Numerical Study of Quantum Hall Bilayers at Total Filling ${\nu }_T=1$: A New Phase at Intermediate Layer Distances

We study the phase diagram of quantum Hall bilayer systems with total filing ${\nu }_T=1/2+1/2$ of the lowest Landau level as a function of layer distances $d$. Based on numerical exact diagonalization calculations, we obtain three distinct phases, including an exciton superfluid phase with spontaneous interlayer coherence at small $d$, a composite Fermi liquid at large $d$, and an intermediate phase for $1.1<d/l_B<1.8$ ($l_B$ is the magnetic length). The transition from the exciton superfluid to the intermediate phase is identified by (i) a dramatic change in the Berry curvature of the ground state under twisted boundary conditions on the two layers and (ii) an energy level crossing of the first excited state. The transition from the intermediate phase to the composite Fermi liquid is identified by the vanishing of the exciton superfluid stiffness. Furthermore, from our finite-size study, the energy cost of transferring one electron between the layers shows an even-odd effect and possibly extrapolates to a finite value in the thermodynamic limit, indicating the enhanced intralayer correlation. Our identification of an intermediate phase and its distinctive features shed new light on the theoretical understanding of the quantum Hall bilayer system at total filling ${\nu }_T=1$.

Figure 1

The phase diagram of $\nu =1/2+1/2$ quantum Hall bilayers with varying layer distance $d/l_B$. We identify three phases: exciton superfluid phase, the intermediate phase, and composite Fermi liquid phase. (a) The transition from exciton superfluid to intermediate phase near $d_{c_1}\approx 1.1$ is identified by the drag Hall conductance ${\sigma }_{xy}^d$ and the energy level crossing. Here, the ground state is in the momentum sector $K_0=\pi$ and $N=16$. (b) The transition from intermediate phase to CFL phase near $d_{c_2}\approx 1.8$ is identified by the exciton superfluid stiffness ${\rho }_s$ [see Eq. (2)].

Figure 2

(a) The energy dispersion curves of lowest-energy excitations at each momentum sector. Here, the ground state is in the momentum sector $K_0=\pi$. (b) and (c) show finite size scaling of the single pseudospin excitation gap ${\mathrm{\Delta }}_{ps}$ by using a parabolic function for layer distance $d/l_B<1.1$ (b) and $d/l_B>1.1$ (c). The inset of (c) indicates the even-odd effect in the intermediate phase up to $N=20$. (d) The energy spectrum gap ${\mathrm{\Delta }}_E\equiv E_1(d)-E_0(d)$ as a function of $d/l_B$. The cusp near $d/l_B\approx 1.1$ indicates the level crossing for the excited states.

Figure 3

The Berry curvature $F({\theta }_x^{\alpha },{\theta }_y^{\beta })$ for $d/l_B=0.8$ (a), $d/l_B=1.2$ (b). Here, $\mathrm{\Delta }{\theta }_x$ and $\mathrm{\Delta }{\theta }_y$ are the interval of mesh in phase space. It has strong fluctuation in the FMLRO phase (a), while it is smooth in the intermediate phase (b). (c) to (e) are the energy spectrum of the $N=16$ system with twisted boundary phases for $d/l_B=0.8$ (c), $d/l_B=1.4$ (d), and $d/l_B=3$ (e). By fitting the energy spectrum with twisted phases, one can get the exciton superfluid stiffness ${\rho }_s$ [see Eq. (2)] (f), which decreases with the layer distance and finally vanishes for $d>d_{c_2}$. (f) From bottom to top, $d/l_B$ increases from $d/l_B=0$ with interval 0.2.