Fractional quantum Hall bilayers at half filling: Tunneling-driven non-Abelian phase

Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here, we investigate the possibility of non-Abelian topological order in the half-filled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the state-of-the-art density-matrix renormalization group, we unveil “fingerprint” evidence of the non-Abelian Moore-Read Pfaffian state emerging in the intermediate-tunneling regime, including the ground-state degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian (331) Halperin state to the non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the ground-state energy. Our results not only provide a “proof-of-principle” demonstration of realizing a non-Abelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral $p\text{−}\text{wave}$ pairing states.

Figure 1

A schematic diagram for the FQH bilayer system. Both double QW and single wide QW systems can be mapped to a bilayer system, where electrons interact with each other through intralayer interaction $V_{\sigma \sigma }$ and interlayer interaction $V_{\sigma \overline{\sigma }}$, and the electron tunneling $t_{\perp }$ is tunable between two separated layers.

Figure 2

Energy spectra as a function of the tunneling strength $t_{\perp }$, obtained on a square torus with $N_e=12$ electrons by DMRG. Different momentum sectors are labeled by different symbols. We highlight the ground-state degeneracy by boxes. Here, we only show two lowest-energy levels in $K=0$ (red cross, orange star) and $K=\pi$ (black square, navy diamond) sectors, and one lowest-energy level in other momentum sectors. Due to the $C_4$ symmetry on the square torus, one ground state in the $K=0$ sector (orange star) has the nearly the same energy with one in the $K=\pi$ sector (black square). The dashed line marks the level crossing between the ground state in the $K=0$ sector (red cross) and high excited states, indicating a quantum phase transition. All calculations are performed at layer distance $d=3.0$ and layer width $w=1.5$.

Figure 3

(a), (d) The entanglement entropy $S\left(l_A\right)$ for (a) the (331) Halperin state and (d) the MR Pfaffian state as a function of $\sqrt{l_A}$. The open circles were discarded in the extrapolation because they either represent very small subsystems violating the area law or suffer from the finite-size saturation effect (shaded by gray). The linear extrapolated $\gamma$ in both cases are in agreement with the predicted value $ln\sqrt{8}$ (blue dashed line). (b), (c), (e), (f) The low-lying orbital ES of (b), (c) the (331) Halperin state and (e), (f) the MR Pfaffian state, with even or odd electrons in the half-cut subsystem. The countings matching the degeneracy patterns given in the text are labeled by red. All calculations are performed at system size $N_e=22$ with layer width $w=1.5$, layer distance $d=3.0$, and tunneling strengths (a)–(c) $t_{\perp }=0.03$ for the (331) state and (d)–(f) $t_{\perp }=0.10$ for the MR Pfaffian state.

Figure 4

The continuous phase transition from the (331) Halperin state to the MR Pfaffian state as a function of $t_{\perp }$ on the sphere. (a) Evolution of ES versus $t_{\perp }$. The expected levels for the MR Pfaffian state are labeled by red, and redundant levels originally from the (331) Halperin state are labeled by green. ${\mathrm{\Delta }}_{\mathrm{\Delta }{L}_z^A}$ measures the entanglement gap of the MR Pfaffian state in the $\mathrm{\Delta }{L}_z^A$ sector. We consider the half-cut subsystem with even number of electrons for $N_e=18$. (b) Partial derivative ${\partial }^2{E}_0/{\partial }^2{t}_{\perp }$ as a function of $t_{\perp }$ for different system sizes $N_e=14$ (red), 16 (blue), and 18 (green). Inset: evolution of the ground-state entropy with $t_{\perp }$. (c) The excitation gap ${\mathrm{\Delta }}_{\mathrm{exc}}$ as a function of $t_{\perp }$ for various system sizes. All calculations are performed at layer width $w=1.5$ and layer distance $d=3.0$.

Figure 5

The phase diagram of the FQH bilayer at ${\nu }_T=\frac{1}{2}$ in terms of the layer distance $d$ and tunneling strength $t_{\perp }$, obtained from $N_e=18$ on the sphere with layer width $w=1.5$. The continuous phase transition from the (331) Halperin state to the MR Pfaffian state is labeled by dashed line, while the solid line marks the transition between the (331) Halperin state and a possible CFL.

Figure 6

The low-lying orbital ES for various bilayer system sizes $N_e=16,18,20,22,24$ at the tunneling strength $t_{\perp }=0.10$, layer width $w=1.5$, and layer distance $d=3.0$, with even (top) or odd (bottom) number of electrons in the half-cut subsystem. The levels whose counting is consistent with the MR Pfaffian edge excitations proposed in Tables 1 and 2 are labeled by red. $\mathrm{\Delta }{L}_z^A=L_z^A-L_{z,min}^A$, where $L_{z,min}^A$ is the total angular momentum of the subsystem $A$ without edge excitations.

Figure 7

The low-lying orbital ES of $N_e=18$ for different tunneling strength $t_{\perp }$ at layer width $w=1.5$ and layer distance $d=3.0$, with even number of electrons in the half-cut subsystem. The levels whose counting is consistent with the MR Pfaffian edge excitations proposed in Tables 1 and 2 are labeled by red. The green levels match the (331) edge excitations proposed in Tables 3 and 4 at small $t_{\perp }$, but are continuously gapped out with increasing the tunneling strength. $\mathrm{\Delta }{L}_z^A=L_z^A-L_{z,min}^A$, where $L_{z,min}^A$ is the total angular momentum of the subsystem $A$ without edge excitations.

Figure 8

Energy spectra of the one-dimensional transverse field Ising model as a function of parameter $J_x/J_z$. Here, the calculation is performed on a spin chain with 18 sites. It is clear that, as increasing $J_x$, one ground state is continuously gapped out (labeled by red cross). Please note that the evolution of the low-energy spectrum versus $J_x$ is very similar to that in our bilayer FQH system [Fig. 2]. Dotted line indicates the transition point predicted by analytic result. Please see Ref. [59] for detailed analysis of the transition in Ising model.

Figure 9

Energy spectra as a function of tunneling $t_{\perp }$ on a square torus with $N_e=8$ (left) and 10 (right) electrons obtained by exact diagonalization. Different momentum sectors are labeled by different symbols. All calculations are performed at layer width $w=1.5$ and layer distance $d=3.0$.

Figure 10

Energy spectra in the strong-tunneling limit for (left) $N_e=12$ and (right) $N_e=16$ on the square torus obtained by ED. Energy eigenstates are labeled by a $\sqrt{K_1^2+N_e{K}_2^2}$ (in unit of $2\pi /N_s)$, where $(K_1,K_2)$ is two-dimensional momentum (see Appendix pp1). The threefold ground-state degeneracy in $(0,\pi ),(\pi ,0),(\pi ,\pi )$ is labeled by red squares. All calculations are performed at layer distance $d=3.0$ and layer width $w=1.5$.

Figure 11

(Left) The experimental observed quasiparticle excitation gaps ${\mathrm{\Delta }}_{\mathrm{exc}}$ (in unit of Kelvin) versus the tunneling strength $t_{\perp }$. The data are obtained from Ref. [21]. (Right) The tunneling current and particle-number fluctuation of each layer as a function of $t_{\perp }$ with layer width $w=1.5$ and layer separation $d=3.0$, which can be used to distinguish a two-component state from a single-component state.