Introduction

Defects, as well as their special case–boundaries, are fundamental elements that inevitably exist in nearly all realistic physical systems. Historically, research on defects has played a pivotal role in shaping modern theoretical physics. This includes contributions to the theory of the renormalization group (RG)1, studies of topological phases2,3,4, investigations into the confinement of gauge theories5,6, explorations of quantum gravity7, and advancements in the understanding of quantum entanglement8,9. An important instance to study defects is in the context of conformal field theory (CFT)10,11, where one considers the situation of deforming a CFT with interactions living on a sub-dimensional defect. The defect may trigger an RG flow towards a non-trivial infrared (IR) fixed point, which can still have an emergent conformal symmetry defined on the space-time dimensions of the defect12,13,14,15,16,17. The theory describing such a conformal defect is called a defect CFT (dCFT) (see refs. 18,19 for recent discussions). Understanding dCFTs is an important step in comprehending CFTs in nature, as most experimental realizations of CFTs necessarily accompany defects (and boundaries). Moreover, dCFTs have a non-trivial interplay with the bulk CFTs, and knowledge of the former will advance the understanding of the latter. For example, the two-point correlators of bulk operators in dCFT constrain and encode the conformal data of the bulk CFT20, similar to the well-known story of four-point correlators of a bulk CFT.

dCFTs are typically richer and more intricate than their bulk CFT counterparts. On one hand, for a given bulk CFT, there exist multiple (even potentially infinite) distinct dCFTs, and their classification remains an open challenge. On the other hand, breaking of the full conformal symmetry group into a subgroup renders the study of dCFTs more challenging, as the space-time conformal symmetry becomes less restrictive, making modern approaches like the conformal bootstrap program21 less powerful20,22,23,24,25. Notably, most of the well-established results concerning dCFTs are confined to 2D CFTs, including the seminal results on the boundary operator contents13 and RG flow26, thanks to the special integrability property of 2D CFTs. In comparison, higher-dimensional CFTs pose greater difficulties, and the knowledge of dCFTs in dimensions beyond two is rather limited. Current studies of dCFTs mainly revolve around perturbative RG computations27,28,29,30,31,32,33,34 and Monte Carlo simulations of lattice models18,35,36,37. An important progress made recently is the non-perturbative proof of RG monotonic g-theorem in 3D and higher dimensions38,39, generalizing the original result in 2D26,40,41.

In the context of dCFTs, many important questions remain to be answered, ranging from basic inquiries such as the existence of conformal defect fixed points to more advanced queries concerning the infrared properties of dCFTs, including their conformal data such as critical exponents. The central aim of this paper is to develop an efficient tool for the non-perturbative analysis of 3D dCFTs. Specifically, we extend the success of the recently proposed fuzzy sphere regularization42 from bulk CFTs42,43,44,45 to the realm of dCFTs. As a concrete example, we explore the properties of the 3D Ising CFT in the presence of a magnetic line defect32,33,34,35,36,46,47,48,49. We directly demonstrate that this line defect indeed flows to an attractive conformal fixed point, and we identify 6 low-lying defect primary operators with their scaling dimensions extracted through the state-operator correspondence. Furthermore, we study the one-point bulk primary correlators and the two-point bulk-defect correlators, both of which are fixed by conformal invariance, up to a set of operator product expansion (OPE) coefficients. As far as we know, most of conformal data of dCFT reported here have never been studied before. In this context, our paper not only presents a comprehensive set of results concerning the magnetic line defect in the 3D Ising CFT, but also lays the foundation for further exploration of 3D dCFTs using the fuzzy sphere regularization technique.

Results

Conformal defect and radial quantization

We consider a 3D CFT deformed by a p-dimensional defect, described by the Hamiltonian

$${H}_{CFT}+h\int\,{d}^{p}r{{{{{{{\mathcal{O}}}}}}}}(r).$$
(1)

Examples include the line defect (p = 1, see Fig. 1a) and the plane defect (p = 2). If the defect is not screened in the IR, the system will flow into a non-trivial fixed point that breaks the original conformal symmetry SO(4, 1) of HCFT. Furthermore, if the non-trivial fixed point is still conformal, such a defect is called a conformal defect described by a dCFT. For such a dCFT, the original conformal group is broken down to a smaller subgroup SO(p + 1, 1) × SO(3 − p)17,18,19, where SO(p + 1, 1) is the conformal symmetry of the defect, and SO(3 − p) is the rotation symmetry around the defect that acts as a global symmetry on the defect.

