Vortex states of a disordered quantum Hall bilayer

We present and solve a model for the vortex configuration of a disordered quantum Hall bilayer in the limit of strong and smooth disorder. We argue that there is a characteristic disorder strength below which vortices will be rare and above which they proliferate. We predict that this can be observed tuning the electron density in a given sample. The ground state in the strong-disorder regime can be understood as an emulsion of vortex-antivortex crystals. Its signatures include a suppression of the spatial decay of counterflow currents. We find an increase of at least an order of magnitude in the length scale for this decay compared to a clean system. This provides a possible explanation of the apparent absence of leakage of counterflow currents through interlayer tunneling, even in experiments performed deep in the coherent phase where enhanced interlayer tunneling is observed.

Figure 1

Schematic of a disordered quantum Hall bilayer with compressible puddles of electron liquid (dark-shaded areas) of size $d_p$ surrounded by channels of incompressible excitonic superfluid of size $l_c$. For smooth disorder $d_p\sim l_c\sim d_d$ and the depicted length scales are larger than the magnetic length $l_0$. In the limit of very strong disorder, $l_c$ can become small and comparable to $l_0$.

Figure 2

Top panel: average ground-state vorticity $\left|v\right|$ as a function of disorder energy scale $E_{\text{cap}}$, obtained by simulated annealing on systems of linear size $L$, with $S$ Monte Carlo sweeps per temperature step (see text). Each point is an average over 25 disorder realizations. ${\epsilon }_i$ is taken from a uniform distribution of width $2E_{\text{cap}}$. Bottom panel: corresponding ground-state energies (crosses and hollow symbols) and interaction energy for $L=18,S=1$ (solid triangles). Solid curves show the corresponding results of the mean-field theory.

Figure 3

Ground states for a typical realization of the disorder at strengths $E_{\text{cap}}=0.8$ (left) and 1.1 (right). Black/white are vorticities $\pm 1$ and gray is 0. $L=18$ and $S=1000$.