Quantum Hall phase diagram of half-filled bilayers in the lowest and the second orbital Landau levels: Abelian versus non-Abelian incompressible fractional quantum Hall states

We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.

Figure 1

(Color online) Wave-function overlap between the exact ground state, ${\Psi }_0$, and the Pf state, ${\Psi }_{\text{Pf}}$ [blue (dark gray)], and 331 state, ${\Psi }_{331}$ [yellow (light gray)], as a function of layer separation $d$ and tunneling amplitude $t$ for the $\nu =1/2$ lowest LL with $N=8$ electrons and single-layer width (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$ ($d\ge w$ necessarily).

Figure 2

(Color online) Wave-function overlap between the exact ground state (a) ${\Psi }_0$ and ${\Psi }_{\text{Pf}}$ at strong tunneling $t=0.2$ and (b) ${\Psi }_{331}$ at zero tunneling $t=0$ as a function of separation $d$ and single-layer well width $w$ (where $d\ge w$) for the $\nu =1/2$ lowest LL with $N=8$ electrons. White corresponds to an overlap of unity while black corresponds to an overlap of zero. The dashed red line is the condition $w=d$ and for $w>d$ the bilayer system is undefined, i.e., the single-layer width cannot be larger than the layer separation.

Figure 3

(Color online) $\left(N_S-N_A\right)/2$ (or $⟨{\left(S_z\right)}_{\text{SAS}}⟩$ when the Hamiltonian $\widehat{H}$ is written in the SAS basis) as a function of layer separation $d$ and tunneling amplitude $t$ for $\nu =1/2$ and $N=8$ electrons for single-layer width (a) $w=0$, (b) $w=0.6$, and (c) $w=1.2$ and $w=2.4$.

Figure 4

(Color online) FQHE energy gap as a function of layer separation $d$ and tunneling amplitude $t$ for the $\nu =1/2$ lowest LL system with $N=8$ electrons. We consider single-quantum-well widths of (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. The energy gap is also color coded such that black is zero gap and white is maximum on a scale from 0 to 0.0425 in Coulomb energy units, $e^2/\left(\kappa l\right)$.

Figure 5

(Color online) Quantum phase diagram (QPD) and FQHE gap (color coded) versus layer separation $d$ and tunneling strength $t$ for widths (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. For the QPD, the 331 and Pf phases (as discussed in the text) are separated by a dashed black line and labeled appropriately. The FQHE gap is given as a contour plot with color coding given by the color bar from dark to light, i.e., white being a largest value of 0.0425 and black being value of 0. The asterisks, triangles, circles, and squares correspond to the different experiments in Refs. 38, 39, 41, 19, 1, respectively. Only experimental points showing FQHE are within the large solid circles in (a) with the lower smaller circles and upper larger circles indicating experiments in double-quantum-well structures and WQW structures, respectively. We note that the single triangle on the Pf side of the QPD does not manifest any experimental FQHE indicating that the theoretical gap may be overestimated for the Pf state.

Figure 6

(Color online) FQHE energy gap versus tunneling strength $t$ for several values of layer separation $d$ for (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. A dashed vertical line of the same color corresponds to the boundary between the Pfaffian phase (right of the line) and the 331 phase (left of the line).

Figure 7

(Color online) FQHE energy gap versus layer separation $d$ for a few values of tunneling strength $t$ for (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. A dashed vertical corresponds to the phase boundary between the Pfaffian phase (left of the line) and the 331 phase (right of the line).

Figure 8

(Color online) Wave-function overlap between the exact ground state ${\Psi }_0$ and the Pf state ${\Psi }_{\text{Pf}}$ [blue (dark gray)] and 331 state ${\Psi }_{331}$ [yellow (light gray)] as a function of distance $d$ and tunneling amplitude $t$ for the half-filled second LL with $N=8$ electrons and width (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$ ($d>w$ necessarily).