Fig. 1: Schematic plot of the defect in 3D.
figure 1

Through a Weyl transformation, a Euclidean flat space-time \({{\mathbb{R}}}^{3}\) is mapped to (b) the cylinder manifold \({S}^{2}\times {\mathbb{R}}\). The line defect before and after the Weyl transformation are shown by the colored line. c The 0 + 1-D impurities (cyan point) located at the north and south pole on two-dimensional sphere S2 in the radial quantization, where the flux at the center represents the magnetic monopole defined in the fuzzy sphere model.

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A dCFT possesses a richer structure compared to its bulk counterpart. Firstly, there is a set of operators living on the defect, forming representations of the defect conformal group SO(p + 1, 1). Furthermore, there are non-trivial correlators between bulk operators and defect operators. (Hereafter, we follow the usual convention and denote the defect operator with a hat \(\hat{O}\), while the bulk operator is represented as O without a hat.) The simplest example is that the bulk primary operator gets a non-vanishing one-point correlator, which is in sharp contrast to the bulk CFT17,18,19:

$$\langle {O}_{1}(x)\rangle=\frac{{a}_{{O}_{1}}}{| {x}_{\perp }{| }^{{\Delta }_{1}}}.$$
(2)

Here, ∣x⊥∣ is the perpendicular distance from the bulk operator to the defect, Δ1 is the scaling dimension of O1, and \({a}_{{O}_{1}}\) is an operator-dependent universal number (we consider the case of O1 to be a Lorentz scalar). Moreover, we can consider a bulk-defect two-point (scalar-scalar) correlator defined as17,18,19:

$$\langle {O}_{1}(x){\hat{O}}_{2}(0)\rangle=\frac{{b}_{{O}_{1}{\hat{O}}_{2}}}{| {x}_{\perp }{| }^{{\Delta }_{1}-{\hat{\Delta }}_{2}}| x{| }^{2{\hat{\Delta }}_{2}}},$$
(3)

where \({b}_{{O}_{1}{\hat{O}}_{2}}\) is the bulk-defect OPE coefficient. Interestingly, the bulk two-point correlator already becomes non-trivial, and its functional form cannot be completely fixed by the conformal symmetry.

Similar to the bulk CFT, we consider the radial quantization of a dCFT. Specifically, we first foliate the Euclidean space \({{\mathbb{R}}}^{3}\) using spheres S2 with their origins situated on the defect, as illustrated in Fig. 1a. Next, we can perform a Weyl transformation to map \({{\mathbb{R}}}^{3}\) to a cylinder \({S}^{2}\times {\mathbb{R}}\), and the p-dimensional defect transforms into a defect intersecting the cylinder. For instance, as shown in Fig. 1, the Weyl transformation maps a line defect (p = 1) in \({{\mathbb{R}}}^{3}\) to 0 + 1D point impurities located at the north and south poles of the sphere S2, forming two continuous line cuts along the time direction from τ = − ∞ to τ = ∞. Similarly, a plane defect (p = 2) in \({{\mathbb{R}}}^{3}\) will be mapped to a 1 + 1D defect with its spatial component located on the equator of the sphere S2.

Akin to the state-operator correspondence in bulk CFT50,51, we have a one-to-one correspondence between the defect operators and the eigenstates of the dCFT quantum Hamiltonian on \({S}^{2}\times {\mathbb{R}}\), where energy gaps of these states are proportional to the scaling dimensions \({\hat{\Delta }}_{n}\) of the defect operators:

$${E}_{n}-{E}_{0}=\frac{v}{R}{\hat{\Delta }}_{n}.$$
(4)

Here, E0 denotes the ground state energy of the defect Hamiltonian, R represents the sphere radius, and v is a model-dependent non-universal velocity that corresponds to the arbitrary normalization of the Hamiltonian. Notably, this velocity v is identical to the velocity of the bulk CFT Hamiltonian (further discussions see Supplementary Note 2 in Supplementary Material).

