$s$-Wave Paired Electron and Hole Composite Fermion Trial State for Quantum Hall Bilayers with $\nu =1$

We introduce a new variational wave function for a quantum Hall bilayer at total filling ${\nu }_T=1$, which is based on $s$-wave BCS pairing between electron composite fermions in one layer and hole composite fermions in the other. In addition, we reexamine a trial wave function based on $p$-wave BCS pairing between electron composite fermions in both layers. We compute the overlap of the optimized trial functions with the ground state from exact diagonalization calculations of up to 14 electrons in a spherical geometry, and we find excellent agreement over the entire range of values of the ratio between the layer separation and the magnetic length. The $s$-wave trial wave function naturally allows for charge imbalance between the layers and provides important insights into how the physics at large interlayer separations crosses over to that at small separations in a fashion analogous to the BEC-BCS crossover.

Figure 1

Exact diagonalization results for a balanced system with $6+6$ electrons on the sphere. (a) We plot the overlap of the trial wave functions with the true ground state $|{\mathrm{\Psi }}_{\mathrm{GS}}⟩$ as a function of the interlayer distance $d$. We compare the overlap of our $s$-wave BCS state with the previously proposed $p$-wave BCS state of Refs. [25, 26]. The $s$- and $p$-wave curves are almost indistinguishable on this plot. For both trial wave functions, we include 5 variational parameters. We also show the overlaps with the composite Fermi liquid (CFL) state and the (111) state, which are accurate descriptions of the state in the large and small $d$ limits, respectively. The error bars denote the errors of the Monte Carlo integration. (b) BCS parameters $\mathrm{\Delta }/E_F$ and $\xi /{\ell }_B$ extracted from the $s$-wave variational wave function from (a). $\mathrm{\Delta }$ is the $s$-wave superconducting order parameter, $E_F$ is the Fermi energy, and $\xi$ is the coherence length. The evolution of the BCS parameters as a function of the interlayer separation $d$ is consistent with a BEC-BCS crossover.

Figure 2

Exact diagonalization results for the overlap of the $s$-wave BCS trial wave functions with the true ground state for imbalanced layers with a total of 14 electrons (using 6 variational parameters). The $L^2=0$ Hilbert space dimensions are 1599, 1319, 614, 205 for $7+7$, $6+8$, $5+9$, and $4+10$, respectively. We note that in the limit $d=0$ the Hamiltonian has an enhanced $SU(2)$ pseudospin symmetry, which our trial states do not exhibit, since the overlaps in different pseudospin sectors are not identical.