Uncovering conformal symmetry in the 3DIsing transition State-operator correspondence from a fuzzy sphere regularization Wei Zhu1Chao Han2Emilie Huffman3Johannes S. Hofmann4and Yin-Chen He3

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Uncovering conformal symmetry in the 3DIsing transition:

State-operator correspondence from a fuzzy sphere regularization

Wei Zhu,1, ∗Chao Han,2Emilie Huﬀman,3Johannes S. Hofmann,4and Yin-Chen He3, †

1School of Science, Westlake University, Hangzhou, 310030, China

2Westlake Institute of Advanced Study, Westlake University, Hangzhou, 310024, China

3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

4Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100, Israel

The 3DIsing 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 2DIsing

transition. Yet, the emergence of conformal invariance in the 3DIsing 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 3DIsing 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 3DIsing transition, a conjecture made by Polyakov half a century ago. More importantly,

our approach opens a new avenue for studying 3DCFTs 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 bro-

ken 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 2DIsing model, for

which Polyakov discovered emergent conformal symme-

try in 1970 [1], 26 years after Onsager’s exact solution [2].

Polyakov’s remarkable discovery of emergent confor-

mal symmetry in the 2DIsing 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 con-

densed matter, string theory and quantum gravity. In

statistical physics, it is a common belief that many uni-

versality classes of (classical and quantum) phase transi-

tions are captured by CFTs, however this has not been

proven for 3Dtransitions. 1The 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 2Dthe (local) confor-

mal symmetry has an inﬁnite-dimensional algebra, and it

∗zhuwei@westlake.edu.cn

†yhe@perimeterinstitute.ca

1For phase transitions in 2D[4] and 4D[5] the combination of

scale symmetry, Lorentz symmetry and unitarity was shown to

lead to conformal symmetry.

makes many 2DCFTs exactly solvable [3,6]. In d > 2 di-

mensions, there is only a ﬁnite-dimensional (global) con-

formal symmetry, i.e. SO(d+ 1,1), with which one is

not able to analytically solve CFTs as in 2D. There-

fore, CFTs beyond 2Dare 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 2Dclassi-

cal phase transitions and their quantum cousins (1 + 1D

quantum phase transitions) played a key role in the dis-

covery and understanding of 2DCFTs [1,2,7]. Sim-

ilar 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 3Dphase transitions on

the lattice, e.g. computing critical exponents. How-

ever, the perspective of conformal symmetry has rarely

been explored [8–13]. The conformal symmetry of a

d−dimensional CFT is most transparent in geometries

such as Rd,Sdas well as Sd−1×R. In particular, CFTs on

Sd−1×Robey a property called state-operator correspon-

dence (i.e. radial quantization), which is a direct con-

sequence of conformal symmetry [7]. Speciﬁcally, for a

quantum Hamiltonian deﬁned on sphere Sd−1, its eigen-

states 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 dimen-

sions of their corresponding scaling operators [14]. This

nice feature can be used to explore various properties of

CFTs, including scaling dimensions of operators, oper-

ator product expansion coeﬃcients, and even operator

algebras [7]. For 2DCFTs, S1×Ris very natural as one

just needs to study a 1 + 1Dquantum lattice model de-

ﬁned on a 1Dperiodic chain (i.e. S1) [15–18]. However,

simulating lattice models of d≥3 dimensional CFTs on

arXiv:2210.13482v3 [cond-mat.stat-mech] 30 Oct 2023

Sd−1×Rwill be problematic, because a regular lattice

cannot be put on a sphere Sd−1≥2due 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 + 1Dquantum

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 surpris-

ingly small system sizes. We use exact diagonalization to

calculate properties of the 2+1DIsing transition for up to

16 eﬀective spins, and we have found its low lying eigen-

states (up to 70 lowest states) split into representations of

the 3Dconformal symmetry (i.e. conformal multiplets),

hence directly demonstrating the emergence of confor-

mal 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 parity-

even 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 3DIsing transition, which was conjectured by

