In the continuum limit (i.e., the monopole flux $q\to \infty$) of the fuzzy-sphere model, any (bare) local operator, as well as its correlators, remain nondiverging. Moreover, the microscopic couplings at the critical point also remain finite, i.e., they converge to finite values in the continuum limit.

On the sphere, the microscopic particles form a spin-$s$ representation of SO(3); as its spin is size dependent, it cannot flow to a local field in the CFT in the IR. On the other hand, on the flat space, the microscopic fermions transform in a projective representation of the continuous translation group, which is incompatible with any local operator of a CFT.

A recent work [47] also studied a boson-fermion mixture fuzzy-sphere model that realizes the confinement transition of $\nu =1/2$ bosonic Laughlin state. Its setup involves two flavors of charge-1 microscopic fermions and a flavor of charge-2 microscopic bosons. Hence, the charge-0 CFT operators are still bosonic.

In a 3D Euclidean signature, the $\gamma$ matrices are identical to the Pauli matrices. We use $\gamma$ to avoid overloading the symbol $\sigma$.

Unitarity and conformal symmetry imply that any local operator of spin-$\frac{1}{2}$ with $\mathrm{\Delta }=1$ is a free fermion, so this scaling dimension also gives a clear criterion to identify the free-fermion CFT.

In this language, the $\nu =1$ fermionic integer quantum Hall phase can be regarded as a MQH phase with ${\nu }_K=2$.

The name captures the parity and time-reversal breaking fermion pairing, although the TSC do not spontaneously break any U(1) symmetry.

An alternative choice without a flux mismatch is to let the bosons carry the two component spinor spherical harmonics ${\mathrm{\Omega }}_{q-1/2,qm}^{(q)}$ [80] with spin-weight $q$ instead of the scalar spherical harmonics $Y_{q-1/2,m}^{(q-1/2)}$ with spin-weight $q-1/2$. These two choices assert the same spatial profile $\parallel {\mathrm{\Omega }}_{q-1/2,qm}^{(q)}(\mathbf{r}){\parallel }^2=|Y_{q-1/2,m}^{(q-1/2)}(\mathbf{r}){|}^2$ and differ only by a local basis transformation of the spinors. The equivalence shows that the physical interpretation of the flux mismatch is the coupling of an additional spin-$1/2$ degree of freedom.

Specifically, $D^{\mu }={\partial }^{\mu }+iq{A}^{\mu }$ and $A^{\varphi }=ctg\theta$. The derivatives act as the raising and lowering operators the spin-weighted spherical harmonics $$D_{\pm }{Y}_{lm}^{(s)}=\pm \sqrt{(l\mp s)(l\pm s+1)}{Y}_{lm}^{(s\pm 1)}.$$The operator $\eta =b^{†}f$ experiences a monopole flux $q=-1/2$.

The module Fuzzifino, which is part of fuzzified, specializes in the ED for boson-fermion mixed systems Sample codes can be found at Refs. [87, 88, 89].

For $t=0$, the number of fermions and bosons are separately conserved, level crossings appear for states in different $N_f$ sectors. The positive and negative $t$’s give exactly the same results, as they are connected by a $2\pi$-rotation along the $z$ axis.

An equivalent way to see this Higgs mechanism is to impose $\mathrm{d}a=0$ and ${\oint }_{\mathcal{C}}a=0$ or $\pi$, where $\mathcal{C}$ is a closed loop.

The correspondence holds more precisely between the field-theory Hamiltonian and fuzzy-sphere Hamiltonian. Since the Lagrangian and Hamiltonian can be easily converted, we pick the Lagrangian as a more familiar representative.

In Eq. (3), we write the Hamiltonian in terms of the pseudopotentials, while it is written in terms of local operators in Eq. (25) for convenience. The difference produces a constant conversion factor.

The spinor indices are raised and lowered by ${\epsilon }^{\alpha \beta }$ and ${\epsilon }_{\alpha \beta }$ tensor. In the following we will omit the $\epsilon$.

The positivity requires $J_3$ to be positive-semi-definite $$⟨\mathrm{\Psi }|J_3|\mathrm{\Psi }⟩=⟨\mathrm{\Psi }|\{V_{+},V_{-}\}|\mathrm{\Psi }⟩=\parallel V_{+}|\mathrm{\Psi }⟩{\parallel }^2+\parallel V_{-}|\mathrm{\Psi }⟩{\parallel }^2\ge 0,$$which is only possible for states in the identity representation of its real part SU(2).

The space-time parity is not accessible in our calculation.