Phase Diagram for Quantum Hall Bilayers at $\nu =1$

We present a phase diagram for a double quantum well bilayer electron gas in the quantum Hall regime at a total filling factor $\nu =1$, based on exact numerical calculations of the topological Chern number matrix and the (interlayer) superfluid density. We find three phases: a quantized Hall state with pseudospin superfluidity, a quantized Hall state with pseudospin “gauge-glass” order, and a decoupled composite Fermi liquid. Comparison with experiments provides a consistent explanation of the observed quantum Hall plateau, Hall drag plateau, and vanishing Hall drag resistance, as well as the zero-bias conductance peak effect, and suggests some interesting points to pursue experimentally.

Figure 1

The solid line is the critical layer separation $d_c$ vs $W$, which separates the IQHE plateau (phase I and II) from the composite Fermi liquid state (phase III). The dashed line is the critical $d_s$ vs $W$, which separates the superfluid state (phase I) from the gauge glass state (phase II).

Figure 2

The effective charge mobility ${\rho }_{\mathrm{e}\mathrm{x}\mathrm{t}}$ at $W=0.16$, defined as the ratio of the number of samples with $C^c\ne 1$ and the total number of samples, as a function of $d$ at $N_e=6$, 10, and 11. $\Delta {\rho }_{\mathrm{e}\mathrm{x}\mathrm{t}}/\Delta d$ vs $d$ is also shown.

Figure 3

The superfluid density $n_s$ vs $d$ at $W=0.16$. At the transition point $d_s=1.2$, $n_s\to 0$.

Figure 4

The charge Hall conductance ${\sigma }_{xy}^c$ (in units of $e^2/h$) vs filling number $\nu =N_e/N_s$. In the inset, the random spin Chern number $C^s$ vs disorder configuration $I$ for 20 samples at $\nu =1$. $W=0.16$ and $d=0.8$ (in phase I) for both data.