Polyakov 50 years ago [1]. Before our results, the most

compelling evidence for the 3DIsing transition being

conformal was from numerical conformal bootstrap [23–

27], which assumes conformal symmetry and found crit-

ical 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

3DIsing by showing that the virial current operator

does not exist. 3Our obtained operator spectrum from

the state-operator correspondence indeed convincingly

shows that the 3DIsing transition does not have the

virial current, which is a structural explanation of the

3DIsing being conformal [32]. A major surprise of our

results is that an incredibly small system size (8 ∼16 to-

tal spins) is already enough to yield accurate conformal

data of the 3DIsing 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 renormal-

ization group (DMRG) and determinantal Monte Carlo.

2In mathematics the problem of tiling a sphere is called spherical

tiling or spherical polyhedron.

3We note that Ref. [31] claimed a proof of the conformal invariance

of 3DIsing transition, but it is unclear if the proof is correct (see

the comment in Appendix B in the ﬁrst arXiv version of Ref.[13]).

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 3DCFT is simulated on a sphere,

such as operator product expansion coeﬃcients, F(of F-

theorem) [33–36], 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 correspon-

dence. The spherical Landau level quantization and re-

lated fuzzy sphere are discussed in Sec. II B. Readers

familiar with these topics can skip some of these subsec-

tions. In Sec. III, we formulate spherical Landau levels

to regularize the 3DIsing 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 correspon-

dence 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.

II. REVIEW OF BACKGROUND

A. Radial quantization of CFTs: state-operator

correspondence

In this subsection we review some basics of radial quan-

tization, and for an elaborated discussion we refer the

readers to CFT lecture notes such as those in [6,38].

The conformal group in ddimensions SO(d+ 1,1) is

generated by d-dimensional translations Pµ=i∂µ,d-

dimensional Lorentz rotations Mµν =i(xµ∂ν−xν∂µ),

dilatations D=ixµ∂µ, and special conformal transfor-

mations Kµ=i(2xµ(xν∂ν)−x2∂µ). 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 {ˆ

Oα}as eigen-operators (i.e. irreducible repre-

Sd−1

Weyl transformation

τ= log r

FIG. 1. Through a Weyl transformation, Euclidean ﬂat space-

time Rdis mapped to the manifold of cylinder Sd−1×R. As

a result, a CFT on Rdquantized on equal radius slices can be

described equivalently in terms of a CFT on Sd−1×Rquan-

tized on equal time slices. The states deﬁned on the Sd−1×R

have well-deﬁned quantum numbers of SO(d) Lorentz rota-

tion and dilatation, and thus they are in one-to-one correspon-

dence 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 scal-

ing 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 descen-

dant operators: 1) primary operators are operators that

are annihilated by the special conformal transformation

Kµ; 2) descendant operators are not annihilated by Kµ,

and all of them can be obtained by applying translations

Pµ(multiple times) to the primary operators. There-

fore, one can organize CFT operators as primary oper-

ators 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 3DCFT. In 3Dthe Lorentz rotation group is the fa-

miliar SO(3) group, all the irreducible representations of

which are rank-ℓsymmetric traceless representations, i.e.,

spin-ℓrepresentations. So all (primary and descendant)

operators have two quantum numbers (∆, ℓ). A primary

operator Owith quantum number ℓ= 0 is called a scalar

operator, and any of its descendants can be written as

∂ν1···∂νj□nO, n, j ≥0,(1)

with quantum number (∆ + 2n+j, j). We note □=∂2.

Here and hereafter all the free indices shall be sym-

metrized with the trace subtracted. The descendants of

a spin-ℓprimary operator Oµ1···µℓare a bit more compli-

cated as there are two diﬀerent types. The ﬁrst type can

be written as,

∂ν1···∂νj∂µ1···∂µi□nOµ1···µℓ,(2)

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 εtensor of SO(3), and can be written as,

εµlρτ ∂ρ∂ν1···∂νj∂µ1···∂µi□nOµ1···µℓ,(3)

with quantum number (∆ + 2n+j+i+ 1, ℓ +j−i) for

ℓ−1≥i≥0, n, j ≥0. We note that the εtensor alters

spacetime parity symmetry of Oµ1···µℓ.

