Free and Interacting Fermionic Conformal Field Theories on the Fuzzy Sphere

Abstract

The fuzzy-sphere regularisation is a powerful tool to study conformal field theories (CFT) in three spacetime dimensions. In this paper, we extend its scope to CFTs with local fermionic operators. We realise the free-Majorana-fermion CFT on a set-up with one flavour of bosons and one flavour of fermions on the lowest Landau level with a $1/2$ angular momentum mismatch and allow conversion between two bosons and two fermions, and use a relative chemical potential as the tuning parameter. On the phase diagram, we observe two continuous transitions described respectively by a free Majorana fermion and a gauged Ising CFT. We numerically confirm the emergent conformal symmetry through the operator spectrum and the two-point correlation function of the local Majorana fermion. We further establish a correspondence between the fuzzy-sphere models and the field-theory Lagrangians, and extend it to an interacting fermionic CFT -- the super-Ising theory with emergent super-conformal symmetry.

Figures (15)

• Figure 1: The phase diagram on the $\mu$-$t$ plane. The colour denotes the average fermion density calculated at $N_{mf}=12$. Selected contours are marked by dashed lines as visual guidance. The circles separate the $\nu=1$ fermionic integer quantum Hall (fIQH) phase and the $\nu_K=3$ Majorana quantum Hall (MQH) phase, and the squares separate MQH and bosonic Pfaffian (bPf) phase. The approximate phase boundary is determined through the scaling of the energy gap from $N_{mf}=12$ and $10$ data.
• Figure 2: (a) The rescaled fermionic gap $\Delta E_f N_{mf}^{1/2}$ and (b) the rescaled bosonic gap $\Delta E_b N_{mf}^{1/2}$ as a function of $\mu$. In the calculation, we vary $N_{mf}$ and fix $t=0.3$. As bPf ground state only exists for even $N_{mf}$, we only keep the data with $\mu<1.5$ for odd $N_{mf}$.
• Figure 3: The real-space entanglement spectrum with the cut at the equator at different phases at different charge sector $Q=(Q_{e,A}-Q_{e,B})/2$: (a) the fermionic integer quantum Hall phase at $\mu=-2$, (b) the $\nu_K=3$ Majorana quantum Hall phase at $\mu=1$ and (c) the bosonic Pfaffian phase at $\mu=5$. Markers are plotted with transparency, so that nearly degenerate levels appear darker. The degeneracies for different charge sectors are labelled beneath. In the calculation we take $N_{mf}=10$.
• Figure 4: The rescaled energy spectrum at the conformal points (a) $t=0.3,\mu=0.0$ described by the free-Majorana-fermion CFT and (b) $t=0.1,\mu=2.25$ described by the Ising$^\ast$ CFT. We identify some multiplets of the lowest primaries in the spectra and the bars denote their expected value in the CFT. The spectra are calibrated by $\Delta_T=3$. The brown circles denote states not identified. In the calculation, we fix $N_{mf}=12$.
• Figure 5: The spectrum with $\Delta\lesssim 4$ for various systems sizes in the free-Majorana-fermion CFT. The scaling dimensions are calibrated by the stress tensor with $\Delta_T=3$. The grey bars denote the theoretical value for comparison. In the calculation, we fix $t=0.3$ and $\mu=0$. The verticle gridlines with interval $0.5$ separates differnt spins, which are either integers or half-integers.