Critical Majorana fermion at a topological quantum Hall bilayer transition

Abstract

Quantum Hall bilayers are a uniquely tunable platform that can realize continuous transitions between distinct topological phases of matter. One prominent example is the transition between the Halperin state and the Moore--Read Pfaffian, long predicted to host a critical theory of Majorana fermions but so far not verified in unbiased microscopic simulations. Using the fuzzy sphere regularization, we identify the low-energy spectrum at this transition with the 3D gauged Majorana conformal field theory. We show that the transition is driven by the closing of the neutral fermion gap, and we directly extract the operator content in both integer and half-integer spin sectors. Our results resolve the long-standing question of the nature of a topological phase transition in a setting relevant to quantum Hall experiments, while also providing the first realization of a fermionic theory on the fuzzy sphere, previously limited to bosonic theories.

Figures (8)

• Figure 1: (a) Quantum Hall bilayer system with the layers $\uparrow$, $\downarrow$. The particles interact via intralayer ($V_\mathrm{intra}$) and interlayer ($V_\mathrm{inter}$) interactions, and they can tunnel between the layers with an amplitude $h$. (b) At a critical tunneling $h_c$, the bosonic particles at filling $\nu=1$ undergo a transition between the 220 Halperin state and the Moore--Read state. (c) At criticality, the energy spectrum on the fuzzy sphere, resolved as a function of angular momentum, matches that of the free 3D Majorana fermion conformal field theory, with the characteristic towers of primary fields (labelled) and their descendants.
• Figure 2: (a) Phase diagram of the model in Eq. (\ref{['eq:hamiltonian']}), calculated using overlaps with the states in Eqs. (\ref{['eq:220 wf']})-(\ref{['eq:pf wf']}). The color intensity represents the maximum overlap, $\max \left(|\langle 0 | \Psi_{220} \rangle|^2, |\langle 0 | \Psi_\mathrm{Pf} \rangle|^2 \right)$, while the blue dashed line is the approximate phase boundary where the two overlaps are equal. The white cross at $(V_0^\text{inter},h) = (0.48, 0.58)$ marks the gap closing point in Fig. \ref{['fig:cost_functions']}. (b) The real-space entanglement spectra for one point inside each phase, denoted by black crosses in (a). The multiplicities of the low-lying entanglement levels $\xi$, resolved by the $z$-component of angular momentum in the subsystem $A$, are indicated by numbers and match those of the respective model states (see text). Data in (a) are obtained by exact diagonalization for $N=12$ bosons, while (b) is for $N=14$, the largest accessible system size with Hlibert space dimension 87,150,620.
• Figure 3: (a) The cost function $\delta_\mathrm{tow}$ quantifying the conformal structure in the low-lying spectrum across the phase diagram at a fixed system size $N=12$. A sharp peak occurs along the dashed line at $V_0^\text{inter}=0.48$. The optimal point (white cross) is further determined by the gap vanishing in (b). (b) Extrapolated gaps of $\{ \bar{\psi}\psi, \partial(\bar{\psi}\psi), T \}$ at the optimal point $(V_0^\text{inter},h) = (0.48, 0.58)$, identified from the minimum of the gaplessness cost function $\delta_\text{gap}$ shown in the inset. Although not explicitly enforced, the gap of the neutral fermion $\psi$ also converges to zero at this point, a signature of a quantum critical point in both even- and odd-particle sectors. Empty markers correspond to exact diagonalization data, while solid markers ($N=15-18$) are obtained from DMRG and are not used in the extrapolations.
• Figure 4: Comparison between CFT data and the Hamiltonian spectrum at the optimal point $(V_0^\text{inter},h) = (0.48, 0.58)$. (a) Even-particle sector for $N=6,8,10,12,14$, containing integer angular momentum states. The lowest energy state in $L=2$ sector is taken to be the stress-energy tensor, whose energy was fixed to $\Delta = 3$. (b) Odd-particle sector for $N=7,9,11,13$, containing half-integer angular momentum states. Energy levels are normalized with respect to the mean energies of the vacuum and stress-energy tensors in the adjacent, $N\pm 1$, even-particle sectors. Both panels show the complete energy spectra for levels with $\Delta \leq 4.5$, $L \leq 7/2$. The expected CFT operators are labeled, with their scaling dimensions shown by dashed lines. Other signatures of the free Majorana CFT in these spectra also include the absence of certain states, such as the conserved vector current, $J^\mu = \bar{\psi}\gamma^\mu\psi$, at $(L=1, \Delta = 2)$, and the state at $(L=1/2, \Delta=2)$, which is absent due to the equation of motion $\gamma_\mu \partial^\mu \psi = 0$.
• Figure S1: Finite-size spectrum of the model in the main text at its critical point, rescaled such that the stress-energy tensor has scaling dimension $\Delta_T=3$. (a) The even-particle sector at system size $N=14$. At low energy and angular momentum, the expected CFT scaling dimensions (shown as black lines) are in good agreement with the microscopic energies of the $\mathbb{Z}_2$-even sector. The $\mathbb{Z}_2$-odd sector is decoupled from the critical theory. The collective excitation branches of the bilayer FQH state are also highlighted by the grey box. (b) Same as (a) but for the odd-particle sector at $N=13$. Similarly to the even-particle sector, the CFT tower of states is well captured by the $\mathbb{Z}_2$-odd sector, while the reverse parity sector does not participate to the transition. In this sector, the collective excitation corresponds to the unpaired fermion mode of the Moore--Read state Moller11Bonderson11.