Phase diagram of $\nu =\frac{1}{2}+\frac{1}{2}$ bilayer bosons with interlayer couplings

We present the quantitative phase diagram of the bilayer bosonic fractional quantum Hall system on the torus geometry at total filling factor $\nu =1$ in the lowest Landau level. We consider short-range interactions within and between the two layers, as well as the interlayer tunneling. In the fully polarized regime, we provide an updated detailed numerical analysis to establish the presence of the Moore-Read phase of both even and odd numbers of particles. In the actual bilayer situation, we find that both interlayer interactions and tunneling can provide the physical mechanism necessary for the low-energy physics to be driven by the fully polarized regime, thus leading to the emergence of the Moore-Read phase. Interlayer interactions favor a ferromagnetic phase when the system is $\mathrm{SU}\left(2\right)$ symmetric, while the interlayer tunneling acts as a Zeeman field polarizing the system. Besides the Moore-Read phase, the $\left(220\right)$ Halperin state and the coupled Moore-Read state are also realized in this model. We study their stability against each other.

Figure 1

(a) Finite-size scaling of the energy gap ${\mathrm{\Delta }}_{\mathrm{MR}}$ for even $N$ (solid line) and odd $N$ (dotted line), and the ground-state splitting ${\delta }_{\mathrm{MR}}$ for even $N$ (solid line). The data from $N=6$ to $N=20$ are included. (b) The ground-state overlap with exact MR states for even $N$ (solid line) and odd $N$ (dotted line) from $N=5$ to $N=20$. The overlap in the $\mathbf{k}=(0,\pi )$ sector is the same as that in the $\mathbf{k}=(\pi ,0)$ sector due to the $C_4$ symmetry of the square torus. For $N=19$, the exact MR state cannot be built; thus we have used the very accurate approximation on the torus involving one Laughlin quasihole and one Laughlin quasiparticle [9]. (c) The finite-size scaling of the entanglement gap ${\mathrm{\Delta }}_{\xi }$ in the $N_A=4,N_A=5$, and $N_A=6$ sectors for even $N$ (solid line) and odd $N$ (dotted line). The data are included only from $N=8$ to $N=16$ due to numerical limitations in the full diagonalization of reduced density matrices.

Figure 2

Energy spectra of $N=12$ bosons as a function of momentum $\left|\mathbf{k}\right|=\sqrt{k_x^2+k_y^2}$ at $U_0=1$ and (a) $U_1=0$, (b) $U_1=0.4$, and (c) $U_1=1$. The spin eigenvalues $S$ of some low-lying states as well as their degeneracy (the spin degeneracy has been excluded) are indicated. One can see that the ferromagnetic levels dominate the low-energy spectrum from the bottom with the increase of $U_1$. The states below the blue lines are the three quasidegenerate ferromagnetic MR states appearing at $\mathbf{k}=(0,0),(\pi ,0)$, and $(0,\pi )$.

Figure 3

Critical value of $U_1$ at which the ferromagnetic MR manifold and its first ferromagnetic excitation dominate the low-energy spectrum for even $N$ (solid line) and odd $N$ (dotted line). The data from $N=6$ to $N=14$ are included.

Figure 4

Schematic $U_0-U_1$ phase diagram for even $N$ in the $S_z=0$ sector. As discussed in the text, we observe three phases: $\left(220\right)$ Halperin, Moore-Read, and coupled Moore-Read state. The rough ranges of these phases are indicated in the figure. The shadowed areas between the three phases are transition regions, whose properties are difficult to be identified based on our present numerical data and could be compressible. It is also difficult to tell if a direct transition between the $\left(220\right)$ Halperin and MR (or between the cMR and MR) could occur.

Figure 5

Nature of the low-energy manifold as a function of $U_0$ and $U_1$ for a system of $N=12$ bosons in the $S_z=0$ sector. (a),(b) The ground-state overlap with the exact MR state in (a) $\mathbf{k}=(0,0)$ and (b) $\mathbf{k}=(\pi ,0)$ sector. (c)–(e) The ground-state overlap with the exact $\left(220\right)$ state in (c) $\mathbf{k}=(0,0)$, (d) $\mathbf{k}=(\pi ,0)$, and (e) $\mathbf{k}=(\pi ,\pi )$ sector. The overlap in the $\mathbf{k}=(0,\pi )$ sector is identical to that in the $\mathbf{k}=(\pi ,0)$ sector due to the $C_4$ symmetry of the square torus.

Figure 6

Energy gap relative to (a) the MR manifold and (b) the $\left(220\right)$ manifold for $N=12$ bosons in the $S_z=0$ sector.

Figure 7

Ground-state overlap with the exact cMR state for $N=12$ bosons in the $S_z=0$ sector at (a) $\mathbf{k}=(0,0)$ and (b) $\mathbf{k}=(\pi ,0)$. In the $U_0-U_1$ region shown here, the overlap in both momentum sectors is maximum around the point $U_0\approx 2.0$ and $U_1\approx 0.35$ (up to $0.91)$.

Figure 8

Polarization ${\mathcal{P}}_z$ along the $z$ spin axis for the absolute ground state and $N=12$ bosons. $S_z$ being a good quantum number, the ${\mathcal{P}}_z$ has only a discrete number of values. The purple region is unpolarized $\left(S_z=0\right)$ while the red region is fully polarized $\left(S_z=N/2=6\right)$. This latest corresponds to $U_0>1.0$ and $U_1\gtrsim 0.3$.

Figure 9

(a) Ground-state overlap with the exact MR state in the $\mathbf{k}=(0,0)$ sector. (b) The energy gap relative to the MR manifold. Here we consider $N=11$ bosons in the $S_z=\frac{1}{2}$ sector.

Figure 10

Low-energy spectrum as a function of $S_z$ for $N=14$ bosons at $U_0=2.0$ and $U_1=0.35$. The red symbols correspond to the cMR like states. We see the alternation of the degeneracy: a unique state when $S_z$ is even and threefold (with one exact degeneracy due to the $C_4$ symmetry) when $S_z$ is odd. The black symbols stand for the lowest-energy states in each momentum sector or the first excited state of the cMR momentum sectors. We clearly observe the important dispersion of the (red) low-energy states.

Figure 11

Nature of the low-energy manifold as a function of $U_0$ and $t$ for a system of $N=10$ bosons with $U_1=0$. The ground-state overlap with the exact MR state in (a) $\mathbf{k}=(0,0)$ and (b) $\mathbf{k}=(\pi ,0)$ sector. (c)–(e) The ground-state overlap with the exact $\left(220\right)$ state in (c) $\mathbf{k}=(0,0)$, (d) $\mathbf{k}=(\pi ,0)$, and (e) $\mathbf{k}=(\pi ,\pi )$ sector.

Figure 12

Properties of the low-energy manifold as a function of $U_0$ and $t$ for a system of $N=10$ bosons with $U_1=0$. (a) The energy gap relative to the MR manifold. (b) The energy gap relative to the $\left(220\right)$ manifold. (c) The $x$ polarization ${\mathcal{P}}_x$ of the absolute ground state.

Figure 13

Properties of the low-energy manifold as a function of $U_0$ and $t$ for a system of $N=9$ bosons with $U_1=0$. (a) The ground-state overlap with the exact MR state in the $\mathbf{k}=(0,0)$ sector. (b) The energy gap relative to the MR state. (c) The $x$ polarization ${\mathcal{P}}_x$ of the absolute ground state.