Bosonization Approach for Bilayer Quantum Hall Systems at ${\nu }_T=1$

We develop a nonperturbative bosonization approach for bilayer quantum Hall systems at ${\nu }_T=1$, which allows us to systematically study the existence of an exciton condensate in these systems. An effective boson model is derived and the excitation spectrum is calculated in both the Bogoliubov and the Popov approximations. In the latter case, we show that the ground state of the system is an exciton condensate only when the distance between the layers is very small compared to the magnetic length, indicating that the system possibly undergoes another phase transition before the incompressible-compressible one. The effect of a finite electron interlayer tunneling is included and a quantitative phase diagram is proposed.

Figure 1

Schematic representation of (a) one-boson state (magnetic exciton) of the quantum Hall ferromagnet with $|\mathbf{q}{l}^2|=1$ and (b) the condensate of $N_{\varphi }/2$ zero-momentum bosons. The guiding center quantum number is denoted by $\mathbf{m}$.

Figure 2

Dispersion relation of the quasiparticles in the Bogoliubov approximation for different values of the ratio $d/l$: 0.2, 0.5, 0.8, 1.0, 1.5, and 2.0 (from top to bottom in the large momentum region). (a) ${\Delta }_{SAS}=0$ and (b) ${\Delta }_{SAS}=0.1e^2/ϵl$.

Figure 3

Phase diagram ($d/l\times {\Delta }_{SAS}$) of the bilayer QHS at ${\nu }_T=1$: the solid circles are the calculated critical ratio $(d/l{)}_c$ (the solid line is a guide to the eyes) and the dashed line is the experimental estimate for $(d/l{)}_c^{\mathrm{I}\mathrm{\text{−}}\mathrm{C}}$ (from Ref. 3), setting the transition to a phase where no quantum Hall effect (QHE) can be observed.