The state-operator correspondence offers distinct advantages for studying CFTs. Firstly, it provides direct access to information regarding whether the conformal symmetry emerges in the IR. Secondly, it enables an efficient extraction of various conformal data, such as scaling dimensions and OPE coefficients of primaries. The key step involves studying a quantum Hamiltonian on the sphere geometry. However, for 3D CFTs, this was challenging as no regular lattice could fit S2. Recently, this fundamental obstacle was overcome through a scheme called “fuzzy sphere regularization"42, and its superior capabilities have been convincingly demonstrated42,43,44,45. Below we discuss how to adapt the fuzzy sphere regularization scheme to solve dCFTs. We will focus on the case of magnetic line defect of the 3D Ising CFT, but the generalizations to other cases should be straightforward.

Magnetic line defect on the fuzzy sphere

The fuzzy sphere regularization42 considers a quantum mechanical model describing fermions moving on a sphere with a 4πs magnetic monopole at the center. The model is generically described by a Hamiltonian H = Hkin + Hint, where Hkin represents the kinetic energy of fermions, and its eigenstates form quantized Landau levels described by the monopole Harmonics \({Y}_{n+s,m}^{(s)}(\theta,\varphi )\)52. Here, n = 0, 1, ⋯   denotes the Landau level index, and (θ, φ) are the spherical coordinates. We consider the limit where Hkin is much larger than the interaction Hint, allowing us to project the system onto the lowest Landau level (i.e. n = 0), resulting in a fuzzy sphere53.

The 3D Ising transition on the fuzzy sphere can be realized by two-flavor fermions \({{{{{{{{\boldsymbol{\psi }}}}}}}}}^{{{{\dagger}}} }=({\psi }_{\uparrow }^{{{{\dagger}}} },{\psi }_{\downarrow }^{{{{\dagger}}} })\) with interactions that mimic a 2+1D transverse Ising model on the sphere,

$$\begin{array}{rcl}{H}_{0}&=&\int\,{R}^{4}d{\Omega }_{a}d{\Omega }_{b}\,U({\Omega }_{ab})\left({n}^{0}({\Omega }_{a}){n}^{0}({\Omega }_{b})\right.\\ &&-\left.{n}^{z}({\Omega }_{a}){n}^{z}({\Omega }_{b})\right)-h\int\,{R}^{2}d\Omega \,{n}^{x}(\Omega ).\end{array}$$
(5)

Here we are using the spherical coordinate Ω = (θ, ϕ) and R is the sphere radius. The density operators are defined as nα(Ω) = ψ†(Ω)σaψ(Ω), where σx,y,z are the Pauli matrices and σ0 is the identity matrix. U(Ωab) encodes the Ising density-density interaction as \(U({\Omega }_{ab})=\frac{{g}_{0}}{{R}^{2}}\delta ({\Omega }_{ab})+\frac{{g}_{1}}{{R}^{4}}{\nabla }^{2}\delta ({\Omega }_{ab})\). One can tune the transverse field h to realize a phase transition which falls into the 2+1D Ising universality class42. In the following, we set U(Ωab) and h the same as the bulk Ising CFT that has been identified in42.

To study the magnetic line defect of 3D Ising CFT, we add 0 + 1D point-like magnetic impurities located at sphere’s north and south pole, modeled by a Hamiltonian term,

$${H}_{d}=2\pi {h}_{d}({n}^{z}(\theta=0,\varphi=0))+{n}^{z}\left.(\theta=\pi,\varphi=0)\right),$$
(6)

where hd controls the strength of the magnetic impurities. This type of defect can be artificially realized in experiments54,55. Crucially, the defect term Hd breaks the Ising \({{\mathbb{Z}}}_{2}\) symmetry, causing the σ field (of the 3D Ising CFT) to be turned on at the defect. This σ deformation is relevant on the line defect (Δσ ≈ 0.518 < 121), driving the system to flow to a nontrivial fixed point, conjectured to be a conformal defect. This fixed point is expected to be an attractive fixed point32,33,34, implying that regardless of the strength of hd, the defect will flow to the same conformal defect fixed point (see Fig. 2a). Next we will provide compelling numerical evidence to support this conjecture.