Figure 9

(Color online) Wave-function overlap between the exact ground state (a) ${\Psi }_0$ and ${\Psi }_{\text{Pf}}$ at strong tunneling $t=0.2$ and (b) ${\Psi }_{331}$ at zero tunneling $t=0$ as a function of separation $d$ and single-layer well width $w$ (where $d\ge w$) for the SLL with $N=8$ electrons. White corresponds to an overlap of unity while black corresponds to an overlap of zero. As in Fig. 2, the dashed red line is the condition $w=d$ and for $w>d$ the bilayer system is undefined, i.e., the single-layer width cannot be larger than the layer separation.

Figure 10

(Color online) $\left(N_S-N_A\right)/2$ as a function of distance $d$ and tunneling amplitude $t$ for the half-filled second Landau level, $N=8$ electrons, and (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$.

Figure 11

(Color online) FQHE energy gap as a function of layer separation $d$ and tunneling amplitude $t$ for the second LL system with $N=8$ electrons. We consider single-quantum-well widths of (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. The energy gap is also color coded such that black is zero gap and white is maximum on a scale from 0 to 0.035.

Figure 12

(Color online) QPD and FQHE gap (color coded) for the second LL for widths (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. For the QPD, the 331 and Pf phases are labeled appropriately and the gap is given as a contour plot with color coding given by the color bar from dark to light, i.e., white being a largest value of 0.035 and black being value of 0.

Figure 13

(Color online) FQHE energy gap versus tunneling strength $t$ for a few values of layer separation $d$ for (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. A dashed vertical line of the same color corresponds to the boundary between the Pfaffian phase (right of the line) and the 331 phase (left of the line).

Figure 14

(Color online) FQHE energy gap versus layer separation $d$ for a few values of tunneling strength $t$ for (a) $w=0$, (b) $w=0.6$, (c) $w=1.2$, and (d) $w=2.4$. A dashed vertical line of the same color corresponds to the boundary between the Pfaffian phase (left of the line) and the 331 phase (right of the line).

Figure 15

(Color online) Schematic depicting the FQHE system at total filling factor (a) $\nu =5/2$ and (b) $\nu =1/2$ being broken up into a bilayer system with each separate layer being at (a) $\tilde{\nu }=5/4$ and (b) $\tilde{\nu }=1/4$. The colored shaded section represents a density of electrons filling up a quantum well.

Figure 16

(Color online) Schematic depicting a spinless FQHE system at total filling factor $\nu =3/2$ becoming a bilayer spinless system with each layer at $\tilde{\nu }=3/4$. The fraction of the density fractionally filling a LL is colored blue (dark gray) while an inert LL (in this case the lowest) is colored red (light gray).

Figure 17

(Color online) Schematic depicting two possible scenarios for the spin-full FQHE system at total filling factor $\nu =5/2$ being broken up into a bilayer system with each separate layer being at $\tilde{\nu }=5/4$. In (a) we consider the small $d$ and strong $t$ limit where the lowest spin-up and spin-down Landau levels are completely occupied leaving a 1/4-filled spin-up Landau level in each layer. In (b) we depict the large $d$ and weak-tunneling $t$ limit with the lowest spin-up LLL being filled in each layer leaving a 1/4-filled lowest Landau level of spin-down electrons in each layer. The notation is such that ${\nu }_{\left(\sigma ,n\right)}$ corresponds to the filling factor $\nu$ for electrons with spin-$\sigma$ electrons with orbital Landau level index $n$, i.e., $1_{\left(\uparrow ,0\right)}$ corresponds to filling factor 1 of spin-$\uparrow$ electrons with LLL index 0.

Figure 18

(Color online) Schematic depicting the two bilayer structures considered in this work. (a) shows a double-quantum-well structure along with a generic density profile in the $z$ direction $n\left(z\right)$ (blue dashed line) for experiments such as Ref. 38 while (b) shows a single wide-quantum-well structure and generic density profile (blue dashed line) typical of experiments such as Ref. 39.