Competing Abelian and non-Abelian topological orders in $\nu =1/3+1/3$ quantum Hall bilayers

Bilayer quantum Hall systems, realized either in two separated wells or in the lowest two subbands of a wide quantum well, provide an experimentally realizable way to tune between competing quantum orders at the same filling fraction. Using newly developed density matrix renormalization group techniques combined with exact diagonalization, we return to the problem of quantum Hall bilayers at filling $\nu =1/3+1/3$. We first consider the Coulomb interaction at bilayer separation $d$, bilayer tunneling energy ${\mathrm{\Delta }}_{\mathrm{SAS}}$, and individual layer width $w$, where we find a phase diagram which includes three competing Abelian phases: a bilayer Laughlin phase (two nearly decoupled $\nu =1/3$ layers), a bilayer spin-singlet phase, and a bilayer symmetric phase. We also study the order of the transitions between these phases. A variety of non-Abelian phases has also been proposed for these systems. While absent in the simplest phase diagram, by slightly modifying the interlayer repulsion we find a robust non-Abelian phase which we identify as the “interlayer-Pfaffian” phase. In addition to non-Abelian statistics similar to the Moore-Read state, it exhibits a novel form of bilayer-spin charge separation. Our results suggest that $\nu =1/3+1/3$ systems merit further experimental study.

Figure 1

Phase diagram of $1/3+1/3$ QHB in terms of dimensionless layer separation $d$ and tunneling energy ${\mathrm{\Delta }}_{\mathrm{SAS}}$. Data were taken with cylinder circumference $L=14{\ell }_B$ and layer width $w=0$. The dashed lines indicate sweeps performed to determine the nature of the phase transitions (see Sec. 4 for details). Later in this work, additional axes will be added to this plot, driving the system into a non-Abelian phase (see Fig. 7). The black dotted line and square mark the region studied experimentally in Ref. [46], and their observed phase transition.

Figure 2

Momentum polarization $M_a$ for the representative points from the phases in Fig. 1, plotted against $\frac{\nu }{{\left(4\pi {\ell }_B\right)}^2}{L}^2$. The coefficient of proportionality is the shift $\mathcal{S}$, which we can read off to be 3, 1, and 0 for the (330), (112), and $\overline{1/3}$ phases, respectively, as expected. Data were taken at $d=1.6, {\mathrm{\Delta }}_{\mathrm{SAS}}=0$; $d=0.2, {\mathrm{\Delta }}_{\mathrm{SAS}}=0$; and $d=2, {\mathrm{\Delta }}_{\mathrm{SAS}}=0.1$ for the (330), (112), and $\overline{1/3}$ phases, respectively. Values for the shift obtained from fitting the data are shown directly on the figure.

Figure 3

Entanglement spectra for the phases in Fig. 1: the $\left(330\right)$ state, with counting of $1,2,5,\cdots$ dispersing to the right; the $\overline{1/3}$ state, with counting $1,1,2,\cdots$ dispersing to the left; the $\left(112\right)$ state, which has a nonchiral spectra (being a convolution of a left and right mover). These results are in agreement with the predicted values for these phases.

Figure 4

Data as a function of tunneling strength, crossing the $\left(112\right):\overline{1/3}$ transition. The correlation length is flat except very close to the transition, where it is discontinuous. There is also a kink in the energy and in $\tilde{g}$. This is all consistent with a first-order transition.

Figure 5

Data as a function of tunneling strength, crossing the $\left(330\right):\overline{1/3}$ transition. The correlation length has a peak near the transition, but this is consistent with both a first- and a second-order transition. The middle panel shows the energy for both the $\left(330\right)$ and $\overline{1/3}$ phases (see text), and as these lines are not parallel the system's energy has a kink. There is also a jump in $g(r=0)$, consistent with a first-order transition.

Figure 6

Data as a function of interlayer separation, crossing the $\left(330\right):\left(112\right)$ transition. The correlation length has a peak, while the energy has a kink and the $g(r=0)$ are jumps across the transition. This is indicative of a first-order transition, though the transition is weaker compared to the others in the phase diagram.

Figure 7

Phase diagram as a function of interlayer separation $d$ and the modification of the Haldane potential $\delta V_0$. We find that as $-\delta V_0$ is increased, a new phase appears which we believe is a bilayer spin-charge separated non-Abelian phase. Data are taken with zero tunneling ${\mathrm{\Delta }}_{\mathrm{SAS}}=0$ and layer width $w=0$.

Figure 8

Correlation length and energy for the spin-charge separated state as a function of $\delta V_0$ for $d=0.5$, showing a clear first-order transition at $\delta V_0=-0.16$. Note that correlation length increases rapidly as $V_0$ is further reduced.

Figure 9

Overlap between the iPf state and the ground state of Coulomb interaction with modified short-range pseudopotentials from ED. The system contains 8 electrons and 12 flux quanta, on a torus with a hexagonal unit cell. The color scale indicates the sum of singular values of the $3\times 3$ overlap matrix defined in the main text. (a) The interaction is varied by changing $d/{\ell }_B$ and adding a $\delta V_0$ pseudopotential. (b) The interaction is varied by changing $d/{\ell }_B$ and adding a $\delta V_1$ pseudopotential. The same amount of $\delta V_1$ is added to both intralayer and interlayer Coulomb. (c) The effect of varying both $V_0$ and $V_1$ at fixed bilayer distance $d=1.5{\ell }_B$. Note that the iPf phase is located in the narrow red strip, and can be stabilized by either the reduction in $V_0$ (a), the increase in $V_1$ (b), or increase of both $\delta V_0$ and $\delta V_1$ (c).

Figure 10

Entanglement for spin-charge separation in the non-Abelian phase. We plot the probability $P_a(Q_L,S_L^z)$ for charges $Q_L,S_L^z$ to fluctuate to the left of the cut in ground states $a={\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2$. The center of this distribution gives the charge and spin of the anyon associated with the ground state. We see that ground state ${\mathrm{\Omega }}_1$ corresponds to a quasiparticle with $S^z=0$ and $Q=0$, consistent with either the $\mathbb{1}$ or $\psi$ sector; in the other ground state there is a quasiparticle with $S^z=1/2$ and $Q=0$, consistent with the ${\varphi }_s$ sector.

Figure 11

Differences in entanglement entropy and momentum polarization for the two degenerate states as a function of circumference. Data were taken at a variety of different interlayer separations, $\delta V_0$ and $\delta V_1$. The blue dashed lines show the expected values for the iPf phase, at $S_{{\varphi }_s}-S_{\psi }=log\sqrt{2}$ and $h_{{\varphi }_s}-h_{\psi }=-\frac{3}{16}$.

Figure 12

Entanglement spectra for the putative iPf state. The left two panels show the entanglement spectra for the $|{\mathrm{\Omega }}^1\rangle$ state; for the charge sectors with the lowest-lying and second-lowest-lying entanglement states, the counting for these states is $1,3,...$ and $1,2,...$, as expected if $|{\mathrm{\Omega }}^1\rangle =|\psi \rangle$. The right panel shows spectra for the $|{\mathrm{\Omega }}^2\rangle$ state. There are two degenerate charge sectors with lowest-lying states. Here we show one example from each of the two sectors with the lowest-lying entanglement states, and we find counting of $1,3,\cdots$ in both, as expected if $|{\mathrm{\Omega }}^2\rangle =|{\varphi }_s\rangle$.