Fig. 2: Defect induced attractive fixed point.
figure 2

a Schematic plot of the RG diagram. b The scaling dimension of displacement operator \({\Delta }_{\hat{D}}\) as a function of defect strength hd. Different colored symbols represent the results based on various system sizes. c Finite-size extrapolation of \({\Delta }_{\hat{D}}\) for various hd (see Supplementary Note 2 and 3 in Supplementary Material). A sufficient large hd gives almost identical \({\Delta }_{\hat{D}}\approx 2\), supporting an attractive RG fixed point at hd = ∞. Different colored symbols represent the results based on various defect strength hd.

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Emergent conformal symmetry and operator spectrum

The energy spectrum of the defect Hamiltonian (H0 + Hd) is expected to be proportional to the defect operators’ scaling dimensions, up to a non-universal velocity in Eq. (4). Here we determine the velocity using the bulk CFT Hamiltonian (H0) by setting the σ state to have Δσ = 0.51814921. The defect term Hd breaks the sphere rotation SO(3) down to SO(2), so each eigenstate has a well defined SO(2) quantum number Lz. Akin to the stress tensor of the bulk CFT, there exists a special primary operator in dCFT due to the broken of translation symmetry, dubbed the displacement operator \(\hat{D}\)17,18,19, which has Lz = ± 1 and a protected scaling dimension \({\Delta }_{\hat{D}}=2\). Fig. 2b, c depicts \({\Delta }_{\hat{D}}\) via the state-operator correspondence (Eq. (4)) for various defect strength hd and system sizes. It clearly shows that the obtained \({\Delta }_{\hat{D}}\) are very close to 2, for different defect strengths hd, which indicates an attractive conformal fixed point at hd = ∞ (see Supplementary Note 3 and 6 in Supplementary Material). In what follows, we present the representative results for hd = 300 and we ensure the conclusions are insensitive to the choice of hd.

We further establish the emergent conformal symmetry by confirming that the excitation spectra form representations of SO(2, 1). The generators of SO(2, 1) are the dilation D, translation P, and special conformal transformation K. It is important to note that P and K do not have any Lorentz index due to the triviality of the Lorentz symmetry, i.e., SO(1). For each primary operator, we have descendants generated by the translation, \({P}^{n}\hat{O}\), whose scaling dimension is \({\Delta }_{{P}^{n}\hat{O}}={\Delta }_{\hat{O}}+n\), and its SO(2) quantum number Lz remains unchanged. Figure 3 displays our numerical data of the low-lying energy spectrum, clearly exhibiting the emergent conformal symmetry, i.e., approximate integer spacing between each primary and its descendants. These observations firmly establish that the magnetic line defect of the 3D Ising CFT flows to a conformal defect with a conformal symmetry of SO(2, 1).

Fig. 3: Conformal tower of defect primaries.
figure 3

Defect primary fields and their descendants with global symmetry (a) Lz = 0 and (b) Lz = 1. The gray horizon lines stand for extrapolated values for primaries and their integer-spaced descendants. Different colored symbols represent the results based on various system sizes. By increasing system size N all of the scaling dimensions approach the theoretical values consistently, supporting an emergent conformal symmetry in the thermodynamic limit.

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From our numerical data, we can identify five low-lying defect primary operators in addition to \(\hat{D}\), as listed in Table 1. Notably, all these operators are found to be irrelevant (i.e., Δ > 1), which is consistent with the observation of an attractive defect fixed point. Our lowest-lying operator \(\hat{\phi }\) has Lz = 0 and \({\Delta }_{\hat{\phi }}\approx 1.63(6)\). This value is in good agreement with Monte Carlo simulations, e.g. 1.60(5)36, 1.52(6)36, and 1.40(3)35, as well as with the perturbative ε-expansion computation of ~ 1.55(14) ref. 34. The second low-lying operator in the Lz = 0 sector has \({\Delta }_{\hat{{\phi }^{{\prime} }}}=3.12(10)\), which significantly deviates from the ε-expansion value of Δ ≈ 4.33 + O(ε2) (it was called \({\hat{s}}_{+}\) in34). This suggests a large sub-leading correction in the ε-expansion. All other primary operators identified in our study have not been computed by any other methods. It is essential to mention that the scaling dimensions in Table 1 are obtained by the finite-size extrapolating (see details in Supplementary Note 2 in Supplementary Material), and the data at finite N is already very close to the extrapolated value (The finite-size extrapolation improve the results by around 2%). One can also improve the accuracy by making use of conformal perturbation56.