We also remark that conserved operators (i.e. global

symmetry current Jµand energy momentum tensor Tµν )

should be treated a bit diﬀerently, because they satisfy

the conservation equations ∂µJµ= 0 and ∂µTµν = 0.

4Here we are talking about primary operators under the global

conformal symmetry SO(d+ 1,1). For 2DCFTs one usually

talks about primary operators under Virasoro symmetries, and

the global conformal primaries are called quasi-primaries.

Therefore, their descendants in Eq. (2) and (3) should

have i= 0. 5

Now we turn to the state perspective of CFTs. To de-

ﬁne 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 Hamilto-

nian 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 transi-

tions. Speciﬁcally, one can foliate d-dimensional space-

time into d−1-dimensional surfaces, and each leaf of the

foliation is endowed with its own Hilbert space. One con-

venient quantization is radial quantization, which has the

d-dimensional Euclidean space Rdfoliated to Sd−1×R,

as shown in the left hand side of Fig. 1. In the radial

quantization, the SO(d) Lorentz rotation acts on the

Sd−1sphere, while the dilatation acts as the scaling of

sphere radius. Therefore, the states deﬁned on the folia-

tion Sd−1have well-deﬁned quantum numbers of SO(d)

rotation and dilatation, and they are indeed in one-to-

one 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 iden-

tical Hilbert space on each leaf of foliation. A quantum

Hamiltonian is usually deﬁned on the Md−1×Rmanifold:

Ris the time direction, while Md−1is a d−1-dimensional

space manifold (e.g. sphere, torus, etc.), the leaf of folia-

tion, 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 Rdto

the cylinder Sd−1×Rusing a Weyl transformation [7,14],

as shown in Fig. 1. Under the Weyl transformation the

dilatation r→eλrof Rdbecomes the translation along

the time direction τ→τ+λof Sd−1×R. If the theory

has conformal symmetry, we can simply relate correlators

and states on Rdto those on Sd−1×R. Moreover, we still

have the state-operator correspondence on the cylinder

Sd−1×R. In particular, the state-operator correspon-

dence on the cylinder has a nice physical interpretation,

namely the eigenstates |ψn⟩of the CFT quantum Hamil-

tonian on Sd−1are in one-to-one correspondence with

the CFT operators, and the energy gaps δEnof these

states are proportional to the scaling dimensions ∆nof

CFT operators [7,14],

δEn=En−E0=v

R∆n,(4)

where Ris the radius of sphere Sd−1and vis the velocity

of light that is model dependent. Also the SO(d) rotation

5The conformal multiplet of a conserved operator is called a short

multiplet.

symmetry of Sd−1is identiﬁed with the SO(d) Lorentz

rotation of the conformal group, so the SO(d) quantum

numbers of |ψn⟩are identical to those of CFT operators.

We emphasize that in contrast to radial quantiza-

tion on Rd, conformal symmetry is indispensable for the

state-operator correspondence of radial quantization on

the cylinder Sd−1×R. Therefore, observing the state-

operator correspondence on the cylinder Sd−1×Rwill be

direct evidence for the conformal symmetry of the theory

or phase transition. For d= 2, the cylinder S1×Rcorre-

sponds to nothing but a quantum Hamiltonian deﬁned on

a periodic chain, and there are very nice results studying

the resulting state-operator correspondence [15–18]. In

higher dimensions, one needs to study a quantum Hamil-

tonian deﬁned on Sd−1, however, it is highly nontrivial

for a discrete lattice model as Sd−1≥2has a curvature.

B. Spherical Landau levels, fuzzy two-sphere and

lowest Landau level projection

As originally shown by Landau, electrons moving in

2Dspace 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-rsphere

with a 4πs monopole (2s∈Z) placed at the origin (Fig.

