy sphere regularization may help to develop a systematic framework that is applicable to any CFT and QFT on various manifolds in arbitrary space-time dimensions.For example, an ambitious question is, can one directly regularize the continuum QFTs on the fuzzy sphere without encountering the infamous UV infiniteness of QFTs?Indeed the similar idea was pursued decades ago in the context of non-commutative field theory [45], but was unsuccessful due to the phenomenon called UV-IR mixing.Our regularization scheme offers a new angle to this question, namely one can introduce auxiliary fields (i.e.electrons in our model) that are living on the fuzzy sphere, and the true low energy quantum fields of the theory (i.e.Ising spins in our model) are living on the normal sphere.We believe this way of thinking may lead to many fruitful results of CFTs and QFTs, and may reveal a new connection between physics and mathematics.

VI. ACKNOWLEGEMENT
this work were initiated, and for the Galileo Galilei Institute and the organizers of the GGI workshop "Boot-strapping Nature: Non-perturbative Approaches to Critical Phenomena" for hospitality during the completion of this work.

Spin excitation gap
In this section, we discuss the spin excitation gap.In the Ising ferromagnet (h < h c ), flipping a spin orientation should cost finite exchange energy, so the spin excitation gap should be nonzero.Similarly, the paramagnetic ground state h > h c is a trivial insulator, which should be separated from all other excited states by a finite energy gap.In contrast, at the critical point, the system becomes gapless, which should be distinct from the other two gapped phases.As shown in Fig. 7, we show three typical plots of excitation gap in Ising ferromagnet phase, paramagnet phase and at the phase transition point.It is clear that, the excitation gaps are finite for ferromagnet and paramagnet phase, but the system becomes gapless at the transition point h ≈ h c .The most interesting thing is, these critical excitations at finite system sizes form a characteristic conformal tower structure as discussed in the main text, which calls for a CFT description of 3D Ising criticality.

Appendix D: Details of numerical data
In this section, we present the data of energy spectra which are organized by the good quantum numbers and conformal multiplet of various primary fields, e.g.ϵ (Tab.II), ϵ ′ (Tab.III), T µ1µ2 (Tab.IV), T ′ µ1µ2 (Tab.V), ϵ µ1,µ2,µ3,µ4 (Tab.VI), σ (Tab.VII), σ ′ (Tab.VIII), σ µ1µ2 (Tab.IX), and σ µ1µ2µ3 (Tab.X).For comparison, we also list the results from conformal boostrap (CB) method [23,24].These data are used for plotting Fig. 4. For the primary fields, the discrepancies are really small ( < 1.6%).Generally, fields with higher energies (conformal weights) have larger discrepancies, which is attributed to the finite-size effect.We add that these detailed data also gives a good quantification for the numerical error without inputting other results such as numerical bootstrap.The idea is that, since the conformal symmetry predicts the integer spacings between primaries and their descendants, we can examine how good is this preserved in our spectrum.Based on this we can give a conservative estimate for numerical errors of primiaries and low lying descendants, which are 3% relative errors.A rigorous error analysis based on the finite size scaling and off-critical behavior will be interesting for the future work.
Another interesting point is that, we identify almost perfect state-operator correspondence in surprisingly small system sizes.In the main text, we only present the numerical data at a given system size, i.e.N = 16, which is the largest system size that we can reach using ED.Here, to further elucidate that the numerical findings indeed reflect the physics in the thermodynamic limit, we show the energy spectra on different system sizes.In Fig. 8, we show the energy spectra obtained on different system sizes from N = 8 to N = 16.As one can see that, the energies on all system sizes match the prediction of 3D CFT quite well.
Finally, as we discussed in the main text, one of the most surprising aspects of the fuzzy sphere scheme is that the IR CFT emerges in incredibly small system sizes.The best illustration is the observation that our model with only N = 4 spins (electrons) (Tab.XI) already produces 6 primaries and the approximate conformal invariance.All calculations can be done on the laptop, where the N = 16 spins requires around 16G memory, and the computation took around 30 minutes on a M1 Macbook.

FIG. 2 .
FIG. 2. (a) Schematic plot of electrons moving on a sphere in the presence of 4π•s monopole.The LLL has 2s+1 degenerate orbitals, which form an SO(3) spin-s irreducible representation.A system projected into the LLL can be equivalently viewed as a fuzzy sphere.(b) Phase diagram of the proposed model consisting of a continuous phase transition from a quantum Hall Ising ferromagnet to a disordered paramagnet.
FIG. 3. (a) Finite size scaling of order parameter ⟨M 2 ⟩/N 2−∆ .∆ = 0.518148 is the scaling dimension of the Ising order parameter field.N = 2s + 1 is the number of electrons (i.e.Ising spins), hence it should be identified as space volume and the length scale is ∼ √ N .The rescaled order parameter perfectly crosses at the same point hc ≈ 3.16.(b) Plot of the RG-invariant binder cumulant U4.The binder cumulant does not stably cross at the same point due to the large finite size effect.We set V0 = 4.75 here.

FIG. 7 .
FIG. 7. Finite-size scaling of the lowest six excitation gap of (a) quantum Hall ferromagnet at h = 1.0 < hc, (b) transition point at h = hc and (c) disordered paramagnet at h = 4.0 > hc.

FIG. 8 .
FIG.8.The energy spectra corresponding to primary fields for various system sizes N = 8 − 16.The dashed color lines denote the numerical values from conformal boostrap method.