Table 1 Scaling dimensions of primary operators in the magnetic line defect of 3D Ising CFT, determined through the state-operator correspondence on the fuzzy sphere
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Additionally, to verify the physics of dCFT presented here is independent of the specific value hd, we directly study the spectrum at hd = ∞ (see Supplementary Note 6 in Supplementary Material). Comparing it with the results at hd = 300, we found them to be consistent with each other. This result indicates that the large hd regime shares the same dCFT and also supports that the fixed point of dCFT indeed resides at hd = ∞.

Correlators and OPE coefficients

Using Weyl transformation, we can map the bulk-defect correlators in Eq. (2), Eq. (3) in \({{\mathbb{R}}}^{3}\) to the correlators on cylinder \({S}^{2}\times {\mathbb{R}}\) (see Supplementary Note 1 in Supplementary Material),

$${G}_{{O}_{1}{\hat{O}}_{2}}\equiv \frac{\langle \hat{1}| {O}_{1}(\tau=0,\theta )| {\hat{O}}_{2}\rangle }{\langle 1| {O}_{1}(\tau=0,\theta )| {O}_{1}\rangle }=\frac{{b}_{{O}_{1}{\hat{O}}_{2}}}{{(\sin \theta )}^{{\Delta }_{1}-{\hat{\Delta }}_{2}}}.$$
(7)

The bulk operator O1 is positioned at a point that has an angle θ with respect to the north pole. In the denominator, we use the states of the bulk CFT, while in the numerator, we use the states of the dCFT. The one-point bulk correlator corresponds to taking \(\left\vert {\hat{O}}_{2}\right\rangle\) to be the ground state of the defect, i.e., \(\left\vert \hat{1}\right\rangle\).

In the fuzzy sphere model, we can use the spin operators nz and nx to approximate the bulk CFT primary operators σ and ϵ43,44. For example, the correlator between the bulk primary σ and a defect primary operator \({\hat{O}}_{2}\) is computed by,

$${G}_{\sigma {\hat{O}}_{2}}\equiv \frac{\langle \hat{1}| {n}^{z}(\tau=0,\theta )| {\hat{O}}_{2}\rangle }{\langle 1| {n}^{z}(\tau=0,\theta )| \sigma \rangle }=\frac{{b}_{\sigma {\hat{O}}_{2}}}{{(\sin \theta )}^{{\Delta }_{\sigma }-{\hat{\Delta }}_{2}}}+O({N}^{-1/2}).$$
(8)

Here, Δσ ≈ 0.518149, and the first-order correction O(N−1/2) comes from the descendant operator ∂μσ contained in nz. Figure 4 illustrates the one-point bulk correlator Gσ(θ) and bulk-defect correlator \({G}_{\sigma \hat{\phi }}(\theta )\) for different system sizes N = 12–36. Both correlators agree perfectly with the CFT prediction Eq.(7), except for the small θ regime. It is worth noting that the one-point correlator Gσ(θ) is divergent at θ = 0, π and reaches a minimum at θ = π/2. In contrast, the bulk-defect correlator \({G}_{\sigma \hat{\phi }}(\theta )\) has an opposite behavior (because \({\Delta }_{\sigma }-{\Delta }_{\hat{\phi }} < 0\)); it vanishes at θ = 0, π and reaches a maximum at θ = π/2. These behaviors are nicely reproduced in our data, which is highly nontrivial because computationally the only difference for the two correlators is the choice of \(\left\vert {\hat{O}}_{2}\right\rangle\) in Eq. (8).