2), the Hamiltonian is

H0=1

2Mer2Λ2

µ,(5)

where Meis the electron’s mass and Λµ=∂µ+iAµis

the covariant angular momentum, Aµis 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 En= [n(n+ 1) + (2n+ 1)s]/(2Mer2),

with n= 0,1,2,··· the Landau level index. The (n+1)th

Landau level is (2s+ 2n+ 1)-fold degenerate, and the

single particle states in each Landau level are called Lan-

dau orbitals. Assuming all interactions are much smaller

than the energy gap between Landau levels, we can just

consider the lowest Landau level (LLL) n= 0, which is

2s+ 1-fold degenerate. The wave-functions for each Lan-

dau orbital on LLL are called monopole harmonics [40]

Φm(θ, φ) = Nmeimφ coss+mθ

2sins−mθ

2,(6)

with m=−s, −s+ 1,··· , s and Nm=q(2s+1)!

4π(s+m)!(s−m)! .

Here (θ, φ) is the spherical coordinate.

These LLL Landau orbitals indeed form a SO(3) spin-

sirreducible representation. This can be understood by

constructing the SO(3) angular momentum operator [41],

Lµ= Λµ+sxµ

r,(7)

which satisﬁes the SO(3) algebra [Lµ, Lν] = iεµνρLρ.

Projecting the system into the LLL, the kinetic energy

of the covariant angular momentum will be quenched, so

eﬀectively we have Lµ∼s˜xµ/r. (˜xµdenotes the coordi-

nates in the projected LLL.) As a result, the coordinates

˜xµof electrons will not actually commute, instead we

have

[˜xµ,˜xν] = ir

sϵµνρ ˜xρ.(8)

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

ψ(θ, φ) = 1

√2s+ 1

m=−s

Φ∗

mˆcm.(9)

ˆcmstands for the annihilation operator of Landau orbital

m, and is independent of coordinates (θ, φ). The density

operator n(θ, φ) = ψ†ψcan be written as,

n(θ, φ) = 1

2s+ 1 X

m1,m2

Φm1Φ∗

m2c†

m1cm2.(10)

Any interaction can be straightforwardly (though per-

haps tediously) written in the second quantized form

using Landau orbital operators c†

m, cm. For example,

the density-density interaction HI=Rd2rad2rbU(ra−

rb)n(ra)n(rb) can be written as,

HI= (2s+ 1)2ZdΩadΩbU(θa, φa;θb, φb)n(θa, φa)n(θb, φb)

m1,m2,m3,m4

Vm1,m2,m3,m4c†

m1c†

m2cm3cm4,(11)

where Vm1,m2,m3,m4can be further expanded using the

so-called Haldane pseudopotential Vl[39], corresponding

to the two-fermion scattering in the spin-2s−lchannel

(see Appendix Sec. A).

In summary, the model we are working with is a fer-

monic Hamiltonian enclosing 2s+1-Landau orbitals with

long-range SO(3) invariant interactions. Interestingly, all

the orbitals form an SO(3) spin-sirrep. Furthermore, the

length scale of the system is √2s+ 1 instead of 2s+ 1

since the spatial dimension is d= 2, and the thermody-

namic limit corresponds to taking sto inﬁnity.

III. MODEL ON A FUZZY-TWO SPHERE

A. Hamiltonian

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Uncoveringconformalsymmetryinthe3DIsingtransition:State-operatorcorrespondencefromafuzzysphereregularizationWeiZhu,1,∗ChaoHan,2EmilieHuffman,3JohannesS.Hofmann,4andYin-ChenHe3,†1SchoolofScience,WestlakeUniversity,Hangzhou,310030,China2WestlakeInstituteofAdvancedStudy,WestlakeUniversity,Hangzhou,3100...

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Uncovering conformal symmetry in the 3DIsing transition State-operator correspondence from a fuzzy sphere regularization Wei Zhu1Chao Han2Emilie Huffman3Johannes S. Hofmann4and Yin-Chen He3

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