Emergence of 3D Superconformal Ising Criticality on the Fuzzy Sphere

TL;DR

This work provides a non-perturbative, microscopic route to a three-dimensional $\mathcal{N}=1$ superconformal Ising fixed point by realizing the Gross–Neveu–Yukawa model on a fuzzy sphere, coupling an Ising CFT to a gauged Majorana fermion CFT via a Yukawa interaction. Using the state–operator correspondence on $\mathbb{S}^2\times\mathbb{R}$, the authors extract operator dimensions and demonstrate conformal multiplet structure along with emergent spacetime supersymmetry, including a protected spin-$\tfrac{3}{2}$ supercurrent $G_\mu$. They locate the SUSY fixed point by exact diagonalization and track the RG flow of operator spectra from the decoupled Ising $\otimes$ MF fixed point to the interacting GNY fixed point, showing how operators mix and reassemble into SUSY multiplets under tuning. The work confirms emergent $\mathcal{N}=1$ SUSY in a controlled setting and opens a path to compute universal data (OPE coefficients, central charges, entanglement features) for 3D SCFTs with a robust microscopic basis. This framework provides a versatile platform for exploring broader classes of 3D SUSY and non-SUSY CFTs beyond the Ising universality class.

Abstract

Supersymmetric conformal field theories (SCFTs) form a unique subset of quantum field theories which provide powerful insights into strongly coupled critical phenomena. Here, we present a microscopic and non-perturbative realization of the three-dimensional $\mathcal{N}=1$ superconformal Ising critical point, based on a Yukawa-type coupling between a 3D Ising CFT and a gauged Majorana fermion. Using the recently developed fuzzy sphere regularization, we directly extract the scaling dimensions of low-lying operators via the state-operator correspondence. At the critical point, we demonstrate conformal multiplet structure together with the hallmark of emergent spacetime supersymmetry through characteristic relations between fermionic and bosonic operators. Moreover, by tuning the Yukawa coupling, we explicitly track the evolution of operator spectra from the decoupled Ising-Majorana fixed point to the interacting superconformal fixed point, revealing renormalization-group flow at the operator level. Our results establish a controlled, non-perturbative microscopic route to 3D SCFTs.

Figures (7)

• Figure 1: (a) Our GNY model, Eq. (\ref{['eq:GNY']}), consists of fermions (red dots) and bosons (blue dots), each residing on their own fuzzy sphere Zhu:2022gjc, and interacting with each other via the Yukawa interaction (green dashed line). (b) A schematic phase diagram of the GNY model by tuning the bosonic field mass $m$, with a critical point separating a paramagnet with a massless fermion from a symmetry-breaking ferromagnet with a massive fermion. SUSY is expected to emerge at the quantum critical point, which exchanges bosonic and fermionic degrees of freedom. (c) RG phase diagram for the 3D $N=1$ GNY model (reproduced using $\beta$ function in Fei:2016sgs), with the horizontal and vertical axes representing the Yukawa coupling $g$ between the boson and fermion and the bosonic quartic coupling $u$, respectively. The diagram encloses a free Gaussian fixed point (green star), an Ising CFT fixed point (red star), an attractive SUSY CFT fixed point (blue star), and an unstable non-SUSY fixed point, also called GNY$^*$Iliesiu:2015qraIliesiu:2017nrv) (purple triangle).
• Figure 2: The low-lying operator spectra for the 3D Ising SCFT. The squares and hollow circles respectively denote the conformal primary and descendant fields, obtained from our model Hamiltonian, Eq. \ref{['eq:ham']}, with $s = 3$ (containing integer angular momentum states) and $s = 5/2$ (containing half-integer angular momentum states). The critical point $\{\lambda_c\}$ is determined by minimizing the associated cost function, which also sets the energy spectrum normalization sm. The expected CFT operators are indicated, along with their scaling dimensions computed via conformal bootstrap and marked by short horizontal lines Rong:2018okzAtanasov:2018kqwAtanasov:2022bpiErramilli:2022kgp. Different conformal multiplets are distinguished by color, with dashed lines indicating the actions of the conformal generators $P_\mu$ and $K_\mu$. Within the three observed supermultiplets, superconformal descendant states are connected by solid black arrows, which align with the emergent $\mathcal{N}=1$ superconformal algebra.
• Figure 3: Evolution of the low-lying spectra with varying coupling $\{\lambda\}$. $\lambda/\lambda_c=0$ refers the 3D Ising$\otimes$MF CFT fixed point (red star in Fig. \ref{['fig:combined']}) and $\lambda/\lambda_c=1$ represents the 3D Ising SCFT fixed point (blue star in Fig. \ref{['fig:combined']}). The calculation is performed on the system size $s=3$ (a) and $s=5/2$ (b). We renormalize the spectra by setting $\Delta_T=3$ for any $\{\lambda\}$. For half-integer momentum sectors, the ground state $E_{GS}$ and $E_T$ are approximated by the average values of the two adjacent integer momentum sectors Voinea:2025iun (see text).
• Figure S1: The distribution of minimized cost function within the $\lambda^{zx}_0$-$\lambda^{zx}_1$ plane with the other two parameters fixed at critical values. Left for the integer momentum sector ($s=3$) and right for the half-integer momentum sector ($s=5/2$) with identified critical points marked by red star.
• Figure S2: Raw data of low-lying operator spectra for 3D super-Ising CFT. (a) $Z_2$-even sector with $s= 3$ (8 bosons and 7 fermions) at $\lambda=\lambda_c^{s=3}$, containing integer angular momentum states. The whole spectrum is rescaled by minimizing the associated cost function. (b) $Z_2$-even sector with $s=5/2$ (7 bosons and 6 fermions), containing half-integer angular momentum states. Both panels show the complete energy spectra for levels with $\Delta \leq 4.5$, $L \leq 3.5$. The expected CFT operators are labeled, with their scaling dimensions computed from conformal bootstrap shown by short lines Rong:2018okzAtanasov:2018kqwAtanasov:2022bpiErramilli:2022kgp. The rounded circles denotes numerical results calculated from our Hamiltonian \ref{['eq:h_coupled']}.