Fig. 4: Correlators involving defect.
figure 4

The angle dependence of (a) correlator Gσ(θ) and (b) \({G}_{\sigma \hat{\phi }}(\theta )\), for system sizes ranging from N = 12–36. The dashed lines correspond to theoretical correlator in Eq. (7) with \({b}_{\sigma {\hat{O}}_{2}}\) and \({\Delta }_{\hat{{O}_{2}}}\) from the N = 36 curves. The insets are finite-size scaling analysis by setting θ = π/2, respectively giving one-point OPE coefficient aσ ≈ 1.37(1) and bulk-defect OPE coefficient \({b}_{\sigma \hat{\phi }}\approx 0.68(1)\).

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We can further extract the bulk-defect OPE coefficients from \({G}_{{O}_{1}{\hat{O}}_{2}}(\theta=\pi /2)={b}_{{O}_{1}{\hat{O}}_{2}}\), and the results are summarized in Table 2. None of these OPE coefficients was computed non-perturbatively before. There are perturbative computations for aσ and aϵ34 from ε expansion, giving \({a}_{\sigma }^{2}\, \approx \, 3.476+O({\varepsilon }^{2})\) (i.e. aσ ≈ 1.86) and aϵ ≈ 1.83 + O(ε2). Our estimates are aσ = 1.37(1) and aϵ = 1.31(19), it will be interesting to compute higher order corrections in the ε-expansion. Moreover, using the Ward identity of any bulk operator (O)19, we can extract Zamolodchikov norm

$$\sqrt{{C}_{\hat{D}}}=\frac{2}{\pi }\frac{{\Delta }_{O}{G}_{O}(\theta=\pi /2)}{{G}_{O\hat{D}}(\theta=\pi /2)}.$$
(9)

The estimates using σ and ϵ gives \({C}_{\hat{D}}=0.53(3)\) and \({C}_{\hat{D}}=0.59(18)\), respectively.

Table 2 Bulk-to-defect OPE coefficients magnetic line defect of 3D Ising CFT
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Discussion

We have outlined a systematic procedure to study defect conformal field theory (dCFT) using the recently proposed fuzzy sphere regularization scheme. As a concrete application, we investigated the magnetic line defect of 3D Ising CFT and provided clear evidence that it flows to a conformal defect. Crucially, we accurately computed a number of conformal data of this dCFT, including defect primaries’ scaling dimensions and bulk-defect OPE coefficients. As far as we know, most of the conformal data of dCFT reported here have never been studied in a microscopic model before, thus this conformal information paves the way for exploring the rich physics in 3D Ising CFTs.

Looking forward, the current setup can be readily applied to the study of various types of defects in distinct 3D CFTs, potentially resolving numerous open questions and offering insights into defects in CFTs. For example, the plane defect (p = 2 in Eq. (1)), which may resemble surface critical phenomena, is interesting to investigate. It is also highly desired to study 3D dCFTs in a broad universality class (e.g. Wilson–Fisher O(N) critical point). Moreover, the results of current work provide a necessary input in the study of infrared data for the dCFT within the numerical conformal bootstrap20,21,22. Taking it further would potentially be very interesting to study the dCFT in holography and string theory.

Methods

The model H0 + Hd for the magnetic line defect of 3D Ising CFT is a continuous model with fully local interaction in the spatial space. In practice, we consider the second quantization form of this model by the projecting H0 + Hd to the lowest Landau level (fuzzy sphere), using \({\psi }_{a}(\Omega )=\frac{1}{\sqrt{N}}\mathop{\sum }\nolimits_{m=-s}^{s}{c}_{m,a}{Y}_{s,m}^{(s)}(\Omega )\) (we are using a slightly different convention compared to ref. 42). Here N = 2s + 1 playing the role of system size N ~ R2, and we simply replace R2 with N during the projection. This lowest Landau level projection leads to a second quantized Hamiltonian defined by fermionic operators cm,a, and similar models have been extensively studied in the context of the quantum Hall effect57. Numerically, this model can be simulated using various techniques such as exact diagonalization and density matrix renormalization group (DMRG)58,59. We perform DMRG calculations with bond dimensions up to D = 5000, and for the largest system size N = 36, the maximum truncation errors for the ground state and the tenth excited state are 1.37 × 10−9 and 1.96 × 10−8, respectively. We explicitly impose two U(1) symmetries, i.e., fermion number and SO(2) angular momentum.